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t,n;if(e.crossDomain||e.scriptAttrs)return{send:function(i,o){t=w(" + + + + + + + diff --git a/docs/ktools/index.html b/docs/ktools/index.html new file mode 100644 index 00000000000..84290dae3bc --- /dev/null +++ b/docs/ktools/index.html @@ -0,0 +1,726 @@ + + + + + + + + + + + + + + +K Tools | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +

+ + + + K + + Semantic Framework + + + Download +
+ + + +
+
+
+ + +
+ +
+

K Tools

+

Here we document how to use some of the most commonly used K tools.

+

Minimizing Output

+

When one is working with kore-repl or the prover in general and looking at +specific configurations using config, sometimes the configurations can be huge.

+

One tool to help print configuration compactly is the pyk print utility:

+
sh
pyk print
+
+

We are going to use --minimize option (which is actually used automatically +when printing with pyk). This will filter out many uninteresting cells for the +current config and make the result more compact.

+

Then, when invoking the prover, you can minimize your output by piping it into +the pyk print ... facility with arguments for controlling the output:

+
sh
kprove --output json --definition DEFN ... \
+    | jq .term                             \
+    | pyk print DEFN /dev/stdin --omit-labels ... --keep-labels ...
+
+

You can also use this in the kore-repl more easily, by making a help script. +In your current directory, save a new script pykprint.sh:

+
sh
#!/bin/bash
+
+kast --input kore --output json --definition $1 /dev/stdin \
+    | jq .term                                             \
+    | pyk print $1 /dev/stdin --omit-labels $2
+
+

Now call config | bash pykprint.sh DEFN in Kore REPL to make the output +smaller.

+

The options you have to control the output are as follows:

+
    +
  • --no-minimize: do not remove uninteresting cells.
    • +
  • --omit-cells: remove the selected cells from the output.
    • +
  • --keep-cells: keep only the selected cells in the output.
    • +
+

Note: Make sure that there is no whitespace around , in the omit list, +otherwise you'll get an error (, is a list separator, so this +requirement is strict).

+

Debugging

+

The LLVM Backend has support for integration with GDB. You can run the debugger +on a particular program by passing the --debugger flag to krun, or by +invoking the llvm backend interpreter directly. Below we provide a simple +tutorial to explain some of the basic commands supported by the LLVM backend.

+

LLDB Support

+

GDB is not well-supported on macOS, particularly on newer OS versions and Apple +Silicon ARM hardware. Consequently, if the --debugger option is passed to krun +on macOS, LLDB[^1] is launched instead of GDB. However, the K-specific debugger +scripts that GDB uses have not been ported to LLDB yet, and so the instructions +in the rest of this section will not work.

+

The K Definition

+

Here is a sample K definition we will use to demonstrate debugging +capabilities:

+
k
module TEST
+  imports INT
+
+  configuration <k> foo(5) </k>
+  rule [test]: I:Int => I +Int 1 requires I <Int 10
+
+  syntax Int ::= foo(Int) [function]
+  rule foo(I) => 0 -Int I
+
+endmodule
+
+

You should compile this definition with --backend llvm --enable-llvm-debug to +use the debugger most effectively.

+

Stepping

+

Important: When you first run krun with option --debugger, GDB / LLDB +will instruct you on how to modify ~/.gdbinit or ~/.lldbinit to enable +printing abstract syntax of K terms in the debugger. If you do not perform this +step, you can still use all the other features, but K terms will be printed as +their raw address in memory.

+

GDB will need the kompiled interpreter in its safe path in order to access the +pretty printing python script within it. A good way to do this would be to pick +a minimum top-level path that covers all of your kompiled semantics (ie. set auto-load safe-path ~/k-semantics). LLDB has slightly different security +policies that do not require fully-arbitrary code execution.

+

This section uses GDB syntax to demonstrate the debugging features. Please +refer to the GDB to LLDB command map on +macOS.

+

You can break before every step of execution is taken by setting a breakpoint +on the k_step function.

+
(gdb) break definition.kore:k_step
+Breakpoint 1 at 0x25e340
+(gdb) run
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("0", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb) continue
+Continuing.
+
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("1", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb) continue 2
+Will ignore next crossing of breakpoint 1.  Continuing.
+
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("3", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb)
+
+

Breaking on a specific rule

+

You can break when a rule is applied by giving the rule a rule label. If the +module name is TEST and the rule label is test, you can break when the rule +applies by setting a breakpoint on the TEST.test.rhs function:

+
(gdb) break TEST.test.rhs
+Breakpoint 1 at 0x25e250: file /home/dwightguth/test/./test.k, line 4.
+(gdb) run
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+
+

Note that the substitution associated with that rule is visible in the +description of the frame.

+

You can also break when a side condition is applied using the TEST.test.sc +function:

+
(gdb) break TEST.test.sc
+Breakpoint 1 at 0x25e230: file /home/dwightguth/test/./test.k, line 4.
+(gdb) run
+Breakpoint 1, TEST.test.sc (VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+
+

Note that every variable used in the side condition can have its value +inspected when stopped at this breakpoint, but other variables are not visible.

+

You can also break on a rule by its location:

+
(gdb) break test.k:4
+Breakpoint 1 at 0x25e230: test.k:4. (2 locations)
+(gdb) run
+Breakpoint 1, TEST.test.sc (VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.sc (VarI=#token("1", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+
+

Note that this sets a breakpoint at two locations: one on the side condition +and one on the right hand side. If the rule had no side condition, the first +would not be set. You can also view the locations of the breakpoints and +disable them individually:

+
(gdb) info breakpoint
+Num     Type           Disp Enb Address            What
+1       breakpoint     keep y   <MULTIPLE>
+        breakpoint already hit 3 times
+1.1                         y     0x000000000025e230 in TEST.test.sc at /home/dwightguth/test/./test.k:4
+1.2                         y     0x000000000025e250 in TEST.test.rhs at /home/dwightguth/test/./test.k:4
+(gdb) disable 1.1
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("1", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("2", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+
+

Now only the breakpoint when the rule applies is enabled.

+

Breaking on a function

+

You can also break when a particular function in your semantics is invoked:

+
(gdb) info functions foo
+All functions matching regular expression "foo":
+
+File /home/dwightguth/test/./test.k:
+struct __mpz_struct *Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int(struct __mpz_struct *);
+(gdb) break Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int
+Breakpoint 1 at 0x25e640: file /home/dwightguth/test/./test.k, line 6.
+(gdb) run
+Breakpoint 1, Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+6         syntax Int ::= foo(Int) [function]
+(gdb)
+
+

In this case, the variables have numbers instead of names because the names of +arguments in functions in K come from rules, and we are stopped before any +specific rule has applied. For example, _1 is the first argument to the +function.

+

You can also set a breakpoint in this location by setting it on the line +associated with its production:

+
(gdb) break test.k:6
+Breakpoint 1 at 0x25e640: file /home/dwightguth/test/./test.k, line 6.
+(gdb) run
+Breakpoint 1, Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+6         syntax Int ::= foo(Int) [function]
+
+

These two syntaxes are equivalent; use whichever is easier for you.

+

You can also view the stack of function applications:

+
(gdb) bt
+#0  Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+#1  0x000000000025e5f8 in apply_rule_111 (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList)) at /home/dwightguth/test/./test.k:9
+#2  0x0000000000268a52 in take_steps ()
+#3  0x000000000026b7b4 in main ()
+(gdb)
+
+

Here we see that foo was invoked while applying the rule on line 9 of test.k, +and we also can see the substitution of that rule. If foo was evaluated while +evaluating another function, we would also be able to see the arguments of that +function as well, unless the function was tail recursive, in which case no +stack frame would exist once the tail call was performed.

+

Breaking on a set of rules or functions

+

Using rbreak <regex> you can set breakpoints on multiple functions.

+
    +
  • +

    rbreak Lbl - sets a breakpoint on all non hooked functions

    +
    • +
  • +

    rbreak Lbl.*TEST - sets a breakpoint on all functions from module TEST

    +
    • +
  • +

    rbreak hook_INT - sets a breakpoint on all hooks from module INT

    +
    • +
+

Other debugger issues

+
    +
  • <optimized out> try kompiling without -O1, -O2, or -O3.
    • +
  • (gdb) break definition.kore:break -> No source file named definition.kore. +send --enable-llvm-debug to kompile in order to generate debug info symbols.
    • +
+

Profiling your K semantics

+

The first thing to be aware of is in order to get meaningful data, +you need to build the semantics and all of its dependencies with +optimizations enabled but without the frame pointer elimination +optimization. For example, for EVM, this means rebuilding GMP, MPFR, +JEMalloc, Crypto++, SECP256K1, etc with the following exports.

+
sh
export CFLAGS="-DNDEBUG -O2 -fno-omit-frame-pointer"
+export CXXFLAGS="-DNDEBUG -O2 -fno-omit-frame-pointer"
+
+

You can skip this step, but if you do, any samples within these +libraries will not have correct stack trace information, which means +you will likely not get a meaningful set of data that will tell you +where the majority of time is really being spent. Don't worry about +rebuilding literally every single dependency though. Just focus on the +ones that you expect to take a non-negligible amount of runtime. You +will be able to tell if you haven't done enough later, and you can go +back and rebuild more. Once this is done, you then build K with +optimizations and debug info enabled, like so:

+
sh
mvn package -Dproject.build.type="FastBuild"
+
+

Next, you build the semantics with optimizations and debug info +enabled (i.e., kompile -ccopt -O2 --iterated -ccopt -fno-omit-frame-pointer).

+

Once all this is done, you should be ready to profile your +application. Essentially, you should run whatever test suite you +usually run, but with perf record -g -- prefixed to the front. For +example, for KEVM it's the following command. (For best data, don't +run this step in parallel.)

+
sh
perf record -g -- make test-conformance
+
+

Finally, you want to filter out just the samples that landed within +the llvm backend and view the report. For this, you need to know the +name of the binary that was generated by your build system. Normally +it is interpreter, but e.g. if you are building the web3 client for +kevm, it would be kevm-client. You will want to run the following +command.

+
sh
perf report -g -c $binary_name
+
+

If all goes well, you should see a breakdown of where CPU time has +been spent executing the application. You will know that sufficient +time was spent rebuilding dependencies with the correct flags when the +total time reported by the main method is close to 100%. If it's not +close to 100%, this is probably because a decent amount of self time +was reported in stack traces that were not built with frame pointers +enabled, meaning that perf was unable to walk the stack. You will have +to go back, rebuild the appropriate libraries, and then record your +trace again.

+

Your ultimate goal is to identify the hotspots that take the most +time, and make them execute faster. Entries like step and +step_1234 like functions refer to the cost of matching. An entry +like side_condition_1234 is a side condition and apply_rule_1234 +is constructing the rhs of a rule. You can convert from this rule +ordinal to a location using the llvm-kompile-compute-loc script in +the bin folder of the llvm backend repo. For example,

+
sh
llvm-kompile-compute-loc 5868 evm-semantics/.build/defn/llvm/driver-kompiled
+
+

spits out the following text.

+
Line: 18529
+/home/dwightguth/evm-semantics/./.build/defn/llvm/driver.k:493:10
+
+

This is the line of definition.kore that the axiom appears on as +well as the original location of the rule in the K semantics. You can +use this information to figure out which rules and functions are +causing the most time and optimize them to be more efficient.

+

Running tests - kserver

+

The kserver is a front-end tool based on Nailgun +which helps to reduce the startup time of the JVM. Calling kserver in a terminal +window will wait for all kompile/kprove calls and force them to run in the same process +and share the same threads. This also reduces the thread contention significantly. kompile +uses all the threads available to do rule parsing. Another benefit is that it saves caches, +and each time you call kprove/kast, you can access those directly w/o extra disk usage. +Running the regression-new integration tests on a powerful machine (32 threads) takes 8m, +with the kserver active it takes 2m. You can start the kserver in two ways.

+
    +
  • blocking: call kserver in the command line. Close it after you are done testing. Useful for quick testing.
    • +
  • non-blocking: call spawn-kserver <log.flie> and close it with stop-kserver - this is used for automation on CI
    • +
+

Because we reuse caches, you should stop and restart the server between runs. +The Nailgun implementation hasn't been updated in the last 3-5 years, and it's not compatible with Java 18 onwards.

+
+
+ + +
+ + + +
+
+ +
+
+ + + + + + + + + + + + diff --git a/docs/user_manual/index.html b/docs/user_manual/index.html new file mode 100644 index 00000000000..9f820c2bbc8 --- /dev/null +++ b/docs/user_manual/index.html @@ -0,0 +1,3945 @@ + + + + + + + + + + + + + + +K User Manual | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
+ + + + K + + Semantic Framework + + + Download +
+ + + +
+
+
+ + +
+ +
+

K User Manual

+

NOTE: The K User Manual is still under construction; some features of K +may have partial or missing documentation.

+

Introduction

+

Why K?

+

The K Framework is a programming language and system design toolkit made for +practioners and researchers alike.

+

K For Practioners: +K is a framework for deriving programming languages tools from their semantic +specifications.

+

Typically, programming language tool development follows a similar pattern. +After a new programming language is designed, separate teams will develop +separate language tools (e.g. a compiler, interpreter, parser, symbolic +execution engine, etc). Code reuse is uncommon. The end result is that for each +new language, the same basic tools and patterns are re-implemented again and +again.

+

K approaches the problem differently -- it generates each of these tools from a single language specification. +The work of programming language design and tool implementation are made separate concerns. +The end result is that the exercise of +designing new languages and their associated tooling is now reduced to +developing a single language specification from which we derive our tooling for +free.

+

K For Researchers: +K is a configuration- and rewrite-based executable semantic framework.

+

In more detail, K specifications are:

+
    +
  1. Executable: compile into runnable and testable programs;
    2. +
  2. Semantic: correspond to a logical theory with a sound and relatively +complete proof system;
    3. +
  3. Configuration-based: organize system states into compositional, +hierarchical, labelled units called cells;
    4. +
  4. Rewrite-based: define system transitions using rewrite rules.
    5. +
+

K specifications are compiled into particular matching logic theories, giving +them a simple and expressive semantics. K semantic rules are implicitly defined +over the entire configuration structure, but omit unused cells, enabling a +highly modular definitional style. Furthermore, K has been used to develop +programming languages, type systems, and formal analysis tools.

+

Manual Objectives

+

As mentioned in the Why K? section above, the K Framework is designed as a +collection of language-generic command-line interface (CLI) tools which revolve +around K specifications. These tools cover a broad range of uses, but they +typically fall into one of the following categories:

+
    +
  1. Transforming K Specs (e.g. compilation)
    2. +
  2. Running K Specs (e.g. concrete and symbolic execution)
    3. +
  3. Analyzing K Specs (e.g. theorem proving)
    4. +
+

The main user-facing K tools include:

+
    +
  • kompile - the K compiler driver
    • +
  • kparse - the stanadlone K parser and abstract syntax tree (AST) +transformation tool
    • +
  • krun - the K interpreter and symbolic execution engine driver
    • +
  • kprove - the K theorem prover
    • +
+

This user manual is designed to be a tool reference. +In particular, it is not desgined to be a tutorial on how to write K +specifications or to teach the logical foundations of K. New K users should +consult our dedicated +K tutorial, +or the more language-design oriented +PL tutorial. +Researchers seeking to learn more about the logic underlying K are encouraged +to peruse the +growing literature on K and matching logic. +We will consider the manual complete when it provides a complete description of +all user-facing K tools and features.

+

Introduction to K

+

Since K specifications are the primary input into the entire system, let us +take a moment to describe them. At the highest level, K specifications describe +a programming language or system using three different pieces:

+
    +
  1. the system primitives, the base datatypes used during system operation, +e.g., numbers, lists, maps, etc;
    2. +
  2. the system state, a tuple or record over system primitives which gives a +complete snapshot of the system at any given moment;
    3. +
  3. the system behavior, a set of rules which defines possible system +evolutions.
    4. +
+

K specifications are then defined by a collection of sentences which +correspond to the three concepts above:

+
    +
  1. syntax declarations encode the system primitives;
    2. +
  2. configuration declarations encode the system state;
    3. +
  3. context and rule declarations encode the system behavior.
    4. +
+

K sentences are then organized into one or modules which are stored in one or +more files. In this scheme, files may require other files and modules may +import other modules, giving rise to a hierarchy of files and modules. We +give an intuitive sketch of the two levels of grouping in the diagram below:

+
   example.k file
+  +=======================+
+  | requires ".." --------|--> File_1
+  | ...                   |
+  | requires ".." --------|--> File_N
+  |                       |
+  |  +-----------------+  |
+  |  | module ..       |  |
+  |  |   imports .. ---|--|--> Module_1
+  |  |   ...           |  |
+  |  |   imports .. ---|--|--> Module_M
+  |  |                 |  |
+  |  |   sentence_1    |  |
+  |  |   ...           |  |
+  |  |   sentence_K    |  |
+  |  | endmodule       |  |
+  |  +-----------------+  |
+  |                       |
+  +=======================+
+
+

where:

+
    +
  • files and modules are denoted by double-bordered and single-borded boxes +respectively;
    • +
  • file or module identifiers are denoted by double dots (..);
    • +
  • potential repititions are denoted by triple dots (...).
    • +
+

In the end, we require that the file and module hierarchies both form a +directed acyclic graph (DAG). This is, no file may recursively require itself, +and likewise, no module may recursively import itself.

+

We now zoom in further to discuss the various kinds of sentences contained in K +specifications:

+
    +
  1. +

    sentences that define our system's primitives, including:

    +
      +
    • sort declarations: define new categories of primitive datatypes
      • +
    • Backus-Naur Form (BNF) grammar declarations: define the +operators that inhabit our primitive datatypes
      • +
    • lexical syntax declarations: define lexemes/tokens for the +lexer/tokenizer
      • +
    • syntax associativity declarations: specify the +associativity/grouping of our declared operators
      • +
    • syntax priority declarations: specify the priority of +potential ambiguous operators
      • +
    +
    2. +
  2. +

    sentences that define our system's state, including:

    +
      +
    • configuration declarations: define labelled, hierarchical records +using an nested XML-like syntax
      • +
    +
    3. +
  3. +

    sentences that define our system's behavior, including:

    +
      +
    • context declarations: describe how primitives and configurations +can simplify
      • +
    • context alias declarations: define templates that can generate new +contexts
      • +
    • rule declarations: define how the system transitions from one state +to the next
      • +
    +
    4. +
+

K Process Overview

+

We now examine how the K tools are generally used. The main input to all of the +K tools is a K specification. For effieciency reasons, this specification is +first compiled into an intermediate representation called Kore. Once we have +obtained this intermediate representation, we can use it to do:

+
    +
  1. parsing/pretty-printing, i.e., converting a K term, whose syntax is defined +by a K specification, into a alternate representation
    2. +
  2. concrete and abstract execution of a K specification
    3. +
  3. theorem proving, i.e., verifying whether a set of claims about a K +specification hold
    4. +
+

We represent the overall process using the graphic below:

+
 K Compilation Process
++============================================================+
+|                     +---------+                            |
+|  K Specification ---| kompile |--> Kore Specification --+  |
+|                     +---------+                         |  |
++=========================================================|==+
+                                                          |
+ K Execution Process                                      |
++=========================================================|==+
+|                                                         |  |
+|             +-------------------------------------------+  |
+|             |                                              |
+|             |       +---------+                            |
+|  K Term ----+-------| kparse  |--> K Term                  |
+|             |       +---------+                            |
+|             |                                              |
+|             |       +---------+                            |
+|  K Term ----+-------|  krun   |--> K Term                  |
+|             |       +---------+                            |
+|             |                                              |
+|             |       +---------+                            |
+|  K Claims --+-------| kprove  |--> K Claims                |
+|                     +---------+                            |
+|                                                            |
++============================================================+
+
+

where:

+
    +
  • process outlines are denoted by boxes with double-lined borders
    • +
  • executables are denoted by boxes with single-lined borders
    • +
  • inputs and outputs are denoted by words attached to lines
    • +
  • K terms typically correspond to programs defined in a particular +language's syntax (which are either parsed using kparse or executed using +krun)
    • +
  • K claims are a notation for describing how certain K programs should +execute (which are checked by our theorem prover kprove)
    • +
+

K Compilation Process: +Let us start with a description of the compilation process. According to the +above diagram, the compiler driver is called kompile. For our purposes, it is +enough to view the K compilation process as a black box that transforms a K +specification into a lower-level Kore specification that encodes the same +information, but that is easier to work with programmatically.

+

K Execution Process: +We now turn our attention to the K execution process. Abstractly, we can divide +the K execution process into the following stages:

+
    +
  1. the kore specification is loaded (which defines a lexer, parser, and +unparser among other things)
    2. +
  2. the input string is lexed into a token stream
    3. +
  3. the token stream is parsed into K terms/claims
    4. +
  4. the K term/claims are transformed according the K tool being used (e.g. +kparse, krun, or kprove)
    5. +
  5. the K term/claims are unparsed into a string form and printed
    6. +
+

Note that all of the above steps performed in K execution process are fully +prescribed by the input K specification. Of course, there are entire languages +devoted to encoding these various stages proces individually, e.g., flex for +lexers, bison for parsers, etc. What K offers is a consistent language to +package the above concepts in a way that we believe is convenient and practical +for a wide range of uses.

+

Module Declaration

+

K modules are declared at the top level of a K file. They begin with the +module keyword and are followed by a module ID and an optional set of +attributes. They continue with zero or more imports and zero or more sentences +until the endmodule keyword is reached.

+

A module ID consists of an optional # at the beginning, followed by one or +more components separated by hyphens. Each component can contain letters, +numbers, or underscores.

+

After the module ID, attributes can be specified in square brackets. See below +for an (incomplete) list of allowed module attributes.

+

Following the attributes, a module can contain zero or more imports. An +import consists of the import or imports keywords followed by a module ID. +An import tells the compiler that this module should contain all the sentences +(recursively) contained by the module being imported.

+

Imports can be public or private. By default, they are public, which +means that all the imported syntax can be used by any module that imports the +module doing the import. However, you can explicitly override the visibility +of the import with the public or private keyword immediately prior to the +module name. A module imported privately does not export its syntax to modules +that import the module doing the import.

+

Following imports, a module can contain zero or more sentences. A sentence can +be a syntax declaration, a rule, a configuration declaration, a context, a +claim, or a context alias. Details on each of these can be found in subsequent +sections.

+

private attribute

+

If the module is given the private attribute, all of its imports and syntax +are private by default. Individual pieces of syntax can be made public with +the public attribute, and individual imports can be made public with the +public keyword. See relevant sections on syntax and modules for more details +on what it means for syntax and imports to be public or private.

+

symbolic and concrete attribute

+

These attributes may be placed on modules to indicate that they should only +be used by the Haskell and LLVM backends respectively. If the definition is +compiled on the opposite backend, they are implicitly removed from the +definition prior to parsing anywhere they are imported. This can be useful when +used in limited capacity in order to provide alternate semantics for certain +features on different backends. It should be used sparingly as it makes it more +difficult to trust the correctness of your semantics, even in the presence of +testing.

+

Syntax Declaration

+

Named Non-Terminals

+

We have added a syntax to Productions which allows non-terminals to be given a +name in productions. This significantly improves the ability to document K, by +providing a way to explicitly explain what a field in a production corresponds +to instead of having to infer it from a comment or from the rule body.

+

The syntax is:

+
k
name: Sort
+
+

This syntax can be used anywhere in a K definition that expects a non-terminal.

+

klabel(_) and symbol attributes

+

By default K generates for each syntax definition a long and obfuscated klabel +string, which serves as a unique internal identifier and also is used in kast +format of that syntax. If we need to reference a certain syntax production +externally, we have to manually define the klabels using the klabel attribute. +One example of where you would want to do this is to be able to refer to a given +symbol via the syntax priorities attribute, or to enable overloading of a +given symbol.

+

If you only provide the klabel attribute, you can use the provided klabel to +refer to that symbol anywhere in the frontend K code. However, the internal +identifier seen by the backend for that symbol will still be the long obfuscated +generated string. Sometimes you want control over the internal identfier used as +well, in which case you use the symbol attribute. This tells the frontend to +use whatever the declared klabel is directly as the internal identfier.

+

For example:

+
k
module MYMODULE
+    syntax FooBarBaz ::= #Foo( Int, Int ) [klabel(#Foo), symbol] // symbol1
+                       | #Bar( Int, Int ) [klabel(#Bar)]         // symbol2
+                       | #Baz( Int, Int )                        // symbol3
+endmodule
+
+

Here, we have that:

+
    +
  • In frontend K, you can refer to "symbol1" as #Foo (from klabel(#Foo)), +and the backend will see 'Hash'Foo as the symbol name.
    • +
  • In frontend K, you can refer to "symbol2" as #Bar (from klabel(#Bar)), +and the backend will see +'Hash'Bar'LParUndsCommUndsRParUnds'MYMODULE'Unds'FooBarBaz'Unds'Int'Unds'Int +as the symbol name.
    • +
  • In frontend K, you can refer to "symbol3" as +#Baz(_,_)_MYMODULE_FooBarBaz_Int_Int (from auto-generated klabel), and +the backend will see +'Hash'Baz'LParUndsCommUndsRParUnds'MYMODULE'Unds'FooBarBaz'Unds'Int'Unds'Int +as the symbol name.
    • +
+

The symbol provided must be unique to this definition. This is enforced by K. +In general, it's recommended to use symbol attribute whenever you use klabel +unless you explicitely have a reason not to (eg. you want to overload symbols, +or you're using a deprecated backend). It can be very helpful use the symbol +attribute for debugging, as many debugging messages are printed in Kast format +which will be more readable with the symbol names you explicitely declare. +In addition, if you are programatically manipulating definitions via the JSON +Kast format, building terms using the user-provided pretty +symbol, klabel(...) is easier and less error-prone when the auto-generation +process for klabels changes.

+

Parametric productions and bracket attributes

+

Some syntax productions, like the rewrite operator, the bracket operator, and +the #if #then #else #fi operator, cannot have their precise type system +expressed using only concrete sorts.

+

Prior versions of K solved this issue by using the K sort in this case, but +this introduces inexactness in which poorly typed terms can be created even +without having a cast operator present in the syntax, which is a design +consideration we would prefer to avoid.

+

It also introduces cases where terms cannot be placed in positions where they +ought to be well sorted unless their return sort is made to be KBott, which in +turn vastly complicates the grammar and makes parsing much slower.

+

In order to introduce this, we provide a new syntax for parametric productions +in K. This allows you to express syntax that has a sort signature based on +parametric polymorphism. We do this by means of an optional curly-brace- +enclosed list of parameters prior to the return sort of a production.

+

Some examples:

+
k
syntax {Sort} Sort ::= "(" Sort ")" [bracket]
+syntax {Sort} KItem ::= Sort
+syntax {Sort} Sort ::= KBott
+syntax {Sort} Sort ::= Sort "=>" Sort
+syntax {Sort} Sort ::= "#if" Bool "#then" Sort "#else" Sort "#fi"
+syntax {Sort1, Sort2} Sort1 ::= "#fun" "(" Sort2 "=>" Sort1 ")" "(" Sort2 ")"
+
+

Here we have:

+
    +
  1. Brackets, which can enclose any sort but should be of the same sort that was +enclosed.
    2. +
  2. Every sort is a KItem.
    3. +
  3. A KBott term can appear inside any sort.
    4. +
  4. Rewrites, which can rewrite a value of any sort to a value of the same sort. +Note that this allows the lhs or rhs to be a subsort of the other.
    5. +
  5. If then else, which can return any sort but which must contain that sort on +both the true and false branches.
    6. +
  6. lambda applications, in which the argument and parameter must be the same +sort, and the return value of the application must be the same sort as the +return value of the function.
    7. +
+

Note the last case, in which two different parameters are specified separated +by a comma. This indicates that we have multiple independent parameters which +must be the same each place they occur, but not the same as the other +parameters.

+

In practice, because every sort is a subsort of K, the Sort2 +parameter in #6 above does nothing during parsing. It cannot +actually reject any parse, because it can always infer that the sort of the +argument and parameter are K, and it has no effect on the resulting sort of +the term. However, it will nevertheless affect the kore generated from the term +by introducing an additional parameter to the symbol generated for the term.

+

function and total attributes

+

Many times it becomes easier to write a semantics if you have "helper" +functions written which can be used in the RHS of rules. The function +attribute tells K that a given symbol should be simplified immediately when it +appears anywhere in the configuration. Semantically, it means that evaluation +of that symbol will result in at most one return value (that is, the symbol is +a partial function).

+

The total attribute indicates that a symbol cannot be equal to matching logic +bottom; in other words, it has at least one value for every possible set of +arguments. It can be added to a production with the function attribute to +indicate to the symbolic reasoning engine that a given symbol is a +total function, that is it has exactly one return value for every possible +input. Other uses of the total attribute (i.e., on multi-valued symbols to +indicate they always have at least one value) are not yet implemented.

+

For example, here we define the _+Word_ total function and the _/Word_ +partial function, which can be used to do addition/division modulo +2 ^Int 256. These functions can be used anywhere in the semantics where +integers should not grow larger than 2 ^Int 256. Notice how _/Word_ is +not defined when the denominator is 0.

+
k
syntax Int ::= Int "+Word" Int [function, total]
+             | Int "/Word" Int [function]
+
+rule I1 +Word I2 => (I1 +Int I2) modInt (2 ^Int 256)
+rule I1 /Word I2 => (I1 /Int I2) modInt (2 ^Int 256) requires I2 =/=Int 0
+
+

freshGenerator attribute

+

In K, you can access "fresh" values in a given domain using the syntax +!VARNAME:VarSort (with the !-prefixed variable name). This is supported for +builtin sorts Int and Id already. For example, you can generate fresh +memory locations for declared identifiers as such:

+
k
rule <k> new var x ; => . ... </k>
+     <env> ENV => ENV [ x <- !I:Int ] </env>
+     <mem> MEM => MEM [ !I <- 0     ] </mem>
+
+

Each time a !-prefixed variable is encountered, a new integer will be used, +so each variable declared with new var _ ; will get a unique position in the +<mem>.

+

Sometimes you want to have generation of fresh constants in a user-defined +sort. For this, K will still generate a fresh Int, but can use a converter +function you supply to turn it into the correct sort. For example, here we can +generate fresh Foos using the freshFoo(_) function annotated with +freshGenerator.

+
k
syntax Foo ::= "a" | "b" | "c" | d ( Int )
+
+syntax Foo ::= freshFoo ( Int ) [freshGenerator, function, total]
+
+rule freshFoo(0) => a
+rule freshFoo(1) => b
+rule freshFoo(2) => c
+rule freshFoo(I) => d(I) [owise]
+
+rule <k> new var x ; => . ... </k>
+     <env> ENV => ENV [ x <- !I:Int  ] </env>
+     <mem> MEM => MEM [ !I <- !F:Foo ] </mem>
+
+

Now each newly allocated memory slot will have a fresh Foo placed in it.

+

token attribute

+

The token attribute signals to the Kore generator that the associated sort +will be inhabited by domain values. Sorts inhabited by domain values must not +have any constructors declared.

+
k
syntax Bytes [hook(BYTES.Bytes), token]
+
+

Converting between [token] sorts

+

You can convert between tokens of one sort via Strings by defining functions +implemented by builtin hooks. +The hook STRING.token2string allows conversion of any token to a string:

+
k
syntax String ::= FooToString(Foo)  [function, total, hook(STRING.token2string)]
+
+

Similarly, the hook STRING.string2Token allows the inverse:

+
k
syntax Bar ::= StringToBar(String) [function, total, hook(STRING.string2token)]
+
+

WARNING: This sort of conversion does NOT do any sort of parsing or validation. +Thus, we can create arbitary tokens of any sort:

+
StringToBar("The sun rises in the west.")
+
+

Composing these two functions lets us convert from Foo to Bar

+
k
syntax Bar ::= FooToBar(Foo) [function]
+rule FooToBar(F) => StringToBar(FooToString(F))
+
+

Parsing comments, and the #Layout sort

+

Productions for the #Layout sort are used to describe tokens that are +considered "whitespace". The scanner removes tokens matching these productions +so they are not even seen by the parser. Below, we use it to define +lines begining with ; (semicolon) as comments.

+
k
syntax #Layout ::= r"(;[^\\n\\r]*)"    // Semi-colon comments
+                 | r"([\\ \\n\\r\\t])" // Whitespace
+
+

prec attribute

+

Consider the following naive attempt at creating a language what syntax that +allows two types of variables: names that contain underbars, and names that +contain sharps/hashes/pound-signs:

+
k
syntax NameWithUnderbar ::= r"[a-zA-Z][A-Za-z0-9_]*"  [token]
+syntax NameWithSharp    ::= r"[a-zA-Z][A-Za-z0-9_#]*" [token]
+syntax Pgm ::= underbar(NameWithUnderbar)
+             | sharp(NameWithSharp)
+
+

Although, it seems that K has enough information to parse the programs +underbar(foo) and sharp(foo) with, the lexer does not take into account +whether a token is being parsed for the sharp or for the underbar +production. It chooses an arbitary sort for the token foo (perhaps +NameWithUnderbar). Thus, during paring it is unable to construct a valid term +for one of those programs (sharp(foo)) and produces the error message: +Inner Parser: Parse error: unexpected token 'foo'.

+

Since calculating inclusions and intersections between regular expressions is +tricky, we must provide this information to K. We do this via the prec(N) +attribute. The lexer will always prefer longer tokens to shorter tokens. +However, when it has to choose between two different tokens of equal length, +token productions with higher precedence are tried first. Note that the default +precedence value is zero when the prec attribute is not specified.

+

For example, the BUILTIN-ID-TOKENS module defines #UpperId and #LowerId with +the prec(2) attribute.

+
k
  syntax #LowerId ::= r"[a-z][a-zA-Z0-9]*"                    [prec(2), token]
+  syntax #UpperId ::= r"[A-Z][a-zA-Z0-9]*"                    [prec(2), token]
+
+

Furthermore, we also need to make sorts with more specific tokens subsorts of ones with more +general tokens. We add the token attribute to this production so that all +tokens of a particular sort are marked with the sort they are parsed as and not a +subsort thereof. e.g. we get underbar(#token("foo", "NameWithUnderbar")) +instead of underbar(#token("foo", "#LowerId"))

+
k
imports BUILTIN-ID-TOKENS
+syntax NameWithUnderbar ::= r"[a-zA-Z][A-Za-z0-9_]*" [prec(1), token]
+                          | #UpperId                [token]
+                          | #LowerId                [token]
+syntax NameWithSharp ::= r"[a-zA-Z][A-Za-z0-9_#]*" [prec(1), token]
+                       | #UpperId                 [token]
+                       | #LowerId                 [token]
+syntax Pgm ::= underbar(NameWithUnderbar)
+             | sharp(NameWithSharp)
+
+

unused attribute

+

K will warn you if you declare a symbol that is not used in any of the rules of +your definition. Sometimes this is intentional, however; in this case, you can +suppress the warning by adding the unused attribute to the production or +cell.

+
k
syntax Foo ::= foo() [unused]
+
+configuration <foo unused=""> .K </foo>
+
+

Symbol priority and associativity

+

Unlike most other parser generators, K combines the task of parsing with AST +generation. A production declared with the syntax keyword in K is both a +piece of syntax used when parsing, and a symbol that is used when rewriting. +As a result, it is generally convenient to describe expression grammars using +priority and associativity declarations rather than explicitly transforming +your grammar into a series of nonterminals, one for each level of operator +precedence. Thus, for example, a simple grammar for addition and multiplication +will look like this:

+
k
syntax Exp ::= Exp "*" Exp
+             | Exp "+" Exp
+
+

However, this grammar is ambiguous. The term x+y*z might refer to x+(y*z) +or to (x+y)*z. In order to differentiate this, we introduce a partial +ordering between productions known as priority. A symbol "has tighter priority" +than another symbol if the first symbol can appear under the second, but the +second cannot appear under the first without a bracket. For example, in +traditional arithmetic, multiplication has tighter priority than addition, +which means that x+y*z cannot parse as (x+y)*z because the addition +operator would appear directly beneath the multiplication, which is forbidden +by the priority filter.

+

Priority is applied individually to each possible ambiguous parse of a term. It +then either accepts or rejects that parse. If there is only a single remaining +parse (after all the other disambiguation steps have happened), this is the +parse that is chosen. If all the parses were rejected, it is a parse error. If +multiple parses remain, they might be resolved by further disambiguation such +as via the prefer and avoid attributes, but if multiple parses remain after +disambiguation finishes, this is an ambiguous parse error, indicating there is +not a unique parse for that term. In the vast majority of cases, this is +an error and indicates that you ought to either change your grammar or add +brackets to the term in question.

+

Priority is specified in K grammars by means of one of two different +mechanisms. The first, and simplest, simply replaces the | operator in a +sequence of K productions with the > operator. This operator indicates that +everything prior to the > operator (including transitively) binds tighter +than what comes after. For example, a more complete grammar for simple +arithmetic might be:

+
k
syntax Exp ::= Exp "*" Exp
+             | Exp "/" Exp
+             > Exp "+" Exp
+             | Exp "-" Exp
+
+

This indicates that multiplication and division bind tigher than addition +and subtraction, but that there is no relationship in priority between +multiplication and division.

+

As you may have noticed, this grammar is also ambiguous. x*y/z might refer to +x*(y/z) or to (x*y)/z. Indeed, if we removed division and subtraction +entirely, the grammar would still be ambiguous: x*y*z might parse as +x*(y*z), or as (x*y)*z. To resolve this, we introduce another feature: +associativity. Roughly, asssociativity tells us how symbols are allowed to nest +within other symbols with the same priority. If a set of symbols is left +associative, then symbols in that set cannot appear as the rightmost child +of other symbols in that set. If a set of symbols is right associative, then +symbols in that set cannot appear as the leftmost child of other symbols in +that set. Finally, if a set of symbols is non-associative, then symbols +in that set cannot appear as the rightmost or leftmost child of other symbols +in that set. For example, in the above example, if addition and subtraction +are left associative, then x+y+z will parse as (x+y)+z and x+y-z will +parse as (x+y)-z (because the other parse will have been rejected).

+

You might notice that this seems to apply only to binary infix operators. In +fact, the real behavior is slightly more complicated. Priority and +associativity (for technical reasons that go beyond the scope of this document) +really only apply when the rightmost or leftmost item in a production is a +nonterminal. If the rightmost nonterminal is followed by a terminal (or +respectively the leftmost preceded), priority and associativity do not apply. +Thus we can generalize these concepts to arbitrary context-free grammars.

+

Note that in some cases, this is not the behavior you want. You may actually +want to reject parses even though the leftmost and rightmost item in a +production are terminals. You can accomplish this by means of the +applyPriority attribute. When placed on a production, it tells the parser +which nonterminals of a production the priority filter ought to reject children +under, overriding the default behavior. For example, I might have a production +like syntax Exp ::= foo(Exp, Exp) [applyPriority(1)]. This tells the parser +to reject terms with looser priority binding under the first Exp, but not +the second. By default, with this production, neither position would apply +to the priority filter, because the first and last items of the production +are both terminals.

+

Associativity is specified in K grammars by means of one of two different +mechanisms. The first, and simplest, adds the associativity of a priority block +of symbols prior to that block. For example, we can remove the remaining +ambiguities in the above grammar like so:

+
k
syntax Exp ::= left:
+               Exp "*" Exp
+             | Exp "/" Exp
+             > right:
+               Exp "+" Exp
+             | Exp "-" Exp
+
+

This indicates that multiplication and division are left-associative, ie, after +symbols with higher priority are parsed as innermost, symbols are nested with +the rightmost on top. Addition and subtraction are right associative, which +is the opposite and indicates that symbols are nested with the leftmost on top. +Note that this is similar but different from evaluation order, which also +concerns itself with the ordering of symbols, which is described in the next +section.

+

You may note we have not yet introduced the second syntax for priority +and associativity. In some cases, syntax for a grammar might be spread across +multiple modules, sometimes for very good reasons with respect to code +modularity. As a result, it becomes infeasible to declare priority and +associativity inline within a set of productions, because the productions +are not contiguous within a single file.

+

For this purpose, we introduce the equivalent syntax priorities, +syntax left, syntax right, and syntax non-assoc declarations. For +example, the above grammar can be written equivalently as:

+
k
syntax Exp ::= Exp "*" Exp [group(mult)]
+             | Exp "/" Exp [group(div)]
+             | Exp "+" Exp [group(add)]
+             | Exp "-" Exp [group(sub)]
+
+syntax priorities mult div > add sub
+syntax left mult div
+syntax right add sub
+
+

Here, the group(_) attribute is used to create user-defined groups of +sentences. A particular group name collectively refers to the whole set of +sentences within that group. The sets are flattened together, so we could +equivalently have written:

+
k
syntax Exp ::= Exp "*" Exp [group(mult)]
+             | Exp "/" Exp [group(mult)]
+             | Exp "+" Exp [group(add)]
+             | Exp "-" Exp [group(add)]
+
+syntax priorities mult > add
+syntax left mult
+syntax right add
+
+

Note that syntax [left|right|non-assoc] should not be used to group together +productions with different priorities. For example, this code would be invalid:

+
k
syntax priorities mult > add
+syntax left mult add
+
+

Note that there is one other way to describe associativity, but it is +prone to a very common mistake. You can apply the attribute left, right, +or non-assoc directly to a production to indicate that it is, by itself, +left-, right-, or non-associative.

+

However, this often does not mean what users think it means. In particular:

+
k
syntax Exp ::= Exp "+" Exp [left]
+             | Exp "-" Exp [left]
+
+

is not equivalent to:

+
k
syntax Exp ::= left:
+               Exp "+" Exp
+             | Exp "-" Exp
+
+

Under the first, each production is associative with itself, but not each +other. Thus, x+y+z will parse unambiguously as (x+y)+z, but x+y-z will +be ambiguous. However, in the second, x+y-z will parse unambiguously as +(x+y)-z.

+

Think carefully about how you want your grammar to parse. In general, if you're +not sure, it's probably best to group associativity together into the same +blocks you use for priority, rather than using left, right, or non-assoc +attributes on the productions.

+

Lexical identifiers

+

Sometimes it is convenient to be able to give a certain regular expression a +name and then refer to it in one or more regular expression terminals. This +can be done with a syntax lexical sentence in K:

+
k
syntax lexical Alphanum = r"[0-9a-zA-Z]"
+
+

This defines a lexical identifier Alphanum which can be expanded in any +regular expression terminal to the above regular expression. For example, I +might choose to then implement the syntax of identifiers as follows:

+
k
syntax Id ::= r"[a-zA-Z]{Alphanum}*" [token]
+
+

Here {Alphanum} expands to the above regular expression, making the sentence +equivalent to the following:

+
k
syntax Id ::= r"[a-zA-Z]([0-9a-zA-Z])*" [token]
+
+

This feature can be used to more modularly construct the lexical syntax of your +language. Note that K does not currently check that lexical identifiers used +in regular expressions have been defined; this will generate an error when +creating the scanner, however, and the user ought to be able to debug what +happened.

+

assoc, comm, idem, and unit attributes

+

These attributes are used to indicate whether a collection or a production +is associative, commutative, idempotent, and/or has a unit. +In general, you should not need to apply these attributes to productions +yourself, however, they do have certain special meaning to K. K will generate +axioms related to each of these concepts into your definition for you +automatically. It will also automatically sort associative-commutative +collections, and flatten the indentation of associative collections, when +unparsing.

+

public and private attribute

+

K allows users to declare certain pieces of syntax as either public or private. +All syntax is public by default. Public syntax can be used from any module that +imports that piece of syntax. A piece of syntax can be declared private with +the private attribute. This means that that syntax can only be used in the +module in which it is declared; it is not visible from modules that import +that module.

+

You can also change the default visibility of a module with the private +attribute, when it is placed directly on a module. A module with the private +attribute has all syntax private by default; this can be overridden on +specific sentences with the public attribute.

+

Note that the private module attribute also changes the default visiblity +of imports; please refer to the appropriate section elsewhere in the manual +for more details.

+

Here is an example usage:

+
k
module WIDGET-SYNTAX
+
+  syntax Widget ::= foo()
+  syntax WidgetHelper ::= bar() [private] // this production is not visible
+                                          // outside this module
+endmodule
+
+module WIDGET [private]
+  imports WIDGET-SYNTAX
+
+  syntax Widget ::= fooImpl() // this production is not visible outside this
+                              // module
+
+  // this production is visible outside this module
+  syntax KItem ::= adjustWidget(Widget) [function, public]
+endmodule
+
+

Configuration Declaration

+

exit attribute

+

A single configuration cell containing an integer may have the "exit" +attribute. This integer will then be used as the return value on the console +when executing the program.

+

For example:

+
k
configuration <k> $PGM:Pgm </k>
+              <status-code exit=""> 1 </status-code>
+
+

declares that the cell status-code should be used as the exit-code for +invocations of krun. Additionally, we state that the default exit-code is 1 +(an error state). One use of this is for writing testing harnesses which assume +that the test fails until proven otherwise and only set the <status-code> cell +to 0 if the test succeeds.

+

Collection Cells: multiplicity and type attributes

+

Sometimes a semantics needs to allow multiple copies of the same cell, for +example if you are making a concurrent multi-threading programming language. +For this purpose, K supports the multiplicity and type attributes on cells +declared in the configuration.

+

multiplicity can take on values * and ?. Declaring multiplicity="*" +indicates that the cell may appear any number of times in a runtime +configuration. Setting multiplicity="?" indicates that the cell may only +appear exactly 0 or 1 times in a runtime configuration. If there are no +configuration variables present in the cell collection, the initial +configuration will start with exactly 0 instances of the cell collection. If +there are configuration variables present in the cell collection, the initial +configuration will start with exactly 1 instance of the cell collection.

+

type can take on values Set, List, and Map. For example, here we declare +several collecion cells:

+
k
configuration <k> $PGM:Pgm </k>
+              <sets>  <set  multiplicity="?" type="Set">  0:Int </set>  </sets>
+              <lists> <list multiplicity="*" type="List"> 0:Int </list> </lists>
+              <maps>
+                <map multiplicity="*" type="Map">
+                  <map-key> 0:Int </map-key>
+                  <map-value-1> "":String </map-value-1>
+                  <map-value-2> 0:Int     </map-value-2>
+                </map>
+              </maps>
+
+

Declaring type="Set" indicates that duplicate occurrences of the cell should +be de-duplicated, and accesses to instances of the cell will be nondeterministic +choices (constrained by any other parts of the match and side-conditions). +Similarly, declaring type="List" means that new instances of the cell can be +added at the front or back, and elements can be accessed from the front or back, +and the order of the cells will be maintained. The following are examples of +introduction and elimination rules for these collections:

+
k
rule <k> introduce-set(I:Int) => . ... </k>
+     <sets> .Bag => <set> I </set> </sets>
+
+rule <k> eliminate-set => I ... </k>
+     <sets> <set> I </set> => .Bag </sets>
+
+rule <k> introduce-list-start(I:Int) => . ... </k>
+     <lists> (.Bag => <list> I </list>) ... </lists>
+
+rule <k> introduce-list-end(I:Int) => . ... </k>
+     <lists> ... (.Bag => <list> I </list>) </lists>
+
+rule <k> eliminate-list-start => I ... </k>
+     <lists> (<list> I </list> => .Bag) ... </lists>
+
+rule <k> eliminate-list-end => I ... </k>
+     <lists> ... (<list> I </list> => .Bag) </lists>
+
+

Notice that for multiplicity="?", we only admit a single <set> instance at +a time. For the type=List cell, we can add/eliminate cells from the from or +back of the <lists> cell. Also note that we use .Bag to indicate the empty +cell collection in all cases.

+

Declaring type="Map" indicates that the first sub-cell will be used as a +cell-key. This means that matching on those cells will be done as a map-lookup +operation if the cell-key is mentioned in the rule (for performance). If the +cell-key is not mentioned, it will fallback to normal nondeterministic +constrained by other parts of the match and any side-conditions. Note that there +is no special meaning to the name of the cells (in this case <map>, +<map-key>, <map-value-1>, and <map-value-2>). Additionally, any number of +sub-cells are allowed, and the entire instance of the cell collection is +considered part of the cell-value, including the cell-key (<map-key> in this +case) and the surrounding collection cell (<map> in this case).

+

For example, the following rules introduce, set, retrieve from, and eliminate +type="Map" cells:

+
k
rule <k> introduce-map(I:Int) => . ... </k>
+     <maps> ... (.Bag => <map> <map-key> I </map-key> ... </map>) ... </maps>
+
+rule <k> set-map-value-1(I:Int, S:String) => . ... </k>
+     <map> <map-key> I </map-key> <map-value-1> _ => S </map-value-1> ... </map>
+
+rule <k> set-map-value-2(I:Int, V:Int) => . ... </k>
+     <map> <map-key> I </map-key> <map-value-2> _ => V </map-value-2> ... </map>
+
+rule <k> retrieve-map-value-1(I:Int) => S ... </k>
+     <map> <map-key> I </map-key> <map-value-1> S </map-value-1> ... </map>
+
+rule <k> retrieve-map-value-2(I:Int) => V ... </k>
+     <map> <map-key> I </map-key> <map-value-2> V </map-value-2> ... </map>
+
+rule <k> eliminate-map(I:Int) => . ... </k>
+     <maps> ... (<map> <map-key> I </map-key> ... </map> => .Bag) ... </maps>
+
+

Note how each rule makes sure that <map-key> cell is mentioned, and we +continue to use .Bag to indicate the empty collection. Also note that +when introducing new map elements, you may omit any of the sub-cells which are +not the cell-key. In case you do omit sub-cells, you must use structural +framing ... to indicate the missing cells, they will receive the default +value given in the configuration ... declaration.

+

Rule Declaration

+

Rule Structure

+

Each K rule follows the same basic structure (given as an example here):

+
k
rule LHS => RHS requires REQ ensures ENS [ATTRS]
+
+

The portion between rule and requires is referred to as the rule body, +and may contain one or more rewrites (though not nested). Here, the rule body is +LHS => RHS, where LHS and RHS are used as placeholders for the pre- and +post- states. Note that we lose no generality referring to the LHS or the +RHS, even in the presence of multiple rewrites, as the rewrites are pulled to +the top-level anyway.

+

Next is the requires clause, represented here as REQ. The requires clause is +an additional predicate (function-like term of sort Bool), which is to be +evaluated before applying the rule. If the requires clause does not evaluate to +true, then the rule does not apply.

+

Finally is the ensures clause, represented here as ENS. The ensures clause +is to be interpreted as a post-condition, and will be automatically added to the +path condition if the rule applies. It may cause the entire term to become +undefined, but the backend will not stop itself from applying the rule in this +case. Note that concrete backends (eg. the LLVM backend) are free to ignore the +ensures clause.

+

Overall, the transition represented by such a rule is from a state +LHS #And REQ ending in a state RHS #And ENS. When backends apply this rule +as a transition/rewrite, they should:

+
    +
  • Check if pattern LHS matches (or unifies) with the current term, giving +substitution alpha.
    • +
  • Check if the instantiation alpha(REQ) is valid (or satisfiable).
    • +
  • Build the new term alpha(RHS #And ENS), and check if it's satisfiable.
    • +
+

Pattern Matching operator

+

Sometimes when you want to express a side condition, you want to say that a +rule matches if a particular term matches a particular pattern, or if it +instead does /not/ match a particular pattern.

+

The syntax in K for this is :=K and :/=K. It has similar meaning to ==K and +=/=K, except that where ==K and =/=K express equality, :=K and =/=K express +model membership. That is to say, whether or not the rhs is a member of the set +of terms expressed by the lhs pattern. Because the lhs of these operators is a +pattern, the user can use variables in the lhs of the operator. However, due to +current limitations, these variables are NOT bound in the rest of the term. +The user is thus encouraged to use anonymous variables only, although this is +not required.

+

This is compiled by the K frontend down to an efficient pattern matching on a +fresh function symbol.

+

Anonymous function applications

+

There are a number of cases in K where you would prefer to be able to take some +term on the RHS, bind it to a variable, and refer to it in multiple different +places in a rule.

+

You might also prefer to take a variable for which you know some of its +structure, and modify some of its internal structure without requiring you to +match on every single field contained inside that structure.

+

In order to do this, we introduce syntax to K that allows you to construct +anonymous functions in the RHS of a rule and apply them to a term.

+

The syntax for this is:

+
#fun(RuleBody)(Argument)
+
+

Note the limitations currently imposed by the implementation. These functions +are not first-order: you cannot bind them to a variable and inject them like +you can with a regular klabel for a function. You also cannot express multiple +rules or multiple parameters, or side conditions. All of these are extensions +we would like to support in the future, however.

+

In the following, we use three examples to illustrate the behavior of #fun. +We point out that the support for #fun is provided by the frontend, not the +backends.

+

The three examples are real examples borrowed or modified from existing language +semantics.

+

Example 1 (A Simple Self-Explained Example).

+
#fun(V:Val => isFoo(V) andBool isBar(V))(someFunctionReturningVal())
+
+

Example 2 (Nested #fun).

+
   #fun(C
+=> #fun(R
+=> #fun(E
+=> foo1(E, R, C)
+  )(foo2(C))
+  )(foo3(0))
+  )(foo4(1))
+
+

This example is from the beacon +semantics:https://github.com/runtimeverification/beacon-chain-spec/blob/master/b +eacon-chain.k at line 302, with some modification for simplicity. Note how +variables C, R, E are bound in the nested #fun.

+

Example 3 (Matching a structure).

+
rule foo(K, RECORD) =>
+  #fun(record(... field: _ => K))(RECORD)
+
+

Unlike previous examples, the LHS of #fun in this example is no longer a +variable, but a structure. It has the same spirit as the first two examples, +but we match the RECORD with a structure record( DotVar, field: X), instead +of a standalone variable. We also use K's local rewrite syntax (i.e., the +rewriting symbol => does not occur at the top-level) to prevent writing +duplicate expressions on the LHS and RHS of the rewriting.

+

Macros and Aliases

+

A production can be tagged with the macro, alias, macro-rec, or alias-rec +attributes. In all cases, what this signifies is that this is a macro production. +Macro rules are rules where the top symbol of the left-hand-side are macro +labels. Macro rules are applied statically during compilation on all terms that +they match, and statically before program execution on the initial configuration. +Currently, macro rules are required to not have side conditions, although they +can contain sort checks.

+

alias rules are also applied statically in reverse prior to unparsing on the +final configuration. Note that a macro rule can have unbound variables in the +right hand side. When such a macro exists, it should be used only on the left +hand side of rules, unless the user is performing symbolic execution and expects +to introduce symbolic terms into the subject being rewritten.

+

However, when used on the left hand side of a rule, it functions similarly to a +pattern alias, and allows the user to concisely express a reusable pattern that +they wish to match on in multiple places.

+

For example, consider the following semantics:

+
k
syntax KItem ::= "foo" [alias] | "foobar"
+syntax KItem ::= bar(KItem) [macro] | baz(Int, KItem)
+rule foo => foobar
+rule bar(I) => baz(?_, I)
+rule bar(I) => I
+
+

This will rewrite baz(0, foo) to foo. First baz(0, foo) will be rewritten +statically to baz(0, foobar). Then the non-macro rule will apply (because +the rule will have been rewritten to rule baz(_, I) => I). Then foobar will +be rewritten statically after rewriting finishes to foo via the reverse form +of the alias.

+

Note that macros do not apply recursively within their own expansion. This is +done so as to ensure that macro expansion will always terminate. If the user +genuinely desires a recursive macro, the macro-rec and alias-rec attributes +can be used to provide this behavior.

+

For example, consider the following semantics:

+
k
syntax Exp ::= "int" Exp ";" | "int" Exps ";" [macro] | Exp Exp | Id
+syntax Exps ::= List{Exp,","}
+
+rule int X:Id, X':Id, Xs:Exps ; => int X ; int X', Xs ;
+
+

This will expand int x, y, z; to int x; int y, z; because the macro does +not apply the second time after applying the substitution of the first +application. However, if the macro attribute were changed to the macro-rec +attribute, it would instead expand (as the user likely intended) to +int x; int y; int z;.

+

The alias-rec attribute behaves with respect to the alias attribute the +same way the macro-rec attribute behaves with respect to macro.

+

anywhere rules

+

Some rules are not functional, but you want them to apply anywhere in the +configuration (similar to functional rules). You can use the anywhere +attribute on a rule to instruct the backends to make sure they apply anywhere +they match in the entire configuration.

+

For example, if you want to make sure that some associative operator is always +right-associated anywhere in the configuration, you can do:

+
k
syntax Stmt ::= Stmt ";" Stmt
+
+rule (S1 ; S2) ; S3 => S1 ; (S2 ; S3) [anywhere]
+
+

Then after every step, all occurrences of _;_ will be re-associated. Note that +this allows the symbol _;_ to still be a constructor, even though it is +simplified similarly to a function.

+

trusted claims

+

You may add the trusted attribute to a given claim for the K prover to +automatically add it to the list of proven circularities, instead of trying to +discharge it separately.

+

Projection and Predicate functions

+

K automatically generates certain predicate and projection functions from the +syntax you declare. For example, if you write:

+
k
syntax Foo ::= foo(bar: Bar)
+
+

It will automatically generate the following K code:

+
k
syntax Bool ::= isFoo(K) [function]
+syntax Foo ::= "{" K "}" ":>Foo" [function]
+syntax Bar ::= bar(Foo) [function]
+
+rule isFoo(F:Foo) => true
+rule isFoo(_) => false [owise]
+
+rule { F:Foo }:>Foo => F
+rule bar(foo(B:Bar)) => B
+
+

The first two types of functions are generated automatically for every sort in +your K definition, and the third type of function is generated automatically +for each named nonterminal in your definition. Essentially, isFoo for some +sort Foo will tell you whether a particular term of sort K is a Foo, +{F}:>Foo will cast F to sort Foo if F is of sort Foo and will be +undefined (i.e., theoretically defined as #Bottom, the bottom symbol in +matching logic) otherwise. Finally, bar will project out the child of a foo +named bar in its production declaration.

+

Note that if another term of equal or smaller sort to Foo exists and has a +child named bar of equal or smaller sort to Bar, this will generate an +ambiguity during parsing, so care should be taken to ensure that named +nonterminals are sufficiently unique from one another to prevent such +ambiguities. Of course, the compiler will generate a warning in this case.

+

simplification attribute

+

The simplification attribute identifies rules outside the main semantics that +are used to simplify function patterns.

+

Conditions: A simplification rule is applied by matching the function +arguments, instead of unification as when applying function definition +rules. This allows function symbols to appear nested as arguments to other +functions on the left-hand side of a simplification rule, which is forbidden in +function definition rules. For example, this rule would not be accepted as a +function definition rule:

+
k
rule (X +Int Y) +Int Z => X +Int (Y +Int Z) [simplification]
+
+

A simplification rule is only applied when the current side condition implies +the requires clause of the rule, like function definition rules.

+

Order: The simplification attribute accepts an optional integer argument +which is the rule's simplification priority; if the optional argument is not +specified, it is equivalent to a simplification priority of 50. Backends +should attempt simplification rules in order of their simplification +priority, but are not required to do so; in fact, the backend is free to apply +simplification rules at any time. Because of this, users must ensure that +simplification rules are sound regardless of their order of application. This +differs from the priority attribute in that rules with the priority +attribute must be applied in their priority order by the backend. It is an +error to have the priority attribute on a simplification rule.

+

For example, for the following definition:

+
k
    syntax WordStack ::= Int ":" WordStack | ".WordStack"
+    syntax Int ::= sizeWordStack    ( WordStack       ) [function]
+                 | sizeWordStackAux ( WordStack , Int ) [function]
+ // --------------------------------------------------------------
+    rule sizeWordStack(WS) => sizeWordStackAux(WS, 0)
+
+    rule sizeWordStackAux(.WordStack, N) => N
+    rule sizeWordStackAux(W : WS    , N) => sizeWordStackAux(WS, N +Int 1)
+
+

We might add the following simplification lemma:

+
k
    rule sizeWordStackAux(WS, N) => N +Int sizeWordStackAux(WS, 0)
+      requires N =/=Int 0
+      [simplification]
+
+

Then this simplification rule will only apply if the Haskell backend can prove +that notBool N =/=Int 0 is unsatisfiable. This avoids an infinite cycle of +applying this simplification lemma.

+

NOTE: The frontend and Haskell backend do not check that supplied +simplification rules are sound, this is the developer's responsibility. In +particular, rules with the simplification attribute must preserve definedness; +that is, if the left-hand side refers to any partial function then:

+
    +
  • the right-hand side must be #Bottom when the left-hand side is #Bottom, or
    • +
  • the rule must have an ensures clause that is false when the left-hand +side is #Bottom, or
    • +
  • the rule must have a requires clause that is false when the left-hand +side is #Bottom.
    • +
+

These conditions are in order of decreasing preference: the best option is to +preserve #Bottom on the right-hand side, the next best option is to have an +ensures clause, and the least-preferred option is to have a requires clause. +The most preferred option is to write total functions and avoid the entire issue.

+

NOTE: The Haskell backend does not attempt to prove claims which right-hand +side is #Bottom. The reason for this is that the general case is undecidable, +and the backend might enter an infinite loop. Therefore, the backend emits a +warning if it encounters such a claim.

+

concrete and symbolic attributes (Haskell backend)

+

Users can control the application of simplification rules using the concrete +and the symbolic attributes by specifying the type of patterns the rule's +arguments are to match.

+

A concrete pattern is a pattern which does not contain variables or unevaluated +functions, otherwise the pattern is symbolic.

+

The semantics of the two attributes is defined as follows:

+
    +
  • If a simplification rule is marked concrete, then all arguments must be +concrete for the rule to match.
    • +
  • If a simplification rule is marked symbolic, then all arguments must be +symbolic for the rule to match.
    • +
  • The following syntax concrete(<variables>) (resp. symbolic(<variables>)), +where <variables> is a list of variable names separated by commas, can be used +to specify the exact arguments the user expects to match concrete (resp. symbolic) +patterns.
    • +
+

For example, the following will only match when all arguments +are concrete:

+
k
rule X +Int (Y +Int Z) => (X +Int Y) +Int Z [simplification, concrete]
+
+

Conversely, the following will only match when all arguments +are symbolic:

+
k
rule X +Int (Y +Int Z) => (X +Int Y) +Int Z [simplification, symbolic]
+
+

In practice, the following rules will re-associate and commute terms to combine +concrete arguments:

+
k
rule (A +Int Y) +Int Z => A +Int (Y +Int Z)
+  [concrete(Y, Z), symbolic(A), simplification]
+
+rule X +Int (B +Int Z) => B +Int (X +Int Z)
+  [concrete(X, Z), symbolic(B), simplification]
+
+

The unboundVariables attribute

+

Normally, K rules are not allowed to contain regular (i.e., not fresh, not +existential) variables in the RHS / requires / ensures clauses which are not +bound in the LHS.

+

However, in certain cases this behavior might be desired, like, for example, +when specifying a macro rule which is to be used in the LHS of other rules. +To allow for such cases, but still be useful and perform the unboundness checks +in regular cases, the unboundVariables attributes allows the user to specify +a comma-separated list of names of variables which can be unbound in the rule.

+

For example, in the macro declaration

+
k
  rule cppEnumType => bar(_, scopedEnum() #Or unscopedEnum() ) [unboundVariables(_)]
+
+

the declaration unboundVariables(_) allows the rule to pass the unbound +variable checks, and this in turn allows for cppEnumType to be used in +the LHS of a rule to mean the pattern above:

+
k
  rule inverseConvertType(cppEnumType, foo((cppEnumType #as T::CPPType => underlyingType(T))))
+
+

The memo attribute

+

The memo attribute is a hint from the user to the backend to memoize a +function. Not all backends support memoization, but when the attribute is used +and the definition is compiled for a memo-supporting backend, then calls to +the function may be cached. At the time of writing, only the Haskell +backend supports memoization.

+

Limitations of memoization with the Haskell backend

+

The Haskell backend will only cache a function call if all arguments are concrete.

+

It is recommended not to memoize recursive functions, as each recursive call +will be stored in the cache, but only the first iteration will be retrieved from +the cache; that is, the cache will be filled with many unreachable +entries. Instead, we recommend to perform a worker-wrapper transformation on +recursive functions, and apply the memo attribute to the wrapper.

+

Warning: A function declared with the memo attribute must not use +uninterpreted functions in the side-condition of any rule. Memoizing such an +impure function is unsound. To see why, consider the following rules:

+
k
syntax Bool ::= impure( Int ) [function]
+
+syntax Int ::= unsound( Int ) [function, memo]
+rule unsound(X:Int) => X +Int 1 requires impure(X)
+rule unsound(X:Int) => X        requires notBool impure(X)
+
+

Because the function impure is not given rules to cover all inputs, unsound +can be memoized incoherently. For example,

+
{unsound(0) #And {impure(0) #Equals true}} #Equals 1
+
+

but

+
{unsound(0) #And {impure(0) #Equals false}} #Equals 0
+
+

The memoized value of unsound(0) would be incoherently determined by which +pattern the backend encounters first.

+

Variable Sort Inference

+

In K, it is not required that users declare the sorts of variables in rules or +in the initial configuration. If the user does not explicitly declare the sort +of a variable somewhere via a cast (see below), the sort of the variable is +inferred from context based on the sort signature of every place the variable +appears in the rule.

+

As an example, consider the rule for addition in IMP:

+
k
    syntax Exp ::= Exp "+" Exp | Int
+
+    rule I1 + I2 => I1 +Int I2
+
+

Here +Int is defined in the INT module with the following signature:

+
k
    syntax Int ::= Int "+Int" Int [function]
+
+

In the rule above, the sort of both I1 and I2 is inferred as Int. This is because +a variable must have the same sort every place it appears within the same rule. +While a variable appearing only on the left-hand-side of the rule could have +sort Exp instead, the same variable appears as a child of +Int, which +constriants the sorts of I1 and I2 more tightly. Since the sort must be a +subsort of Int or equal to Int, and Int has no subsorts, we infer Int +as the sorts of I1 and I2. This means that the above rule will not match +until I1 and I2 become integers (i.e., have already been evaluated).

+

More complex examples are possible, however:

+
k
    syntax Exp ::= Exp "+" Int | Int
+    rule _ + _ => 0
+
+

Here we have two anonymous variables. They do not refer to the same variable +as one another, so they can have different sorts. The right side is constrained +by + to be of sort Int, but the left side could be either Exp or Int. +When this occurs, we have multiple solutions to the sorts of the variables in +the rule. K will only choose solutions which are maximal, however. To be +precise, if two different solutions exist, but the sorts of one solution are +all greater than or equal to the sorts of the other solution, K will discard +the smaller solution. Thus, in the case above, the variable on the left side +of the + is inferred of sort Exp, because the solution (Exp, Int) is +strictly greater than the solution (Int, Int).

+

It is possible, however, for terms to have multiple maximal solutions:

+
k
    syntax Exp ::= Exp "+" Int | Int "+" Exp | Int
+    rule I1 + I2 => 0
+
+

In this example, there is an ambiguous parse. This could parse as either +the first + or the second. In the first case, the maximal solution chosen is +(Exp, Int). In the second, it is (Int, Exp). Neither of these solutions is +greater than the other, so both are allowed by K. As a result, this program +will emit an error because the parse is ambiguous. To pick one solution over +the other, a cast or a prefer or avoid attribute can be used.

+

Casting

+

There are three main types of casts in K: the semantic cast, the strict cast, +and the projection cast.

+

Semantic casts

+

For every sort S declared in your grammar, K will define the following +production for you for use in rules:

+
k
    syntax S ::= S ":S"
+
+

The meaning of this cast is that the term inside the cast must be less than +or equal to Sort. This can be used to resolve ambiguities, but its principle +purpose is to guide execution by telling K what sort variables must match in +order for the rule to apply. When compiled, it will generate a pattern that +matches on an injection into Sort.

+

Strict casts

+

K also introduces the strict cast:

+
k
    syntax S ::= S "::S"
+
+

The meaning at runtime is exactly the same as the semantic cast; however, it +restricts the sort of the term inside the cast to exactly Sort. That is +to say, if you use it on something that is a strictly smaller sort, it will +generate a type error. This is useful in certain circumstances to help +disambiguate terms, when a semantic cast would not have resolved the ambiguity. +As such, it is primarily used to solve ambiguities rather than to guide +execution.

+

Projection casts

+

K also introduces the projection cast:

+
k
    syntax {S2} S ::= "{" S2 "}" ":>S"
+
+

The meaning of this cast at runtime is that if the term inside is of sort +Sort, it should have it injection stripped away and the value inside is +returned as a term of static sort Sort. However, if the term is of a +different sort, it is an error and execution will get stuck. Thus the primary +usefulness of this cast is to cast the return value of a function with a +greater sort down to a strictly smaller sort that you expect the return value +of the function to have. For example:

+
k
    syntax Exp ::= foo(Exp) [function] | bar(Int) | Int
+    rule foo(I:Int) => I
+    rule bar(I) => bar({foo(I +Int 1)}:>Int)
+
+

Here we know that foo(I +Int 1) will return an Int, but the return sort of +foo is Exp. So we project the result into the Int sort so that it can +be placed as the child of a bar.

+

owise and priority attributes.

+

Sometimes, it is simply not convenient to explicitly describe every +single negative case under which a rule should not apply. Instead, +we simply wish to say that a rule should only apply after some other set of +rules have been tried. K introduces two different attributes that can be +added to rules which will automatically generate the necessary matching +conditions in a manner which is performant for concrete execution (indeed, +it generally outperforms during concrete execution code where the conditions +are written explicitly).

+

The first is the owise attribute. Very roughly, rules without an attribute +indicating their priority apply first, followed by rules with the owise +attribute only if all the other rules have been tried and failed. For example, +consider the following function:

+
k
syntax Int ::= foo(Int) [function]
+rule foo(0) => 0
+rule foo(_) => 1 [owise]
+
+

Here foo(0) is defined explicitly as 0. Any other integer yields the +integer 1. In particular, the second rule above will only be tried after the +first rule has been shown not to apply.

+

This is because the first rule has a lower number assigned for its priority +than the second rule. In practice, each rule in your semantics is implicitly +or explicitly assigned a numerical priority. Rules are tried in increasing +order of priority, starting at zero and trying each increasing numerical value +successively.

+

You can specify the priority of a rule with the priority attribute. For +example, I could equivalently write the second rule above as:

+
k
rule foo(_) => 1 [priority(200)]
+
+

The number 200 is not chosen at random. In fact, when you use the owise +attribute, what you are doing is implicitly setting the priority of the rule +to 200. This has a couple of implications:

+
    +
  1. Multiple rules with the owise attribute all have the same priority and thus +can apply in any order.
    2. +
  2. Rules with priority higher than 200 apply after all rules with the +owise attribute have been tried.
    3. +
+

There is one more rule by which priorities are assigned: a rule with no +attributes indicating its priority is assigned the priority 50. Thus, +with each priority explicitly declared, the above example looks like:

+
k
syntax Int ::= foo(Int) [function]
+rule foo(0) => 0 [priority(50)]
+rule foo(_) => 1 [owise]
+
+

One final note: the llvm backend reserves priorities between 50 and 150 +inclusive for certain specific purposes. Because of this, explicit +priorities which are given within this region may not behave precisely as +described above. This is primarily in order that it be possible where necessary +to provide guidance to the pattern matching algorithm when it would otherwise +make bad choices about which rules to try first. You generally should not +give any rule a priority within this region unless you know exactly what the +implications are with respect to how the llvm backend orders matches.

+

Evaluation Strategy

+

strict and seqstrict attributes

+

The strictness attributes allow defining evaluation strategies without having +to explicitly make rules which implement them. This is done by injecting +heating and cooling rules for the subterms. For this to work, you need to +define what a result is for K, by extending the KResult sort.

+

For example:

+
k
syntax AExp ::= Int
+              | AExp "+" AExp [strict, klabel(addExp)]
+
+

This generates two heating rules (where the hole syntaxes "[]" "+" AExp and +AExp "+" "[]" is automatically added to create an evaluation context):

+
k
rule [addExp1-heat]: <k> HOLE:AExp +  AE2:AExp => HOLE ~>  [] + AE2 ... </k> [heat]
+rule [addExp2-heat]: <k>  AE1:AExp + HOLE:AExp => HOLE ~> AE1 +  [] ... </k> [heat]
+
+

And two corresponding cooling rules:

+
k
rule [addExp1-cool]: <k> HOLE:AExp ~>  [] + AE2 => HOLE +  AE2 ... </k> [cool]
+rule [addExp2-cool]: <k> HOLE:AExp ~> AE1 +  [] =>  AE1 + HOLE ... </k> [cool]
+
+

Note that the rules are given labels based on the klabel of the production, which +nonterminal is the hole, and whether it's the heating or the cooling rule.

+

You will note that these rules can apply one after another infinitely. In +practice, the KResult sort is used to break this cycle by ensuring that only +terms that are not part of the KResult sort will be heated. The heat and +cool attributes are used to tell the compiler that these are heating and +cooling rules and should be handled in the manner just described. Nothing stops +the user from writing such heating and cooling rules directly if they wish, +although we describe other more convenient syntax for most of the advanced +cases below.

+

One other thing to note is that in the above sentences, HOLE is just a +variable, but it has special meaning in the context of sentences with the +heat or cool attribute. In heating or cooling rules, the variable named +HOLE is considered to be the term being heated or cooled and the compiler +will generate isKResult(HOLE) and notBool isKResult(HOLE) side conditions +appropriately to ensure that the backend does not loop infinitely. The module +BOOL will also be automatically and privately included for semantic +purposes. The syntax for parsing programs will not be affected.

+

In order for this functionality to work, you need to define the KResult sort. +For instance, we tell K that a term is fully evaluated once it becomes an Int +here:

+
k
syntax KResult ::= Int
+
+

Note that you can also say that a given expression is only strict only in +specific argument positions. Here we use this to define "short-circuiting" +boolean operators.

+
k
syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [strict(1)]
+              | BExp "&&" BExp [strict(1)]
+
+rule <k> true  || _    => true ... </k>
+rule <k> false || REST => REST ... </k>
+
+rule <k> true  && REST => REST  ... </k>
+rule <k> false && _    => false ... </k>
+
+

If you want to force a specific evaluation order of the arguments, you can use +the variant seqstrict to do so. For example, this would make the boolean +operators short-circuit in their second argument first:

+
k
syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [seqstrict(2,1)]
+              | BExp "&&" BExp [seqstrict(2,1)]
+
+rule <k> _    || true  => true ... </k>
+rule <k> REST || false => REST ... </k>
+
+rule <k> REST && true  => REST  ... </k>
+rule <k> _    && false => false ... </k>
+
+

This will generate rules like this in the case of _||_ (note that BE1 will +not be heated unless isKResult(BE2) is true, meaning that BE2 must be +evaluated first):

+
k
rule <k>  BE1:BExp || HOLE:BExp => HOLE ~> BE1 ||  [] ... </k> [heat]
+rule <k> HOLE:BExp ||  BE2:BExp => HOLE ~>  [] || BE2 ... </k> requires isKResult(BE2) [heat]
+
+rule <k> HOLE:BExp ~>  [] || BE2 => HOLE ||  BE2 ... </k> [cool]
+rule <k> HOLE:BExp ~> BE1 ||  [] =>  BE1 || HOLE ... </k> [cool]
+
+

Context Declaration

+

Sometimes more advanced evaluation strategies are needed. By default, the +strict and seqstrict attributes are limited in that they cannot describe +the context in which heating or cooling should occur. When this type of +control over the evaluation strategy is required, context sentences can be +used to simplify the process of declaring heating and cooling when it would be +unnecessarily verbose to write heating and cooling rules directly.

+

For example, if the user wants to heat a term if it exists under a foo +constructor if the term to be heated is of sort bar, one might write the +following context (with the optional label):

+
k
context [foo]: foo(HOLE:Bar)
+
+

Once again, note that HOLE is just a variable, but one that has special +meaning to the compiler indicating the position in the context that should +be heated or cooled.

+

This will automatically generate the following sentences:

+
k
rule [foo-heat]: <k> foo(HOLE:Bar) => HOLE ~> foo([]) ... </k> [heat]
+rule [foo-cool]: <k> HOLE:Bar ~> foo([]) => foo(HOLE) ... </k> [cool]
+
+

The user may also write the K cell explicitly in the context declaration +if they want to match on another cell as well, for example:

+
k
context <k> foo(HOLE:Bar) ... </k> <state> .Map </state>
+
+

This context will now only heat or cool if the state cell is empty.

+

Side conditions in context declarations

+

The user is allowed to write a side condition in a context declaration, like +so:

+
k
context foo(HOLE:Bar) requires baz(HOLE)
+
+

This side condition will be appended verbatim to the heating rule that is +generated, however, it will not affect the cooling rule that is generated:

+
k
rule <k> foo(HOLE:Bar) => HOLE ~> foo([]) ... </k> requires baz(HOLE) [heat]
+rule <k> HOLE:Bar ~> foo([]) => foo(HOLE) ... </k> [cool]
+
+

Rewrites in context declarations

+

The user can also include exactly one rewrite operation in a context +declaration if that rule rewrites the variable HOLE on the left hand side +to a term containing HOLE on the right hand side. For exampl;e:

+
k
context foo(HOLE:Bar => bar(HOLE))
+
+

In this case, the code generated will be as follows:

+
k
rule <k> foo(HOLE:Bar) => bar(HOLE) ~> foo([]) ... </k> [heat]
+rule <k> bar(HOLE:Bar) ~> foo([]) => foo(HOLE) ... </k> [cool]
+
+

This can be useful if the user wishes to evaluate a term using a different +set of rules than normal.

+

result attribute

+

Sometimes it is necessary to be able to evaluate a term to a different sort +than KResult. This is done by means of adding the result attribute to +a strict production, a context, or an explicit heating or cooling rule:

+
k
syntax BExp ::= Bool
+              | BExp "||" BExp [seqstrict(2,1), result(Bool)]
+
+

In this case, the sort check used by seqstrict and by the heat and cool +attributes will be isBool instead of isKResult. This particular example +does not really require use of the result attribute, but if the user wishes +to evaluate a term of sort KResult further, the result attribute would be +required.

+

hybrid attribute

+

In certain situations, it is desirable to treat a particular production which +has the strict attribute as a result if the term has had its arguments fully +evaluated. This can be accomplished by means of the hybrid attribute:

+
k
syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [strict(1), hybrid]
+
+

This attribute is equivalent in this case to the following additional axiom +being added to the definition of isKResult:

+
k
rule isKResult(BE1:BExp || BE2:BExp) => true requires isKResult(BE1)
+
+

Sometimes you wish to declare a production hybrid with respect to a predicate +other than isKResult. You can do this by specifying a sort as the body of the +hybrid attribute, e.g.:

+
k
syntax BExp ::= BExp "||" BExp [strict(1), hybrid(Foo)]
+
+

generates the rule:

+
k
rule isFoo(BE1:BExp || BE2:BExp) => true requires isFoo(BE1)
+
+

Properly speaking, hybrid takes an optional comma-separated list of sort +names. If the list is empty, the attribute is equivalent to hybrid(KResult). +Otherwise, it generates hybrid predicates for exactly the sorts named.

+

Context aliases

+

Sometimes it is necessary to define a fairly complicated evaluation strategy +for a lot of different operators. In this case, the user could simply write +a number of complex context declarations, however, this quickly becomes +tedious. For this purpose, K has a concept called a context alias. A context +alias is a bit like a template for describing contexts. The template can then +be instantiated against particular productions using the strict and +seqstrict attributes.

+

Here is a (simplified) example taken from the K semantics of C++:

+
k
context alias [c]: <k> HERE:K ... </k> <evaluate> false </evaluate>
+context alias [c]: <k> HERE:K ... </k> <evaluate> true </evaluate> [result(ExecResult)]
+
+syntax Expr ::= Expr "=" Init [strict(c; 1)]
+
+

This defines the evaluation strategy during the translation phase of a C++ +program for the assignment operator. It is equivalent to writing the following +context declarations:

+
k
context <k> HOLE:Expr = I:Init ... </k> <evaluate> false </evaluate>
+context <k> HOLE:Expr = I:Init ... </k> <evaluate> true </evaluate> [result(ExecResult)]
+
+

What this is saying is, if the evaluate cell is false, evaluate the term +like normal to a KResult. But if the evaluate cell is true, instead +evaluate it to the ExecResult sort.

+

Essentially, we have given a name to this evaluation strategy in the form of +the rule label on the context alias sentences (in this case, c). We can +then say that we want to use this evaluation strategy to evaluate particular +arguments of particular productions by referring to it by name in a strict +attribute. For example, strict(c) will instantiate these contexts once for +each argument of the production, whereas strict(c; 1) will instantiate it +only for the first argument. The special variable HERE is used to tell the +compiler where you want to place the production that is to be heated or cooled.

+

You can also specify multiple context aliases for different parts of a production, +for example:

+
k
syntax Exp ::= foo(Exp, Exp) [strict(left; 1; right; 2)]
+
+

This says that we can evaluate the left and right arguments in either order, but to evaluate +the left using the left context alias and the right using the right context alias.

+

We can also say seqstrict(left; 1; right; 2), in which case we additionally must evaluate +the left argument before the right argument. Note, all strict positions are considered collectively +when determining the evaluation order of seqstrict or the hybrid predicates.

+

A strict attribute with no rule label associated with it is equivalent to +a strict attribute given with the following context alias:

+
k
context alias [default]: <k> HERE:K ... </k>
+
+

One syntactic convenience that is provided is that if you wish to declare the following context:

+
k
context foo(HOLE => bar(HOLE))
+
+

you can simply write the following:

+
k
syntax Foo ::= foo(Bar) [strict(alias)]
+
+context alias [alias]: HERE [context(bar)]
+
+

Pattern Matching

+

As Patterns

+

New syntax has been added to K for matching a pattern and binding the resulting +match in its entirety to a variable.

+

The syntax is:

+
Pattern #as V::Var
+
+

In this case, Pattern, including any variables, is matched and the resulting +variables are added to the substitution if matching succeeds. Furthermore, the +term matched by Pattern is added to the substitution as V.

+

This code can also be used outside of any rewrite, in which case matching +occurs as if it appeared on the left hand side, and the right hand side becomes +a variable corresponding to the alias.

+

It is an error to use an as pattern on the right hand side of a rule.

+

Record-like KApply Patterns

+

We have added a syntax for matching on KApply terms which mimics the record +syntax in functional languages. This allows us to more easily express patterns +involving a KApply term in which we don't care about some or most of the +children, without introducing a dependency into the code on the number of +arguments which could be changed by a future refactoring.

+

The syntax is:

+
record(... field1: Pattern1, field2: Pattern2)
+
+

Note that this only applies to productions that are prefix productions. +A prefix production is considered by the implementation to be any production +whose production items match the following regular expression:

+
(Terminal(_)*) Terminal("(")
+(NonTerminal (Terminal(",") NonTerminal)* )?
+Terminal(")")
+
+

In other words, any sequence of terminals followed by an open parenthesis, an +optional comma separated list of non-terminals, and a close parenthesis.

+

If a prefix production has no named nonterminals, a record(...) syntax is +allowed, but in order to reference specific fields, it is necessary to give one +or more of the non-terminals in the production names.

+

Note: because the implementation currently creates one production per possible +set of fields to match on, and because all possible permutations of all +possible subsets of a list of n elements is a number that scales factorially +and reaches over 100 thousand productions at n=8, we currently do not allow +fields to be matched in any order like a true record, but only in the same +order as appears in the production itself.

+

Given that this only reduces the number of productions to the size of the power +set, this will still explode the parsing time if we create large productions of +10 or more fields that all have names. This is something that should probably +be improved, however, productions with that large of an arity are rare, and +thus it has not been viewed as a priority.

+

Or Patterns

+

Sometimes you wish to express that a rule should match if one out of multiple +patterns should match the same subterm. We can now express this in K by means +of using the #Or ML connective on the left hand side of a rule.

+

For example:

+
k
rule foo #Or bar #Or baz => qux
+
+

Here any of foo, bar, or baz will match this rule. Note that the behavior is +ill-defined if it is not the case that all the clauses of the or have the same +bound variables.

+

Matching global context in function rules

+

On occasion it is highly desirable to be able to look up information from the +global configuration and match against it when evaluating a function. For this +purpose, we introduce a new syntax for function rules.

+

This syntax allows the user to match on function context from within a +function rule:

+
k
syntax Int ::= foo(Int) [function]
+
+rule [[ foo(0) => I ]]
+     <bar> I </bar>
+
+rule something => foo(0)
+
+

This is completely desugared by the K frontend and does not require any special +support in the backend. It is an error to have a rewrite inside function +context, as we do not currently support propagating such changes back into the +global configuration. It is also an error if the context is not at the top +level of a rule body.

+

Desugared code:

+
k
syntax Int ::= foo(Int, GeneratedTopCell) [function]
+
+rule foo(0, <generatedTop>
+              <bar> I </bar>
+              ...
+            </generatedTop> #as Configuration) => I
+rule <generatedTop>
+       <k> something ... </k>
+       ...
+     </generatedTop> #as Configuration
+  => <generatedTop>
+       <k> foo(0, Configuration> ... </k>
+       ...
+     </generatedTop>
+
+

Collection patterns

+

It is allowed to write patterns on the left hand side of rules which refer to +complex terms of sort Map, List, and Set, despite these patterns ostensibly +breaking the rule that terms which are functions should not appear on the left +hand side of rules. Such terms are destructured into pattern matching +operations.

+

The following forms are allowed:

+
// 0 or more elements followed by 0 or 1 variables of sort List followed by
+// 0 or more elements
+ListItem(E1) ListItem(E2) L:List ListItem(E3) ListItem(E4)
+
+// the empty list
+.List
+
+// 0 or more elements in any order plus 0 or 1 variables of sort Set
+// in any order
+SetItem(K1) SetItem(K2) S::Set SetItem(K3) SetItem(K4)
+
+// the empty set
+.Set
+
+// 0 or more elements in any order plus by 0 or 1 variables of sort Map
+// in any order
+K1 |-> E1 K2 |-> E2 M::Map K3 |-> E3 K4 |-> E4
+
+// the empty map
+.Map
+
+

Here K1, K2, K3, K4 etc can be any pattern except a pattern containing both +function symbols and unbound variables. An unbound variable is a variable whose +binding cannot be determined by means of decomposing non-set-or-map patterns or +map elements whose keys contain no unbound variables.

+

This is determined recursively, ie, the term K1 |-> E2 E2 |-> E3 E3 |-> E4 is +considered to contain no unbound variables.

+

Note that in the pattern K1 |-> E2 K3 |-> E4 E4 |-> E5, K1 and K3 are +unbound, but E4 is bound because it is bound by deconstructing the key E3, even +though E3 is itself unbound.

+

In the above examples, E1, E2, E3, and E4 can be any pattern that is normally +allowed on the lhs of a rule.

+

When a map or set key contains function symbols, we know that the variables in +that key are bound (because of the above restriction), so it is possible to +evaluate the function to a concrete term prior to performing the lookup.

+

Indeed, this is the precise semantics which occurs; the function is evaluated +and the result is looked up in the collection.

+

For example:

+
k
syntax Int ::= f(Int) [function]
+rule f(I:Int) => I +Int 1
+rule <k> I:Int => . ... </k> <state> ... SetItem(f(I)) ... </state>
+
+

This will rewrite I to . if and only if the state cell contains +I +Int 1.

+

Note that in the case of Set and Map, one guarantee is that K1, K2, K3, and K4 +represent /distinct/ elements. Pattern matching fails if the correct number of +distinct elements cannot be found.

+

Matching on cell fragments

+

K allows matching fragments of the configuration and using them to construct +terms and use as function parameters.

+
k
configuration <t>
+                <k> #init ~> #collectOdd ~> $PGM </k>
+                <fs>
+                  <f multiplicity="*" type="Set"> 1 </f>
+                </fs>
+              </t>
+
+

The #collectOdd construct grabs the entire content of the <fs> cell. +We may also match on only a portion of its content. Note that the fragment +must be wrapped in a <f> cell at the call site.

+
k
syntax KItem ::= "#collectOdd"
+rule <k> #collectOdd => collectOdd(<fs> Fs </fs>) ... </k>
+     <fs> Fs </fs>
+
+

The collectOdd function collects the items it needs

+
k
syntax Set ::= collectOdd(FsCell) [function]
+rule collectOdd(<fs> <f> I </f> REST </fs>) => SetItem(I) collectOdd(<fs> REST </fs>) requires I %Int 2 ==Int 1
+rule collectOdd(<fs> <f> I </f> REST </fs>) =>            collectOdd(<fs> REST </fs>) requires I %Int 2 ==Int 0
+rule collectOdd(<fs> .Bag </fs>) => .Set
+
+

all-path and one-path attributes to distinguish reachability claims

+

As the Haskell backend can handle both one-path and all-path reachability +claims, but both these are encoded as rewrite rules in K, these attributes can +be used to clarify what kind of claim a rule is.

+

In addition of being able to annotate a rule with one of them +(if annotating with more at the same time, only one of them would be chosen), +one can also annotate whole modules, to give a default claim type for all rules +in that module.

+

Additionally, the Haskell backend introduces an extra command line option +for the K frontend, --default-claim-type, with possible values +all-path and one-path to allow choosing a default type for all +claims.

+

Set Variables

+

Motivation

+

Set variables were introduced as part of Matching Mu Logic, the mathematical +foundations for K. In Matching Mu Logic, terms evaluate to sets of values. +This is useful for both capturing partiality (as in 3/0) and capturing +non-determinism (as in 3 #Or 5). Consequently, symbol interpretation is +extended to have a collective interpretation over sets of input values.

+

Usually, K rules are given using regular variables, which expect that the term +they match is both defined and has a unique interpretation.

+

However, it is sometimes useful to have simplification rules which work over +any kind of pattern, be it undefined or non-deterministic. This behavior can be +achieved by using set variables to stand for any kind of pattern.

+

Syntax

+

Any variable prefixed by @ will be considered a set variable.

+

Example

+

Below is a simplification rule which motivated this extension:

+
  rule #Ceil(@I1:Int /Int @I2:Int) =>
+    {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2)
+    [anywhere]
+
+

This rule basically says that @I1:Int /Int @I2:Int is defined if @I1 and +@I2 are defined and @I2 is not 0. Using sets variables here is important as +it allows the simplification rule to apply any symbolic patterns, without +caring whether they are defined or not.

+

This allows simplifying the expression #Ceil((A:Int /Int B:Int) / C:Int) to:

+
{(C =/=Int 0) #Equals true} #And #Ceil(C) #And ({(B =/=Int 0) #Equals true}
+#And #Ceil(B) #And #Ceil(A)`
+
+

See kframework/kore#729 for +more details.

+

SMT Translation

+

K makes queries to an SMT solver (Z3) to discharge proof obligations when doing +symbolic execution. You can control how these queries are made using the +attributes smtlib, smt-hook, and smt-lemma on declared productions. +These attributes guide the prover when it tries to apply rules to discharge a +proof obligation.

+
    +
  • smt-hook(...) allows you to specify a term in SMTLIB2 format which should +be used to encode that production, and assumes that all symbols appearing in +the term are already declared by the SMT solver.
    • +
  • smtlib(...) allows you to declare a new SMT symbol to be used when that +production is sent to Z3, and gives it uninterpreted function semantics.
    • +
  • smt-lemma can be applied to a rule to encode it as a conditional equality +when sending queries to Z3. A rule rule LHS => RHS requires REQ will be +encoded as the conditional equality (=> REQ (= (LHS RHS)). Every symbol +present in the rule must have an smt-hook(...) or smtlib(...) attribute.
    • +
+
k
syntax Int ::= "~Int" Int          [function, klabel(~Int_), symbol,
+                                    smtlib(notInt)]
+             | Int "^%Int" Int Int [function, klabel(_^%Int__), symbol,
+                                    smt-hook((mod (^ #1 #2) #3))]
+
+

In the example above, we declare two productions ~Int_ and _^%Int__, and +tell the SMT solver to:

+
    +
  • use uninterpreted function semantics for ~Int_ via SMTLIB2 symbol +notInt, and
    • +
  • use the SMTLIB2 term (mod (^ #1 #2) #3) (where #N marks the Nth +production non-terminal argument positions) for _^%Int__, where mod and +^ already are declared by the SMT solver.
    • +
+

Caution

+

Set variables are currently only supported by the Haskell backend. +The use of rules with set variables should be sound for all other backends +which just execute by rewriting, however it might not be safe for backends +which want to guarantee coverage.

+

Variables occurring only in the RHS of a rule

+

This section presents possible scenarios requiring variables to only appear in +the RHS of a rule.

+

Summary

+

Except for ? variables and ! (fresh) variables, which are +required to only appear in the RHS of a rule, all other variables must +also appear in the LHS of a rule. This restriction also applies to anonymous +variables; in particular, for claims, ?_ (not _) should be used in the RHS +to indicate that something changes but we don't care to what value.

+

To support specifying random-like behavior, the above restriction can be relaxed +by annotating a rule with the unboundVariables attribute whenever the rule +intentionally contains regular variables only occurring in the RHS.

+

Introduction

+

K uses question mark variables of the form ?X to refer to +existential variables, and uses ensures to specify logical constraints on +those variables. +These variables are only allowed to appear in the RHS of a K rule.

+

If the rules represent rewrite (semantic) steps or verification claims, +then the ? variables are existentially quantified at the top of the RHS; +otherwise, if they represent equations, the ? variables are quantified at the +top of the entire rule.

+

Note that when both ?-variables and regular variables are present, +regular variables are (implicitly) universally quantified on top of the rule +(already containing the existential quantifications). +This essentially makes all ? variables depend on all regular variables.

+

All examples below are intended more for program verification / +symbolic execution, and thus concrete implementations might choose to ignore +them altogether or to provide ad-hoc implementations for them.

+

Example: Verification claims

+

Consider the following definition of a (transition) system:

+
k
module A
+  rule foo => true
+  rule bar => true
+  rule bar => false
+endmodule
+
+

Consider also, the following specification of claims about the definition above:

+
k
module A-SPEC
+  rule [s1]: foo => ?X:Bool
+  rule [s2]: foo =>  X:Bool  [unboundVariables(X)]
+  rule [s3]: bar => ?X:Bool
+  rule [s4]: bar =>  X:Bool  [unboundVariables(X)]
+endmodule
+
+
One-path interpretation
+
    +
  • (s1) says that there exists a path from foo to some boolean, which is +satisfied easily using the foo => true rule
    • +
  • (s3) says the same thing about bar and can be satisfied by either of +bar => true and bar => false rules
    • +
  • (s2) and (s4) can be better understood by replacing them with instances for +each element of type Bool, which can be interpreted that +both true and false are reachable from foo for (s2), or bar for (s4), +respectively. +
      +
    • (s2) cannot be verified as we cannot find a path from foo to false.
      • +
    • (s4) can be verified by using bar => true to show true is reachable and +bar => false to achieve the same thing for false
      • +
    +
    • +
+
All-path interpretation
+
    +
  • +

    (s1) says that all paths from foo will reach some boolean, which is +satisfied by the foo => true rule and the lack of other rules for foo

    +
    • +
  • +

    (s3) says the same thing about bar and can be satisfied by checking that +both bar => true and bar => false end in a boolean, and there are no +other rules for bar

    +
    • +
  • +

    (s2) and (s4) can be better understood by replacing them with instances for +each element of type Bool, which can be interpreted that +both true and false are reachable in all paths originating in +foo for (s2), or bar for (s4), respectively. +This is a very strong claim, requiring that all paths originating in +foo (bar) pass through both true and false, +so neither (s2) nor (s4) can be verified.

    +

    Interestingly enough, adding a rule like false => true would make both +(s2) and (s4) hold.

    +
    • +
+

Example: Random Number Construct rand()

+

The random number construct rand() is a language construct which could be +easily conceived to be part of the syntax of a programming language:

+
k
Exp ::= "rand" "(" ")"
+
+

The intended semantics of rand() is that it can rewrite to any integer in +a single step. This could be expressed as the following following infinitely +many rules.

+
k
rule  rand() => 0
+rule  rand() => 1
+rule  rand() => 2
+  ...    ...
+rule rand() => (-1)
+rule rand() => (-2)
+  ...    ...
+
+

Since we need an instance of the rule for every integer, one could summarize +the above infinitely many rules with the rule

+
rule rand() => I:Int [unboundVariables(I)]
+
+

Note that I occurs only in the RHS in the rule above, and thus the rule +needs the unboundVariables(I) attribute to signal that this is intentionally.

+

One can define variants of rand() by further constraining the output variable +as a precondition to the rule.

+
Rand-like examples
+
    +
  1. +

    randBounded(M,N) can rewrite to any integer between M and N

    +
    k
    syntax Exp ::= randBounded(Int, Int)
+rule randBounded(M, N) => I
+  requires M <=Int I andBool I <=Int N
+  [unboundVariables(I)]
+
    +
    2. +
  2. +

    randInList(Is) takes a list Is of items +and can rewrite in one step to any item in Is.

    +
    k
    syntax Exp ::= randInList (List)
+rule randInList(Is) => I
+  requires I inList Is
+  [unboundVariables(I)]
+
    +
    3. +
  3. +

    randNotInList(Is) takes a list Is of items +and can rewrite in one step to any item not in Is.

    +
    k
    syntax Exp ::= randNotInList (List)
+rule randNotInList(Is) => I
+  requires notBool(I inList Is)
+  [unboundVariables(I)]
+
    +
    4. +
  4. +

    randPrime(), can rewrite to any prime number.

    +
    k
    syntax Exp ::= randPrime ()
+rule randPrime() => X:Int
+  requires isPrime(X)
+  [unboundVariables(X)]
+
    +

    where isPrime(_) is a predicate that can be defined in the usual way.

    +
    5. +
+

Note 1: all above are not function symbols, but language constructs.

+

Note 2: Currently the frontend does not allow rules with universally quantified +variables in the RHS which are not bound in the LHS.

+

Note 3. Allowing these rules in a concrete execution engine would require an +algorithm for generating concrete instances for such variables, satisfying the +given constraints; thus the unboundVariables attribute serves two purposes:

+
    +
  • to allow such rules to pass the variable checks, and
    • +
  • to signal (concrete execution) backends that specialized algorithm would be +needed to instantiate these variables.
    • +
+

Example: Fresh Integer Construct fresh(Is)

+

The fresh integer construct fresh(Is) is a language construct.

+
Exp ::= ... | "fresh" "(" List{Int} ")"
+
+

The intended semantics of fresh(Is) is that it can always rewrite to an +integer that in not in Is.

+

Note that fresh(Is) and randNotInList(Is) are different; the former +does not need to be able to rewrite to every integers not in Is, +while the latter requires so.

+

For example, it is correct to implement fresh(Is) so it always returns the +smallest positive integer that is not in Is, but same implementation for +randNotInList(Is) might be considered inadequate. +In other words, there exist multiple correct implementations of fresh(Is), +some of which may be deterministic, but there only exists a unique +implementation of randNotInList(Is). +Finally, note that randNotInList(Is) is a correct implementation +for fresh(Is); Hence, concrete execution engines can choose to handle +such rules accordingly.

+

We use the following K syntax to define fresh(Is)

+
k
syntax Exp ::= fresh (List{Int})
+rule fresh(Is:List{Int}) => ?I:Int
+  ensures notBool (?I inList{Int} Is)
+
+

A variant of this would be a choiceInList(Is) language construct which would +choose some number from a list:

+
k
syntax Exp ::= choiceInList (List{Int})
+rule choiceInList(Is:List{Int}) => ?I:Int
+  ensures ?I inList{Int} Is
+
+

Note: This definition is different from one using a ! variable to indicate +freshness because using ! is just syntactic sugar for generating globally +unique instances and relies on a special configuration cell, and cannot be +constrained, while the fresh described here is local and can be constrained. +While the first is more appropriate for concrete execution, this might be +better for symbolic execution / program verification.

+

Example: Arbitrary Number (Unspecific Function) arb()

+

The function arb() is not a PL construct, but a mathematical function. +Therefore, its definition should not be interpreted as an execution step, but +rather as an equality.

+

The intended semantics of arb() is that it is an unspecified nullary function. +The exact return value of arb() is unspecified in the semantics but up to the +implementations. +However, being a mathematical function, arb() must return the same value in +any one implementation.

+

We do not need special frontend syntax to define arb(). +We only need to define it in the usual way as a function +(instead of a language construct), and provide no axioms for it. +The total attribute ensures that the function is total, i.e., +that it evaluates to precisely one value for each input.

+
Variants
+

There are many variants of arb(). For example, arbInList(Is) is +an unspecified function whose return value must be an element from Is.

+

Note that arbInList(Is) is different from choiceInList(Is), because +choiceInList(Is) transitions to an integer in Is (could be a different one +each time it is used), while arbInList(Is) is equal to a (fixed) +integer not in Is.

+

W.r.t. the arb variants, we can use ? variables and the function +annotation to signal that we're defining a function and the value of the +function is fixed, but non-determinate.

+
k
syntax Int ::= arbInList(List{Int}) [function]
+rule arbInList(Is:List{Int}) => ?I:Int
+  ensures ?I inList{Int} Is
+
+

If elimination of existentials in equational rules is needed, one possible +approach would be through Skolemization, +i.e., replacing the ? variable with a new uninterpreted function depending +on the regular variables present in the function.

+

Example: Interval (Non-function Symbols) interval()

+

The symbol interval(M,N) is not a PL construct, nor a function in the +first-order sense, but a proper matching-logic symbol, whose interpretation is +in the powerset of its domain. +Its axioms will not use rewrites but equalities.

+

The intended semantics of interval(M,N) is that it equals the set of +integers that are larger than or equal to M and smaller than or equal to N.

+

Since expressing the axiom for interval requires an an existential +quantification on the right-hand-side, thus making it a non-total symbol +defined through an equation, using ? variables might be confusing since their +usage would be different from that presented in the previous sections.

+

Hence, the proposal to support this would be to write this as a proper ML rule. +A possible syntax for this purpose would be:

+
eq  interval(M,N)
+    ==
+    #Exists X:Int .
+        (X:Int #And { X >=Int M #Equals true } #And { X <=Int N #Equals true })
+
+

Additionally, the symbol declaration would require a special attribute to +signal the fact that it is not a constructor but a defined symbol.

+

Since this feature is not clearly needed by K users at the moment, it is only +presented here as an example; its implementation will be postponed for such time +when its usefulness becomes apparent.

+

Parser Generation

+

In addition to on-the-fly parser generation using kast, K is capable of +ahead-of-time parser generation of LR(1) or GLR parsers using Flex and Bison. +This can be done one of two different ways.

+
    +
  1. You can explicitly request for a particular parser to be generated by +invoking kast --gen-parser <outputFile> or +kast --gen-glr-parser <outputFile> respectively. kast will then create a +parser based on the same command line flags that govern on-the-fly parsing, +like -s to specify the starting sort, and -m to specify the module to +parse under. By default, this generates a parser for the sort of the $PGM +configuration variable in the main syntax module of the definition.
    2. +
  2. You can request that a specific set of parsers be generated for all the +configuration variables of your definition by passing the +--gen-bison-parser or --gen-glr-bison-parser flags to kompile. +kompile will decide the sorts to use as start symbols based on the sorts +in the configuration declaration for the configuration variables. The $PGM +configuration variable will be generated based on the main syntax module +of the definition. The user must explicitly annotate the configuration +declaration with the other modules to use to parse the other configuration +variables as attributes. For example, if I have the following cell in the +configuration declaration: <cell> foo($FOO:Foo, $BAR:Bar) </cell>, +One might annotate it with the attribute pair parser="FOO, TEST; BAR, TEST2" +to indicate that configuration variable $FOO should be parsed in the +TEST module, and configuration variable $BAR should be parsed in the +TEST2 module. If the user forgets to annotate the declaration with the +parser attribute, only the $PGM parser will be generated.
    3. +
+

Bison-generated parsers are extremely fast compared to kast, but they have +some important limitations:

+
    +
  • Bison parsers will always output Kore. You can then pass the resulting AST +directly to llvm-krun or kore-exec and bypass the krun frontend, making +them very fast, but lower-level.
    • +
  • Bison parsers do not yet support macros. This may change in a future release. +Note that you can use anywhere rules instead of macros in most cases to get +around this limitation, although they will not benefit from unparsing via the +alias attribute.
    • +
  • Obligation falls on the user to ensure that the grammar they write is LR(1) +if they choose to use LR(1) parsing. If this does not happen, the parser +generated will have shift/reduce or reduce/reduce conflicts and the parser +may behave differently than kast would (kast is a GLL parser, ie, it +is based on LL parsers and parses all unambiguous context-free grammars). +K provides an attribute, not-lr1, which can be applied to modules known to +not be LR(1), and will trigger a warning if the user attempts to generate an +LR(1) parser which recursively imports that module.
    • +
  • If you are using LR(1) based parsing, the prefer and avoid attributes are +ignored. It is only possible to implement these attributes by means of +generalized LL or LR parsing and a postprocessing on the AST to remove the +undesirable ambiguity.
    • +
  • Obligation falls on the user to ensure that the grammar they write has as +few conflicts as possible if they are using GLR parsing. Bison's GLR support +is quite primitive, and in the worst case it can use exponential space and +time to parse a program, which generally leads the generated parser to report +"memory exhausted", indicating that the parse could not be completed within +the stack space allocated by Bison. It's best to ensure that the grammar is +as close to LR(1) as possible and only utilizes conflicts where absolutely +necessary. One tool that can be used to facilitate this is to pass +--bison-lists to kompile. This will disable support for the List{Sort} +syntax production, and it will make NeList{Sort} left associative, but the +resulting productions generated for NeList{Sort} will be LR(1) and use bounded +stack space.
    • +
  • If the grammar you are parsing is context-sensitive (for example, because +it requires a symbol table to parse), one thing you can do to make this +language parse in K is to implement the language as an ambiguous grammar. +Bison's GLR parser will generate an amb production that is parametric in +the sort of the ambiguity. You can then import the K-AMBIGUITIES module +and use rewriting to resolve the ambiguities using whatever preprocessing +mechanisms you prefer.
    • +
+

Location Information

+

K is able to insert file, line, and column metadata into the parse tree on a +per-sort basis when parsing using a bison-generated parser. To enable this, +mark the sort with the locations attribute.

+
k
  syntax Exp [locations]
+  syntax Exp ::= Exp "/" Exp | Int
+
+

K implicitly wraps productions of these sorts in a #location term (see the +K-LOCATIONS module in kast.md). The metadata can thus be accessed with +ordinary rewrite rules:

+
k
  rule #location(_ / 0, File, StartLine, _StartColumn, _EndLine, _EndColumn) =>
+  "Error: Division by zero at " +String File +String ":" Int2String(StartLine)
+
+

Sometimes it is desirable to allow code to be written in a file which +overwrites the current location information provided by the parser. This can be +done via a combination of the #LineMarker sort and the --bison-file flag to +the parser generator. If you declare a production of sort #LineMarker which +contains a regular expression terminal, this will be treated as a +line marker by the bison parser. The user will then be expected to provide +an implementation of the parser for the line marker in C. The function expected +by the parser has the signature void line_marker(char *, yyscan_t), where +yyscan_t is a +reentrant flex scanner. +The string value of the line marker token as specified by your regular +expression can be found in the first parameter of the function, and you can +set the line number used by the scanner using yyset_lineno(int, yyscan_t). If +you declare the variable extern char *filename, you can also set the current +file name by writing a malloc'd, zero-terminated string to that variable.

+

Unparsing

+

A number of factors go into how terms are unparsed in K. Here we describe some +of the features the user can use to control how unparsing happens.

+

Brackets

+

One of the phases that the unparser goes through is to insert productions +tagged with the bracket attribute where it believes this is necessary +in order to create a correct string that will be parsed back into the original +AST. The most common case of this is in expression grammars. For example, +consider the following grammar:

+
k
syntax Exp ::= Int
+             | Exp "*" Exp
+             > Exp "+" Exp
+
+

Here we have declared that expressions can contain integer addition and +multiplication, and that multiplication binds tighter than addition. As a +result, when writing a program, if we want to write an expression that first +applies addition, then multiplication, we must use brackets: (1 + 2) * 3. +Similarly, if we have such an AST, we must insert brackets into the AST +in order to faithfully unparse the term in a manner that will be parsed back +into the same ast, because if we do not, we end up unparsing the term as +1 + 2 * 3, which will be parsed back as 1 + (2 * 3) because of the priority +declaration in the grammar.

+

You can control how the unparser will insert such brackets by adding a +production with the bracket attribute and the correct sort. For example, if, +instead of parentheses, you want to use curly braces, you could write:

+
k
syntax Exp ::= "{" Exp "}" [bracket]
+
+

This would signal to the unparser how brackets should look for terms of sort +Exp, and it will use this syntax when unparsing terms of sort Exp.

+

Commutative collections

+

One thing that K will do (unless you pass the --no-sort-collections flag to +krun) is to sort associative, commutative collections (such as Set and Map) +alphanumerically. For example, if I have a collection whose keys are sort Id +and they have the values a, b, c, and d, then unparsing will always print +first the key a, then b, then c, then d, because this is the alphabetic order +of these keys when unparsed.

+

Furthermore, K will sort numeric keys numerically. For example, if I have a +collection whose keys are 1, 2, 5, 10, 30, it will first display 1, then 2, +then 5, then 10, then 30, because it will sort these keys numerically. Note +that this is different than an alphabetic sort, which would sort them as +1, 10, 2, 30, 5. We believe the former is more intuitive to users.

+

Substitution filtering

+

K will remove substitution terms corresponding to anonymous variables when +using the --pattern flag if those anonymous variables provide no information +about the named variables in your serach pattern. You can disable this behavior +by passing --no-substitution-filtering to krun. When this flag is not passed, +and you are using the Haskell backend, any equality in a substitution (ie, an +#Equals under an #And under an #Or), will be hidden from the user if the +left hand side is a variable that was anonymous in the --pattern passed by +the user, unless that variable appears elsewhere in the substitution. If you +want to see that variable in the substitution, you can either disable this +filtering, or give that variable a name in the original search pattern.

+

Variable alpha renaming

+

K will automatically rename variables that appear in the output configuration. +Similar to commutative collections, this is done to normalize the resulting +configuration so that equivalent configurations will be printed identically +regardless of how they happen to be reached. This pass can be disabled by +passing --no-alpha-renaming to krun.

+

Macro expansion

+

K will apply macros in reverse on the output configuration if the macro was +created with the alias or alias-rec attribute. See the section on macro +expansion for more details.

+

Formatting

+

format attribute

+

K allows you to control how terms are unparsed using the format attribute. +By default, a domain value is unparsed by printing its string value verbatim, +and an application pattern is unparsed by printing its terminals and children +in the sequence implied by its concrete syntax, separated by spaces. However, +K gives you complete control over how you want to unparse the symbol.

+

A format attribute is a string containing zero or more escape sequences that +tell K how to unparse the symbol. Escape sequences begin with a '%' and are +followed by either an integer, or a single non-digit character. Below is a +list of escape sequences recognized by the formatter:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Escape SequenceMeaning
nInsert '\n' followed by the current indentation level
iIncrease the current indentation level by 1
dDecrease the current indentation level by 1
cMove to the next color in the list of colors for this production
rReset color to the default foreground color for the terminal (See below for more information on how colors work)
an integerPrint a terminal or nonterminal from the production (See below for more information)
any other charPrint that character verbatim
+

Using the integer escape sequence

+

In the integer escape sequence %a, the integer a is treated as a 1-based +index into the terminals and nonterminals of the production.

+
    +
  • +

    If the offset refers to a terminal, move to the next color in the list of +colors for this production, print the value of that terminal, then reset the +color to the default foreground color for the terminal.

    +
    • +
  • +

    If the offset refers to a regular expression terminal, it is an error.

    +
    • +
  • +

    If the offset refers to a nonterminal, print the unparsed representation of +the corresponding child of the current term.

    +
    • +
+

color and colors attributes

+

K allows you to take advantage of ANSI terminal codes for foreground color +in order to colorize output pretty-printed by the unparser. This is controlled +via the color and colors attributes of productions. These attributes +combine with the format attribute to control how a term is colorized.

+

The first thing to understand about how colorization works is that the color +and colors attributes are used to construct a list of colors associated +with each production, and the format attribute then uses that list to choose +the color for each part of the production. For more information on how the +format attribute chooses a color from the list, see above, but essentially, +each terminal or %c in the format attribute advances the pointer in the list +by one element, and terminals and %r reset the current color to the default +foreground color of the terminal afterwards.

+

There are two ways you can construct a list of colors associated with a +production:

+
    +
  • +

    The color attribute creates the entire list all with the same color, as +specified by the value of the attribute. When combined with the default format +attribute, this will color all the terminals in that production that color, but +more advanced techniques can be used as well.

    +
    • +
  • +

    The colors attribute creates the list from a manual, comma-separated list +of colors. The attribute is invalid if the length of the list is not equal to +the number of terminals in the production plus the number of %c substrings in +the format attribute.

    +
    • +
+

Attributes Reference

+

Attribute Syntax Overview

+

In K, many different syntactic categories accept an optional trailing list of +keywords known as attributes. Attribute lists have two different syntaxes, +depending on where they occur. Each attribute also has a type which describes +where it may occur.

+

The first syntax is a square-bracketed ([]) list of words. This syntax is +available for following attribute types:

+
    +
  1. module attributes - may appear immediately after the module keyword
    2. +
  2. sort attributes - may appear immediately after a sort declaration
    3. +
  3. production attributes - may appear immediately after a BNF production +alternative
    4. +
  4. rule attributes - may appear immediately after a rule
    5. +
  5. context attributes - may appear immediately after a context or context +alias
    6. +
  6. context alias attributes - may appear immediately after a context alias
    7. +
  7. claim attributes - may appear immediately after a claim
    8. +
+

The second syntax is the XML attribute syntax, i.e., a space delemited list of +key-and-quoted-value pairs appearing inside the start tag of an XML element: +<element key1="value" key2="value2" ... > </element>. This syntax is +available for the following attribute types:

+
    +
  1. cell attributes - may appear inside of the cell start tag in +configuration declarations
    2. +
+

Unrecognized attributes are reported as an error. When we talk about +the type of an attribute, we mean a syntactic category to which an attribute +can be attached where the attribute has some semantic effect.

+

Attribute Index

+

We now provide an index of available attributes organized alphabetically with a +brief description of each. Note that the same attribute may appear in the index +multiple times to indicate its effect in different contexts or with/without +arguments. A legend describing how to interpret the index follows.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeBackendReference
alias-recprodallMacros and Aliases
aliasprodallMacros and Aliases
all-pathclaimhaskellall-path and one-path attributes to distinguish reachability claims
anywhereruleallanywhere rules
applyPriority(_)prodallSymbol priority and associativity
avoidprodallSymbol priority and associativity
binderprodallNo reference yet.
bracketprodallParametric productions and bracket attributes
color(_)prodallcolor and colors attributes
colors(_)prodallcolor and colors attributes
concretemodllvmsymbolic and concrete attribute
concrete(_)rulehaskellconcrete and symbolic attributes (Haskell backend)
concreterulehaskellconcrete and symbolic attributes (Haskell backend)
context(_)aliasallContext aliases
exit = ""cellallexit attribute
formatprodallformat attribute
freshGeneratorprodallfreshGenerator attribute
functionprodallfunction and total attributes
group(_)allallSymbol priority and associativity
hook(_)prodallNo reference yet
hybrid(_)prodallhybrid attribute
hybridprodallhybrid attribute
klabel(_)prodallklabel(_) and symbol attributes
latex(_)prodallNo reference yet
leftprodallSymbol priority and associativity
locationssortallLocation Information
macro-recprodallMacros and Aliases
macroprodallMacros and Aliases
memorulehaskellThe memo attribute
multiplicity = "_"cellallCollection Cells: multiplicity and type attributes
non-assocprodallSymbol priority and associativity
one-pathclaimhaskellall-path and one-path attributes to distinguish reachability claims
owiseruleallowise and priority attributes
prec(_)tokenallprec attribute
preferprodallSymbol priority and associativity
priority(_)ruleallowise and priority attributes
privatemodallprivate attribute
privateprodallpublic and private attribute
publicmodallNo reference yet.
publicprodallpublic and private attribute
result(_)ctxtallresult attribute
result(_)ruleallresult attribute
rightprodallSymbol priority and associativity
seqstrict(_)prodallstrict and seqstrict attributes
seqstrictprodallstrict and seqstrict attributes
simplificationrulehaskellsimplification attribute (Haskell backend)
simplification(_)rulehaskellsimplification attribute (Haskell backend)
smt-hook(_)prodhaskellSMT Translation
smtlib(_)prodhaskellSMT Translation
smt-lemmarulehaskellSMT Translation
strictprodallstrict and seqstrict attributes
strict(_)prodallstrict and seqstrict attributes
symbolicmodhaskellsymbolic and concrete attribute
symbolicrulehaskellconcrete and symbolic attributes (Haskell backend)
symbolic(_)rulehaskellconcrete and symbolic attributes (Haskell backend)
symbolprodallklabel(_) and symbol attributes
tokenprodalltoken attribute
tokensortalltoken attribute
totalprodallfunction and total attributes
trustedclaimhaskelltrusted attribute
type = "_"cellallCollection Cells: multiplicity and type attributes
unboundVariables(_)ruleallThe unboundVariables attribute
unusedprodallunused attribute
kastmodallSpecify that this module should only be included in KAST backends (Java backend).
koremodallSpecify that this module should only be included in Kore backends (Haskell/LLVM backend).
concretemodallSpecify that this module should only be included in concrete backends (LLVM backend).
symbolicmodallSpecify that this module should only be included in symbolic backends (Haskell/Java backend).
stream = "_"cellallSpecify that this cell should be hooked up to a stream, either stdin, stdout, or stderr.
+

Internal Attribute Index

+

Some attributes should not generally appear in user code, except in some +unusual or complex examples. Such attributes are typically generated by the +compiler and used internally. We list these attributes below as a reference for +interested readers:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameTypeBackendReference
assocprodallassoc, comm, idem and unit attributes
commprodallassoc, comm, idem and unit attributes
digestmodallContains the hash of the textual contents of the module.
idemprodallassoc, comm, idem and unit attributes
unitprodallassoc, comm, idem and unit attributes
userListprodallIdentifies the desugared form of Lst ::= List{Elm,"delim"}
predicateprodallSpecifies the sort of a predicate label
elementprodallSpecifies the label of the elements in a list
bracketLabelprodallKeep track of the label of a bracket production since it can't have a klabel
injectiveprodallLabel a given production as injective (unique output for each input)
internalprodallProduction is reserved for internal use by the compiler
coolruleallstrict and seqstrict attributes
heatruleallstrict and seqstrict attributes
+

Index Legend

+
    +
  • +

    Name - the attribute's name (optionally followed by an underscore _ to indicate the attribute takes arguments)

    +
    • +
  • +

    Type - the syntactic categories where this attribute is not ignored; +the possible values are the types mentioned above or shorthands:

    +
      +
    1. all - short for any type except cell
      2. +
    2. mod - short for module
      3. +
    3. sort
      4. +
    4. prod - short for production
      5. +
    5. rule
      6. +
    6. ctxt - short for context or context alias
      7. +
    7. claim
      8. +
    8. cell
      9. +
    +
    • +
  • +

    Backend - the backends that do not ignore this attribute; possible values:

    +
      +
    1. all - all backends
      2. +
    2. llvm - the LLVM backend
      3. +
    3. haskell - the Haskell backend
      4. +
    +
    • +
  • +

    Effect - the attribute's effect (when it applies)

    +
    • +
+

Pending Documentation

+

Backend features not yet given documentation:

+
    +
  • Parser of KORE terms and definitions
    • +
  • Term representation of K terms
    • +
  • Hooked sorts and symbols
    • +
  • Substituting a substitution into the RHS of a rule +
      +
    • domain values
      • +
    • functions
      • +
    • variables
      • +
    • symbols
      • +
    • polymorphism
      • +
    • hooks
      • +
    • injection compaction
      • +
    • overload compaction
      • +
    +
    • +
  • Pattern Matching / Unification of subject and LHS of rule +
      +
    • domain values
      • +
    • symbols
      • +
    • side conditions
      • +
    • and/or patterns
      • +
    • list patterns
      • +
    • nonlinear variables
      • +
    • map/set patterns +
        +
      • deterministic
        • +
      • nondeterministic
        • +
      +
      • +
    • modulo injections
      • +
    • modulo overloads
      • +
    +
    • +
  • Stepping +
      +
    • initialization
      • +
    • termination
      • +
    +
    • +
  • Print kore terms
    • +
  • Equality/comparison of terms
    • +
  • Owise rules
    • +
  • Strategy #STUCK axiom
    • +
  • User substitution +
      +
    • binders
      • +
    • kvar
      • +
    +
    • +
+

To get a complete list of hooks supported by K, you can run:

+
grep -P -R "(?<=[^-])hook\([^)]*\)" k-distribution/include/kframework/builtin/ \
+     --include "*.k" -ho | \
+sed 's/hook(//' | sed 's/)//' | sort | uniq | grep -v org.kframework
+
+

All of these hooks will also eventually need documentation.

+
+
+ + +
+ +
+
+ + K User Manual + +
+
+ + Introduction + + +
+
+ + Introduction to K + + +
+
+ + Module Declaration + + +
+
+ + Syntax Declaration + + +
+
+ + Configuration Declaration + + +
+
+ + Rule Declaration + + +
+
+ + Evaluation Strategy + + +
+
+ + Pattern Matching + +
+
+ + As Patterns +
+ + Record-like KApply Patterns +
+ + Or Patterns +
+ + Matching global context in function rules +
+ + Collection patterns +
+ + Matching on cell fragments +
+ + all-path and one-path attributes to distinguish reachability claims +
+ + Set Variables + + +
+
+ + Variables occurring only in the RHS of a rule + + +
+ +
+
+
+ + Parser Generation +
+ + Location Information +
+ + Unparsing + + +
+
+ + Attributes Reference + + +
+
+ + Pending Documentation +
+
+
+ +
+ +
+
+ +
+
+ + + + + + + + + + + + diff --git a/editor_support/index.html b/editor_support/index.html new file mode 100644 index 00000000000..08ac4b5cf08 --- /dev/null +++ b/editor_support/index.html @@ -0,0 +1,455 @@ + + + + + + + + + + + + + + +Editor Support | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
+ + + + K + + Semantic Framework + + + Download +
+ + + +
+
+
+ + +
+ +
+

Editor Support

+

This page lists (alphabetically) known Editor/IDE plugins for K.

+

Please feel free to contribute to any of the projects below (via pull requests) or to suggest new entries.

+

Atom

+

K/Maude syntax highlighter for Atom based on the BBEdit grammar described below

+
    +
  • Source: https://github.com/traiansf/language-k
    • +
+

BBEdit/TextWrangler

+

K/Maude syntax highlighter for TextWrangler/BBEdit

+
    +
  • Source: https://github.com/kframework/k-editor-support/tree/master/BBEdit
    • +
+

Emacs

+

Emacs mode for K

+
    +
  • Source: https://github.com/kframework/k-editor-support/tree/master/emacs
    • +
+

IntelliJ Idea

+
    +
  • Comprehensive plugin for IntelliJ Idea
    • +
  • Binary: https://github.com/kframework/k-editor-support/blob/master/k-idea-plugin.jar
    • +
  • Source: https://github.com/kframework/k-editor-support/tree/master/k-idea-plugin-src
    • +
+

Notepad++

+

K syntax highlighter for Notepad++

+
    +
  • Source: https://github.com/kframework/k-editor-support/tree/master/notepad%2B%2B
    • +
+

Pygments

+

Support for https://pygments.org/ Pygments

+
    +
  • Source: https://github.com/kframework/k-editor-support/tree/master/pygments
    • +
  • Note: the lexer is far from being complete.
    • +
+

Vim

+

K/Maude syntax highlighter for vim

+
    +
  • Source: https://github.com/kframework/k-editor-support/tree/master/vim
    • +
+

Visual Studio Code

+

K extension for Visual Studio Code

+
    +
  • Extension page on Visual Studio Marketplace: https://marketplace.visualstudio.com/items?itemName=clv.kframework
    • +
  • Source code: https://github.com/LucianCumpata/K-VSCode
    • +
+
+
+ + +
+ + + +
+
+ +
+
+ + + + + + + + + + + + diff --git a/events/k-a-rewriting-based-language-definitional-framework/index.html b/events/k-a-rewriting-based-language-definitional-framework/index.html new file mode 100644 index 00000000000..4034444e9ce --- /dev/null +++ b/events/k-a-rewriting-based-language-definitional-framework/index.html @@ -0,0 +1,401 @@ + + + + + + + + + + + + + + +K: A Rewriting-Based Language Definitional Framework | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
+ + + + K + + Semantic Framework + + + Download +
+ + + +
+
+
+ + +
+ +
+

K: A Rewriting-Based Language Definitional Framework

+

Tutorial at the 33rd ACM SIGPLAN International Conference on Programming Language Design and Implementation (PLDI)

+
    +
  • June 16, 2012 – Beijing (China)
    • +
  • Presenter: Grigore Rosu (the main designer of K)
    • +
  • Duration: Half a day
    • +
  • Expected participants: ~20
    • +
+

Description

+

K is an executable semantic framework in which programming languages, calculi, as well as type systems or formal analysis tools can be defined. K is a suitable framework for defining truly concurrent languages or calculi, even in the presence of sharing. Since computations can be handled like any other terms in a rewriting environment, that is, they can be matched, moved from one place to another in the original term, modified, or even deleted, K is also suitable for defining control-intensive language features such as abrupt termination, exceptions, or call/cc. K has been used to define real world languages like C.

+

This tutorial will provide participants with a basic knowledge of the framework, as well as hands-on experience with using K to define a real programming language. Definitional techniques available in K, as well as comparisons of such techniques with other formalisms will be described. Time will be spent showing how one can automatically generate an interpreter, debugger, state space search, and a model checker from a single semantic definition. After attending the tutorial, participants will be able to use K to define their own languages or calculi and then derive similar tools from their semantics for free.

+ +
    +
  • http://k-framework.org: The main page for the K framework (see the Quick Overview section for a movie, demo and slides).
    • +
  • http://k-framework.googlecode.com: The Googlecode page for the K tool.
    • +
+

Tutorial format

+

Material and instructions will be provided to participants to load software and examples on their laptops. The presenter will give background material and an introduction to K, then the majority of the time will be spent working through examples in the K tool. The examples will be used to demonstrate both features of K, as well as design decisions that must take place when defining a language. Participants will be encouraged to examine and understand the example languages, then guided through making their own changes/improvements to those languages.

+

Expected audience

+

The audience should be interested in practical aspects of programming language semantics. This includes interest in semantics as objects to be created/studied, as well as interest in the using such semantics for different program analyses. They need no previous knowledge, although a basic understanding of other definitional styles (such as SOS or evaluation contexts) may be helpful.

+
+
+ + +
+ + + +
+
+ +
+
+ + + + + + + + + + + + diff --git a/exports/K.epub b/exports/K.epub new file mode 100644 index 0000000000000000000000000000000000000000..1c0679c93a58fe69ed6c8ee214ef6ce9911c3ff2 GIT binary patch literal 1820070 zcmb5VQ;=p+v#wiqmu>Sa+qP}nwr$(CZC96V+vu{obb76|FV4TO)}9eLBj$`87h}W| z7ej`Rzm&v%74y4ApiCH_dx$MSy2^1T1h!EdQhPMbWF&HaLM}T@bu5n|NoB0 z_I56YmUgC2bRM=gs&RwxLxhMTw|?Lo)s_o1fiK!>U8=+lU=<7f7}hqIj^^g4vH1Ce z(U2#H9M~MM6qw<`ZMZaD67N4`S)Ayg(s5CZ9Yw?t)Kyx)BM(R1pi!%}Y^`Vl4+ul% zGdpKzl{M#jiBErj&>8k(y_IN*&sjnsSvak)s1t)`B)OwoYm;QsRkj559;L?pf_zZ# zDmsoUJz+b(@&x_Q-DnQ-Wh{dO0cGO>0RjH)#@5x=l!<|fnS+UwmVmtEE*fmh^x{_FIWiK$5T{!)#|eZ3 zQ6>w)lbf01KvmT(-04sb%1m zlFTrauIv);$Io}z%R(^Ae5xrLv%Vzko3$saR=1{rPwwY$Fy!Ookq`mmx?f029GIA! z70c{ONSOgfYDls%q@j&0K{)&(bT@Tf6ii;SM!UxW(994i^R7a(V2LoI?Gk|^dD96< zh`@-5w8kGS#KNd~u8cr>NqH*L`?8d**IUE9SfWlT6ZMSNHVO!g6^dde{6N!jHubSE zk}VS;=vf$#^m~sHl8VMMWMJ@uz(e$gb3s6&V%r%Exso%WQ!QhPot4il3lPD>T}C6a z854$&(eDI%U`bM~Z!8#Hu z;u=vvU6usC0&ph3h5R*P 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      +

      K

      + +

      Table of Contents

      + +       +
      +

      K is a rewrite-based
      +executable semantic framework in which programming languages, type
      +systems and formal analysis tools can be defined using configurations
      +and rules. Configurations organize the state in units called cells,
      +which are labeled and can be nested. K rewrite rules make it explicit
      +which parts of the term are read-only, write-only, read-write, or
      +unused. This makes K suitable for defining truly concurrent languages
      +even in the presence of sharing. Computations are represented as
      +syntactic extensions of the original language abstract syntax, using a
      +nested list structure which sequentializes computational tasks, such
      +as program fragments. Computations are like any other terms in a
      +rewriting environment: they can be matched, moved from one place to
      +another, modified, or deleted. This makes K suitable for defining
      +control-intensive features such as abrupt termination, exceptions, or
      +call/cc.

      +

      K Tool Download

      + +
        +
      • Install from the latest K GitHub Release.
        • +
      • Try our Editor Support page for links to K syntax highlighting definitions for various popular editors/IDEs. Please feel free to contribute.
        • +
      • Build or browse the code on GitHub, where you can also report bugs.
        • +
      +

      Learn K

      + + +

      Support

      + + +

      Resources

      + + +

      K Tutorial

      + +

      The purpose of this series of lessons is to teach developers how to program in
      +K. While the primary use of K is in the specification of operational semantics
      +of programming languages, this tutorial is agnostic on how the knowledge of K
      +is used. For a more detailed tutorial explaining the basic principles of
      +programming language design, refer to the
      +K PL Tutorial. Note that that tutorial is somewhat
      +out of date presently.

      +

      This K tutorial is a work in progress. Many lessons are currently simply
      +placeholders for future content.

      +

      To start the K tutorial, begin with
      +Section 1: Basic Programming in K.

      +

      Section 1: Basic K Concepts

      + +

      The goal of this first section of the K tutorial is to teach the basic
      +principles of K to someone with no prior experience with K as a programming
      +language. However, this is not written with the intended audience of someone
      +who is a complete beginner to programming. We are assuming that the reader
      +has a firm grounding in computer science broadly, as well as that they have
      +experience writing code in functional programming languages before.

      +

      By the end of this section, the reader ought to be able to write specifications
      +of simple languages in K, use these specifications to generate a fast
      +interpreter for their programming language, as well as write basic deductive
      +program verification proofs over programs in their language. This should give
      +them the theoretical grounding they need to begin expanding their knowledge
      +of K in Section 2: Intermediate K Concepts.

      +

      To begin this section, refer to
      +Lesson 1.1: Setting up a K Environment.

      +

      Lesson 1.1: Setting up a K Environment

      + +

      The first step to learning K is to install K on your system, and configure your
      +editor for K development.

      +

      Installing K

      + +

      You have two options for how to install K, depending on how you intend to
      +interact with the K codebase. If you are solely a user of K, and have no
      +interest in developing or making changes to K, you most likely will want to
      +install one of our binary releases of K. However, if you are going to be a K
      +developer, or simply want to build K from source, you should follow the
      +instructions for a source build of K.

      +

      Installing K from a binary release

      + +

      K is developed as a rolling release, with each change to K that passes our
      +CI infrastructure being deployed on GitHub for download. The latest release of
      +K can be downloaded here.
      +This page also contains information on how to install K. It is recommended
      +that you fully uninstall the old version of K prior to installing the new one,
      +as K does not maintain entries in package manager databases, with the exception
      +of Homebrew on MacOS.

      +

      Installing K from source

      + +

      You can clone K from GitHub with the following Git command:

      +
      git clone https://github.com/runtimeverification/k --recursive
+

      Instructions on how to build K from source can be found
      +here.

      +

      Configuring your editor

      + +

      K maintains a set of scripts for a variety of text editors, including vim and
      +emacs, in various states of maintenance. You can download these scripts with
      +the following Git command:

      +
      git clone https://github.com/kframework/k-editor-support
+

      Because K allows users to define their own grammars for parsing K itself,
      +not all features of K can be effectively highlighted. However, at the cost of
      +occasionally highlighting things incorrectly, you can get some pretty good
      +results in many cases. With that being said, some of the editor scripts in the
      +above repository are pretty out of date. If you manage to improve them, we
      +welcome pull requests into the repository.

      +

      Troubleshooting

      + +

      If you have problems installing K, we encourage you to reach out to us. If you
      +follow the above install instructions and run into a problem, you can
      +Create a bug report on GitHub

      +

      Next lesson

      + +

      Once you have set up K on your system to your satisfaction, you can continue to
      +Lesson 1.2: Basics of Functional K.

      +

      Lesson 1.2: Basics of Functional K

      + +

      The purpose of this lesson is to explain the basics of productions and
      +rules in K. These are two types of K sentences. A K file consists of
      +one or more requires or modules in K. Each module consists of one or
      +more imports or sentences. For more information on requires, modules, and
      +sentences, refer to Lesson 1.5. However, for the time
      +being, just think of a module as a container for sentences, and don't worry
      +about requires or imports just yet.

      +

      Our first K program

      + +

      To start with, input the following program into your editor as file
      +lesson-02-a.k:

      +
      module LESSON-02-A
+
+  syntax Color ::= Yellow() | Blue()
+  syntax Fruit ::= Banana() | Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+

      Save this file and then run:

      +
      kompile lesson-02-a.k
+

      kompile is K's compiler. By default, it takes a program or specification
      +written in K and compiles it into an interpreter for that input. Right now we
      +are compiling a single file. A set of K files that are compiled together are
      +called a K definition. We will cover multiple file K definitions later on.
      +kompile will output a directory containing everything needed to execute
      +programs and perform proofs using that definition. In this case, kompile will
      +(by default) create the directory lesson-02-a-kompiled under the current
      +directory.

      +

      Now, save the following input file in your editor as banana.color in the same
      +directory as lesson-02-a.k:

      +
      colorOf(Banana())
+

      We can now evaluate this K term by running (from the same directory):

      +
      krun banana.color
+

      krun will use the interpreter generated by the first call to kompile to
      +execute this program.

      +

      You will get the following output:

      +
      <k>
+  Yellow ( ) ~> .
+</k>
+

      For now, don't worry about the <k>, </k>, or ~> . portions of this
      +output file.

      +

      You can also execute small programs directly by specifying them on the command
      +line instead of putting them in a file. For example, the same program above
      +could also have been executed by running the following command:

      +
      krun -cPGM='colorOf(Banana())'
+

      Now, let's look at what this definition and program did.

      +

      Productions, Constructors, and Functions

      + +

      The first thing to realize is that this K definition contains 5 productions.
      +Productions are introduced with the syntax keyword, followed by a sort,
      +followed by the operator ::= followed by the definition of one or more
      +productions themselves, separated by the | operator. There are different
      +types of productions, but for now we only care about constructors and
      +functions. Each declaration separated by the | operator is individually
      +a single production, and the | symbol simply groups together productions that
      +have the same sort. For example, we could equally have written an identical K
      +definition lesson-02-b.k like so:

      +
      module LESSON-02-B
+
+  syntax Color ::= Yellow()
+  syntax Color ::= Blue()
+  syntax Fruit ::= Banana()
+  syntax Fruit ::= Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+

      You can try compiling and running lesson-02-b.k to see that it produces the same output as lesson-02-a.k:

      +
      kompile lesson-02-b.k
+krun -cPGM='colorOf(Banana())' --definition 'lesson-02-b-kompiled'
+

      where the --definition attribute points to the directory containing a compiled version of LESSON-02-B.
      +Even the following definition is equivalent:

      +
      module LESSON-02-C
+
+  syntax Color ::= Yellow()
+                 | Blue()
+                 | colorOf(Fruit) [function]
+  syntax Fruit ::= Banana()
+                 | Blueberry()
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+

      Each of these types of productions named above has the same underlying syntax,
      +but context and attributes are used to distinguish between the different
      +types. Tokens, brackets, lists, macros, aliases, and anywhere productions will
      +be covered in a later lesson, but this lesson does introduce us to constructors
      +and functions. Yellow(), Blue(), Banana(), and Blueberry() are
      +constructors. You can think of a constructor like a constructor for an
      +algebraic data type, if you're familiar with a functional language. The data
      +type itself is the sort that appears on the left of the ::= operator. Sorts
      +in K consist of uppercase identifiers.

      +

      Constructors can have arguments, but these ones do not. We will cover the
      +syntax of productions in detail in the next lesson, but for now, you can write
      +a production with no arguments as an uppercase or lowercase identifier followed
      +by the () operator.

      +

      A function is distinguished from a constructor by the presence of the
      +function attribute. Attributes appear in a comma separated list between
      +square brackets after any sentence, including both productions and rules.
      +Various attributes with built-in meanings exist in K and will be discussed
      +throughout the tutorial.

      +

      Exercise

      + +

      Use krun to compute the return value of the colorOf function on a
      +Blueberry().

      +

      Rules, Matching, and Variables

      + +

      Functions in K are given definitions using rules. A rule begins with the rule
      +keyword and contains at least one rewrite operator. The rewrite operator
      +is represented by the syntax =>. The rewrite operator is one of the built-in
      +productions in K, and we will discuss in more detail how it can be used in
      +future lessons, but for now, you can think of a rule as consisting of a
      +left-hand side and a right-hand side, separated by the rewrite
      +operator. On the left-hand side is the name of the function and zero or more
      +patterns corresponding to the parameters of the function. On the right-hand
      +side is another pattern. The meaning of the rule is relatively simple, having
      +defined these components. If the function is called with arguments that
      +match the patterns on the left-hand side, then the return value of the
      +function is the pattern on the right-hand side.

      +

      For example, in the above example, if the argument of the colorOf function
      +is Banana(), then the return value of the function is Yellow().

      +

      So far we have introduced that a constructor is a type of pattern in K. We
      +will introduce more complex patterns in later lessons, but there is one other
      +type of basic pattern: the variable. A variable, syntactically, consists
      +of an uppercase identifier. However, unlike a constructor, a variable will
      +match any pattern with one exception: Two variables with the same name
      +must match the same pattern.

      +

      Here is a more complex example (lesson-02-d.k):

      +
      module LESSON-02-D
+
+  syntax Container ::= Jar(Fruit)
+  syntax Fruit ::= Apple() | Pear()
+
+  syntax Fruit ::= contentsOfJar(Container) [function]
+
+  rule contentsOfJar(Jar(F)) => F
+
+endmodule
+

      Here we see that Jar is a constructor with a single argument. You can write a
      +production with multiple arguments by putting the sorts of the arguments in a
      +comma-separated list inside the parentheses.

      +

      In this example, F is a variable. It will match either Apple() or Pear().
      +The return value of the function is created by substituting the matched
      +values of all of the variables into the variables on the right-hand side of
      +the rule.

      +

      To demonstrate, compile this definition and execute the following program with
      +krun:

      +
      contentsOfJar(Jar(Apple()))
+

      You will see when you run it that the program returns Apple(), because that
      +is the pattern that was matched by F.

      +

      Exercises

      + +
        +
      1. Extend the definition in lesson-02-a.k with the addition of blackberries
        +and kiwis. For simplicity, blackberries are black and kiwis are green. Then
        +compile your definition and test that your additional fruits are correctly
        +handled by the colorOf function.
        2. +
      2. Create a new definition which defines an outfit as a multi-argument
        +constructor consisting of a hat, shirt, pants, and shoes. Define a new sort,
        +Boolean, with two constructors, true and false. Each of hat, shirt, pants,
        +and shoes will have a single argument (a color), either black or
        +white. Then define an outfitMatching function that will return true if all
        +the pieces of the outfit are the same color. You do not need to define the
        +case that returns false. Write some tests that your function behaves the way
        +you expect.
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.3: BNF Syntax and Parser Generation.

      +

      Lesson 1.3: BNF Syntax and Parser Generation

      + +

      The purpose of this lesson is to explain the full syntax and semantics of
      +productions in K as well as how productions and other syntactic
      +sentences can be used to define grammars for use parsing both rules as well
      +as programs.

      +

      K's approach to parsing

      + +

      K's grammar is divided into two components: the outer syntax of K and the
      +inner syntax of K. Outer syntax refers to the parsing of requires,
      +modules, imports, and sentences in a K definition. Inner syntax
      +refers to the parsing of rules and programs. Unlike the outer syntax of
      +K, which is predetermined, much of the inner syntax of K is defined by you, the
      +developer. When rules or programs are parsed, they are parsed within the
      +context of a module. Rules are parsed in the context of the module in which
      +they exist, whereas programs are parsed in the context of the
      +main syntax module of a K definition. The productions and other syntactic
      +sentences in a module are used to construct the grammar of the module, which
      +is then used to perform parsing.

      +

      Basic BNF productions

      + +

      To illustrate how this works, we will consider a simple K definition which
      +defines a relatively basic calculator capable of evaluating Boolean expressions
      +containing and, or, not, and xor.

      +

      Input the following program into your editor as file lesson-03-a.k:

      +
      module LESSON-03-A
+
+  syntax Boolean ::= "true" | "false"
+                   | "!" Boolean [function]
+                   | Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+

      You will notice that the productions in this file look a little different than
      +the ones from the previous lesson. In point of fact, K has two different
      +mechanisms for defining productions. We have previously been focused
      +exclusively on the first mechanism, where the ::= symbol is followed by an
      +alphanumeric identifier followed by a comma-separated list of sorts in
      +parentheses. However, this is merely a special case of a more generic mechanism
      +for defining the syntax of productions using a variant of
      +BNF Form.

      +

      For example, in the previous lesson, we had the following set of productions:

      +
      module LESSON-03-B
+  syntax Color ::= Yellow() | Blue()
+  syntax Fruit ::= Banana() | Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+endmodule
+

      It turns out that this is equivalent to the following definition which defines
      +the same grammar, but using BNF notation:

      +
      module LESSON-03-C
+  syntax Color ::= "Yellow" "(" ")" | "Blue" "(" ")"
+  syntax Fruit ::= "Banana" "(" ")" | "Blueberrry" "(" ")"
+  syntax Color ::= "colorOf" "(" Fruit ")" [function]
+endmodule
+

      In this example, the sorts of the argument to the function are unchanged, but
      +everything else has been wrapped in double quotation marks. This is because
      +in BNF notation, we distinguish between two types of production items:
      +terminals and non-terminals. A terminal represents simply a literal
      +string of characters that is verbatim part of the syntax of that production.
      +A non-terminal, conversely, represents a sort name, where the syntax of that
      +production accepts any valid term of that sort at that position.

      +

      This is why, when we wrote the program colorOf(Banana()), krun was able to
      +execute that program: because it represented a term of sort Color that was
      +parsed and interpreted by K's interpreter. In other words, krun parses and
      +interprets terms according to the grammar defined by the developer. It is
      +automatically converted into an AST of that term, and then the colorOf
      +function is evaluated using the function rules provided in the definition.

      +

      You can ask yourself: How does K match the strings between the double quotes?
      +The answer is that K uses Flex to generate a scanner for the grammar. Flex looks
      +for the longest possible match of a regular expression in the input. If there
      +are ambiguities between 2 or more regular expressions, it will pick the one with
      +the highest prec attribute. You can learn more about how Flex matching works
      +here.

      +

      Bringing us back to the file lesson-03-a.k, we can see that this grammar
      +has given a simple BNF grammar for expressions over Booleans. We have defined
      +constructors corresponding to the Boolean values true and false, and functions
      +corresponding to the Boolean operators for and, or, not, and xor. We have also
      +given a syntax for each of these functions based on their syntax in the C
      +programming language. As such, we can now write programs in the simple language
      +we have defined.

      +

      Input the following program into your editor as and.bool in the same
      +directory:

      +
      true && false
+

      We cannot interpret this program yet, because we have not given rules defining
      +the meaning of the && function yet, but we can parse it. To do this, you can
      +run (from the same directory):

      +
      kast --output kore and.bool
+

      kast is K's just-in-time parser. It will generate a grammar from your K
      +definition on the fly and use it to parse the program passed on the command
      +line. The --output flag controls how the resulting AST is represented; don't
      +worry about the possible values yet, just use kore.

      +

      You ought to get the following AST printed on standard output, minus the
      +formatting:

      +
      inj{SortBoolean{}, SortKItem{}}(
+  Lbl'UndsAnd-And-UndsUnds'LESSON-03-A'Unds'Boolean'Unds'Boolean'Unds'Boolean{}(
+    Lbltrue'Unds'LESSON-03-A'Unds'Boolean{}(),
+    Lblfalse'Unds'LESSON-03-A'Unds'Boolean{}()
+  )
+)
+

      Don't worry about what exactly this means yet, just understand that it
      +represents the AST of the program that you just parsed. You ought to be able
      +to recognize the basic shape of it by seeing the words true, false, and
      +And in there. This is Kore, the intermediate representation of K, and we
      +will cover it in detail later.

      +

      Note that you can also tell kast to print the AST in other formats. For a
      +more direct representation of the original K, while still maintaining the
      +structure of an AST, you can say kast --output kast and.bool. This will
      +yield the following output:

      +
      `_&&__LESSON-03-A_Boolean_Boolean_Boolean`(
+  `true_LESSON-03-A_Boolean`(.KList),
+  `false_LESSON-03-A_Boolean`(.KList)
+)
+

      Note how the first output is largely a name-mangled version of the second
      +output. The one difference is the presence of the inj symbol in the KORE
      +output. We will talk more about this in later lessons.

      +

      Exercise

      + +

      Parse the expression false || true with --output kast. See if you can
      +predict approximately what the corresponding output would be with
      +--output kore, then run the command yourself and compare it to your
      +prediction.

      +

      Ambiguities

      + +

      Now let's try a slightly more advanced example. Input the following program
      +into your editor as and-or.bool:

      +
      true && false || false
+

      When you try and parse this program, you ought to see the following error:

      +
      [Error] Inner Parser: Parsing ambiguity.
+1: syntax Boolean ::= Boolean "||" Boolean [function]
+
+`_||__LESSON-03-A_Boolean_Boolean_Boolean`(`_&&__LESSON-03-A_Boolean_Boolean_Boolean`(`true_LESSON-03-A_Boolean`(.KList),`false_LESSON-03-A_Boolean`(.KList)),`false_LESSON-03-A_Boolean`(.KList))
+2: syntax Boolean ::= Boolean "&&" Boolean [function]
+
+`_&&__LESSON-03-A_Boolean_Boolean_Boolean`(`true_LESSON-03-A_Boolean`(.KList),`_||__LESSON-03-A_Boolean_Boolean_Boolean`(`false_LESSON-03-A_Boolean`(.KList),`false_LESSON-03-A_Boolean`(.KList)))
+        Source(./and-or.bool)
+        Location(1,1,1,23)
+

      This error is saying that kast was unable to parse this program because it is
      +ambiguous. K's just-in-time parser is a GLL parser, which means it can handle
      +the full generality of context-free grammars, including those grammars which
      +are ambiguous. An ambiguous grammar is one where the same string can be parsed
      +as multiple distinct ASTs. In this example, it can't decide whether it should
      +be parsed as (true && false) || false or as true && (false || false). As a
      +result, it reports the error to the user.

      +

      Brackets

      + +

      Currently there is no way of resolving this ambiguity, making it impossible
      +to write complex expressions in this language. This is obviously a problem.
      +The standard solution in most programming languages to this problem is to
      +use parentheses to indicate the appropriate grouping. K generalizes this notion
      +into a type of production called a bracket. A bracket production in K
      +is any production with the bracket attribute. It is required that such a
      +production only have a single non-terminal, and the sort of the production
      +must equal the sort of that non-terminal. However, K does not otherwise
      +impose restrictions on the grammar the user provides for a bracket. With that
      +being said, the most common type of bracket is one in which a non-terminal
      +is surrounded by terminals representing some type of bracket such as
      +(), [], {}, <>, etc. For example, we can define the most common
      +type of bracket, the type used by the vast majority of programming languages,
      +quite simply.

      +

      Consider the following modified definition, which we will save to
      +lesson-03-d.k:

      +
      module LESSON-03-D
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   | "!" Boolean [function]
+                   | Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+

      In this definition, if the user does not explicitly define parentheses, the
      +grammar remains ambiguous and K's just-in-time parser will report an error.
      +However, you are now able to parse more complex programs by means of explicitly
      +grouping subterms with the bracket we have just defined.

      +

      Consider and-or-left.bool:

      +
      (true && false) || false
+

      Now consider and-or-right.bool:

      +
      true && (false || false)
+

      If you parse these programs with kast, you will once again get a single
      +unique AST with no error. If you look, you might notice that the bracket itself
      +does not appear in the AST. In fact, this is a property unique to brackets:
      +productions with the bracket attribute are not represented in the parsed AST
      +of a term, and the child of the bracket is folded immediately into the parent
      +term. This is the reason for the requirement that a bracket production have
      +a single non-terminal of the same sort as the production itself.

      +

      Exercise

      + +

      Write out what you expect the AST to be arising from parsing these two programs
      +above with --output kast, then parse them yourself and compare them to the
      +AST you expected. Confirm for yourself that the bracket production does not
      +appear in the AST.

      +

      Tokens

      + +

      So far we have seen how we can define the grammar of a language. However,
      +the grammar is not the only relevant part of parsing a language. Also relevant
      +is the lexical syntax of the language. Thus far, we have implicitly been using
      +K's automatic lexer generation to generate a token in the scanner for each
      +terminal in our grammar. However, sometimes we wish to define more complex
      +lexical syntax. For example, consider the case of integers in C: an integer
      +consists of a decimal, octal, or hexadecimal number followed by an optional
      +suffix indicating the type of the literal.

      +

      In theory it would be possible to define this syntax via a grammar, but not
      +only would it be cumbersome and tedious, you would also then have to deal with
      +an AST generated for the literal which is not convenient to work with.

      +

      Instead of doing this, K allows you to define token productions, where
      +a production consists of a regular expression followed by the token
      +attribute, and the resulting AST consists of a typed string containing the
      +value recognized by the regular expression.

      +

      For example, the builtin integers in K are defined using the following
      +production:

      +
      syntax Int ::= r"[\\+-]?[0-9]+" [token]
+

      Here we can see that we have defined that an integer is an optional sign
      +followed by a nonzero sequence of digits. The r preceding the terminal
      +indicates that what appears inside the double quotes is a regular expression,
      +and the token attribute indicates that terms which parse as this production
      +should be converted into a token by the parser.

      +

      It is also possible to define tokens that do not use regular expressions. This
      +can be useful when you wish to declare particular identifiers for use in your
      +semantics later. For example:

      +
      syntax Id ::= "main" [token]
+

      Here, we declare that main is a token of sort Id. Instead of being parsed
      +as a symbol, it gets parsed as a token, generating a typed string in the AST.
      +This is useful in a semantics of C because the parser generally does not treat
      +the main function in C specially; only the semantics treats it specially.

      +

      Of course, languages can have more complex lexical syntax. For example, if we
      +wish to define the syntax of integers in C, we could use the following
      +production:

      +
      syntax IntConstant ::= r"(([1-9][0-9]*)|(0[0-7]*)|(0[xX][0-9a-fA-F]+))(([uU][lL]?)|([uU]((ll)|(LL)))|([lL][uU]?)|(((ll)|(LL))[uU]?))?" [token]
+

      As you may have noted above, long and complex regular expressions
      +can be hard to read. They also suffer from the problem that unlike a grammar,
      +they are not particularly modular.

      +

      We can get around this restriction by declaring explicit regular expressions,
      +giving them a name, and then referring to them in productions.

      +

      Consider the following (equivalent) way to define the lexical syntax of
      +integers in C:

      +
      syntax IntConstant ::= r"({DecConstant}|{OctConstant}|{HexConstant})({IntSuffix}?)" [token]
+syntax lexical DecConstant = r"{NonzeroDigit}({Digit}*)"
+syntax lexical OctConstant = r"0({OctDigit}*)"
+syntax lexical HexConstant = r"{HexPrefix}({HexDigit}+)"
+syntax lexical HexPrefix = r"0x|0X"
+syntax lexical NonzeroDigit = r"[1-9]"
+syntax lexical Digit = r"[0-9]"
+syntax lexical OctDigit = r"[0-7]"
+syntax lexical HexDigit = r"[0-9a-fA-F]"
+syntax lexical IntSuffix = r"{UnsignedSuffix}({LongSuffix}?)|{UnsignedSuffix}{LongLongSuffix}|{LongSuffix}({UnsignedSuffix}?)|{LongLongSuffix}({UnsignedSuffix}?)"
+syntax lexical UnsignedSuffix = r"[uU]"
+syntax lexical LongSuffix = r"[lL]"
+syntax lexical LongLongSuffix = r"ll|LL"
+

      As you can see, this is rather more verbose, but it has the benefit of both
      +being much easier to read and understand, and also increased modularity.
      +Note that we refer to a named regular expression by putting the name in curly
      +brackets. Note also that only the first sentence actually declares a new piece
      +of syntax in the language. When the user writes syntax lexical, they are only
      +declaring a regular expression. To declare an actual piece of syntax in the
      +grammar, you still must actually declare an explicit token production.

      +

      One final note: K uses Flex to implement
      +its lexical analysis. As a result, you can refer to the
      +Flex Manual
      +for a detailed description of the regular expression syntax supported. Note
      +that for performance reasons, Flex's regular expressions are actually a regular
      +language, and thus lack some of the syntactic convenience of modern
      +"regular expression" libraries. If you need features that are not part of the
      +syntax of Flex regular expressions, you are encouraged to express them via
      +a grammar instead.

      +

      Ahead-of-time parser generation

      + +

      So far we have been entirely focused on K's support for just-in-time parsing,
      +where the parser is generated on the fly prior to being used. This benefits
      +from being faster to generate the parser, but it suffers in performance if you
      +have to repeatedly parse strings with the same parser. For this reason, it is
      +generally encouraged that when parsing programs, you use K's ahead-of-time
      +parser generation. K makes use of
      +GNU Bison to generate parsers.

      +

      By default, you can enable ahead-of-time parsing via the --gen-bison-parser
      +flag to kompile. This will make use of Bison's LR(1) parser generator. As
      +such, if your grammar is not LR(1), it may not parse exactly the same as if
      +you were to use the just-in-time parser, because Bison will automatically pick
      +one of the possible branches whenever it encounters a shift-reduce or
      +reduce-reduce conflict. In this case, you can either modify your grammar to be
      +LR(1), or you can enable use of Bison's GLR support by instead passing
      +--gen-glr-bison-parser to kompile. Note that if your grammar is ambiguous,
      +the ahead-of-time parser will not provide you with particularly readable error
      +messages at this time.

      +

      If you have a K definition named foo.k, and it generates a directory when
      +you run kompile called foo-kompiled, you can invoke the ahead-of-time
      +parser you generated by running foo-kompiled/parser_PGM <file> on a file.

      +

      Exercises

      + +
        +
      1. +

        Compile lesson-03-d.k with ahead-of-time parsing enabled. Then compare
        +how long it takes to run kast --output kore and-or-left.bool with how long it
        +takes to run lesson-03-d-kompiled/parser_PGM and-or-left.bool. Confirm for
        +yourself that both produce the same result, but that the latter is faster.

        +
        2. +
      2. +

        Define a simple grammar consisting of integers, brackets, addition,
        +subtraction, multiplication, division, and unary negation. Integers should be
        +in decimal form and lexically without a sign, whereas negative numbers can be
        +represented via unary negation. Ensure that you are able to parse some basic
        +arithmetic expressions using a generated ahead-of-time parser. Do not worry
        +about disambiguating the grammar or about writing rules to implement the
        +operations in this definition.

        +
        3. +
      3. +

        Write a program where the meaning of the arithmetic expression based on
        +the grammar you defined above is ambiguous, and then write programs that
        +express each individual intended meaning using brackets.

        +
        4. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.4: Disambiguating Parses.

      +

      Lesson 1.4: Disambiguating Parses

      + +

      The purpose of this lesson is to teach how to use K's builtin features for
      +disambiguation to transform an ambiguous grammar into an unambiguous one that
      +expresses the intended ASTs.

      +

      Priority blocks

      + +

      In practice, very few formal languages outside the domain of natural language
      +processing are ambiguous. The main reason for this is that parsing unambiguous
      +languages is asymptotically faster than parsing ambiguous languages.
      +Programming language designers instead usually use the notions of operator
      +precedence and associativity to make expression grammars unambiguous. These
      +mechanisms work by instructing the parser to reject certain ASTs in favor of
      +others in case of ambiguities; it is often possible to remove all ambiguities
      +in a grammar with these techniques.

      +

      While it is sometimes possible to explicitly rewrite the grammar to remove
      +these parses, because K's grammar specification and AST generation are
      +inextricably linked, this is generally discouraged. Instead, we use the
      +approach of explicitly expressing the relative precedence of different
      +operators in different situations in order to resolve the ambiguity.

      +

      For example, in C, && binds tighter in precedence than ||, meaning that
      +the expression true && false || false has only one valid AST:
      +(true && false) || false.

      +

      Consider, then, the third iteration on the grammar of this definition
      +(lesson-04-a.k):

      +
      module LESSON-04-A
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > Boolean "&&" Boolean [function]
+                   > Boolean "^" Boolean [function]
+                   > Boolean "||" Boolean [function]
+
+endmodule
+

      In this example, some of the | symbols separating productions in a single
      +block have been replaced with >. This serves to describe the
      +priority groups associated with this block of productions.
      +The first priority group consists of the atoms of the
      +language: true, false, and the bracket operator. In general, a priority
      +group starts either at the ::= or > operator and extends until either the
      +next > operator or the end of the production block. Thus, we can see that the
      +second, third, fourth, and fifth priority groups in this grammar all consist
      +of a single production.

      +

      The meaning of these priority groups becomes apparent when parsing programs:
      +A symbol with a lesser priority, (i.e., one that binds looser), cannot
      +appear as the direct child of a symbol with a greater priority (i.e.,
      +one that binds tighter. In this case, the > operator can be seen as a
      +greater-than operator describing a transitive partial ordering on the
      +productions in the production block, expressing their relative priority.

      +

      To see this more concretely, let's look again at the program
      +true && false || false. As noted before, previously this program was
      +ambiguous because the parser could either choose that && was the child of ||
      +or vice versa. However, because a symbol with lesser priority (i.e., ||)
      +cannot appear as the direct child of a symbol with greater priority
      +(i.e., &&), the parser will reject the parse where || is under the
      +&& operator. As a result, we are left with the unambiguous parse
      +(true && false) || false. Similarly, true || false && false parses
      +unambiguously as true || (false && false). Conversely, if the user explicitly
      +wants the other parse, they can express this using brackets by explicitly
      +writing true && (false || false). This still parses successfully because the
      +|| operator is no longer the direct child of the && operator, but is
      +instead the direct child of the () operator, and the && operator is an
      +indirect parent, which is not subject to the priority restriction.

      +

      Astute readers, however, will already have noticed what seems to be a
      +contradiction: we have defined () as also having greater priority than ||.
      +One would think that this should mean that || cannot appear as a direct
      +child of (). This is a problem because priority groups are applied to every
      +possible parse separately. That is to say, even if the term is unambiguous
      +prior to this disambiguation rule, we still reject that parse if it violates
      +the rule of priority.

      +

      In fact, however, we do not reject this program as a parse error. Why is that?
      +Well, the rule for priority is slightly more complex than previously described.
      +In actual fact, it applies only conditionally. Specifically, it applies in
      +cases where the child is either the first or last production item in the
      +parent's production. For example, in the production Bool "&&" Bool, the
      +first Bool non-terminal is not preceded by any terminals, and the last Bool
      +non-terminal is not followed by any terminals. As a result of this, we apply
      +the priority rule to both children of &&. However, in the () operator,
      +the sole non-terminal is both preceded by and followed by terminals. As a
      +result, the priority rule is not applied when () is the parent. Because of
      +this, the program we mentioned above successfully parses.

      +

      Exercise

      + +

      Parse the program true && false || false using kast, and confirm that the AST
      +places || as the top level symbol. Then modify the definition so that you
      +will get the alternative parse.

      +

      Associativity

      + +

      Even having broken the expression grammar into priority blocks, the resulting
      +grammar is still ambiguous. We can see this if we try to parse the following
      +program (assoc.bool):

      +
      true && false && false
+

      Priority blocks will not help us here: the problem comes between two parses
      +where both possible parses have a direct parent and child which is within a
      +single priority block (in this case, && is in the same block as itself).

      +

      This is where the notion of associativity comes into play. Associativity
      +applies the following additional rules to parses:

      +
        +
      • a left-associative symbol cannot appear as a direct rightmost child of a
        +symbol with equal priority;
        • +
      • a right-associative symbol cannot appear as a direct leftmost child of a
        +symbol with equal priority; and
        • +
      • a non-associative symbol cannot appear as a direct leftmost or rightmost
        +child of a symbol with equal priority.
        • +
      +

      In C, binary operators are all left-associative, meaning that the expression
      +true && false && false parses unambiguously as (true && false) && false,
      +because && cannot appear as the rightmost child of itself.

      +

      Consider, then, the fourth iteration on the grammar of this definition
      +(lesson-04-b.k):

      +
      module LESSON-04-B
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > left: Boolean "&&" Boolean [function]
+                   > left: Boolean "^" Boolean [function]
+                   > left: Boolean "||" Boolean [function]
+
+endmodule
+

      Here each priority group, immediately after the ::= or > operator, can
      +be followed by a symbol representing the associativity of that priority group:
      +either left: for left associativity, right: for right associativity, or
      +non-assoc: for non-associativity. In this example, each priority group we
      +apply associativity to has only a single production, but we could equally well
      +write a priority block with multiple productions and an associativity.

      +

      For example, consider the following, different grammar (lesson-04-c.k):

      +
      module LESSON-04-C
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > left:
+                     Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+

      In this example, unlike the one above, &&, ^, and || have the same
      +priority. However, viewed as a group, the entire group is left associative.
      +This means that none of &&, ^, and || can appear as the right child of
      +any of &&, ^, or ||. As a result of this, this grammar is also not
      +ambiguous. However, it expresses a different grammar, and you are encouraged
      +to think about what the differences are in practice.

      +

      Exercise

      + +

      Parse the program true && false && false yourself, and confirm that the AST
      +places the rightmost && at the top of the expression. Then modify the
      +definition to generate the alternative parse.

      +

      Explicit priority and associativity declarations

      + +

      Previously we have only considered the case where all of the productions
      +which you wish to express a priority or associativity relation over are
      +co-located in the same block of productions. However, in practice this is not
      +always feasible or desirable, especially as a definition grows in size across
      +multiple modules.

      +

      As a result of this, K provides a second way of declaring priority and
      +associativity relations.

      +

      Consider the following grammar, which we will name lesson-04-d.k and which
      +will express the exact same grammar as lesson-04-b.k

      +
      module LESSON-04-D
+
+  syntax Boolean ::= "true" [group(literal)] | "false" [group(literal)]
+                   | "(" Boolean ")" [group(atom), bracket]
+                   | "!" Boolean [group(not), function]
+                   | Boolean "&&" Boolean [group(and), function]
+                   | Boolean "^" Boolean [group(xor), function]
+                   | Boolean "||" Boolean [group(or), function]
+
+  syntax priorities literal atom > not > and > xor > or
+  syntax left and
+  syntax left xor
+  syntax left or
+endmodule
+

      This introduces a couple of new features of K. First, the group(_) attribute
      +is used to conceptually group together sets of sentences under a common
      +user-defined name. For example, literal in the syntax priorities sentence is
      +used to refer to all the productions marked with the group(literal) attribute,
      +i.e., true and false. A production can belong to multiple groups using
      +syntax such as group(myGrp1,myGrp2).

      +

      Once we understand this, it becomes relatively straightforward to understand
      +the meaning of this grammar. Each syntax priorities sentence defines a
      +priority relation where > separates different priority groups. Each priority
      +group is defined by a list of one or more group names, and consists of all
      +productions which are members of at least one of those named groups.

      +

      In the same way, a syntax left, syntax right, or syntax non-assoc sentence
      +defines an associativity relation among left-, right-, or non-associative
      +groups. Specifically, this means that:

      +
      syntax left a b
+

      is different to:

      +
      syntax left a
+syntax left b
+

      As a consequence of this, syntax [left|right|non-assoc] should not be used to
      +group together labels with different priority.

      +

      Prefer/avoid

      + +

      Sometimes priority and associativity prove insufficient to disambiguate a
      +grammar. In particular, sometimes it is desirable to be able to choose between
      +two ambiguous parses directly while still not rejecting any parses if the term
      +parsed is unambiguous. A good example of this is the famous "dangling else"
      +problem in imperative C-like languages.

      +

      Consider the following definition (lesson-04-E.k):

      +
      module LESSON-04-E
+
+  syntax Exp ::= "true" | "false"
+  syntax Stmt ::= "if" "(" Exp ")" Stmt
+                | "if" "(" Exp ")" Stmt "else" Stmt
+                | "{" "}"
+endmodule
+

      We can write the following program (dangling-else.if):

      +
      if (true) if (false) {} else {}
+

      This is ambiguous because it is unclear whether the else clause is part of
      +the outer if or the inner if. At first we might try to resolve this with
      +priorities, saying that the if without an else cannot appear as a child of
      +the if with an else. However, because the non-terminal in the parent symbol
      +is both preceded and followed by a terminal, this will not work.

      +

      Instead, we can resolve the ambiguity directly by telling the parser to
      +"prefer" or "avoid" certain productions when ambiguities arise. For example,
      +when we parse this program, we see the following ambiguity as an error message:

      +
      [Error] Inner Parser: Parsing ambiguity.
+1: syntax Stmt ::= "if" "(" Exp ")" Stmt
+
+`if(_)__LESSON-04-E_Stmt_Exp_Stmt`(`true_LESSON-04-E_Exp`(.KList),`if(_)_else__LESSON-04-E_Stmt_Exp_Stmt_Stmt`(`false_LESSON-04-E_Exp`(.KList),`;_LESSON-04-E_Stmt`(.KList),`;_LESSON-04-E_Stmt`(.KList)))
+2: syntax Stmt ::= "if" "(" Exp ")" Stmt "else" Stmt
+
+`if(_)_else__LESSON-04-E_Stmt_Exp_Stmt_Stmt`(`true_LESSON-04-E_Exp`(.KList),`if(_)__LESSON-04-E_Stmt_Exp_Stmt`(`false_LESSON-04-E_Exp`(.KList),`;_LESSON-04-E_Stmt`(.KList)),`;_LESSON-04-E_Stmt`(.KList))
+        Source(./dangling-else.if)
+        Location(1,1,1,30)
+

      Roughly, we see that the ambiguity is between an if with an else or an if
      +without an else. Since we want to pick the first parse, we can tell K to
      +"avoid" the second parse with the avoid attribute. Consider the following
      +modified definition (lesson-04-f.k):

      +
      module LESSON-04-F
+
+  syntax Exp ::= "true" | "false"
+  syntax Stmt ::= "if" "(" Exp ")" Stmt
+                | "if" "(" Exp ")" Stmt "else" Stmt [avoid]
+                | "{" "}"
+endmodule
+

      Here we have added the avoid attribute to the else production. As a result,
      +when an ambiguity occurs and one or more of the possible parses has that symbol
      +at the top of the ambiguous part of the parse, we remove those parses from
      +consideration and consider only those remaining. The prefer attribute behaves
      +similarly, but instead removes all parses which do not have that attribute.
      +In both cases, no action is taken if the parse is not ambiguous.

      +

      Exercises

      + +
        +
      1. +

        Parse the program if (true) if (false) {} else {} using lesson-04-f.k
        +and confirm that else clause is part of the innermost if statement. Then
        +modify the definition so that you will get the alternative parse.

        +
        2. +
      2. +

        Modify your solution from Lesson 1.3, Exercise 2 so that unary negation should
        +bind tighter than multiplication and division, which should bind tighter than
        +addition and subtraction, and each binary operator should be left associative.
        +Write these priority and associativity declarations explicitly, and then
        +try to write them inline.

        +
        3. +
      3. +

        Write a simple grammar containing at least one ambiguity that cannot be
        +resolved via priority or associativity, and then use the prefer attribute to
        +resolve that ambiguity.

        +
        4. +
      4. +

        Explain why the following grammar is not labeled ambiguous by the K parser when parsing abb, then make the parser realize the ambiguity.

        +
        5. +
      +
      module EXERCISE4
+
+syntax Expr ::= "a" Expr "b"
+              | "abb"
+              | "b"
+
+endmodule
+

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.5: Modules, Imports, and Requires.

      +

      Lesson 1.5: Modules, Imports, and Requires

      + +

      The purpose of this lesson is to explain how K definitions can be broken into
      +separate modules and files and how these distinct components combine into a
      +complete K definition.

      +

      K's outer syntax

      + +

      Recall from Lesson 1.3 that K's grammar is broken
      +into two components: the outer syntax of K and the inner syntax of K.
      +Outer syntax, as previously mentioned, consists of requires, modules,
      +imports, and sentences. A K semantics is expressed by the set of
      +sentences contained in the definition. The scope of what is considered
      +contained in that definition is determined both by the main semantics
      +module       of a K definition, as well as the requires and imports present
      +in the file that contains that module.

      +

      Basic module syntax

      + +

      The basic unit of grouping sentences in K is the module. A module consists
      +of a module name, an optional list of attributes, a list of
      +imports, and a list of sentences.

      +

      A module name consists of one or more groups of letters, numbers, or
      +underscores, separated by a hyphen. Here are some valid module names: FOO,
      +FOO-BAR, foo0, foo0_bar-Baz9. Here are some invalid module names: -,
      +-FOO, BAR-, FOO--BAR. Stylistically, modules names are usually all
      +uppercase with hyphens separating words, but this is not strictly enforced.

      +

      Some example modules include an empty module:

      +
      module LESSON-05-A
+
+endmodule
+

      A module with some attributes:

      +
      module LESSON-05-B [group(attr1,attr2), private]
+
+endmodule
+

      A module with some sentences:

      +
      module LESSON-05-C
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+  rule not true => false
+  rule not false => true
+endmodule
+

      Imports

      + +

      Thus far we have only discussed definitions containing a single module.
      +Definitions can also contain multiple modules, in which one module imports
      +others.

      +

      An import in K appears at the top of a module, prior to any sentences. It can
      +be specified with the imports keyword, followed by a module name.

      +

      For example, here is a simple definition with two modules (lesson-05-d.k):

      +
      module LESSON-05-D-1
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+endmodule
+
+module LESSON-05-D
+  imports LESSON-05-D-1
+
+  rule not true => false
+  rule not false => true
+endmodule
+

      This K definition is equivalent to the definition expressed by the single module
      +LESSON-05-C. Essentially, by importing a module, we include all of the
      +sentences in the module being imported into the module that we import from.
      +There are a few minor differences between importing a module and simply
      +including its sentences in another module directly, but we will cover these
      +differences later. Essentially, you can think of modules as a way of
      +conceptually grouping sentences in a larger K definition.

      +

      Exercise

      + +

      Modify lesson-05-d.k to include four modules: one containing the syntax, two
      +with one rule each that imports the first module, and a final module
      +LESSON-05-D containing no sentences that imports the second and third module.
      +Check to make sure the definition still compiles and that you can still evaluate
      +the not function.

      +

      Parsing in the presence of multiple modules

      + +

      As you may have noticed, each module in a definition can express a distinct set
      +of syntax. When parsing the sentences in a module, we use the syntax
      +of that module, enriched with the basic syntax of K, in order to parse
      +rules in that module. For example, the following definition is a parser error
      +(lesson-05-e.k):

      +
      module LESSON-05-E-1
+  rule not true => false
+  rule not false => true
+endmodule
+
+module LESSON-05-E-2
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+endmodule
+

      This is because the syntax referenced in module LESSON-05-E-1, namely, not,
      +true, and false, is not imported by that module. You can solve this problem
      +by simply importing the modules containing the syntax you want to use in your
      +sentences.

      +

      Main syntax and semantics modules

      + +

      When we are compiling a K definition, we need to know where to start. We
      +designate two specific entry point modules: the main syntax module
      +and the main semantics module. The main syntax module, as well as all the
      +modules it imports recursively, are used to create the parser for programs that
      +you use to parse programs that you execute with krun. The main semantics
      +module, as well as all the modules it imports recursively, are used to
      +determine the rules that can be applied at runtime in order to execute a
      +program. For example, in the above example, if the main semantics module is
      +module LESSON-05-D-1, then not is an uninterpreted function (i.e., has no
      +rules associated with it), and the rules in module LESSON-05-D are not
      +included.

      +

      While you can specify the entry point modules explicitly by passing the
      +--main-module and --syntax-module flags to kompile, by default, if you
      +type kompile foo.k, then the main semantics module will be FOO and the
      +main syntax module will be FOO-SYNTAX.

      +

      Splitting a definition into multiple files

      + +

      So far, while we have discussed ways to break definitions into separate
      +conceptual components (modules), K also provides a mechanism for combining
      +multiple files into a single K definition, namely, the requires directive.

      +

      In K, the requires keyword has two meanings. The first, the requires
      +statement, appears at the top of a K file, prior to any module declarations. It
      +consists of the keyword requires followed by a double-quoted string. The
      +second meaning of the requires keyword will be covered in a later lesson,
      +but it is distinguished because the second case occurs only inside modules.

      +

      The string passed to the requires statement contains a filename. When you run
      +kompile on a file, it will look at all of the requires statements in that
      +file, look up those files on disk, parse them, and then recursively process all
      +the requires statements in those files. It then combines all the modules in all
      +of those files together, and uses them collectively as the set of modules to
      +which imports statements can refer.

      +

      Putting it all together

      + +

      Putting it all together, here is one possible way in which we could break the
      +definition lesson-02-c.k from Lesson 1.2 into
      +multiple files and modules:

      +

      colors.k:

      +
      module COLORS
+  syntax Color ::= Yellow()
+                 | Blue()
+endmodule
+

      fruits.k:

      +
      module FRUITS
+  syntax Fruit ::= Banana()
+                 | Blueberry()
+endmodule
+

      colorOf.k:

      +
      requires "fruits.k"
+requires "colors.k"
+
+module COLOROF-SYNTAX
+  imports COLORS
+  imports FRUITS
+
+  syntax Color ::= colorOf(Fruit) [function]
+endmodule
+
+module COLOROF
+  imports COLOROF-SYNTAX
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+endmodule
+

      You would then compile this definition with kompile colorOf.k and use it the
      +same way as the original, single-module definition.

      +

      Exercise

      + +

      Modify the name of the COLOROF module, and then recompile the definition.
      +Try to understand why you now get a compiler error. Then, resolve this compiler
      +error by passing the --main-module and --syntax-module flags to kompile.

      +

      Include path

      + +

      One note can be made about how paths are resolved in requires statements.

      +

      By default, the path you specify is allowed to be an absolute or a relative
      +path. If the path is absolute, that exact file is imported. If the path is
      +relative, a matching file is looked for within all of the
      +include directories specified to the compiler. By default, the include
      +directories include the current working directory, followed by the
      +include/kframework/builtin directory within your installation of K. You can
      +also pass one or more directories to kompile via the -I command line flag,
      +in which case these directories are prepended to the beginning of the list.

      +

      Exercises

      + +
        +
      1. +

        Take the solution to Lesson 1.4, Exercise 2 which included the explicit
        +priority and associativity declarations, and modify the definition so that
        +the syntax of integers and brackets is in one module, the syntax of addition,
        +subtraction, and unary negation is in another module, and the syntax of
        +multiplication and division is in a third module. Make sure you can still parse
        +the same set of expressions as before. Place priority declarations in the main
        +module.

        +
        2. +
      2. +

        Modify lesson-02-d.k from Lesson 1.2 so that the rules and syntax are in
        +separate modules in separate files.

        +
        3. +
      3. +

        Place the file containing the syntax from Exercise 2 in another directory,
        +then recompile the definition. Observe why a compilation error occurs. Then
        +fix the compiler error by passing -I to kompile.

        +
        4. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.6: Integers and Booleans.

      +

      Lesson 1.6: Integers and Booleans

      + +

      The purpose of this lesson is to explain the two most basic types of builtin
      +sorts in K, the Int sort and the Bool sort, representing
      +arbitrary-precision integers and Boolean algebra.

      +

      Builtin sorts in K

      + +

      K provides definitions of some useful sorts in
      +domains.md, found in the
      +include/kframework/builtin directory of the K installation. This file is
      +defined via a
      +Literate programming
      +style that we will discuss in a future lesson. We will not cover all of the
      +sorts found there immediately, however, this lesson discusses some of the
      +details surrounding integers and Booleans, as well as providing information
      +about how to look up more detailed knowledge about builtin functions in K's
      +documentation.

      +

      Booleans in K

      + +

      The most basic builtin sort K provides is the Bool sort, representing
      +Boolean values (i.e., true and false). You have already seen how we were
      +able to create this type ourselves using K's parsing and disambiguation
      +features. However, in the vast majority of cases, we prefer instead to import
      +the version of Boolean algebra defined by K itself. Most simply, you can do
      +this by importing the module BOOL in your definition. For example
      +(lesson-06-a.k):

      +
      module LESSON-06-A
+  imports BOOL
+
+  syntax Fruit ::= Blueberry() | Banana()
+  syntax Bool ::= isBlue(Fruit) [function]
+
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+endmodule
+

      Here we have defined a simple predicate, i.e., a function returning a
      +Boolean value. We are now able to perform the usual Boolean operations of
      +and, or, and not over these values. For example (lesson-06-b.k):"

      +
      module LESSON-06-B
+  imports BOOL
+
+  syntax Fruit ::= Blueberry() | Banana()
+  syntax Bool ::= isBlue(Fruit) [function]
+
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+
+  syntax Bool ::= isYellow(Fruit) [function]
+                | isBlueOrYellow(Fruit) [function]
+
+  rule isYellow(Banana()) => true
+  rule isYellow(Blueberry()) => false
+
+  rule isBlueOrYellow(F) => isBlue(F) orBool isYellow(F)
+endmodule
+

      In the above example, Boolean inclusive or is performed via the orBool
      +function, which is defined in the BOOL module. As a matter of convention,
      +many functions over builtin sorts in K are suffixed with the name of the
      +primary sort over which those functions are defined. This happens so that the
      +syntax of K does not (generally) conflict with the syntax of any other
      +programming language, which would make it harder to define that programming
      +language in K.

      +

      Exercise

      + +

      Write a function isBlueAndNotYellow which computes the appropriate Boolean
      +expression. If you are unsure what the appropriate syntax is to use, you
      +can refer to the BOOL module in
      +domains.md. Add a term of
      +sort Fruit for which isBlue and isYellow both return true, and test that
      +the isBlueAndNotYellow function behaves as expected on all three Fruits.

      +

      Syntax Modules

      + +

      For most sorts in domains.md, K defines more than one module that can be
      +imported by users. For example, for the Bool sort, K defines the BOOL
      +module that has previously already been discussed, but also provides the
      +BOOL-SYNTAX module. This module, unlike the BOOL module, only declares the
      +values true and false, but not any of the functions that operate over the
      +Bool sort. The rationale is that you may want to import this module into the
      +main syntax module of your definition in some cases, whereas you generally do
      +not want to do this with the version of the module that includes all the
      +functions over the Bool sort. For example, if you were defining the semantics
      +of C++, you might import BOOL-SYNTAX into the syntax module of your
      +definition, because true and false are part of the grammar of C++, but
      +you would only import the BOOL module into the main semantics module, because
      +C++ defines its own syntax for and, or, and not that is different from the
      +syntax defined in the BOOL module.

      +

      Here, for example, is how we might redefine our Boolean expression calculator
      +to use the Bool sort while maintaining an idiomatic structure of modules
      +and imports, for the first time including the rules to calculate the values of
      +expressions themselves (lesson-06-c.k):

      +
      module LESSON-06-C-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Bool ::= "(" Bool ")" [bracket]
+                > "!" Bool [function]
+                > left:
+                  Bool "&&" Bool [function]
+                | Bool "^" Bool [function]
+                | Bool "||" Bool [function]
+endmodule
+
+module LESSON-06-C
+  imports LESSON-06-C-SYNTAX
+  imports BOOL
+
+  rule ! B => notBool B
+  rule A && B => A andBool B
+  rule A ^ B => A xorBool B
+  rule A || B => A orBool B
+endmodule
+

      Note the encapsulation of syntax: the LESSON-06-C-SYNTAX module contains
      +exactly the syntax of our Boolean expressions, and no more, whereas any other
      +syntax needed to implement those functions is in the LESSON-06-C module
      +instead.

      +

      Exercise

      + +

      Add an "implies" function to the above Boolean expression calculator, using the
      +-> symbol to represent implication. You can look up K's builtin "implies"
      +function in the BOOL module in domains.md.

      +

      Integers in K

      + +

      Unlike most programming languages, where the most basic integer type is a
      +fixed-precision integer type, the most commonly used integer sort in K is
      +the Int sort, which represents the mathematical integers, ie,
      +arbitrary-precision integers.

      +

      K provides three main modules for import when using the Int sort. The first,
      +containing all the syntax of integers as well as all of the functions over
      +integers, is the INT module. The second, which provides just the syntax
      +of integer literals themselves, is the INT-SYNTAX module. However, unlike
      +most builtin sorts in K, K also provides a third module for the Int sort:
      +the UNSIGNED-INT-SYNTAX module. This module provides only the syntax of
      +non-negative integers, i.e., natural numbers. The reasons for this involve
      +lexical ambiguity. Generally speaking, in most programming languages, -1 is
      +not a literal, but instead a literal to which the unary negation operator is
      +applied. K thus provides this module to ease in specifying the syntax of such
      +languages.

      +

      For detailed information about the functions available over the Int sort,
      +refer to domains.md. Note again how we append Int to the end of most of the
      +integer operations to ensure they do not collide with the syntax of other
      +programming languages.

      +

      Exercises

      + +
        +
      1. +

        Extend your solution from Lesson 1.4, Exercise 2 to implement the rules
        +that define the behavior of addition, subtraction, multiplication, and
        +division. Do not worry about the case when the user tries to divide by zero
        +at this time. Use /Int to implement division. Test your new calculator
        +implementation by executing the arithmetic expressions you wrote as part of
        +Lesson 1.3, Exercise 2. Check to make sure each computes the value you expected.

        +
        2. +
      2. +

        Combine the Boolean expression calculator from this lesson with your
        +solution to Exercise 1, and then extend the combined calculator with the <,
        +<=, >, >=, ==, and != expressions. Write some Boolean expressions
        +that combine integer and Boolean operations, and test to ensure that these
        +expressions return the expected truth value.

        +
        3. +
      3. +

        Compute the following expressions using your solution from Exercise 2:
        +7 / 3, 7 / -3, -7 / 3, -7 / -3. Then replace the /Int function in
        +your definition with divInt instead, and observe how the value of the above
        +expressions changes. Why does this occur?

        +
        4. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.7: Side Conditions and Rule Priority.

      +

      Lesson 1.7: Side Conditions and Rule Priority

      + +

      The purpose of this lesson is to explain how to write conditional rules in K,
      +and to explain how to control the order in which rules are tried.

      +

      Side Conditions

      + +

      So far, all of the rules we have discussed have been unconditional rules.
      +If the left-hand side of the rule matches the arguments to the function, the
      +rule applies. However, there is another type of rule, a conditional rule.
      +A conditional rule consists of a rule body containing the patterns to
      +match, and a side condition representing a Boolean expression that must
      +evaluate to true in order for the rule to apply.

      +

      Side conditions in K are introduced via the requires keyword immediately
      +following the rule body. For example, here is a rule with a side condition
      +(lesson-07-a.k):

      +
      module LESSON-07-A
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90
+endmodule
+

      In this case, the gradeFromPercentile function takes a single integer
      +argument. The function evaluates to letter-A if the argument passed is
      +greater than 90. Note that the side condition is allowed to refer to variables
      +that appear on the left-hand side of the rule. In the same manner as variables
      +appearing on the right-hand side, variables that appear in the side condition
      +evaluate to the value that was matched on the left-hand side. Then the
      +functions in the side condition are evaluated, which returns a term of sort
      +Bool. If the term is equal to true, then the rule applies. Bear in mind
      +that the side condition is only evaluated at all if the patterns on the
      +left-hand side of the rule match the term being evaluated.

      +

      Exercise

      + +

      Write a rule that evaluates gradeFromPercentile to letter-B if the argument
      +to the function is in the range [80,90). Test that the function correctly
      +evaluates various numbers between 80 and 100.

      +

      owise Rules

      + +

      So far, all the rules we have introduced have had the same priority. What
      +this means is that K does not necessarily enforce an order in which the rules
      +are tried. We have only discussed functions so far in K, so it is not
      +immediately clear why this choice was made, given that a function is not
      +considered well-defined if multiple rules for evaluating it are capable of
      +evaluating the same arguments to different results. However, in future lessons
      +we will discuss other types of rules in K, some of which can be
      +non-deterministic. What this means is that if more than one rule is capable
      +of matching, then K will explore both possible rules in parallel, and consider
      +each of their respective results when executing your program. Don't worry too
      +much about this right now, but just understand that because of the potential
      +later for nondeterminism, we don't enforce a total ordering on the order in
      +which rules are attempted to be applied.

      +

      However, sometimes this is not practical; It can be very convenient to express
      +that a particular rule applies if no other rules for that function are
      +applicable. This can be expressed by adding the owise attribute to a rule.
      +What this means, in practice, is that this rule has lower priority than other
      +rules, and will only be tried to be applied after all the other,
      +higher-priority rules have been tried and they have failed.

      +

      For example, in the above exercise, we had to add a side condition containing
      +two Boolean comparisons to the rule we wrote to handle letter-B grades.
      +However, in practice this meant that we compare the percentile to 90 twice. We
      +can more efficiently and more idiomatically write the letter-B case for the
      +gradeFromPercentile rule using the owise attribute (lesson-07-b.k):

      +
      module LESSON-07-B
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [owise]
+endmodule
+

      This rule is saying, "if all the other rules do not apply, then the grade is a
      +B if the percentile is greater than or equal to 80." Note here that we use both
      +a side condition and an owise attribute on the same rule. This is not
      +required (as we will see later), but it is allowed. What this means is that the
      +side condition is only tried if the other rules did not apply and the
      +left-hand side of the rule matched. You can even use more complex matching on
      +the left-hand side than simply a variable. More generally, you can also have
      +multiple higher-priority rules, or multiple owise rules. What this means in
      +practice is that all of the non-owise rules are tried first, in any order,
      +followed by all the owise rules, in any order.

      +

      Exercise

      + +

      The grades D and F correspond to the percentile ranges [60, 70) and [0, 60)
      +respectively. Write another implementation of gradeFromPercentile which
      +handles only these cases, and uses the owise attribute to avoid redundant
      +Boolean comparisons. Test that various percentiles in the range [0, 70) are
      +evaluated correctly.

      +

      Rule Priority

      + +

      As it happens, the owise attribute is a specific case of a more general
      +concept we call rule priority. In essence, each rule is assigned an integer
      +priority. Rules are tried in increasing order of priority, starting with a
      +rule with priority zero, and trying each increasing numerical value
      +successively.

      +

      By default, a rule is assigned a priority of 50. If the rule has the owise
      +attribute, it is instead given the priority 200. You can see why this will
      +cause owise rules to be tried after regular rules.

      +

      However, it is also possible to directly assign a numerical priority to a rule
      +via the priority attribute. For example, here is an alternative way
      +we could express the same two rules in the gradeFromPercentile function
      +(lesson-07-c.k):

      +
      module LESSON-07-C
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90 [priority(50)]
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [priority(200)]
+endmodule
+

      We can, of course, assign a priority equal to any non-negative integer. For
      +example, here is a more complex example that handles the remaining grades
      +(lesson-07-d.k):

      +
      module LESSON-07-D
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90 [priority(50)]
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [priority(51)]
+  rule gradeFromPercentile(I) => letter-C requires I >=Int 70 [priority(52)]
+  rule gradeFromPercentile(I) => letter-D requires I >=Int 60 [priority(53)]
+  rule gradeFromPercentile(_) => letter-F                     [priority(54)]
+endmodule
+

      Note that we have introduced a new piece of syntax here: _. This is actually
      +just a variable. However, as a special case, when a variable is named _, it
      +does not bind a value that can be used on the right-hand side of the rule, or
      +in a side condition. Effectively, _ is a placeholder variable that means "I
      +don't care about this term."

      +

      In this example, we have explicitly expressed the order in which the rules of
      +this function are tried. Since rules are tried in increasing numerical
      +priority, we first try the rule with priority 50, then 51, then 52, 53, and
      +finally 54.

      +

      As a final note, remember that if you assign a rule a priority higher than 200,
      +it will be tried after a rule with the owise attribute, and if you assign
      +a rule a priority less than 50, it will be tried before a rule with no
      +explicit priority.

      +

      Exercises

      + +
        +
      1. +

        Write a function isEven that returns whether an integer is an even number.
        +Use two rules and one side condition. The right-hand side of the rules should
        +be Boolean literals. Refer back to
        +domains.md for the relevant
        +integer operations.

        +
        2. +
      2. +

        Modify the calculator application from Lesson 1.6, Exercise 2, so that division
        +by zero will no longer make krun crash with a "Divison by zero" exception.
        +Instead, the / function should not match any of its rules if the denominator
        +is zero.

        +
        3. +
      3. +

        Write your own implementation of ==, <, <=, >, >= for integers and modify your solution from Exercise 2 to use it.
        +You can use any arithmetic operations in the INT module, but do not use any built-in boolean functions for comparing integers.

        +

        Hint: Use pattern matching and recursive definitions with rule priorities.

        +
        4. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.8: Literate Programming with Markdown.

      +

      Lesson 1.8: Literate Programming with Markdown

      + +

      The purpose of this lesson is to teach a paradigm for performing literate
      +programming in K, and explain how this can be used to create K definitions
      +that are also documentation.

      +

      Markdown and K

      + +

      The K tutorial so far has been written in
      +Markdown. Markdown,
      +for those not already familiar, is a lightweight plain-text format for styling
      +text. From this point onward, we assume you are familiar with Markdown and how
      +to write Markdown code. You can refer to the above link for a tutorial if you
      +are not already familiar.

      +

      What you may not necessarily realize, however, is that the K tutorial is also
      +a sequence of K definitions written in the manner of
      +Literate Programming.
      +For detailed information about Literate Programming, you can read the linked
      +Wikipedia article, but the short summary is that literate programming is a way
      +of intertwining documentation and code together in a manner that allows
      +executable code to also be, simultaneously, a documented description of that
      +code.

      +

      K is provided with built-in support for literate programming using Markdown.
      +By default, if you pass a file with the .md file extension to kompile, it
      +will look for any code blocks containing k code in that file, extract out
      +that K code into pure K, and then compile it as if it were a .k file.

      +

      A K code block begins with a line of text containing the keyword ```k,
      +and ends when it encounters another ``` keyword.

      +

      For example, if you view the markdown source of this document, this is a K
      +code block:

      +
      module LESSON-08
+  imports INT
+

      Only the code inside K code blocks will actually be sent to the compiler. The
      +rest, while it may appear in the document when rendered by a markdown viewer,
      +is essentially a form of code comment.

      +

      When you have multiple K code blocks in a document, K will append each one
      +together into a single file before passing it off to the outer parser.

      +

      For example, the following code block contains sentences that are part of the
      +LESSON-08 module that we declared the beginning of above:

      +
        syntax Int ::= Int "+" Int [function]
+  rule I1 + I2 => I1 +Int I2
+

      Exercise

      + +

      Compile this file with kompile README.md --main-module LESSON-08. Confirm
      +that you can use the resulting compiled definition to evaluate the +
      +function.

      +

      Markdown Selectors

      + +

      On occasion, you may want to generate multiple K definitions from a single
      +Markdown file. You may also wish to include a block of syntax-highlighted K
      +code that nonetheless does not appear as part of your K definition. It is
      +possible to accomplish this by means of the built-in support for syntax
      +highlighting in Markdown. Markdown allows a code block that was begun with
      +``` to be immediately followed by a string which is used to signify what
      +programming language the following code is written in. However, this feature
      +actually allows arbitrary text to appear describing that code block. Markdown
      +parsers are able to parse this text and render the code block differently
      +depending on what text appears after the backticks.

      +

      In K, you can use this functionality to specify one or more
      +Markdown selectors which are used to describe the code block. A Markdown
      +selector consists of a sequence of characters containing letters, numbers, and
      +underscores. A code block can be designated with a single selector by appending
      +the selector immediately following the backticks that open the code block.

      +

      For example, here is a code block with the foo selector:

      +
      foo bar
+

      Note that this is not K code. By convention, K code should have the k
      +selector on it. You can express multiple selectors on a code block by putting
      +them between curly braces and prepending each with the . character. For
      +example, here is a code block with the foo and k selectors:

      +
        syntax Int ::= foo(Int) [function]
+  rule foo(0) => 0
+

      Because this code block contains the k Markdown selector, by default it is
      +included as part of the K definition being compiled.

      +

      Exercise

      + +

      Confirm this fact by using krun to evaluate foo(0).

      +

      Markdown Selector Expressions

      + +

      By default, as previously stated, K includes in the definition any code block
      +with the k selector. However, this is merely a specific instance of a general
      +principle, namely, that K allows you to control which selectors get included
      +in your K definition. This is done by means of the --md-selector flag to
      +kompile. This flag accepts a Markdown selector expression, which you
      +can essentially think of as a kind of Boolean algebra over Markdown selectors.
      +Each selector becomes an atom, and you can combine these atoms via the &,
      +|, !, and () operators.

      +

      Here is a grammar, written in K, of the language of Markdown selector
      +expressions:

      +
        syntax Selector ::= r"[0-9a-zA-Z_]+" [token]
+  syntax SelectorExp ::= Selector
+                       | "(" SelectorExp ")" [bracket]
+                       > right:
+                         "!" SelectorExp
+                       > right:
+                         SelectorExp "&" SelectorExp
+                       > right:
+                         SelectorExp "|" SelectorExp
+

      Here is a selector expression that selects all the K code blocks in this
      +definition except the one immediately above:

      +
      k & (! selector)
+

      Addendum

      + +

      This code block exists in order to make the above lesson a syntactically valid
      +K definition. Consider why it is necessary.

      +
      endmodule
+

      Exercises

      + +
        +
      1. +

        Compile this lesson with the selector expression k & (! foo) and confirm
        +that you get a parser error if you try to evaluate the foo function with the
        +resulting definition.

        +
        2. +
      2. +

        Compile Lesson 1.3
        +as a K definition. Identify why it fails to compile. Then pass an appropriate
        +--md-selector to the compiler in order to make it compile.

        +
        3. +
      3. +

        Modify your calculator application from Lesson 1.7, Exercise 2, to be written
        +in a literate style. Consider what text might be appropriate to turn the
        +resulting markdown file into documentation for your calculator.

        +
        4. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.9: Unparsing and the format and color attributes.

      +

      Lesson 1.9: Unparsing and the format and color attributes

      + +

      The purpose of this lesson is to teach the user about how terms are
      +pretty-printed in K, and how the user can make adjustments to the default
      +settings for how to print specific terms.

      +

      Parsing, Execution, and Unparsing

      + +

      When you use krun to interpret a program, the tool passes through three major
      +phases. In the first, parsing, the program itself is parsed using either kast
      +or an ahead-of-time parser generated via Bison, and the resulting AST becomes
      +the input to the interpreter. In the second phase, execution, K evaluates
      +functions and (as we will discuss in depth later) performs rewrite steps to
      +iteratively transform the program state. The third and final phase is called
      +unparsing, because it consists of taking the final state of the application
      +after the program has been interpreted, and converting it from an AST back into
      +text that (in theory, anyway) could be parsed back into the same AST that was
      +the output of the execution phase.

      +

      In practice, parsing is not always precisely reversible. It turns out
      +(although we are not going to cover exactly why this is here), that
      +constructing a sound algorithm that takes a grammar and an AST and emits text
      +that could be parsed via that grammar to the original AST is an
      +NP-hard problem. As a result, in the interests of avoiding exponential time
      +algorithms when users rarely care about unparsing being completely sound, we
      +take certain shortcuts that provide a linear-time algorithm that approximates
      +a sound solution to the problem while sacrificing the notion that the result
      +can be parsed into the exact original term in all cases.

      +

      This is a lot of theoretical explanation, but at root, the unparsing process
      +is fairly simple: it takes a K term that is the output of execution and pretty
      +prints it according to the syntax defined by the user in their K definition.
      +This is useful because the original AST is not terribly user-readable, and it
      +is difficult to visualize the entire term or decipher information about the
      +final state of the program at a quick glance. Of course, in rare cases, the
      +pretty-printed configuration loses information of relevance, which is why K
      +allows you to obtain the original AST on request.

      +

      As an example of all of this, consider the following K definition
      +(lesson-09-a.k):

      +
      module LESSON-09-A
+  imports BOOL
+
+  syntax Exp ::= "(" Exp ")" [bracket]
+               | Bool
+               > "!" Exp
+               > left:
+                 Exp "&&" Exp
+               | Exp "^" Exp
+               | Exp "||" Exp
+
+  syntax Exp ::= id(Exp) [function]
+  rule id(E) => E
+endmodule
+

      This is similar to the grammar we defined in LESSON-06-C, with the difference
      +that the Boolean expressions are now constructors of sort Exp and we define a
      +trivial function over expressions that returns its argument unchanged.

      +

      We can now parse a simple program in this definition and use it to unparse some
      +Boolean expressions. For example (exp.bool):

      +
      id(true&&false&&!true^(false||true))
+

      Here is a program that is not particularly legible at first glance, because all
      +extraneous whitespace has been removed. However, if we run krun exp.bool, we
      +see that the result of the unparser will pretty-print this expression rather
      +nicely:

      +
      <k>
+  true && false && ! true ^ ( false || true ) ~> .
+</k>
+

      Notably, not only does K insert whitespace where appropriate, it is also smart
      +enough to insert parentheses where necessary in order to ensure the correct
      +parse. For example, without those parentheses, the expression above would parse
      +equivalent to the following one:

      +
      (((true && false) && ! true) ^ false) || true
+

      Indeed, you can confirm this by passing that exact expression to the id
      +function and evaluating it, then looking at the result of the unparser:

      +
      <k>
+  true && false && ! true ^ false || true ~> .
+</k>
+

      Here, because the meaning of the AST is the same both with and without
      +parentheses, K does not insert any parentheses when unparsing.

      +

      Exercise

      + +

      Modify the grammar of LESSON-09-A above so that the binary operators are
      +right associative. Try unparsing exp.bool again, and note how the result is
      +different. Explain the reason for the difference.

      +

      Custom unparsing of terms

      + +

      You may have noticed that right now, the unparsing of terms is not terribly
      +imaginative. All it is doing is taking each child of the term, inserting it
      +into the non-terminal positions of the production, then printing the production
      +with a space between each terminal or non-terminal. It is easy to see why this
      +might not be desirable in some cases. Consider the following K definition
      +(lesson-09-b.k):

      +
      module LESSON-09-B
+  imports BOOL
+
+  syntax Stmt ::= "{" Stmt "}" | "{" "}"
+                > right:
+                  Stmt Stmt
+                | "if" "(" Bool ")" Stmt
+                | "if" "(" Bool ")" Stmt "else" Stmt [avoid]
+endmodule
+

      This is a statement grammar, simplified to the point of meaninglessness, but
      +still useful as an object lesson in unparsing. Consider the following program
      +in this grammar (if.stmt):

      +
      if (true) {
+  if (true) {}
+  if (false) {}
+  if (true) {
+    if (false) {} else {}
+  } else {
+    if (false) {}
+  }
+}
+

      This is how that term would be unparsed if it appeared in the output of krun:

      +
      if ( true ) { if ( true ) { } if ( false ) { } if ( true ) { if ( false ) { } else { } } else { if ( false ) { } } }
+

      This is clearly much less legible than we started with! What are we to do?
      +Well, K provides an attribute, format, that can be applied to any production,
      +which controls how that production gets unparsed. You've seen how it gets
      +unparsed by default, but via this attribute, the developer has complete control
      +over how the term is printed. Of course, the user can trivially create ways to
      +print terms that would not parse back into the same term. Sometimes this is
      +even desirable. But in most cases, what you are interested in is controlling
      +the line breaking, indentation, and spacing of the production.

      +

      Here is an example of how you might choose to apply the format attribute
      +to improve how the above term is unparsed (lesson-09-c.k):

      +
      module LESSON-09-C
+  imports BOOL
+
+  syntax Stmt ::= "{" Stmt "}" [format(%1%i%n%2%d%n%3)] | "{" "}" [format(%1%2)]
+                > right:
+                  Stmt Stmt [format(%1%n%2)]
+                | "if" "(" Bool ")" Stmt [format(%1 %2%3%4 %5)]
+                | "if" "(" Bool ")" Stmt "else" Stmt [avoid, format(%1 %2%3%4 %5 %6 %7)]
+endmodule
+

      If we compile this new definition and unparse the same term, this is the
      +result we get:

      +
      if (true) {
+  if (true) {}
+  if (false) {}
+  if (true) {
+    if (false) {} else {}
+  } else {
+    if (false) {}
+  }
+}
+

      This is the exact same text we started with! By adding the format attributes,
      +we were able to indent the body of code blocks, adjust the spacing of if
      +statements, and put each statement on a new line.

      +

      How exactly was this achieved? Well, each time the unparser reaches a term,
      +it looks at the format attribute of that term. That format attribute is a
      +mix of characters and format codes. Format codes begin with the %
      +character. Each character in the format attribute other than a format code is
      +appended verbatim to the output, and each format code is handled according to
      +its meaning, transformed (possibly recursively) into a string of text, and
      +spliced into the output at the position the format code appears in the format
      +string.

      +

      Provided for reference is a table with a complete list of all valid format
      +codes, followed by their meaning:

      + + + + + + + + + +
      Format Code Meaning
      n Insert '\n' followed by the current indentation + level
      i Increase the current indentation level by 1
      d Decrease the current indentation level by 1
      c Move to the next color in the list of colors for + this production (see next section)
      r Reset color to the default foreground color for + the terminal (see next section)
      an integer Print a terminal or non-terminal from the + production. The integer is treated as a 1-based + index into the terminals and non-terminals of + the production. +
      +
      If the offset refers to a terminal, move to the + next color in the list of colors for this + production, print the value of that terminal, + then reset the color to the default foreground + color for the terminal. +
      +
      If the offset refers to a regular expression + terminal, it is an error. +
      +
      If the offset refers to a non-terminal, unparse + the corresponding child of the current term + (starting with the current indentation level) + and print the resulting text, then set the + current color and indentation level to the color + and indentation level following unparsing that + term.
      other char Print that character verbatim
      +

      Exercise

      + +

      Change the format attributes for LESSON-09-C so that if.stmt will unparse
      +as follows:

      +
      if (true)
+{
+  if (true)
+  {
+  }
+  if (false)
+  {
+  }
+  if (true)
+  {
+    if (false)
+    {
+    }
+    else
+    {
+    }
+  }
+  else
+  {
+    if (false)
+    {
+    }
+  }
+}
+

      Output coloring

      + +

      When the output of unparsing is displayed on a terminal supporting colors, K
      +is capable of coloring the output, similar to what is possible with a syntax
      +highlighter. This is achieved via the color and colors attributes.

      +

      Essentially, both the color and colors attributes are used to construct a
      +list of colors associated with each production, and then the format attribute
      +is used to control how those colors are used to unparse the term. At its most
      +basic level, you can set the color attribute to color all the terminals in
      +the production a certain color, or you can use the colors attribute to
      +specify a comma-separated list of colors for each terminal in the production.
      +At a more advanced level, the %c and %r format codes control how the
      +formatter interacts with the list of colors specified by the colors
      +attribute. You can essentially think of the color attribute as a way of
      +specifying that you want all the colors in the list to be the same color.

      +

      Note that the %c and %r format codes are relatively primitive in nature.
      +The color and colors attributes merely maintain a list of colors, whereas
      +the %c and %r format codes merely control how to advance through that list
      +and how individual text is colored.

      +

      It is an error if the colors attribute does not provide all the colors needed
      +by the terminals and escape codes in the production. %r does not change the
      +position in the list of colors at all, so the next %c will advance to the
      +following color.

      +

      As a complete example, here is a variant of LESSON-09-A which colors the
      +various boolean operators:

      +
      module LESSON-09-D
+  imports BOOL
+
+  syntax Exp ::= "(" Exp ")" [bracket]
+               | Bool
+               > "!" Exp [color(yellow)]
+               > left:
+                 Exp "&&" Exp [color(red)]
+               | Exp "^" Exp [color(blue)]
+               | Exp "||" Exp [color(green)]
+
+  syntax Exp ::= id(Exp) [function]
+  rule id(E) => E
+endmodule
+

      For a complete list of allowed colors, see
      +here.

      +

      Exercises

      + +
        +
      1. +

        Use the color attribute on LESSON-09-C to color the keywords true and
        +false one color, the keywords if and else another color, and the operators
        +(, ), {, and } a third color.

        +
        2. +
      2. +

        Use the format, color, and colors attributes to tell the unparser to
        +style the expression grammar from Lesson 1.8, Exercise 3 according to your own
        +personal preferences for syntax highlighting and code formatting. You can
        +view the result of the unparser on a function term without evaluating that
        +function by means of the command kparse <file> | kore-print -.

        +
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.10: Strings.

      +

      Lesson 1.10: Strings

      + +

      The purpose of this lesson is to explain how to use the String sort in K to
      +represent sequences of characters, and explain where to find additional
      +information about builtin functions over strings.

      +

      The String Sort

      + +

      In addition to the Int and Bool sorts covered in
      +Lesson 1.6, K provides, among others, the
      +String sort to represent sequences of characters. You can import this
      +functionality via the STRING-SYNTAX module, which contains the syntax of
      +string literals in K, and the STRING module, which contains all the functions
      +that operate over the String type.

      +

      Strings in K are double-quoted. The following list of escape sequences is
      +supported:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      Escape SequenceMeaning
      \"The literal character "
      \\The literal character \
      \nThe newline character (ASCII code 0x0a)
      \rThe carriage return character (ASCII code 0x0d)
      \tThe tab character (ASCII code 0x09)
      \fThe form feed character (ASCII code 0x0c)
      \x00\x followed by 2 hexadecimal digits indicates a code point between 0x00 and 0xFF
      \u0000\u followed by 4 hexadecimal digits indicates a code point between 0x0000 and 0xFFFF
      \U00000000\U followed by 8 hexadecimal digits indicates a code point between 0x000000 and 0x10FFFF
      +

      Please note that as of the current moment, K's unicode support is not fully
      +complete, so you may run into errors using code points greater than 0xff.

      +

      As an example, you can construct a string literal containing the following
      +block of text:

      +
      This is an example block of text.
+Here is a quotation: "Hello world."
+	This line is indented.
+ÁÉÍÓÚ
+

      Like so:

      +
      "This is an example block of text.\nHere is a quotation: \"Hello world.\"\n\tThis line is indented.\n\xc1\xc9\xcd\xd3\xda\n"
+

      Basic String Functions

      + +

      The full list of functions provided for the String sort can be found in
      +domains.md, but here we
      +describe a few of the more basic ones.

      +

      String concatenation

      + +

      The concatenation operator for strings is +String. For example, consider
      +the following K rule that constructs a string from component parts
      +(lesson-10.k):

      +
      module LESSON-10
+  imports STRING
+
+  syntax String ::= msg(String) [function]
+  rule msg(S) => "The string you provided: " +String S +String "\nHave a nice day!"
+endmodule
+

      Note that this operator is O(N), so repeated concatenations are inefficient.
      +For information about efficient string concatenation, refer to
      +Lesson 2.14.

      +

      String length

      + +

      The function to return the length of a string is lengthString. For example,
      +lengthString("foo") will return 3, and lengthString("") will return 0.
      +The return value is the length of the string in code points.

      +

      Substring computation

      + +

      The function to compute the substring of a string is substrString. It
      +takes two string indices, starting from 0, and returns the substring within the
      +range [start..end). It is only defined if end >= start, start >= 0, and
      +end <= length of string. Here, for example, we return the first 5 characters
      +of a string:

      +
      substrString(S, 0, 5)
+

      Here we return all but the first 3 characters:

      +
      substrString(S, 3, lengthString(S))
+

      Exercises

      + +
        +
      1. Write a function that takes a paragraph of text (i.e., a sequence of
        +sentences, each ending in a period), and constructs a new (nonsense) sentence
        +composed of the first word of each sentence, followed by a period. Do not
        +worry about capitalization or periods within the sentence which do not end the
        +sentence (e.g. "Dr."). You can assume that all whitespace within the paragraph
        +are spaces. For more information about the functions over strings required to
        +implement such a function, refer to domains.md.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.11: Casting Terms.

      +

      Lesson 1.11: Casting Terms

      + +

      The purpose of this lesson is to explain how to use cast expressions in
      +order to disambiguate terms using sort information. We also explain how the
      +variable sort inference algorithm works in K, and how to change the default
      +behavior by casting variables to a particular sort.

      +

      Casting in K

      + +

      Sometimes the grammar you write for your rules in K can be a little bit
      +ambiguous on purpose. While grammars for programming languages may be
      +unambiguous when considered in their entirety, K allows you to write rules
      +involving arbitrary fragments of that grammar, and those fragments can
      +sometimes be ambiguous by themselves, or similar enough to other fragments
      +of the grammar to trigger ambiguity. As a result, in addition to the tools
      +covered in Lesson 1.4, K provides one
      +additional powerful tool for disambiguation: cast expressions.

      +

      K provides three main types of casts: the semantic cast, the strict cast, and
      +the projection cast. We will cover each of them, and their similarities and
      +differences, in turn.

      +

      Semantic casts

      + +

      The most basic, and most common, type of cast in K is called the
      +semantic cast. For every sort S declared in a module, K provides the
      +following (implicit) production for use in sentences:

      +
        syntax S ::= S ":S"
+

      Note that S simply represents the name of the sort. For example, if we
      +defined a sort Exp, the actual production for that sort would be:

      +
        syntax Exp ::= Exp ":Exp"
+

      At runtime, this expression will not actually exist; it is merely an annotation
      +to the compiler describing the sort of the term inside the cast. It is telling
      +the compiler that the term inside the cast must be of sort Exp. For example,
      +if we had the following grammar:

      +
      module LESSON-11-A
+  imports INT
+
+  syntax Exp ::= Int | Exp "+" Exp
+  syntax Stmt ::= "if" "(" Exp ")" Stmt | "{" "}"
+endmodule
+

      Then we would be able to write 1:Exp, or (1 + 2):Exp, but not {}:Exp.

      +

      You can also restrict the sort that a variable in a rule will match by casting
      +it. For example, consider the following additional module:

      +
      module LESSON-11-B
+  imports LESSON-11-A
+  imports BOOL
+
+  syntax Term ::= Exp | Stmt
+  syntax Bool ::= isExpression(Term) [function]
+
+  rule isExpression(_E:Exp) => true
+  rule isExpression(_) => false [owise]
+endmodule
+

      Here we have defined a very simple function that decides whether a term is
      +an expression or a statement. It does this by casting the variable inside the
      +isExpression rule to sort Exp. As a result, that variable will only match terms
      +of sort Exp. Thus, isExpression(1) will return true, as will isExpression(1 + 2), but
      +isExpression({}) will return false.

      +

      Exercise

      + +

      Verify this fact for yourself by running isExpression on the above examples. Then
      +write an isStatement function, and test that it works as expected.

      +

      Strict casts

      + +

      On occasion, a semantic cast is not strict enough. It might be that you want
      +to, for disambiguation purposes, say exactly what sort a term is. For
      +example, consider the following definition:

      +
      module LESSON-11-C
+  imports INT
+
+  syntax Exp ::= Int | Exp "+" Exp [group(exp)]
+  syntax Exp2 ::= Exp | Exp2 "+" Exp2 [group(exp2)]
+endmodule
+

      This grammar is a little ambiguous and contrived, but it serves to demonstrate
      +how a semantic cast might be insufficient to disambiguate a term. If we were
      +to write the term (I1:Int + I2:Int):Exp2, the term would be ambiguous,
      +because the cast is not sufficiently strict to determine whether you mean
      +to derive the "+" production in the group exp or the one in the group exp2.

      +

      In this situation, there is a solution: the strict cast. For every sort
      +S in your grammar, K also defines the following production:

      +
        syntax S ::= S "::S"
+

      This may at first glance seem the same as the previous cast. And indeed,
      +from the perspective of the grammar and from the perspective of rewriting,
      +they are in fact identical. However, the second variant has a unique meaning
      +in the type system of K: namely, the term inside the cast cannot be a
      +subsort, i.e., a term of another sort S2 such that the production
      +syntax S ::= S2 exists.

      +

      As a result, if we were to write in the above grammar the term
      +(I1:Int + I2:Int)::Exp2, then we would know that the second derivation above
      +should be chosen, whereas if we want the first derivation, we could write
      +(I1:Int + I2:Int)::Exp.

      +

      Projection casts

      + +

      Thus far we have focused entirely on casts which exist solely to inform the
      +compiler about the sort of terms. However, sometimes when dealing with grammars
      +containing subsorts, it can be desirable to reason with the subsort production
      +itself, which injects one sort into another. Remember from above that such
      +a production looks like syntax S ::= S2. This type of production, called a
      +subsort production, can be thought of as a type of inheritance involving
      +constructors. If we have the above production in our grammar, we say that S2
      +is a subsort of S, or that any S2 is also an S. K implicitly maintains a
      +symbol at runtime which keeps track of where such subsortings occur; this
      +symbol is called an injection.

      +

      Sometimes, when one sort is a subsort of another, it can be the case that
      +a function returns one sort, but you actually want to cast the result of
      +calling that function to another sort which is a subsort of the first sort.
      +This is similar to what happens with inheritance in an object-oriented
      +language, where you might cast a superclass to a subclass if you know for
      +sure the object at runtime is in fact an instance of that class.

      +

      K provides something similar for subsorts: the projection cast.

      +

      For each pair of sorts S and S2, K provides the following production:

      +
        syntax S ::= "{" S2 "}" ":>S"
+

      What this means is that you take any term of sort S2 and cast it to sort
      +S. If the term of sort S2 consists of an injection containing a term of sort
      +S, then this will return that term. Otherwise, an error occurs and rewriting
      +fails, returning the projection function which failed to apply. The sort is
      +not actually checked at compilation time; rather, it is a runtime check
      +inserted into the code that runs when the rule applies.

      +

      For example, here is a module that makes use of projection casts:

      +
      module LESSON-11-D
+  imports INT
+  imports BOOL
+
+  syntax Exp ::= Int | Bool | Exp "+" Exp | Exp "&&" Exp
+
+  syntax Exp ::= eval(Exp) [function]
+  rule eval(I:Int) => I
+  rule eval(B:Bool) => B
+  rule eval(E1 + E2) => {eval(E1)}:>Int +Int {eval(E2)}:>Int
+  rule eval(E1 && E2) => {eval(E1)}:>Bool andBool {eval(E2)}:>Bool
+endmodule
+

      Here we have defined constructors for a simple expression language over
      +Booleans and integers, as well as a function eval that evaluates these
      +expressions to a value. Because that value could be an integer or a Boolean,
      +we need the casts in the last two rules in order to meet the type signature of
      ++Int and andBool. Of course, the user can write ill-formed expressions like
      +1 && true or false + true, but these will cause errors at runtime, because
      +the projection cast will fail.

      +

      Exercises

      + +
        +
      1. +

        Extend the eval function in LESSON-11-D to include Strings and add a .
        +operator which concatenates them.

        +
        2. +
      2. +

        Modify your solution from Lesson 1.9, Exercise 2 by using an Exp sort to
        +express the integer and Boolean expressions that it supports, in the same style
        +as LESSON-11-D. Then write an eval function that evaluates all terms of
        +sort Exp to either a Bool or an Int.

        +
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.12: Syntactic Lists.

      +

      Lesson 1.12: Syntactic Lists

      + +

      The purpose of this lesson is to explain how K provides support for syntactic
      +repetition through the use of the List{} and NeList{} constructs,
      +generally called syntactic lists.

      +

      The List{} construct

      + +

      Sometimes, when defining a grammar in K, it is useful to define a syntactic
      +construct consisting of an arbitrary-length sequence of items. For example,
      +you might wish to define a function call construct, and need to express a way
      +of passing arguments to the function. You can in theory simply define these
      +productions using ordinary constructors, but it can be tricky to get the syntax
      +exactly right in K without a lot of tedious glue code.

      +

      For this reason, K provides a way of specifying that a non-terminal represents
      +a syntactic list (lesson-12-a.k):

      +
      module LESSON-12-A-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= List{Int,","}
+endmodule
+
+module LESSON-12-A
+  imports LESSON-12-A-SYNTAX
+endmodule
+

      Note that instead of a sequence of terminals and non-terminals, the right hand
      +side of the Ints production contains the symbol List followed by two items
      +in curly braces. The first item is the non-terminal which is the element type
      +of the list, and the second item is a terminal representing the separator of
      +the list. As a special case, lists which are separated only by whitespace can
      +be specified with a separator of "".

      +

      This List{} construct is roughly equivalent to the following definition
      +(lesson-12-b.k):

      +
      module LESSON-12-B-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= Int "," Ints | ".Ints"
+endmodule
+
+module LESSON-12-B
+  imports LESSON-12-B-SYNTAX
+endmodule
+

      As you can see, the List{} construct represents a cons-list with an element
      +at the head and another list at the tail. The empty list is represented by
      +a . followed by the sort of the list.

      +

      However, the List{} construct provides several key syntactic conveniences
      +over the above definition. First of all, when writing a list in a rule,
      +explicitly writing the terminator is not always required. For example, consider
      +the following additional module (lesson-12-c.k):

      +
      module LESSON-12-C
+  imports LESSON-12-A
+  imports INT
+
+  syntax Int ::= sum(Ints) [function]
+  rule sum(I:Int) => I
+  rule sum(I1:Int, I2:Int, Is:Ints) => sum(I1 +Int I2, Is)
+endmodule
+

      Here we see a function that sums together a non-empty list of integers. Note in
      +particular the first rule. We do not explicitly mention .Ints, but in fact,
      +the rule in question is equivalent to the following rule:

      +
        rule sum(I:Int, .Ints) => I
+

      The reason for this is that K will automatically insert a list terminator
      +anywhere a syntactic list is expected, but an element of that list appears
      +instead. This works even with lists of more than one element:

      +
        rule sum(I1:Int, I2:Int) => I1 +Int I2
+

      This rule is redundant, but here we explicitly match a list of exactly two
      +elements, because the .Ints is implicitly added after I2.

      +

      Parsing Syntactic Lists in Programs

      + +

      An additional syntactic convenience takes place when you want to express a
      +syntactic list in the input to krun. In this case, K will automatically
      +transform the grammar in LESSON-12-B-SYNTAX into the following
      +(lesson-12-d.k):

      +
      module LESSON-12-D
+  imports INT-SYNTAX
+
+  syntax Ints ::= #NonEmptyInts | #IntsTerminator
+  syntax #NonEmptyInts ::= Int "," #NonEmptyInts
+                         | Int #IntsTerminator
+  syntax #IntsTerminator ::= ""
+endmodule
+

      This allows you to express the usual comma-separated list of arguments where
      +an empty list is represented by the empty string, and you don't have to
      +explicitly terminate the list. Because of this, we can write the syntax
      +of function calls in C very easily (lesson-12-e.k):

      +
      module LESSON-12-E
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id | Exp "(" Exps ")"
+  syntax Exps ::= List{Exp,","}
+endmodule
+

      Exercise

      + +

      Write a function concat which takes a list of String and concatenates them
      +all together. Do not worry if the function is O(n^2).
      +Test your implementation using the syntactic sugar for lists added by the parser.

      +

      Then write some function call expressions using identifiers in C and verify with
      +kast that the above grammar captures the intended syntax. Make sure to test
      +with function calls with zero, one, and two or more arguments.

      +

      The NeList{} construct

      + +

      One limitation of the List{} construct is that it is always possible to
      +write a list of zero elements where a List{} is expected. While this is
      +desirable in a number of cases, it is sometimes not what the grammar expects.

      +

      For example, in C, it is not allowable for an enum definition to have zero
      +members. In other words, if we were to write the grammar for enumerations like
      +so (lesson-12-f.k):

      +
      module LESSON-12-F
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id
+
+  syntax EnumSpecifier ::= "enum" Id "{" Ids "}"
+  syntax Ids ::= List{Id,","}
+endmodule
+

      Then we would be syntactically allowed to write enum X {}, which instead,
      +ought to be a syntax error.

      +

      For this reason, we introduce the additional NeList{} construct. The syntax
      +is identical to List{}, except with NeList instead of List before the
      +curly braces. When parsing rules, it behaves identically to the List{}
      +construct. However, when parsing inputs to krun, the above grammar, if we
      +replaced syntax Ids ::= List{Id,","} with syntax Ids ::= NeList{Id,","},
      +would become equivalent to the following (lesson-12-g.k):

      +
      module LESSON-12-G
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id
+
+  syntax EnumSpecifier ::= "enum" Id "{" Ids "}"
+  syntax Ids ::= Id | Id "," Ids
+endmodule
+

      In other words, only non-empty lists of Id would be allowed.

      +

      Exercises

      + +
        +
      1. +

        Modify the sum function in LESSON-12-C so that the Ints sort is an
        +NeList{}. Verify that calling sum() with no arguments is now a syntax
        +error.

        +
        2. +
      2. +

        Write a modified sum function with the List construct that can also sum
        +up an empty list of arguments. In such a case, the sum ought to be 0.

        +
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.13: Basics of K Rewriting.

      +

      Lesson 1.13: Basics of K Rewriting

      + +

      The purpose of this lesson is to explain how rewrite rules that are not the
      +definition of a function behave, and how, using these rules, you can construct
      +a semantics of programs in a programming language in K.

      +

      Recap: Function rules in K

      + +

      Recall from Lesson 1.2 that we have, thus far,
      +introduced two types of productions in K: constructors and functions.
      +A function is identified by the function attribute placed on the
      +production. As you may recall, when we write a rule with a function on the
      +left-hand side of the => operator, we are defining the meaning of that
      +function for inputs which match the patterns on the left-hand side of the rule.
      +If the argument to the function match the patterns, then the function is
      +evaluated to the value constructed by substituting the bindings for the
      +variables into the right-hand side of the rule.

      +

      Top-level rules

      + +

      However, function rules are not the only type of rule permissible in K, nor
      +even the most frequently used. K also has a concept of a
      +top-level rewrite rule. The simplest way to ensure that a rule is treated
      +as a top-level rule is for the left-hand side of the rule to mention one or
      +more cells. We will cover how cells work and are declared in more detail
      +in a later lesson, but for now, what you should know is that when we ran krun
      +in our very first example in Lesson 1.2 and got the following output:

      +
      <k>
+  Yellow ( ) ~> .
+</k>
+

      <k> is a cell, known by convention as the K cell. This cell is available
      +by default in any definition without needing to be explicitly declared.

      +

      The K cell contains a single term of sort K. K is a predefined sort in K
      +with two constructors, that can be roughly represented by the following
      +grammar:

      +
        syntax K ::= KItem "~>" K
+             | "."
+

      As a syntactic convenience, K allows you to treat ~> like it is an
      +associative list (i.e., as if it were defined as syntax K ::= K "~>" K).
      +When a definition is compiled, it will automatically transform the rules you
      +write so that they treat the K sort as a cons-list. Another syntactic
      +convenience is that, for disambiguation purposes, you can write .K anywhere
      +you would otherwise write . and the meaning is identical.

      +

      Now, you may notice that the above grammar mentions the sort KItem. This is
      +another built-in sort in K. For every sort S declared in a definition (with
      +the exception of K and KItem), K will implicitly insert the following
      +production:

      +
        syntax KItem ::= S
+

      In other words, every sort is a subsort of the sort KItem, and thus a term
      +of any sort can be injected as an element of a term of sort K, also called
      +a K sequence.

      +

      By default, when you krun a program, the AST of the program is inserted as
      +the sole element of a K sequence into the <k> cell. This explains why we
      +saw the output we did in Lesson 1.2.

      +

      With these preliminaries in mind, we can now explain how top-level rewrite
      +rules work in K. Put simply, any rule where there is a cell (such as the K
      +cell) at the top on the left-hand side will be a top-level rewrite rule. Once
      +the initial program has been inserted into the K cell, the resulting term,
      +called the configuration, will be matched against all the top-level
      +rewrite rules in the definition. If only one rule matches, the substitution
      +generated by the matching will be applied to the right-hand side of the rule
      +and the resulting term is rewritten to be the new configuration. Rewriting
      +proceeds by iteratively applying rules, also called taking steps, until
      +no top-level rewrite rule can be applied. At this point the configuration
      +becomes the final configuration and is output by krun.

      +

      If more than one top-level rule applies, by default, K will pick just one
      +of those rules, apply it, and continue rewriting. However, it is
      +non-deterministic which rule applies. In theory, it could be any of them.
      +By passing the --search flag to krun, you are able to tell krun to
      +explore all possible non-deterministic choices, and generate a complete list of
      +all possible final configurations reachable by each nondeterminstic choice that
      +can be made. Note that the --search flag to krun only works if you pass
      +--enable-search to kompile first.

      +

      Unlike top-level rewrite rules, function rules are not associated with any
      +particular set of cells in the configuration (although they can contain cells
      +in their function arguments and return value). While top-level rewrite rules
      +apply to the entire term being rewritten, function rules apply anywhere a
      +function application for that function appears, and are immediately rewritten
      +to their return value in that position.

      +

      Another key distinction between top-level rules and function rules is that
      +function symbols, i.e., productions with the function attribute, are
      +mathematical functions rather than constructors. While a constructor is
      +logically distinct from any other constructor of the same sort, and can be
      +matched against unconditionally, a function does not necessaraily have the
      +same restriction unless it happens to be an injective function. Thus, two
      +function symbols with different arguments may still ultimately produce the
      +same value and thus compare equal to one another. Due to this, concrete
      +execution (i.e., all K definitions introduced thus far; see Lesson 1.21)
      +introduces the restriction that you cannot match on a function symbol on the
      +left-hand side of a rule, except as the top symbol on the left-hand side of
      +a function rule. This restriction will be later lifted when we introduce the
      +Haskell Backend which performs symbolic execution.

      +

      Exercise

      + +

      Pass a program containing no functions to krun. You can use a term of sort
      +Exp from LESSON-11-D. Observe the output and try to understand why you get
      +the output you do. Then write two rules that rewrite that program to another.
      +Run krun --search on that program and observe both results. Then add a third
      +rule that rewrites one of those results again. Test that that rule applies as
      +well.

      +

      Using top-level rules to evaluate expressions

      + +

      Thus far, we have focused primarily on defining functions over constructors
      +in K. However, now that we have a basic understanding of top-level rules,
      +it is possible to introduce a rewrite system to our definitions. A rewrite
      +system is a collection of top-level rewrite rules which performs an organized
      +transformation of a particular program into a result which expresses the
      +meaning of that program. For example, we might rewrite an expression in a
      +programming language into a value representing the result of evaluating that
      +expression.

      +

      Recall in Lesson 1.11, we wrote a simple grammar of Boolean and integer
      +expressions that looked roughly like this (lesson-13-a.k):

      +
      module LESSON-13-A
+  imports INT
+
+  syntax Exp ::= Int
+               | Bool
+               | Exp "+" Exp
+               | Exp "&&" Exp
+endmodule
+

      In that lesson, we defined a function eval which evaluated such expressions
      +to either an integer or Boolean.

      +

      However, it is more idiomatic to evaluate such expressions using top-level
      +rewrite rules. Here is how one might do so in K (lesson-13-b.k):

      +
      module LESSON-13-B-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Val ::= Int | Bool
+  syntax Exp ::= Val
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-13-B
+  imports LESSON-13-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int ~> K:K </k> => <k> I1 +Int I2 ~> K </k>
+  rule <k> B1:Bool && B2:Bool ~> K:K </k> => <k> B1 andBool B2 ~> K </k>
+
+  syntax KItem ::= freezer1(Val) | freezer2(Exp)
+                 | freezer3(Val) | freezer4(Exp)
+
+  rule <k> E1:Val + E2:Exp ~> K:K </k> => <k> E2 ~> freezer1(E1) ~> K </k> [priority(51)]
+  rule <k> E1:Exp + E2:Exp ~> K:K </k> => <k> E1 ~> freezer2(E2) ~> K </k> [priority(52)]
+  rule <k> E1:Val && E2:Exp ~> K:K </k> => <k> E2 ~> freezer3(E1) ~> K </k> [priority(51)]
+  rule <k> E1:Exp && E2:Exp ~> K:K </k> => <k> E1 ~> freezer4(E2) ~> K </k> [priority(52)]
+
+  rule <k> E2:Val ~> freezer1(E1) ~> K:K </k> => <k> E1 + E2 ~> K </k>
+  rule <k> E1:Val ~> freezer2(E2) ~> K:K </k> => <k> E1 + E2 ~> K </k>
+  rule <k> E2:Val ~> freezer3(E1) ~> K:K </k> => <k> E1 && E2 ~> K </k>
+  rule <k> E1:Val ~> freezer4(E2) ~> K:K </k> => <k> E1 && E2 ~> K </k>
+endmodule
+

      This is of course rather cumbersome currently, but we will soon introduce
      +syntactic convenience which makes writing definitions of this type considerably
      +easier. For now, notice that there are roughly 3 types of rules here: the first
      +matches a K cell in which the first element of the K sequence is an Exp whose
      +arguments are values, and rewrites the first element of the sequence to the
      +result of that expression. The second also matches a K cell with an Exp in
      +the first element of its K sequence, but it matches when one or both arguments
      +of the Exp are not values, and replaces the first element of the K sequence
      +with two new elements: one being an argument to evaluate, and the other being
      +a special constructor called a freezer. Finally, the third matches a K
      +sequence where a Val is first, and a freezer is second, and replaces them
      +with a partially evaluated expression.

      +

      This general pattern is what is known as heating an expression,
      +evaluating its arguments, cooling the arguments into the expression
      +again, and evaluating the expression itself. By repeatedly performing
      +this sequence of actions, we can evaluate an entire AST containing a complex
      +expression down into its resulting value.

      +

      Exercise

      + +

      Write an addition expression with integers. Use krun --depth 1 to see the
      +result of rewriting after applying a single top-level rule. Gradually increase
      +the value of --depth to see successive states. Observe how this combination
      +of rules is eventually able to evaluate the entire expression.

      +

      Simplifying the evaluator: Local rewrites and cell ellipses

      + +

      As you saw above, the definition we wrote is rather cumbersome. Over the
      +remainder of Lessons 1.13 and 1.14, we will greatly simplify it. The first step
      +in doing so is to teach a bit more about the rewrite operator, =>. Thus far,
      +all the rules we have written look like rule LHS => RHS. However, this is not
      +the only way the rewrite operator can be used. It is actually possible to place
      +a constructor or function at the very top of the rule, and place rewrite
      +operators inside that term. While a rewrite operator cannot appear nested
      +inside another rewrite operator, by doing this, we can express that some parts
      +of what we are matching are not changed by the rewrite operator. For
      +example, consider the following rule from above:

      +
        rule <k> I1:Int + I2:Int ~> K:K </k> => <k> I1 +Int I2 ~> K </k>
+

      We can equivalently write it like following:

      +
        rule <k> (I1:Int + I2:Int => I1 +Int I2) ~> _:K </k>
+

      When you put a rewrite inside a term like this, in essence, you are telling
      +the rule to only rewrite part of the left-hand side to the right-hand side.
      +In practice, this is implemented by lifting the rewrite operator to the top of
      +the rule by means of duplicating the surrounding context.

      +

      There is a way that the above rule can be simplified further, however. K
      +provides a special syntax for each cell containing a term of sort K, indicating
      +that we want to match only on some prefix of the K sequence. For example, the
      +above rule can be simplified further like so:

      +
        rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+

      Here we have placed the symbol ... immediately prior to the </k> which ends
      +the cell. What this tells the compiler is to take the contents of the cell,
      +treat it as the prefix of a K sequence, and insert an anonymous variable of
      +sort K at the end. Thus we can think of ... as a way of saying we
      +don't care about the part of the K sequence after the beginning, leaving
      +it unchanged.

      +

      Putting all this together, we can rewrite LESSON-13-B like so
      +(lesson-13-c.k):

      +
      module LESSON-13-C-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Val ::= Int | Bool
+  syntax Exp ::= Val
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-13-C
+  imports LESSON-13-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax KItem ::= freezer1(Val) | freezer2(Exp)
+                 | freezer3(Val) | freezer4(Exp)
+
+  rule <k> E1:Val + E2:Exp => E2 ~> freezer1(E1) ...</k> [priority(51)]
+  rule <k> E1:Exp + E2:Exp => E1 ~> freezer2(E2) ...</k> [priority(52)]
+  rule <k> E1:Val && E2:Exp => E2 ~> freezer3(E1) ...</k> [priority(51)]
+  rule <k> E1:Exp && E2:Exp => E1 ~> freezer4(E2) ...</k> [priority(52)]
+
+  rule <k> E2:Val ~> freezer1(E1) => E1 + E2 ...</k>
+  rule <k> E1:Val ~> freezer2(E2) => E1 + E2 ...</k>
+  rule <k> E2:Val ~> freezer3(E1) => E1 && E2 ...</k>
+  rule <k> E1:Val ~> freezer4(E2) => E1 && E2 ...</k>
+endmodule
+

      This is still rather cumbersome, but it is already greatly simplified. In the
      +next lesson, we will see how additional features of K can be used to specify
      +heating and cooling rules much more compactly.

      +

      Exercises

      + +
        +
      1. Modify LESSON-13-C to add rules to evaluate integer subtraction.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.14: Defining Evaluation Order.

      +

      Lesson 1.14: Defining Evaluation Order

      + +

      The purpose of this lesson is to explain how to use the heat and cool
      +attributes, context and context alias sentences, and the strict and
      +seqstrict attributes to more compactly express heating and cooling in K,
      +and to express more advanced evaluation strategies in K.

      +

      The heat and cool attributes

      + +

      Thus far, we have been using rule priority and casts to express when to heat
      +an expression and when to cool it. For example, the rules for heating have
      +lower priority, so they do not apply if the term could be evaluated instead,
      +and the rules for heating are expressly written only to apply if the argument
      +of the expression is a value.

      +

      However, K has built-in support for deciding when to heat and when to cool.
      +This support comes in the form of the rule attributes heat and cool as
      +well as the specially named function isKResult.

      +

      Consider the following definition, which is equivalent to LESSON-13-C
      +(lesson-14-a.k):

      +
      module LESSON-14-A-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-14-A
+  imports LESSON-14-A-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax KItem ::= freezer1(Exp) | freezer2(Exp)
+                 | freezer3(Exp) | freezer4(Exp)
+
+  rule <k> E:Exp + HOLE:Exp => HOLE ~> freezer1(E) ...</k>
+    requires isKResult(E) [heat]
+  rule <k> HOLE:Exp + E:Exp => HOLE ~> freezer2(E) ...</k> [heat]
+  rule <k> E:Exp && HOLE:Exp => HOLE ~> freezer3(E) ...</k>
+    requires isKResult(E) [heat]
+  rule <k> HOLE:Exp && E:Exp => HOLE ~> freezer4(E) ...</k> [heat]
+
+  rule <k> HOLE:Exp ~> freezer1(E) => E + HOLE ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer2(E) => HOLE + E ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer3(E) => E && HOLE ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer4(E) => HOLE && E ...</k> [cool]
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      We have introduced three major changes to this definition. First, we have
      +removed the Val sort. We replace it instead with a function isKResult.
      +The function in question must have the same signature and attributes as seen in
      +this example. It ought to return true whenever a term should not be heated
      +(because it is a value) and false when it should be heated (because it is not
      +a value). We thus also insert isKResult calls in the side condition of two
      +of the heating rules, where the Val sort was previously used.

      +

      Second, we have removed the rule priorities on the heating rules and the use of
      +the Val sort on the cooling rules, and replaced them with the heat and
      +cool attributes. These attributes instruct the compiler that these rules are
      +heating and cooling rules, and thus should implicitly apply only when certain
      +terms on the LHS either are or are not a KResult (i.e., isKResult returns
      +true versus false).

      +

      Third, we have renamed some of the variables in the heating and cooling rules
      +to the special variable HOLE. Syntactically, HOLE is just a special name
      +for a variable, but it is treated specially by the compiler. By naming a
      +variable HOLE, we have informed the compiler which term is being heated
      +or cooled. The compiler will automatically insert the side condition
      +requires isKResult(HOLE) to cooling rules and the side condition
      +requires notBool isKResult(HOLE) to heating rules.

      +

      Exercise

      + +

      Modify LESSON-14-A to add rules to evaluate integer subtraction.

      +

      Simplifying further with Contexts

      + +

      The above example is still rather cumbersome to write. We must explicitly write
      +both the heating and the cooling rule separately, even though they are
      +essentially inverses of one another. It would be nice to instead simply
      +indicate which terms should be heated and cooled, and what part of them to
      +operate on.

      +

      To do this, K introduces a new type of sentence, the context. Contexts
      +begin with the context keyword instead of the rule keyword, and usually
      +do not contain a rewrite operator.

      +

      Consider the following definition which is equivalent to LESSON-14-A
      +(lesson-14-b.k):

      +
      module LESSON-14-B-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-14-B
+  imports LESSON-14-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  context <k> E:Exp + HOLE:Exp ...</k>
+    requires isKResult(E)
+  context <k> HOLE:Exp + _:Exp ...</k>
+  context <k> E:Exp && HOLE:Exp ...</k>
+    requires isKResult(E)
+  context <k> HOLE:Exp && _:Exp ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      In this example, the heat and cool rules have been removed entirely, as
      +have been the productions defining the freezers. Don't worry, they still exist
      +under the hood; the compiler is just generating them automatically. For each
      +context sentence like above, the compiler generates a #freezer production,
      +a heat rule, and a cool rule. The generated form is equivalent to the
      +rules we wrote manually in LESSON-14-A. However, we are now starting to
      +considerably simplify the definition. Instead of 3 sentences, we just have one.

      +

      context alias sentences and the strict and seqstrict attributes

      + +

      Notice that the contexts we included in LESSON-14-B still seem rather
      +similar in form. For each expression we want to evaluate, we are declaring
      +one context for each operand of that expression, and they are each rather
      +similar to one another. We would like to be able to simplify further by
      +simply annotating each expression production with information about how
      +it is to be evaluated instead. We can do this with the seqstrict attribute.

      +

      Consider the following definition, once again equivalent to those above
      +(lesson-14-c.k):

      +
      module LESSON-14-C-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp [seqstrict(exp; 1, 2)]
+               > left: Exp "&&" Exp [seqstrict(exp; 1, 2)]
+endmodule
+
+module LESSON-14-C
+  imports LESSON-14-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  context alias [exp]: <k> HERE ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      This definition has two important changes from the one above. The first is
      +that the individual context sentences have been removed and have been
      +replaced with a single context alias sentence. You may notice that this
      +sentence begins with an identifier in square brackets followed by a colon. This
      +syntax is a way of naming individual sentences in K for reference by the tool
      +or by other sentences. The context alias sentence also has a special variable
      +HERE.

      +

      The second is that the productions in LESSON-14-C-SYNTAX have been given a
      +seqstrict attribute. The value of this attribute has two parts. The first
      +is the name of a context alias sentence. The second is a comma-separated list
      +of integers. Each integer represents an index of a non-terminal in the
      +production, counting from 1. For each integer present, the compiler implicitly
      +generates a new context sentence according to the following rules:

      +
        +
      1. The compiler starts by looking for the context alias sentence named. If
        +there is more than one, then one context sentence is created per
        +context alias sentence with that name.
        2. +
      2. For each context created, the variable HERE in the context alias is
        +substituted with an instance of the production the seqstrict attribute is
        +attached to. Each child of that production is a variable. The non-terminal
        +indicated by the integer offset of the seqstrict attribute is given the name
        +HOLE.
        3. +
      3. For each integer offset prior in the list to the one currently being
        +processed, the predicate isKResult(E) is conjuncted together and included
        +as a side condition, where E is the child of the production term with that
        +offset, starting from 1. For example, if the attribute lists 1, 2, then
        +the rule generated for the 2 will include isKResult(E1) where E1 is the
        +first child of the production.
        4. +
      +

      As you can see if you work through the process, the above code will ultimately
      +generate the same contexts present in LESSON-14-B.

      +

      Finally, note that there are a few minor syntactic conveniences provided by the
      +seqstrict attribute. First, in the special case of the context alias sentence
      +being <k> HERE ...</k>, you can omit both the context alias sentence
      +and the name from the seqstrict attribute.

      +

      Second, if the numbered list of offsets contains every non-terminal in the
      +production, it can be omitted from the attribute value.

      +

      Thus, we can finally produce the idiomatic K definition for this example
      +(lesson-14-d.k):

      +
      module LESSON-14-D-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp [seqstrict]
+               > left: Exp "&&" Exp [seqstrict]
+endmodule
+
+module LESSON-14-D
+  imports LESSON-14-D-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      Exercise

      + +

      Modify LESSON-14-D to add a production and rule to evaluate integer
      +subtraction.

      +

      Nondeterministic evaluation order with the strict attribute

      + +

      Thus far, we have focused entirely on deterministic evaluation order. However,
      +not all languages are deterministic in the order they evaluate expressions.
      +For example, in C, the expression a() + b() + c() is guaranteed to parse
      +to (a() + b()) + c(), but it is not guaranteed that a will be called before
      +b before c. In fact, this evaluation order is non-deterministic.

      +

      We can express non-deterministic evaluation orders with the strict attribute.
      +Its behavior is identical to the seqstrict attribute, except that step 3 in
      +the above list (with the side condition automatically added) does not take
      +place. In other words, if we wrote syntax Exp ::= Exp "+" Exp [strict]
      +instead of syntax Exp ::= Exp "+" Exp [seqstrict], it would generate the
      +following two contexts instead of the ones found in LESSON-14-B:

      +
        context <k> _:Exp + HOLE:Exp ...</k>
+  context <k> HOLE:Exp + _:Exp ...</k>
+

      As you can see, these contexts will generate heating rules that can both
      +apply to the same term. As a result, the choice of which heating rule
      +applies first is non-deterministic, and as we saw in Lesson 1.13, we can
      +get all possible behaviors by passing --search to krun.

      +

      Exercises

      + +
        +
      1. +

        Add integer division to LESSON-14-D. Make division and addition strict
        +instead of seqstrict, and write a rule evaluating integer division with a
        +side condition that the denominator is non-zero. Run krun --search on the
        +program 1 / 0 + 2 / 1 and observe all possible outputs of the program. How
        +many are there total, and why?

        +
        2. +
      2. +

        Rework your solution from Lesson 1.9, Exercise 2 to evaluate expressions from left to right using the seqstrict attribute.

        +
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.15: Configuration Declarations and Cell Nesting.

      +

      Lesson 1.15: Configuration Declarations and Cell Nesting

      + +

      The purpose of this lesson is to explain how to store additional information
      +about the state of your interpreter by declaring cells using the
      +configuration sentence, as well as how to add additional inputs to your
      +definition.

      +

      Cells and Configuration Declarations

      + +

      We have already covered the absolute basics of cells in K by looking at the
      +<k> cell. As explained in Lesson 1.13, the
      +<k> cell is available without being explicitly declared. It turns out this is
      +because, if the user does not explicitly specify a configuration sentence
      +anywhere in the main module of their definition, the configuration sentence
      +from the DEFAULT-CONFIGURATION module of
      +kast.md is imported
      +automatically. Here is what that sentence looks like:

      +
        configuration <k> $PGM:K </k>
+

      This configuration declaration declares a single cell, the <k> cell. It also
      +declares that at the start of rewriting, the contents of that cell should be
      +initialized with the value of the $PGM configuration variable.
      +Configuration variables function as inputs to krun. These terms are supplied
      +to krun in the form of ASTs parsed using a particular module. By default, the
      +$PGM configuration variable uses the main syntax module of the definition.

      +

      The cast on the configuration variable also specifies the sort that is used as
      +the entry point to the parser, in this case the K sort. It is often
      +useful to cast to other sorts there as well for better control over the accepted
      +language. The sort used for the $PGM variable is referred to as the start
      +symbol. During parsing, the default start symbol K subsumes all user-defined
      +sorts except for syntactic lists. These are excluded because they will always
      +produce an ambiguity error when parsing a single element.

      +

      Note that we did not explicitly specify the $PGM configuration variable when
      +we invoked krun on a file. This is because krun handles the $PGM variable
      +specially, and allows you to pass the term for that variable via a file passed
      +as a positional argument to krun. We did, however, specify the PGM name
      +explicitly when we called krun with the -cPGM command line argument in
      +Lesson 1.2. This is the other, explicit, way of
      +specifying an input to krun.

      +

      This explains the most basic use of configuration declarations in K. We can,
      +however, declare multiple cells and multiple configuration variables. We can
      +also specify the initial values of cells statically, rather than dynamically
      +via krun.

      +

      For example, consider the following definition (lesson-15-a.k):

      +
      module LESSON-15-A-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= List{Int,","}
+endmodule
+
+module LESSON-15-A
+  imports LESSON-15-A-SYNTAX
+  imports INT
+
+  configuration <k> $PGM:Ints </k>
+                <sum> 0 </sum>
+
+  rule <k> I:Int, Is:Ints => Is ...</k>
+       <sum> SUM:Int => SUM +Int I </sum>
+endmodule
+

      This simple definition takes a list of integers as input and sums them
      +together. Here we have declared two cells: <k> and <sum>. Unlike <k>,
      +<sum> does not get initialized via a configuration variable, but instead
      +is initialized statically with the value 0.

      +

      Note the rule in the second module: we have explicitly specified multiple
      +cells in a single rule. K will expect each of these cells to match in order for
      +the rule to apply.

      +

      Here is a second example (lesson-15-b.k):

      +
      module LESSON-15-B-SYNTAX
+  imports INT-SYNTAX
+endmodule
+
+module LESSON-15-B
+  imports LESSON-15-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  configuration <k> . </k>
+                <first> $FIRST:Int </first>
+                <second> $SECOND:Int </second>
+
+  rule <k> . => FIRST >Int SECOND </k>
+       <first> FIRST </first>
+       <second> SECOND </second>
+endmodule
+

      This definition takes two integers as command-line arguments and populates the
      +<k> cell with a Boolean indicating whether the first integer is greater than
      +the second. Notice that we have specified no $PGM configuration variable
      +here. As a result, we cannot invoke krun via the syntax krun $file.
      +Instead, we must explicitly pass values for each configuration variable via the
      +-cFIRST and -cSECOND command line flags. For example, if we invoke
      +krun -cFIRST=0 -cSECOND=1, we will get the value false in the K cell.

      +

      You can also specify both a $PGM configuration variable and other
      +configuration variables in a single configuration declaration, in which case
      +you would be able to initialize $PGM with either a positional argument or the
      +-cPGM command line flag, but the other configuration variables would need
      +to be explicitly initialized with -c.

      +

      Exercise

      + +

      Modify your solution to Lesson 1.14, Exercise 2 to add a new cell with a
      +configuration variable of sort Bool. This variable should determine whether
      +the / operator is evaluated using /Int or divInt. Test that by specifying
      +different values for this variable, you can change the behavior of rounding on
      +division of negative numbers.

      +

      Cell Nesting

      + +

      It is possible to nest cells inside one another. A cell that contains other
      +cells must contain only other cells, but in doing this, you are able to
      +create a hierarchical structure to the configuration. Consider the following
      +definition (lesson-15-c.k), which is equivalent to the one in LESSON-15-B:

      +
      module LESSON-15-C-SYNTAX
+  imports INT-SYNTAX
+endmodule
+
+module LESSON-15-C
+  imports LESSON-15-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  configuration <T>
+                  <k> . </k>
+                  <state>
+                    <first> $FIRST:Int </first>
+                    <second> $SECOND:Int </second>
+                  </state>
+                </T>
+
+  rule <k> . => FIRST >Int SECOND </k>
+       <first> FIRST </first>
+       <second> SECOND </second>
+endmodule
+

      Note that we have added some new cells to the configuration declaration:
      +the <T> cell wraps the entire configuration, and the <state> cell is
      +introduced around the <first> and <second> cells.

      +

      However, we have not changed the rule in this definition. This is because of
      +a concept in K called configuration abstraction. K allows you to specify
      +any number of cells in a rule (except zero) in any order you want, and K will
      +compile the rules into a form that matches the structure of the configuration
      +specified by the configuration declaration.

      +

      Here then, is how this rule would look after the configuration abstraction
      +has been resolved:

      +
        rule <T>
+         <k> . => FIRST >Int SECOND </k>
+         <state>
+           <first> FIRST </first>
+           <second> SECOND </second>
+         </state>
+       </T>
+

      In other words, K will complete cells to the top of the configuration by
      +inserting parent cells where appropriate based on the declared structure of
      +the configuration. This is useful because as a definition evolves, the
      +configuration may change, but you don't want to have to modify every single
      +rule each time. Thus, K follows the principle that you should only mention the
      +cells in a rule that are actually needed in order to accomplish its specific
      +goal. By following this best practice, you can significantly increase the
      +modularity of the definition and make it easier to maintain and modify.

      +

      Note that unlike top-level rewrite rules, cells that appear inside function
      +rules are not necessarily completed to the top of the configuration. They still
      +participate in cell ccompletion in the sense that you can mention cell
      +structure loosely inside a function rule and it will be completed into the
      +correct cell structure specified by the configuration declaration. However,
      +they do not complete all the way to the top, instead completing only up to
      +the top-most cell mentioned in the rule.

      +

      For example, if I write the following function rule in the above definition:

      +
        rule doStuff(<first> FIRST </first>) => FIRST
+

      The function will only match on the first cell, rather than the entire
      +configuration. However, if we had mentioned a parent cell in the rule, it still
      +would have completed the children of that parent cell as needed to ensure that
      +the resulting term is well formed.

      +

      Exercise

      + +

      Modify your definition from the previous exercise in this lesson to wrap the
      +two cells you have declared in a top cell <T>. You should not have to change
      +any other rules in the definition.

      +

      Cell Variables

      + +

      Sometimes it is desirable to explicitly match a variable against certain
      +fragments of the configuration. Because K's configuration is hierarchical,
      +we can grab subsets of the configuration as if they were just another term.
      +However, configuration abstraction applies here as well.
      +In particular, for each cell you specify in a configuration declaration, a
      +unique sort is assigned for that cell with a single constructor (the cell
      +itself). The sort name is taken by removing all special characters,
      +capitalizing the first letter and each letter after a hyphen, and adding the
      +word Cell at the end. For example, in the above example, the cell sorts are
      +TCell, KCell, StateCell, FirstCell, and SecondCell. If we had declared
      +a cell as <first-number>, then the cell sort name would be FirstNumberCell.

      +

      You can explicitly reference a variable of one of these sorts anywhere you
      +might instead write that cell. For example, consider the following rule:

      +
        rule <k> true => S </k>
+       (S:StateCell => <state>... .Bag ...</state>)
+

      Here we have introduced two new concepts. The first is the variable of sort
      +StateCell, which matches the entire <state> part of the configuration. The
      +second is that we have introduced the concept of ... once again. When a cell
      +contains other cells, it is also possible to specify ... on either the left,
      +right or both sides of the cell term. Each of these three syntaxes are
      +equivalent in this case. When they appear on the left-hand side of a rule, they
      +indicate that we don't care what value any cells not explicitly named might
      +have. For example, we might write <state>... <first> 0 </first> ...</state> on
      +the left-hand side of a rule in order to indicate that we want to match the
      +rule when the <first> cell contains a zero, regardless of what the <second>
      +cell contains. If we had not included this ellipsis, it would have been a
      +syntax error, because K would have expected you to provide a value for each of
      +the child cells.

      +

      However, if, as in the example above, the ... appeared on the right-hand side
      +of a rule, this instead indicates that the cells not explicitly mentioned under
      +the cell should be initialized with their default value from the configuration
      +declaration. In other words, that rule will set the value of <first> and
      +<second> to zero.

      +

      You may note the presence of the phrase .Bag here. You can think of this as
      +the empty set of cells. It is used as the child of a cell when you want to
      +indicate that no cells should be explicitly named. We will cover other uses
      +of this term in later lessons.

      +

      Exercises

      + +
        +
      1. Modify the definition from the previous exercise in this lesson so that the
        +Boolean cell you created is initialized to false. Then add a production
        +syntax Stmt ::= Bool ";" Exp, and a rule that uses this Stmt to set the
        +value of the Boolean flag. Then add another production
        +syntax Stmt ::= "reset" ";" Exp which sets the value of the Boolean flag back
        +to its default value via a ... on the right-hand side. You will need to add
        +an additional cell around the Boolean cell to make this work.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.16: Maps, Semantic Lists, and Sets.

      +

      Lesson 1.16: Maps, Semantic Lists, and Sets

      + +

      The purpose of this lesson is to explain how to use the data structure sorts
      +provided by K: maps, lists, and sets.

      +

      Maps

      + +

      The most frequently used type of data structure in K is the map. The sort
      +provided by K for this purpose is the Map sort, and it is provided in
      +domains.md in the MAP
      +module. This type is not (currently) polymorphic. All Map terms are maps that
      +map terms of sort KItem to other terms of sort KItem. A KItem can contain
      +any sort except a K sequence. If you need to store such a term in a
      +map, you can always use a wrapper such as syntax KItem ::= kseq(K).

      +

      A Map pattern consists of zero or more map elements (as represented by the
      +symbol syntax Map ::= KItem "|->" KItem), mixed in any order, separated by
      +whitespace, with zero or one variables of sort Map. The empty map is
      +represented by .Map. If all of the bindings for the variables in the keys
      +of the map can be deterministically chosen, these patterns can be matched in
      +O(1) time. If they cannot, then each map element that cannot be
      +deterministically constructed contributes a single dimension of polynomial
      +time to the cost of the matching. In other words, a single such element is
      +linear, two are quadratic, three are cubic, etc.

      +

      Patterns like the above are the only type of Map pattern that can appear
      +on the left-hand-side of a rule. In other words, you are not allowed to write
      +a Map pattern on the left-hand-side with more than one variable of sort Map
      +in it. You are, however, allowed to write such patterns on the right-hand-side
      +of a rule. You can also write a function pattern in the key of a map element
      +so long as all the variables in the function pattern can be deterministically
      +chosen.

      +

      Note the meaning of matching on a Map pattern: a map pattern with no
      +variables of sort Map will match if the map being matched has exactly as
      +many bindings as |-> symbols in the pattern. It will then match if each
      +binding in the map pattern matches exactly one distinct binding in the map
      +being matched. A map pattern with one Map variable will also match any map
      +that contains such a map as a subset. The variable of sort Map will be bound
      +to whatever bindings are left over (.Map if there are no bindings left over).

      +

      Here is an example of a simple definition that implements a very basic
      +variable declaration semantics using a Map to store the value of variables
      +(lesson-16-a.k):

      +
      module LESSON-16-A-SYNTAX
+  imports INT-SYNTAX
+  imports ID-SYNTAX
+
+  syntax Exp ::= Id | Int
+  syntax Decl ::= "int" Id "=" Exp ";" [strict(2)]
+  syntax Pgm ::= List{Decl,""}
+endmodule
+
+module LESSON-16-A
+  imports LESSON-16-A-SYNTAX
+  imports BOOL
+
+  configuration <T>
+                  <k> $PGM:Pgm </k>
+                  <state> .Map </state>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // variable declaration
+  rule <k> int X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+
+  // variable lookup
+  rule <k> X:Id => I ...</k>
+       <state>... X |-> I ...</state>
+
+  syntax Bool ::= isKResult(K) [symbol, function]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      There are several new features in this definition. First, note we import
      +the module ID-SYNTAX. This module is defined in domains.md and provides a
      +basic syntax for identifiers. We are using the Id sort provided by this
      +module in this definition to implement the names of program variables. This
      +syntax is only imported when parsing programs, not when parsing rules. Later in
      +this lesson we will see how to reference specific concrete identifiers in a
      +rule.

      +

      Second, we introduce a single new function over the Map sort. This function,
      +which is represented by the symbol
      +syntax Map ::= Map "[" KItem "<-" KItem "]", represents the map update
      +operation. Other functions over the Map sort can be found in domains.md.

      +

      Finally, we have used the ... syntax on a cell containing a Map. In this
      +case, the meaning of <state>... Pattern ...</state>,
      +<state>... Pattern </state>, and <state> Pattern ...</state> are the same:
      +it is equivalent to writing <state> (Pattern) _:Map </state>.

      +

      Consider the following program (a.decl):

      +
      int x = 0;
+int y = 1;
+int a = x;
+

      If we run this program with krun, we will get the following result:

      +
      <T>
+  <k>
+    .
+  </k>
+  <state>
+    a |-> 0
+    x |-> 0
+    y |-> 1
+  </state>
+</T>
+

      Note that krun has automatically sorted the collection for you. This doesn't
      +happen at runtime, so you still get the performance of a hash map, but it will
      +help make the output more readable.

      +

      Exercise

      + +

      Create a sort Stmt that is a subsort of Decl. Create a production of sort
      +Stmt for variable assignment in addition to the variable declaration
      +production. Feel free to use the syntax syntax Stmt ::= Id "=" Exp ";". Write
      +a rule that implements variable assignment using a map update function. Then
      +write the same rule using a map pattern. Test your implementations with some
      +programs to ensure they behave as expected.

      +

      Semantic Lists

      + +

      In a previous lesson, we explained how to represent lists in the AST of a
      +program. However, this is not the only context where lists can be used. We also
      +frequently use lists in the configuration of an interpreter in order to
      +represent certain types of program state. For this purpose, it is generally
      +useful to have an associative-list sort, rather than the cons-list sorts
      +provided in Lesson 1.12.

      +

      The type provided by K for this purpose is the List sort, and it is also
      +provided in domains.md, in the LIST module. This type is also not
      +(currently) polymorphic. Like Map, all List terms are lists of terms of the
      +KItem sort.

      +

      A List pattern in K consists of zero or more list elements (as represented by
      +the ListItem symbol), followed by zero or one variables of sort List,
      +followed by zero or more list elements. An empty list is represented by
      +.List. These patterns can be matched in O(log(N)) time. This is the only
      +type of List pattern that can appear on the left-hand-side of a rule. In
      +other words, you are not allowed to write a List pattern on the
      +left-hand-side with more than one variable of sort List in it. You are,
      +however, allowed to write such patterns on the right-hand-side of a rule.

      +

      Note the meaning of matching on a List pattern: a list pattern with no
      +variables of sort List will match if the list being matched has exactly as
      +many elements as ListItem symbols in the pattern. It will then match if each
      +element in sequence matches the pattern contained in the ListItem symbol. A
      +list pattern with one variable of sort List operates the same way, except
      +that it can match any list with at least as many elements as ListItem
      +symbols, so long as the prefix and suffix of the list match the patterns inside
      +the ListItem symbols. The variable of sort List will be bound to whatever
      +elements are left over (.List if there are no elements left over).

      +

      The ... syntax is allowed on cells containing lists as well. In this case,
      +the meaning of <cell>... Pattern </cell> is the same as
      +<cell> _:List (Pattern) </cell>, the meaning of <cell> Pattern ...</cell>
      +is the same as <cell> (Pattern) _:List</cell>. Because list patterns with
      +multiple variables of sort List are not allowed, it is an error to write
      +<cell>... Pattern ...</cell>.

      +

      Here is an example of a simple definition that implements a very basic
      +function-call semantics using a List as a function stack (lesson-16-b.k):

      +
      module LESSON-16-B-SYNTAX
+  imports INT-SYNTAX
+  imports ID-SYNTAX
+
+  syntax Exp ::= Id "(" ")" | Int
+  syntax Stmt ::= "return" Exp ";" [strict]
+  syntax Decl ::= "fun" Id "(" ")" "{" Stmt "}"
+  syntax Pgm ::= List{Decl,""}
+  syntax Id ::= "main" [token]
+endmodule
+
+module LESSON-16-B
+  imports LESSON-16-B-SYNTAX
+  imports BOOL
+  imports LIST
+
+  configuration <T>
+                  <k> $PGM:Pgm ~> main () </k>
+                  <functions> .Map </functions>
+                  <fstack> .List </fstack>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // function definitions
+  rule <k> fun X:Id () { S } => . ...</k>
+       <functions>... .Map => X |-> S ...</functions>
+
+  // function call
+  syntax KItem ::= stackFrame(K)
+  rule <k> X:Id () ~> K => S </k>
+       <functions>... X |-> S ...</functions>
+       <fstack> .List => ListItem(stackFrame(K)) ...</fstack>
+
+  // return statement
+  rule <k> return I:Int ; ~> _ => I ~> K </k>
+       <fstack> ListItem(stackFrame(K)) => .List ...</fstack>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      Notice that we have declared the production syntax Id ::= "main" [token].
      +Since we use the ID-SYNTAX module, this declaration is necessary in order to
      +be able to refer to the main identifier directly in the configuration
      +declaration. Our <k> cell now contains a K sequence initially: first we
      +process all the declarations in the program, then we call the main function.

      +

      Consider the following program (foo.func):

      +
      fun foo() { return 5; }
+fun main() { return foo(); }
+

      When we krun this program, we should get the following output:

      +
      <T>
+  <k>
+    5 ~> .
+  </k>
+  <functions>
+    foo |-> return 5 ;
+    main |-> return foo ( ) ;
+  </functions>
+  <fstack>
+    .List
+  </fstack>
+</T>
+

      Note that we have successfully put on the <k> cell the value returned by the
      +main function.

      +

      Exercise

      + +

      Add a term of sort Id to the stackFrame operator to keep track of the
      +name of the function in that stack frame. Then write a function
      +syntax String ::= printStackTrace(List) that takes the contents of the
      +<fstack> cell and pretty prints the current stack trace. You can concatenate
      +strings with +String in the STRING module in domains.md, and you can
      +convert an Id to a String with the Id2String function in the ID module.
      +Test this function by creating a new expression that returns the current stack
      +trace as a string. Make sure to update isKResult and the Exp sort as
      +appropriate to allow strings as values.

      +

      Sets

      + +

      The final primary data structure sort in K is a set, i.e., an idempotent
      +unordered collection where elements are deduplicated. The sort provided by K
      +for this purpose is the Set sort and it is provided in domains.md in the
      +SET module. Like maps and lists, this type is not (currently) polymorphic.
      +Like Map and List, all Set terms are sets of terms of the KItem sort.

      +

      A Set pattern has the exact same restrictions as a Map pattern, except that
      +its elements are treated like keys, and there are no values. It has the same
      +performance characteristics as well. However, syntactically it is more similar
      +to the List sort: An empty Set is represented by .Set, but a set element
      +is represented by the SetItem symbol.

      +

      Matching behaves similarly to the Map sort: a set pattern with no variables
      +of sort Set will match if the set has exactly as many bindings as SetItem
      +symbols, and if each element pattern matches one distinct element in the set.
      +A set with a variable of sort Set also matches any superset of such a set.
      +As with map, the elements left over will be bound to the Set variable (or
      +.Set if no elements are left over).

      +

      Like Map, the ... syntax on a set is syntactic sugar for an anonymous
      +variable of sort Set.

      +

      Here is an example of a simple modification to LESSON-16-A which uses a Set
      +to ensure that variables are never declared more than once. In practice, you
      +would likely just use the in_keys symbol over maps to test for this, but
      +it's still useful as an example of sets in practice:

      +
      module LESSON-16-C-SYNTAX
+  imports LESSON-16-A-SYNTAX
+endmodule
+
+module LESSON-16-C
+  imports LESSON-16-C-SYNTAX
+  imports BOOL
+  imports SET
+
+  configuration <T>
+                  <k> $PGM:Pgm </k>
+                  <state> .Map </state>
+                  <declared> .Set </declared>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // variable declaration
+  rule <k> int X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+       <declared> D => D SetItem(X) </declared>
+    requires notBool X in D
+
+  // variable lookup
+  rule <k> X:Id => I ...</k>
+       <state>... X |-> I ...</state>
+       <declared>... SetItem(X) ...</declared>
+
+  syntax Bool ::= isKResult(K) [symbol, function]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      Now if we krun a program containing duplicate declarations, it will get
      +stuck on the declaration.

      +

      Exercises

      + +
        +
      1. Modify your solution to Lesson 1.14, Exercise 2 and introduce the sorts
        +Decls, Decl, and Stmt which include variable and function declaration
        +(without function parameters), and return and assignment statements, as well
        +as call expressions. Use List and Map to implement these operators, making
        +sure to consider the interactions between components, such as saving and
        +restoring the environment of variables at each call site. Don't worry about
        +local function definitions or global variables for now. Make sure to test the
        +resulting interpreter.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.17: Cell Multiplicity and Cell Collections.

      +

      Lesson 1.17: Cell Multiplicity and Cell Collections

      + +

      The purpose of this lesson is to explain how you can create optional cells
      +and cells that repeat multiple times in a configuration using a feature called
      +cell multiplicity.

      +

      Cell Multiplicity

      + +

      K allows you to specify attributes for cell productions as part of the syntax
      +of configuration declarations. Unlike regular productions, which use the []
      +syntax for attributes, configuration cells use an XML-like attribute syntax:

      +
      configuration <k color="red"> $PGM:K </k>
+

      This configuration declaration gives the <k> cell the color red during
      +unparsing using the color attribute as discussed in
      +Lesson 1.9.

      +

      However, in addition to the usual attributes for productions, there are some
      +other attributes that can be applied to cells with special meaning. One such
      +attribute is the multiplicity attribute. By default, each cell that is
      +declared occurs exactly once in every configuration term. However, using the
      +multiplicity attribute, this default behavior can be changed. There are two
      +values that this attribute can have: ? and *.

      +

      Optional cells

      + +

      The first cell multiplicity we will discuss is ?. Similar to a regular
      +expression language, this attribute tells the compiler that this cell can
      +appear 0 or 1 times in the configuration. In other words, it is an
      +optional cell. By default, K does not create optional cells in the initial
      +configuration, unless that optional cell has a configuration variable inside
      +it. However, it is possible to override the default behavior and create that
      +cell initially by adding the additional cell attribute initial="".

      +

      K uses the .Bag symbol to represent the absence of any cells in a particular
      +rule. Consider the following module:

      +
      module LESSON-17-A
+  imports INT
+
+  configuration <k> $PGM:K </k>
+                <optional multiplicity="?"> 0 </optional>
+
+  syntax KItem ::= "init" | "destroy"
+
+  rule <k> init => . ...</k>
+       (.Bag => <optional> 0 </optional>)
+  rule <k> destroy => . ...</k>
+       (<optional> _ </optional> => .Bag)
+
+endmodule
+

      In this definition, when the init symbol is executed, the <optional> cell
      +is added to the configuration, and when the destroy symbol is executed, it
      +is removed. Any rule that matches on that cell will only match if that cell is
      +present in the configuration.

      +

      Exercise

      + +

      Create a simple definition with a Stmts sort that is a List{Stmt,""} and
      +a Stmt sort with the constructors
      +syntax Stmt ::= "enable" | "increment" | "decrement" | "disable". The
      +configuration should have an optional cell that contains an integer that
      +is created with the enable command, destroyed with the disable command,
      +and its value is incremented or decremented by the increment and decrement
      +command.

      +

      Cell collections

      + +

      The second type of cell multiplicity we will discuss is *. Simlar to a
      +regular expression language, this attribute tells the compiler that this cell
      +can appear 0 or more times in the configuration. In other words, it is a
      +cell collection. Cells with multiplicity * must be the only child of
      +their parent cell. As a convention, the inner cell is usually named with the
      +singular form of what it contains, and the outer cell with the plural form, for
      +example, "thread" and "threads".

      +

      All cell collections are required to have the type attribute set to either
      +Set or Map. A Set cell collection is represented as a set and behaves
      +internally the same as the Set sort, although it actually declares a new
      +sort. A Map cell collection is represented as a Map in which the first
      +subcell of the cell collection is the key and the remaining cells are the
      +value.

      +

      For example, consider the following module:

      +
      module LESSON-17-B
+  imports INT
+  imports BOOL
+  imports ID-SYNTAX
+
+  syntax Stmt ::= Id "=" Exp ";" [strict(2)]
+                | "return" Exp ";" [strict]
+  syntax Stmts ::= List{Stmt,""}
+  syntax Exp ::= Id
+               | Int
+               | Exp "+" Exp [seqstrict]
+               | "spawn" "{" Stmts "}"
+               | "join" Exp ";" [strict]
+
+  configuration <threads>
+                  <thread multiplicity="*" type="Map">
+                    <id> 0 </id>
+                    <k> $PGM:K </k>
+                  </thread>
+                </threads>
+                <state> .Map </state>
+                <next-id> 1 </next-id>
+
+  rule <k> X:Id => I:Int ...</k>
+       <state>... X |-> I ...</state>
+  rule <k> X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+  rule <k> S:Stmt Ss:Stmts => S ~> Ss ...</k>
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+
+  rule <thread>...
+         <k> spawn { Ss } => NEXTID ...</k>
+       ...</thread>
+       <next-id> NEXTID => NEXTID +Int 1 </next-id>
+       (.Bag =>
+       <thread>
+         <id> NEXTID </id>
+         <k> Ss </k>
+       </thread>)
+
+  rule <thread>...
+         <k> join ID:Int ; => I ...</k>
+       ...</thread>
+       (<thread>
+         <id> ID </id>
+         <k> return I:Int ; ...</k>
+       </thread> => .Bag)
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+

      This module implements a very basic fork/join semantics. The spawn expression
      +spawns a new thread to execute a sequence of statements and returns a thread
      +id, and the join statement waits until a thread executes return and then
      +returns the return value of the thread.

      +

      Note something quite novel here: the <k> cell is inside a cell of
      +multiplicity *. Since the <k> cell is just a regular cell (mostly), this
      +is perfectly allowable. Rules that don't mention a specific thread are
      +automatically completed to match any thread.

      +

      When you execute programs in this language, the cells in the cell collection
      +get sorted and printed like any other collection, but they still display like
      +cells. Rules in this language also benefit from all the structural power of
      +cells, allowing you to omit cells you don't care about or complete the
      +configuration automatically. This allows you to have the power of cells while
      +still being a collection under the hood.

      +

      Exercises

      + +
        +
      1. Modify the solution from Lesson 1.16, Exercise 1 so that the cell you use to
        +keep track of functions in a Map is now a cell collection. Run some programs
        +and compare how they get unparsed before and after this change.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.18: Term Equality and the Ternary Operator.

      +

      Lesson 1.18: Term Equality and the Ternary Operator

      + +

      The purpose of this lesson is to introduce how to compare equality of terms in
      +K, and how to put conditional expressions directly into the right-hand side of
      +rules.

      +

      Term Equality

      + +

      One major way you can compare whether two terms are equal in K is to simply
      +match both terms with a variable with the same name. This will only succeed
      +in matching if the two terms are equal structurally. However, sometimes this
      +is impractical, and it is useful to have access to a way to actually compare
      +whether two terms in K are equal. The operator for this is found in
      +domains.md in the K-EQUAL
      +module. The operator is ==K and takes two terms of sort K and returns a
      +Bool. It returns true if they are equal. This includes equality over builtin
      +types such as Map and Set where equality is not purely structural in
      +nature. However, it does not include any notion of semantic equality over
      +user-defined syntax. The inverse symbol for inequality is =/=K.

      +

      Ternary Operator

      + +

      One way to introduce conditional logic in K is to have two separate rules,
      +each with a side condition (or one rule with a side condition and another with
      +the owise attribute). However, sometimes it is useful to explicitly write
      +a conditional expression directly in the right-hand side of a rule. For this
      +purpose, K defines one more operator in the K-EQUAL module, which corresponds
      +to the usual ternary operator found in many languages. Here is an example of its
      +usage (lesson-18.k):

      +
      module LESSON-18
+  imports INT
+  imports BOOL
+  imports K-EQUAL
+
+  syntax Exp ::= Int | Bool | "if" "(" Exp ")" Exp "else" Exp [strict(1)]
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+
+  rule if (B:Bool) E1:Exp else E2:Exp => #if B #then E1 #else E2 #fi
+endmodule
+

      Note the symbol on the right-hand side of the final rule. This symbol is
      +polymorphic: B must be of sort Bool, but E1 and E2 could have been
      +any sort so long as both were of the same sort, and the sort of the entire
      +expression becomes equal to that sort. K supports polymorphic built-in
      +operators, but does not yet allow users to write their own polymorphic
      +productions.

      +

      The behavior of this function is to evaluate the Boolean expression to a
      +Boolean, then pick one of the two children and return it based on whether the
      +Boolean is true or false. Please note that it is not a good idea to use this
      +symbol in cases where one or both of the children is potentially undefined
      +(for example, an integer expression that divides by zero). While the default
      +implementation is smart enough to only evaluate the branch that happens to be
      +picked, this will not be true when we begin to do program verification. If
      +you need short circuiting behavior, it is better to use a side condition.

      +

      Exercises

      + +
        +
      1. +

        Write a function in K that takes two terms of sort K and returns an
        +Int: the Int should be 0 if the terms are equal and 1 if the terms are
        +unequal.

        +
        2. +
      2. +

        Modify your solution to Lesson 1.16, Exercise 1 and introduce an if
        +Stmt to the syntax of the language, then implement it using the #if symbol.
        +Make sure to write tests for the resulting interpreter.

        +
        3. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.19: Debugging with GDB.

      +

      Lesson 1.19: Debugging with GDB or LLDB

      + +

      The purpose of this lesson is to teach how to debug your K interpreter using
      +the K-language support provided in GDB or
      +LLDB.

      +

      Caveats

      + +

      This lesson has been written with GDB support on Linux in mind. Unfortunately,
      +on macOS, GDB has limited support. To address this, we have introduced early
      +experimental support for debugging with LLDB on macOS. In some cases, the
      +features supported by LLDB are slightly different to those supported by GDB; the
      +tutorial text will make this clear where necessary. If you use a macOS with an
      +LLVM version older than 15, you may need to upgrade it to use the LLDB
      +correctly. If you encounter an issue on either operating system, please open an
      +issue against the K repository.

      +

      Getting started

      + +

      On Linux, you will need GDB in order to complete this lesson. If you do not
      +already have GDB installed, then do so. Steps to install GDB are outlined in
      +this GDB Tutorial.

      +

      On macOS, LLDB should already have been installed with K's build dependencies
      +(whether you have built K from source, or installed it using kup or Homebrew).

      +

      The first thing neccessary in order to debug a K interpreter is to build the
      +interpreter with full debugging support enabled. This can be done relatively
      +simply. First, run kompile with the command line flag --enable-llvm-debug.
      +The resulting compiled K definition will be ready to support debugging.

      +

      Once you have a compiled K definition and a program you wish to debug, you can
      +start the debugger by passing the --debugger flag to krun. This will
      +automatically load the program you are executing into GDB and drop you into a
      +GDB shell ready to start executing the program.

      +

      As an example, consider the following K definition (lesson-19-a.k):

      +
      module LESSON-19-A
+  imports INT
+
+  rule I => I +Int 1
+    requires I <Int 100
+endmodule
+

      If we compile this definition with kompile lesson-19-a.k --enable-llvm-debug,
      +and run the program 0 in the debugger with krun -cPGM=0 --debugger, we will
      +see the following output (roughly, and depending on which platform you are
      +using):

      +

      GDB / Linux

      + +
      GNU gdb (Ubuntu 9.2-0ubuntu1~20.04) 9.2
+Copyright (C) 2020 Free Software Foundation, Inc.
+License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>
+This is free software: you are free to change and redistribute it.
+There is NO WARRANTY, to the extent permitted by law.
+Type "show copying" and "show warranty" for details.
+This GDB was configured as "x86_64-linux-gnu".
+Type "show configuration" for configuration details.
+For bug reporting instructions, please see:
+<http://www.gnu.org/software/gdb/bugs/>.
+Find the GDB manual and other documentation resources online at:
+    <http://www.gnu.org/software/gdb/documentation/>.
+
+For help, type "help".
+Type "apropos word" to search for commands related to "word"...
+Reading symbols from ./lesson-19-a-kompiled/interpreter...
+warning: File "/home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter" auto-loading has been declined by your `auto-load safe-path' set to "$debugdir:$datadir/auto-load".
+To enable execution of this file add
+        add-auto-load-safe-path /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter
+line to your configuration file "/home/dwightguth/.gdbinit".
+To completely disable this security protection add
+        set auto-load safe-path /
+line to your configuration file "/home/dwightguth/.gdbinit".
+For more information about this security protection see the
+"Auto-loading safe path" section in the GDB manual.  E.g., run from the shell:
+        info "(gdb)Auto-loading safe path"
+(gdb)
+

      To make full advantage of the GDB features of K, you should follow the first
      +command listed in this output message and add the corresponding
      +add-auto-load-safe-path command to your ~/.gdbinit file as prompted.
      +Please note that the path will be different on your machine than the one
      +listed above. Adding directories to the "load safe path" effectively tells GDB
      +to trust those directories. All content under a given directory will be recursively
      +trusted, so if you want to avoid having to add paths to the "load safe path" every
      +time you kompile a different K definition, then you can just trust a minimal
      +directory containing all your kompiled files; however, do not choose a top-level directory containing arbitrary files as this amounts to trusting arbitrary files and is a security risk. More info on the load safe path
      +can be found here.

      +

      LLDB / macOS

      + +
      (lldb) target create "./lesson-19-a-kompiled/interpreter"
+warning: 'interpreter' contains a debug script. To run this script in this debug session:
+
+    command script import "/Users/brucecollie/code/scratch/lesson-19-a-kompiled/interpreter.dSYM/Contents/Resources/Python/interpreter.py"
+
+To run all discovered debug scripts in this session:
+
+    settings set target.load-script-from-symbol-file true
+
+Current executable set to '/Users/brucecollie/code/scratch/lesson-19-a-kompiled/interpreter' (x86_64).
+(lldb) settings set -- target.run-args  ".krun-2023-03-20-11-22-46-TcYt9ffhb2/tmp.in.RupiLwHNfn" "-1" ".krun-2023-03-20-11-22-46-TcYt9ffhb2/result.kore"
+(lldb) 
+

      LLDB applies slightly different security policies to GDB. To load K's debugging
      +scripts for this session only, you can run the command script import line at
      +the LLDB prompt. The loaded scripts will not persist across debugging sessions
      +if you do this. It is also possible to configure LLDB to automatically load the
      +K scripts when an interpreter is started in LLDB; doing so requires a slightly
      +less broad permission than GDB.

      +

      On macOS, the .dSYM directory that contains debugging symbols for an
      +executable can also contain Python scripts in Contents/Resources/Python. If
      +there is a Python script with a name matching the name of the current executable
      +(here, interpreter and interpreter.py), it will be automatically loaded if
      +the target.load-script-from-symbol-file setting is set). You can therefore add
      +the settings set command to your ~/.lldbinit without enabling full arbitrary
      +code execution, but you should be aware of the paths from which code can be
      +executed if you do so.

      +

      Basic commands

      + +
      +

      LLDB Note: the k start and k step commands are currently not
      +implemented in the K LLDB scripts. To work around this limitation temporarily,
      +you can run process launch --stop-at-entry instead of k start. To emulate
      +k step, first run rbreak k_step once, then continue instead of each k step. We hope to address these limitations soon.

      +
      +

      The most basic commands you can execute in the K GDB session are to run your
      +program or to step through it. The first can be accomplished using GDB's
      +built-in run command. This will automatically start the program and begin
      +executing it. It will continue until the program aborts or finishes, or the
      +debugger is interrupted with Ctrl-C.

      +

      Sometimes you want finer-grained control over how you proceed through the
      +program you are debugging. To step through the rule applications in your
      +program, you can use the k start and k step GDB commands.

      +

      k start is similar to the built-in start command in that it starts the
      +program and then immediately breaks before doing any work. However, unlike
      +the start command which will break immediately after the main method of
      +a program is executed, the K start program will initialize the rewriter,
      +evaluate the initial configuration, and break immediately prior to applying
      +any rewrite steps.

      +

      In the example above, here is what we see when we run the k start command:

      +
      Temporary breakpoint 1 at 0x239210
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter .krun-2021-08-13-14-10-50-sMwBkbRicw/tmp.in.01aQt85TaA -1 .krun-2021-08-13-14-10-50-sMwBkbRicw/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Temporary breakpoint 1, 0x0000000000239210 in main ()
+0x0000000000231890 in step (subject=<k>
+  0 ~> .
+</k>)
+(gdb)
+

      As you can see, we are stopped at the step function in the interpreter.
      +This function is responsible for taking top-level rewrite steps. The subject
      +parameter to this function is the current K configuration.

      +

      We can step through K rewrite steps one at a time by running the k step
      +command. By default, this takes a single rewrite step (including any function
      +rule applications that are part of that step).

      +

      Here is what we see when we run that command:

      +
      Continuing.
+
+Temporary breakpoint -22, 0x0000000000231890 in step (subject=<k>
+  1 ~> .
+</k>)
+(gdb)
+

      As we can see, we have taken a single rewrite step. We can also pass a number
      +to the k step command which indicates the number of rewrite steps to take.

      +

      Here is what we see if we run k step 10:

      +
      Continuing.
+
+Temporary breakpoint -23, 0x0000000000231890 in step (subject=<k>
+  11 ~> .
+</k>)
+(gdb)
+

      As we can see, ten rewrite steps were taken.

      +

      Breakpoints

      + +

      The next important step in debugging an application in GDB is to be able to
      +set breakpoints. Generally speaking, there are three types of breakpoints we
      +are interested in a K semantics: Setting a breakpoint when a particular
      +function is called, setting a breakpoint when a particular rule is applied,
      +and setting a breakpoint when a side condition of a rule is evaluated.

      +

      The easiest way to do the first two things is to set a breakpoint on the
      +line of code containing the function or rule.

      +

      For example, consider the following K definition (lesson-19-b.k):

      +
      module LESSON-19-B
+  imports BOOL
+
+  syntax Bool ::= isBlue(Fruit) [function]
+  syntax Fruit ::= Blueberry() | Banana()
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+
+  rule F:Fruit => isBlue(F)
+endmodule
+

      Once this program has been compiled for debugging, we can run the program
      +Blueberry(). We can then set a breakpoint that stops when the isBlue
      +function is called with the following command in GDB:

      +
      break lesson-19-b.k:4
+

      Similarly, in LLDB, run:

      +
      breakpoint set --file lesson-19-b.k --line 4
+

      Here is what we see if we set this breakpoint and then run the interpreter:

      +
      (gdb) break lesson-19-b.k:4
+Breakpoint 1 at 0x231040: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k, line 4.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-20-27-vXOQmV6lwS/tmp.in.fga98yqXlc -1 .krun-2021-08-13-14-20-27-vXOQmV6lwS/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit (_1=Blueberry ( )) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:4
+4         syntax Bool ::= isBlue(Fruit) [function]
+(gdb)
+
      (lldb) breakpoint set --file lesson-19-b.k --line 4
+Breakpoint 1: where = interpreter`LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit + 20 at lesson-19-b.k:4:19, address = 0x0000000100003ff4
+(lldb) run
+Process 50546 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50546 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100003ff4 interpreter`LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit(_1=Blueberry ( )) at lesson-19-b.k:4:19
+   1   	module LESSON-19-B
+   2   	  imports BOOL
+   3   	
+-> 4   	  syntax Bool ::= isBlue(Fruit) [function]
+   5   	  syntax Fruit ::= Blueberry() | Banana()
+   6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+(lldb)
+

      As we can see, we have stopped at the point where we are evaluating that
      +function. The value _1 that is a parameter to that function shows the
      +value passed to the function by the caller.

      +

      We can also break when the isBlue(Blueberry()) => true rule applies by simply
      +changing the line number to the line number of that rule:

      +
      (gdb) break lesson-19-b.k:6
+Breakpoint 1 at 0x2af710: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-32-36-7kD0ic7XwD/tmp.in.8JNH5Qtmow -1 .krun-2021-08-13-14-32-36-7kD0ic7XwD/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, apply_rule_138 () at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:6
+6         rule isBlue(Blueberry()) => true
+(gdb)
+
      (lldb) breakpoint set --file lesson-19-b.k --line 6
+Breakpoint 1: where = interpreter`apply_rule_140 at lesson-19-b.k:6:8, address = 0x0000000100004620
+(lldb) run
+Process 50681 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50681 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100004620 interpreter`apply_rule_140 at lesson-19-b.k:6:8
+   3   	
+   4   	  syntax Bool ::= isBlue(Fruit) [function]
+   5   	  syntax Fruit ::= Blueberry() | Banana()
+-> 6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+   8   	
+   9   	  rule F:Fruit => isBlue(F)
+(lldb) 
+

      We can also do the same with a top-level rule:

      +
      (gdb) break lesson-19-b.k:9
+Breakpoint 1 at 0x2aefa0: lesson-19-b.k:9. (2 locations)
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-33-13-9fC8Sz4aO3/tmp.in.jih1vtxSiQ -1 .krun-2021-08-13-14-33-13-9fC8Sz4aO3/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, apply_rule_107 (Var'Unds'DotVar0=<generatedCounter>
+  0
+</generatedCounter>, Var'Unds'DotVar1=., VarF=Blueberry ( )) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:9
+9         rule F:Fruit => isBlue(F)
+(gdb)
+
      (lldb) breakpoint set --file lesson-19-b.k --line 9
+Breakpoint 1: 2 locations.
+(lldb) run
+Process 50798 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50798 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100003f2e interpreter`apply_rule_109(Var'Unds'DotVar0=<generatedCounter>
+  0
+</generatedCounter>, Var'Unds'DotVar1=., VarF=Blueberry ( )) at lesson-19-b.k:9:8
+   6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+   8   	
+-> 9   	  rule F:Fruit => isBlue(F)
+   10  	endmodule
+(lldb)  
+

      Unlike the function rule above, we see several parameters to this function.
      +These are the substitution that was matched for the function. Variables only
      +appear in this substitution if they are actually used on the right-hand side
      +of the rule.

      +

      Advanced breakpoints

      + +

      Sometimes it is inconvenient to set the breakpoint based on a line number.

      +

      It is also possible to set a breakpoint based on the rule label of a particular
      +rule. Consider the following definition (lesson-19-c.k):

      +
      module LESSON-19-C
+  imports INT
+  imports BOOL
+
+  syntax Bool ::= isEven(Int) [function]
+  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+
+endmodule
+

      We will run the program isEven(4). We can set a breakpoint for when a rule
      +applies by means of the MODULE-NAME.label.rhs syntax:

      +
      (gdb) break LESSON-19-C.isEven.rhs
+Breakpoint 1 at 0x2afda0: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c-kompiled/interpreter .krun-2021-08-13-14-40-29-LNNT8YEZ61/tmp.in.ZG93vWCGGC -1 .krun-2021-08-13-14-40-29-LNNT8YEZ61/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LESSON-19-C.isEven.rhs () at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+6         rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+(gdb)
+
      (lldb) breakpoint set --name LESSON-19-C.isEven.rhs
+Breakpoint 1: where = interpreter`LESSON-19-C.isEven.rhs at lesson-19-c.k:6:18, address = 0x00000001000038e0
+(lldb) run
+Process 51205 launched: '/Users/brucecollie/code/scratch/lesson-19-c-kompiled/interpreter' (x86_64)
+Process 51205 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x00000001000038e0 interpreter`LESSON-19-C.isEven.rhs at lesson-19-c.k:6:18
+   3   	  imports BOOL
+   4   	
+   5   	  syntax Bool ::= isEven(Int) [function]
+-> 6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8   	
+   9   	endmodule
+(lldb) 
+

      We can also set a breakpoint for when a rule's side condition is evaluated
      +by means of the MODULE-NAME.label.sc syntax:

      +
      (gdb) break LESSON-19-C.isEven.sc
+Breakpoint 1 at 0x2afd70: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c-kompiled/interpreter .krun-2021-08-13-14-41-48-1BoGfJRbYc/tmp.in.kg4F8cwfCe -1 .krun-2021-08-13-14-41-48-1BoGfJRbYc/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LESSON-19-C.isEven.sc (VarI=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+6         rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+(gdb) finish
+Run till exit from #0  LESSON-19-C.isEven.sc (VarI=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+0x00000000002b2662 in LblisEven'LParUndsRParUnds'LESSON-19-C'Unds'Bool'Unds'Int (_1=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:5
+5         syntax Bool ::= isEven(Int) [function]
+Value returned is $1 = true
+(gdb)
+
      (lldb) breakpoint set --name LESSON-19-C.isEven.sc
+Breakpoint 1: where = interpreter`LESSON-19-C.isEven.sc + 1 at lesson-19-c.k:6:18, address = 0x00000001000038c1
+(lldb) run
+Process 52530 launched: '/Users/brucecollie/code/scratch/lesson-19-c-kompiled/interpreter' (x86_64)
+Process 52530 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x00000001000038c1 interpreter`LESSON-19-C.isEven.sc(VarI=0x0000000101800088) at lesson-19-c.k:6:18
+   3   	  imports BOOL
+   4   	
+   5   	  syntax Bool ::= isEven(Int) [function]
+-> 6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8   	
+   9   	endmodule
+(lldb) finish
+Process 52649 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = step out
+Return value: (bool) $0 = true
+
+    frame #0: 0x00000001000069e5 interpreter`LblisEven'LParUndsRParUnds'LESSON-19-C'Unds'Bool'Unds'Int(_1=0x0000000101800088) at lesson-19-c.k:5:19
+   2   	  imports INT
+   3   	  imports BOOL
+   4   	
+-> 5   	  syntax Bool ::= isEven(Int) [function]
+   6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8
+(lldb)
+

      Here we have used the built-in command finish to tell us whether the side
      +condition returned true or not. Note that once again, we see the substitution
      +that was matched from the left-hand side. Like before, a variable will only
      +appear here if it is used in the side condition.

      +

      Debugging rule matching

      + +

      Sometimes it is useful to try to determine why a particular rule did or did
      +not apply. K provides some basic debugging commands which make it easier
      +to determine this.

      +

      Consider the following K definition (lesson-19-d.k):

      +
      module LESSON-19-D
+
+  syntax Foo ::= foo(Bar)
+  syntax Bar ::= bar(Baz) | bar2(Baz)
+  syntax Baz ::= baz() | baz2()
+
+  rule [baz]: foo(bar(baz())) => .K
+
+endmodule
+

      Suppose we try to run the program foo(bar(baz2())). It is obvious from this
      +example why the rule in this definition will not apply. However, in practice,
      +such cases are not always obvious. You might look at a rule and not immediately
      +spot why it didn't apply on a particular term. For this reason, it can be
      +useful to get the debugger to provide a log about how it tried to match that
      +term. You can do this with the k match command. If you are stopped after
      +having run k start or k step, you can obtain this log for any rule after
      +any step by running the command k match MODULE.label subject for a particular
      +top-level rule label.

      +

      For example, with the baz rule above, we get the following output:

      +
      (gdb) k match LESSON-19-D.baz subject
+Subject:
+baz2 ( )
+does not match pattern:
+baz ( )
+
      (lldb) k match LESSON-19-D.baz subject
+Subject:
+baz2 ( )
+does not match pattern:
+baz ( )
+

      As we can see, it provided the exact subterm which did not match against the
      +rule, as well as the particular subpattern it ought to have matched against.

      +

      This command does not actually take any rewrite steps. In the event that
      +matching actually succeeds, you will still need to run the k step command
      +to advance to the next step.

      +

      Final notes

      + +

      In addition to the functionality provided above, you have the full power of
      +GDB or LLDB at your disposal when debugging. Some features are not particularly
      +well-adapted to K code and may require more advanced knowledge of the
      +term representation or implementation to use effectively, but anything that
      +can be done in GDB or LLDB can in theory be done using this debugging functionality.
      +We suggest you refer to the
      +GDB Documentation or
      +LLDB Tutorial if you
      +want to try to do something and are unsure as to how.

      +

      Exercises

      + +
        +
      1. Compile your solution to Lesson 1.18, Exercise 2 with debugging support
        +enabled and step through several programs you have previously used to test.
        +Then set a breakpoint on the isKResult function and observe the state of the
        +interpreter when stopped at that breakpoint. Set a breakpoint on the rule for
        +addition and run a program that causes it to be stopped at that breakpoint.
        +Finally, step through the program until the addition symbol is at the top
        +of the K cell, and then use the k match command to report the reason why
        +the subtraction rule does not apply. You may need to modify the definition
        +to insert some rule labels.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.20: K Backends and the Haskell Backend.

      +

      Lesson 1.20: K Backends and the Haskell Backend

      + +

      The purpose of this lesson is to teach about the multiple backends of K,
      +in particular the Haskell Backend which is the complement of the backend we
      +have been using so far.

      +

      K Backends

      + +

      Thus far, we have not discussed the distinction between the K frontend and
      +the K backends at all. We have simply assumed that if you run kompile on a
      +K definition, there will be a compiler backend that will allow you to execute
      +the K definition you have compiled.

      +

      K actually has multiple different backends. The one we have been using so far
      +implicitly, the default backend, is called the LLVM Backend. It is
      +designed to support efficient, optimized concrete execution and search. It
      +does this by compiling your K definition to LLVM bitcode and then using LLVM
      +to generate machine code for it that is compiled and linked and executed.
      +However, K is a formal methods toolkit at the end of the day, and the primary
      +goal many people have when defining a programming language in K is to
      +ultimately be able to perform more advanced verification on programs in their
      +programming language.

      +

      It is for this purpose that K also provides the Haskell Backend, so called
      +because it is implemented in Haskell. While we will cover the features of the
      +Haskell Backend in more detail in the next two lessons, the important thing to
      +understand is that it is a separate backend which is optimized for more formal
      +reasoning about programming languages. While it is capable of performing
      +concrete execution, it does not do so as efficiently as the LLVM Backend.
      +In exchange, it provides more advanced features.

      +

      Choosing a backend

      + +

      You can choose which backend to use to compile a K definition by means of the
      +--backend flag to kompile. By default, if you do not specify this flag, it
      +is equivalent to if you had specified --backend llvm. However, to use the
      +Haskell Backend instead, you can simply say kompile --backend haskell on a
      +particular K definition.

      +

      As an example, here is a simple K definition that we have seen before in the
      +previous lesson (lesson-20.k):

      +
      module LESSON-20
+  imports INT
+
+  rule I => I +Int 1
+    requires I <Int 100
+endmodule
+

      Previously we compiled this definition using the LLVM Backend, but if we
      +instead execute the command kompile lesson-20.k --backend haskell, we
      +will get an interpreter for this K definition that is implemented in Haskell
      +instead. Unlike the default LLVM Backend, the Haskell Backend is not a
      +compiler per se. It does not generate new Haskell code corresponding to your
      +programming language and then compile and execute it. Instead, it is an
      +interpreter which reads the generated IR from kompile and implements in
      +Haskell an interpreter that is capable of interpreting any K definition.

      +

      Note that on arm64 macOS (Apple Silicon), there is a known issue with the Compact
      +library that causes crashes in the Haskell backend. Pass the additional flag
      +--no-haskell-binary to kompile to resolve this.
      +This flag is also needed when using krun.

      +

      Exercise

      + +

      Try running the program 0 in this K definition on the Haskell Backend and
      +compare the final configuration to what you would get compiling the same
      +definition with the LLVM Backend.

      +

      Legacy backends

      + +

      As a quick note, K does provide one other backend, which exists primarily as
      +legacy code which should be considered deprecated. This is the
      +Java Backend. The Java Backend is essentially a precursor to the Haskell
      +Backend. We will not cover this backend in any detail since it is deprecated,
      +but we still mention it here for the purposes of understanding.

      +

      Exercises

      + +
        +
      1. Compile your solution to Lesson 1.18, Exercise 2 with the Haskell Backend
        +and execute some programs. Compare the resulting configurations with the
        +output of the same program on the LLVM Backend. Note that if you are getting
        +different behaviors on the Haskell backend, you might have some luck debugging
        +by passing --search to krun when using the LLVM backend.
        2. +
      +

      Next lesson

      + +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.21: Unification and Symbolic Execution.

      +

      Lesson 1.21: Unification and Symbolic Execution

      + +

      The purpose of this lesson is to teach the basic concepts of symbolic execution
      +in order to introduce the unique capabilities of the Haskell Backend at a
      +conceptual level.

      +

      Symbolic Execution

      + +

      Thus far, all of the programs we have run using K have been concrete
      +configurations. What this means is that the configuration we use to initialize
      +the K rewrite engine is concrete; in other words, contains no logical
      +variables. The LLVM Backend is a concrete execution engine, meaning that
      +it is only capable of rewriting concrete configurations.

      +

      By contrast, the Haskell Backend performs symbolic execution, which is
      +capable of rewriting any configuration, including those where parts of the
      +configuration are symbolic, ie, contain variables or uninterpreted
      +functions.

      +

      Unification

      + +

      Previously, we have introduced the concept that K rewrite rules operate by
      +means of pattern matching: the current configuration being rewritten is pattern
      +matched against the left-hand side of the rewrite rule, and the substitution
      +is used in order to construct a new term from the right-hand side. In symbolic
      +execution, we use
      +unification
      +instead of pattern matching. To summarize, unification behaves akin to a
      +two-way pattern matching where both the configuration and the left-hand side
      +of the rule can contain variables, and the algorithm generates a
      +most general unifier containing substitutions for the variables in both
      +which will make both terms equal.

      +

      Feasibility

      + +

      Unification by itself cannot completely solve the problem of symbolic
      +execution. One task symbolic execution must perform is to identify whether
      +a particular symbolic term is feasible, that is to say, that there actually
      +exists a concrete instantiation of that term such that all the logical
      +constraints on that term can actually be satisfied. The Haskell Backend
      +delegates this task to Z3, an
      +SMT solver.
      +This solver is used to periodically trim configurations that are determined
      +to be mathematically infeasible.

      +

      Symbolic terms

      + +

      The final component of symbolic execution consists of the task of introducing
      +symbolic terms into the configuration. This can be done one of two different
      +ways. First, the term being passed to krun can actually be symbolic. This
      +is less frequently used because it requires the user to construct an AST
      +that contains variables, something which our current parsing capabilities are
      +not well-equipped to do. The second, more common, way of introducing symbolic
      +terms into a configuration consists of writing rules where there exists an
      +existentially qualified variable on the right-hand side of the rule that does
      +not exist on the left-hand side of the rule.

      +

      In order to prevent users from writing such rules by accident, K requires
      +that such variables begin with the ? prefix. For example, here is a rule
      +that rewrites a constructor foo to a symbolic integer:

      +
      rule <k> foo => ?X:Int ...</k>
+

      When this rule applies, a fresh variable is introduced to the configuration, which
      +then is unified against the rules that might apply in order to symbolically
      +execute that configuration.

      +

      ensures clauses

      + +

      We also introduce here a new feature of K rules that applies when a rule
      +has this type of variable on the right-hand side: the ensures clause.
      +An ensures clause is similar to a requires clause and can appear after
      +a rule body, or after a requires clause. The ensures clause is used to
      +introduce constraints that might apply to the variable that was introduced by
      +that rule. For example, we could write the rule above with the additional
      +constraint that the symbolic integer that was introduced must be less than
      +five, by means of the following rule:

      +
      rule <k> foo => ?X:Int ...</k> ensures ?X <Int 5
+

      Putting it all together

      + +

      Putting all these pieces together, it is possible to use the Haskell Backend
      +to perform symbolic reasoning about a particular K module, determining all the
      +possible states that can be reached by a symbolic configuration.

      +

      For example, consider the following K definition (lesson-21.k):

      +
      module LESSON-21
+    imports INT
+
+    rule <k> 0 => ?X:Int ... </k> ensures ?X =/=Int 0
+    rule <k> X:Int => 5  ... </k> requires X >=Int 10
+endmodule
+

      When we symbolically execute the program 0, we get the following output
      +from the Haskell Backend:

      +
          <k>
+      5 ~> .
+    </k>
+  #And
+    {
+      true
+    #Equals
+      ?X:Int >=Int 10
+    }
+  #And
+    #Not ( {
+      ?X:Int
+    #Equals
+      0
+    } )
+#Or
+    <k>
+      ?X:Int ~> .
+    </k>
+  #And
+    #Not ( {
+      true
+    #Equals
+      ?X:Int >=Int 10
+    } )
+  #And
+    #Not ( {
+      ?X:Int
+    #Equals
+      0
+    } )
+

      Note some new symbols introduced by this configuration: #And, #Or, and
      +#Equals. While andBool, orBool, and ==K represent functions of sort
      +Bool, #And, #Or, and #Equals are matching logic connectives. We
      +will discuss matching logic in more detail later in the tutorial, but the basic
      +idea is that these symbols represent Boolean operators over the domain of
      +configurations and constraints, as opposed to over the Bool sort.

      +

      Notice that the configuration listed above is a disjunction of conjunctions.
      +This is the most common form of output that can be produced by the Haskell
      +Backend. In this case, each conjunction consists of a configuration and a set
      +of constraints. What this conjunction describes, essentially, is a
      +configuration and a set of information that was derived to be true while
      +rewriting that configuration.

      +

      Similar to how we saw --search in a previous lesson, the reason we have
      +multiple disjuncts is because there are multiple possible output states
      +for this program, depending on whether or not the second rule applied. In the
      +first case, we see that ?X is greater than or equal to 10, so the second rule
      +applied, rewriting the symbolic integer to the concrete integer 5. In the
      +second case, we see that the second rule did not apply because ?X is less
      +than 10. Moreover, because of the ensures clause on the first rule, we know
      +that ?X is not zero, therefore the first rule will not apply a second time.
      +If we had omitted this constraint, we would have ended up infinitely applying
      +the first rule, leading to krun not terminating.

      +

      In the next lesson, we will cover how symbolic execution forms the backbone
      +of deductive program verification in K and how we can use K to prove programs
      +correct against a specification.

      +

      Exercises

      + +
        +
      1. Create another rule in LESSON-21 that rewrites odd integers greater than
        +ten to a symbolic even integer less than 10 and greater than 0. This rule will
        +now apply nondeterministically along with the existing rules. Predict what the
        +resulting output configuration will be from rewriting 0 after adding this
        +rule. Then run the program and see whether your prediction is correct.
        2. +
      +

      Once you have completed the above exercises, you can continue to
      +Lesson 1.22: Basics of Deductive Program Verification using K.

      +

      Lesson 1.22: Basics of Deductive Program Verification using K

      + +

      In this lesson, you will familiarize yourself with the basics of using K for
      +deductive program verification.

      +

      1. Setup: Simple Programming Language with Function Calls

      + +

      We base this lesson on a simple programming language with functions,
      +assignment, if conditionals, and while loops. Take your time to study its
      +formalization below (lesson-22.k):

      +
      module LESSON-22-SYNTAX
+    imports INT-SYNTAX
+    imports BOOL-SYNTAX
+    imports ID-SYNTAX
+
+    syntax Exp ::= IExp | BExp
+
+    syntax IExp ::= Id | Int
+
+    syntax KResult ::= Int | Bool | Ints
+
+    // Take this sort structure:
+    //
+    //     IExp
+    //    /    \
+    // Int      Id
+    //
+    // Through the List{_, ","} functor.
+    // Must add a `Bot`, for a common subsort for the empty list.
+
+    syntax Bot
+    syntax Bots ::= List{Bot, ","} [klabel(exps)]
+    syntax Ints ::= List{Int, ","} [klabel(exps)]
+                  | Bots
+    syntax Ids  ::= List{Id, ","}  [klabel(exps)]
+                  | Bots
+    syntax Exps ::= List{Exp, ","} [klabel(exps), seqstrict]
+                  | Ids | Ints
+
+    syntax IExp ::= "(" IExp ")" [bracket]
+                  | IExp "+" IExp [seqstrict]
+                  | IExp "-" IExp [seqstrict]
+                  > IExp "*" IExp [seqstrict]
+                  | IExp "/" IExp [seqstrict]
+                  > IExp "^" IExp [seqstrict]
+                  | Id "(" Exps ")" [strict(2)]
+
+    syntax BExp ::= Bool
+
+    syntax BExp ::= "(" BExp ")" [bracket]
+                  | IExp "<=" IExp [seqstrict]
+                  | IExp "<"  IExp [seqstrict]
+                  | IExp ">=" IExp [seqstrict]
+                  | IExp ">"  IExp [seqstrict]
+                  | IExp "==" IExp [seqstrict]
+                  | IExp "!=" IExp [seqstrict]
+
+    syntax BExp ::= BExp "&&" BExp
+                  | BExp "||" BExp
+
+    syntax Stmt ::=
+         Id "=" IExp ";" [strict(2)]                        // Assignment
+       | Stmt Stmt [left]                                   // Sequence
+       | Block                                              // Block
+       | "if" "(" BExp ")" Block "else" Block [strict(1)]   // If conditional
+       | "while" "(" BExp ")" Block                         // While loop
+       | "return" IExp ";"                    [seqstrict]   // Return statement
+       | "def" Id "(" Ids ")" Block                         // Function definition
+
+    syntax Block ::=
+         "{" Stmt "}"    // Block with statement
+       | "{" "}"         // Empty block
+endmodule
+
+module LESSON-22
+    imports INT
+    imports BOOL
+    imports LIST
+    imports MAP
+    imports LESSON-22-SYNTAX
+
+    configuration
+      <k> $PGM:Stmt </k>
+      <store> .Map </store>
+      <funcs> .Map </funcs>
+      <stack> .List </stack>
+
+ // -----------------------------------------------
+    rule <k> I1 + I2 => I1 +Int I2 ... </k>
+    rule <k> I1 - I2 => I1 -Int I2 ... </k>
+    rule <k> I1 * I2 => I1 *Int I2 ... </k>
+    rule <k> I1 / I2 => I1 /Int I2 ... </k>
+    rule <k> I1 ^ I2 => I1 ^Int I2 ... </k>
+
+    rule <k> I:Id => STORE[I] ... </k>
+         <store> STORE </store>
+
+ // ------------------------------------------------
+    rule <k> I1 <= I2 => I1  <=Int I2 ... </k>
+    rule <k> I1  < I2 => I1   <Int I2 ... </k>
+    rule <k> I1 >= I2 => I1  >=Int I2 ... </k>
+    rule <k> I1  > I2 => I1   >Int I2 ... </k>
+    rule <k> I1 == I2 => I1  ==Int I2 ... </k>
+    rule <k> I1 != I2 => I1 =/=Int I2 ... </k>
+
+    rule <k> B1 && B2 => B1 andBool B2 ... </k>
+    rule <k> B1 || B2 => B1  orBool B2 ... </k>
+
+    rule <k> S1:Stmt S2:Stmt => S1 ~> S2 ... </k>
+
+    rule <k> ID = I:Int ; => . ... </k>
+         <store> STORE => STORE [ ID <- I ] </store>
+
+    rule <k> { S } => S ... </k>
+    rule <k> {   } => . ... </k>
+
+    rule <k> if (true)   THEN else _ELSE => THEN ... </k>
+    rule <k> if (false) _THEN else  ELSE => ELSE ... </k>
+
+    rule <k> while ( BE ) BODY => if ( BE ) { BODY while ( BE ) BODY } else { } ... </k>
+
+    rule <k> def FNAME ( ARGS ) BODY => . ... </k>
+         <funcs> FS => FS [ FNAME <- def FNAME ( ARGS ) BODY ] </funcs>
+
+    rule <k> FNAME ( IS:Ints ) ~> CONT => #makeBindings(ARGS, IS) ~> BODY </k>
+         <funcs> ... FNAME |-> def FNAME ( ARGS ) BODY ... </funcs>
+         <store> STORE => .Map </store>
+         <stack> .List => ListItem(state(CONT, STORE)) ... </stack>
+
+    rule <k> return I:Int ; ~> _ => I ~> CONT </k>
+         <stack> ListItem(state(CONT, STORE)) => .List ... </stack>
+         <store> _ => STORE </store>
+
+    rule <k> return I:Int ; ~> . => I </k>
+         <stack> .List </stack>
+
+    syntax KItem ::= #makeBindings(Ids, Ints)
+                   | state(continuation: K, store: Map)
+ // ----------------------------------------------------
+    rule <k> #makeBindings(.Ids, .Ints) => . ... </k>
+    rule <k> #makeBindings((I:Id, IDS => IDS), (IN:Int, INTS => INTS)) ... </k>
+         <store> STORE => STORE [ I <- IN ] </store>
+endmodule
+

      Next, compile this example using kompile lesson-22.k --backend haskell. If
      +your processor is an Apple Silicon processor, add the --no-haskell-binary
      +flag if the compilation fails.

      +

      2. Setup: Proof Environment

      + +

      Next, take the following snippet of K code and save it in lesson-22-spec.k.
      +This is a skeleton of the proof environment, and we will complete it as the
      +lesson progresses.

      +
      requires "lesson-22.k"
+requires "domains.md"
+
+module LESSON-22-SPEC-SYNTAX
+    imports LESSON-22-SYNTAX
+
+endmodule
+
+module VERIFICATION
+    imports K-EQUAL
+    imports LESSON-22-SPEC-SYNTAX
+    imports LESSON-22
+    imports MAP-SYMBOLIC
+
+endmodule
+
+module LESSON-22-SPEC
+    imports VERIFICATION
+
+endmodule
+

      3. Claims

      + +
        +
      1. The first claim we will ask K to prove is that 3 + 4, in fact, equals 7.
        +Claims are stated using the claim keyword, followed by the claim
        +statement:
        2. +
      +
      claim <k> 3 + 4 => 7 ... </k>
+

      Add this claim to the LESSON-22-SPEC module and run the K prover using the
      +command kprove lesson-22-spec.k. You should get back the output #Top,
      +which denotes the Matching Logic equivalent of true and means, in this
      +context, that all claims have been proven correctly.

      +
        +
      1. The second claim reasons about the if statement that has a concrete condition:
        2. +
      +
      claim <k> if ( 3 + 4 == 7 ) {
+            $a = 1 ;
+            } else {
+            $a = 2 ;
+            }
+        => . ... </k>
+        <store> STORE => STORE [ $a <- 1 ] </store>
+

      stating that the given program terminates (=> .), and when it does, the value
      +of the variable $a is set to 1, meaning that the execution will have taken
      +the then branch. Add this claim to the LESSON-22-SPEC module, but also add

      +
      syntax Id ::= "$a" [token]
+

      to the LESSON-22-SPEC-SYNTAX module in order to declare $a as a token so
      +that it can be used as a program variable. Re-run the K prover, which should
      +again return #Top.

      +
        +
      1. Our third claim demonstrates how to reason about both branches of an if
        +statement at the same time:
        2. +
      +
      claim <k> $a = A:Int ; $b = B:Int ;
+          if ($a < $b) {
+            $c = $b ;
+          } else {
+            $c = $a ;
+          }
+        => . ... </k>
+        <store> STORE => STORE [ $a <- A ] [ $b <- B ] [ $c <- ?C:Int ] </store>
+    ensures (?C ==Int A) orBool (?C ==Int B)
+

      The program in question first assigns symbolic integers A and B to program
      +variables $a and $b, respectively, and then executes the given if
      +statement, which has a symbolic condition (A < B), updating the value of the
      +program variable $c in both branches. The specification we give states that
      +the if statement terminates, with $a and $b updated, respectively, to A
      +and B, and $c updated to some symbolic integer value ?C. Via the
      +ensures clause, which is used to specify additional constraints that hold
      +after execution, we also state that this existentially quantified ?C equals
      +either A or B.

      +

      Add the productions declaring $b and $c as tokens to the
      +LESSON-22-SPEC-SYNTAX module, the claim to the LESSON-22-SPEC module, run
      +the K prover again, and observe the output, which should not be #Top this
      +time. This means that K was not able to prove the claim, and we now need to
      +understand why. We do so by examining the output, which should look as follows:

      +
          (InfoReachability) while checking the implication:
+    The configuration's term unifies with the destination's term,
+    but the implication check between the conditions has failed.
+
+  #Not (
+    #Exists ?C . {
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- ?C:Int ]
+      #Equals
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    }
+  #And
+    {
+      true
+    #Equals
+      ?C ==Int A orBool ?C ==Int B
+    }
+  )
+#And
+  <generatedTop>
+    <k>
+      _DotVar1
+    </k>
+    <store>
+      STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    </store>
+    <funcs>
+      _Gen3
+    </funcs>
+    <stack>
+      _Gen5
+    </stack>
+  </generatedTop>
+#And
+  {
+    true
+  #Equals
+    A <Int B
+  }
+

      This output starts with a message telling us at which point the proof failed,
      +followed by the final state, which consists of three parts: some negative
      +Matching Logic (ML) constraints, the final configuration (<generatedTop> ... </generatedTop>), and some positive ML constraints. Generally speaking,
      +these positive and the negative constraints could arise from various sources,
      +such as (but not limited to) branches taken by the execution
      +(e.g. { true #Equals A <Int B } or #Not ( { true #Equals A <Int B } )),
      +or ensures constraints.

      +

      First, we examine the message:

      +
      (InfoReachability) while checking the implication:
+The configuration's term unifies with the destination's term,
+but the implication check between the conditions has failed.
+

      which tells us that the structure of the final configuration is as expected,
      +but that some of the associated constraints cannot be proven. We next look at
      +the final configuration, in which the relevant item is the <store> ... </store> cell, because it is the only one that we are reasoning about. By
      +inspecting its contents:

      +
      STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+

      we see that we should be within the constraints of the ensures, since the
      +value of $c in the store equals B in this branch. We next examine the
      +negative and positive constraints of the output and, more often than not, the
      +goal is to instruct K how to use the information from the final configuration
      +and the positive constraints to falsify one of the negative constraints. This
      +is done through simplifications.

      +

      So, the positive constraint that we have is

      +
      { true #Equals A <Int B }
+

      meaning that A <Int B holds. Given the analysed program, this tells us that
      +we are in the then branch of the if. The negative constraint is

      +
        #Not (
+    #Exists ?C . {
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- ?C:Int ]
+      #Equals
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    }
+  #And
+    { true #Equals ?C ==Int A orBool ?C ==Int B }
+  )
+

      and we observe, from the first equality, that the existential ?C should be
      +instantiated with B. This would make both branches of the #And true,
      +falsifying the outside #Not. We just need to show K how to conclude that
      +?C ==Int B. We do so by introducing the following simplification into the
      +VERIFICATION module:

      +
      rule { M:Map [ K <- V ] #Equals M [ K <- V' ] } => { V #Equals V' } [simplification]
+

      which formalizes our internal understanding of ?C ==Int B. The rule states
      +that when we update the same key in the same map with two values, and the
      +resulting maps are equal, then the two values must be equal as well. The
      +[simplification] attribute indicates to K to use this rule to simplify the
      +state when trying to prove claims. Like function rules, simplification rules
      +do not complete to the top of the configuration, but instead apply anywhere
      +their left-hand-side matches. Re-run the K prover, which should now return
      +#Top, indicating that K was able to use the simplification and prove the
      +required claims.

      +
        +
      1. Next, we show how to state and prove properties of while loops. In
        +particular, we consider the following loop
        2. +
      +
      claim
+    <k>
+        while ( 0 < $n ) {
+            $s = $s + $n;
+            $n = $n - 1;
+            } => . ...
+    </k>
+    <store>
+        $s |-> (S:Int => S +Int ((N +Int 1) *Int N /Int 2))
+        $n |-> (N:Int => 0)
+    </store>
+    requires N >=Int 0
+

      which adds the sum of the first $n integers to $s, assuming the value of $n
      +is non-negative to begin with. This is reflected in the store by stating that,
      +after the execution of the loop, the original value of $s (which is set to
      +equal some symbolic integer S) is incremented by ((N +Int 1) *Int N /Int 2), and the value of $n always equals 0. Add $n and $s as tokens in
      +the LESSON-22-SPEC-SYNTAX module, the above claim to the LESSON-22-SPEC
      +module, and run the K prover, which should return #Top.

      +
        +
      1. Finally, our last claim is about a program that uses function calls:
        2. +
      +
      claim
+    <k>
+        def $sum($n, .Ids) {
+            $s = 0 ;
+            while (0 < $n) {
+                $s = $s + $n;
+                $n = $n - 1;
+            }
+            return $s;
+        }
+
+        $s = $sum(N:Int, .Ints);
+    => . ... </k>
+    <funcs> .Map => ?_ </funcs>
+    <store> $s |-> (_ => ((N +Int 1) *Int N /Int 2)) </store>
+    <stack> .List </stack>
+    requires N >=Int 0
+

      Essentially, we have wrapped the while loop from claim 3.4 into a function
      +$sum, and then called that function with a symbolic integer N, storing the
      +return value in the variable $s. The specification states that this program
      +ends up storing the sum of the first N integers in the variable $n. Add $sum
      +to the LESSON-22-SPEC-SYNTAX module, the above claim to the
      +LESSON-22-SPEC module, and run the K prover, which should again return
      +#Top.

      +

      Exercises

      + +
        +
      1. +

        Change the condition of the if statement in part 3.2 to take the else
        +branch and adjust the claim so that the proof passes.

        +
        2. +
      2. +

        The post-condition of the specification in part 3.3 loses some information.
        +In particular, the value of ?C is in fact the maximum of A and B.
        +Prove the same claim as in 3.2, but with the post-condition ensures (?C ==Int maxInt(A, B)). For this, you will need to extend the VERIFICATION
        +module with two simplifications that capture the meaning of maxInt(A:Int, B:Int). Keep in mind that any rewriting rule can be used as a
        +simplification; in particular, that simplifications can have requires
        +clauses.

        +
        3. +
      3. +

        Following the pattern shown in part 3.4, assuming a non-negative initial
        +value of $b, specify and verify the following while loop:

        +
        4. +
      +
      while ( 0 < $b ) {
+    $a = $a + $c;
+    $b = $b - 1;
+    $c = $c - 1;
+}
+

      Hint: You will not need additional simplifications---once you've got the
      +specification right, the proof will go through.

      +
        +
      1. Write an arbitrary yet not-too-complex function (or several functions
        +interacting with each other), and try to specify and verify it (them) in K.
        2. +
      +

      Section 2: Intermediate K Concepts

      + +

      The goal of this second section is to supplement a beginning developer's
      +knowledge of K after they have gained a basic understanding of K. Each lesson
      +in this section can be completed independently in order to learn about a
      +particular facet of the K language. The lessons are written to provide basic
      +understanding of less commonly-used features of K to someone who is still
      +learning K. For more complete references of these features, the reader ought to
      +consult the User Manual.

      +

      The reader ought to be able to complete lessons in this section as needed in
      +order to learn about specific features of interest, but if desired, can also
      +complete the entire section in one go. Someone who has completed this entire
      +section ought to be able to read and understand most K specifications, as well
      +as write their own specifications of some complexity, and use them to perform
      +most common K-related tasks. They can then read about specific lessons in
      +Section 3: Advanced K Concepts if they want to
      +learn more.

      +

      Table of Contents

      + +
        +
      1. Macros, Aliases, and Anywhere Rules
        2. +
      2. Fresh Constants
        3. +
      3. KLabels and Abstract Syntax
        4. +
      4. Overloaded Symbols
        5. +
      5. Matching Logic Connectives and #Or Patterns
        6. +
      6. Function Context
        7. +
      7. Record Productions and Named Nonterminals
        8. +
      8. #fun and #let
        9. +
      9. #as patterns
        10. +
      10. The Matching Operators, :=K and :/=K
        11. +
      11. Uncommon Evaluation Order Concepts
        12. +
      12. IEEE 754 Floating Point and Fixed Width Integers
        13. +
      13. Alpha-renaming-aware Substitution
        14. +
      14. File I/O
        15. +
      15. String Buffers and Byte Sequences
        16. +
      16. The Intermediate Language of K, KORE
        17. +
      17. Debugging Proofs using the Haskell Backend REPL
        18. +
      +

      Lesson 2.1: Macros, Aliases, and Anywhere Rules

      + +

      The purpose of this lesson is to explain the behavior of the macro,
      +macro-rec, alias, and alias-rec production attributes, as well as the
      +anywhere rule attribute. These attributes control the meaning of how rules
      +associated with them are applied.

      +

      Macros

      + +

      Thus far in the K tutorial, we have described three different types of rules:

      +
        +
      1. Top-level rewrite rules, which rewrite a configuration composed of cells to
        +another configuration;
        2. +
      2. Function rules, which define the behavior of a function written over
        +arbitrary input and output types; and
        3. +
      3. Simplification rules, which describe ways in which the symbolic execution
        +engine ought to simplify terms containing symbolic values.
        4. +
      +

      This lesson introduces three more types of rules, the first of which are
      +macros. A production is a macro if it has the macro attribute, and all
      +rules whose top symbol on the left hand side is a macro are macro rules
      +which define the behavior of the macro. Like function rules and simplification
      +rules, macro rules do not participate in cell completion. However, unlike
      +function rules and simplification rules, macro rules are applied statically
      +before rewriting begins, and the macro symbol is expected to no longer appear
      +in the initial configuration for rewriting once all macros in that
      +configuration are rewritten.

      +

      The rationale behind macros is they allow you to define one piece of syntax
      +in terms of another piece of syntax without any runtime overhead associated
      +with the cost of rewriting one to the other. This process is a common one in
      +programming language design and specification and is referred to as
      +desugaring; The syntax that is transformed is typically also referred to as
      +syntactic sugar for another type of syntax. For example, in a language with
      +if statements and curly braces, you could write the following fragment
      +(lesson-01.k):

      +
      module LESSON-01
+  imports BOOL
+
+  syntax Stmt ::= "if" "(" Exp ")" Stmt             [macro]
+                | "if" "(" Exp ")" Stmt "else" Stmt
+                | "{" Stmts "}"
+  syntax Stmts ::= List{Stmt,""}
+  syntax Exp ::= Bool
+
+  rule if ( E ) S => if ( E ) S else { .Stmts }
+endmodule
+

      In this example, we see that an if statement without an else clause is
      +defined in terms of one with an else clause. As a result, we would only
      +need to give a single rule for how to rewrite if statements, rather than
      +two separate rules for two types of if statements. This is a common pattern
      +for dealing with program syntax that contains an optional component to it.

      +

      It is worth noting that by default, macros are not applied recursively. To be
      +more precise, by default a macro that arises as a result of the expansion of
      +the same macro is not rewritten further. This is primarily to simplify the
      +macro expansion process and reduce the risk that improperly defined macros will
      +lead to non-terminating behavior.

      +

      It is possible, however, to tell K to expand a macro recursively. To do this,
      +simply replace the macro attribute with the macro-rec attribute. Note that
      +K does not do any kind of checking to ensure termination here, so it is
      +important that rules be defined correctly to always terminate, otherwise the
      +macro expansion phase will run forever. Fortunately, in practice it is very
      +simple to ensure this property for most of the types of macros that are
      +typically used in real-world semantics.

      +

      Exercise

      + +

      Using a Nat sort containing the constructors 0 and S (i.e., a
      +Peano-style axiomatization of the
      +natural numbers where S(N) = N + 1, S(S(N)) = N + 2, etc), write a macro
      +that will compute the sum of two numbers.

      +

      Aliases

      + +

      NOTE: This lesson introduces the concept of "aliases", which are a variant
      +of macros. While similar, this is different from the concept of "aliases" in
      +matching logic, which is introduced in Lesson 2.16.

      +

      Macros can be very useful in helping you define a programming language.
      +However, they can be disruptive while pretty printing a configuration. For
      +example, you might write a set of macros that transforms the code the user
      +wrote into equivalent code that is slightly harder to read. This can make it
      +more difficult to understand the code when it is pretty printed as part of the
      +output of rewriting.

      +

      K defines a relatively straightforward but novel solution to this problem,
      +which is known as a K alias. An alias in K is very similar to a macro,
      +with the exception that the rewrite rule will also be applied backwards
      +during the pretty-printing process.

      +

      It is very simple to make a production be an alias instead of a macro: simply
      +use the alias or alias-rec attributes instead of the macro or macro-rec
      +attributes. For example, if the example involving if statements above was
      +declared using an alias instead of a macro, the Stmt term if (E) {} else {}
      +would be pretty-printed as if (E) {}. This is because during pretty-printing,
      +the term participates in another macro-expansion pass. However, this macro
      +expansion step will only apply rules with the alias or alias-rec attribute,
      +and, critically, it will reverse the rule by treating the left-hand side as if
      +it were the right-hand side, and vice versa.

      +

      This can be very useful to allow you to define one construct in terms of
      +another while still being able to pretty-print the result as if it were
      +the original term in question. This can be especially useful for applications
      +of K where we are taking the output of rewriting and attempting to use it as
      +a code fragment that we then execute, such as with test generation.

      +

      Exercise

      + +

      Modify LESSON-01 above to use an alias instead of a macro and experiment
      +with how various terms are pretty-printed by invoking krun on them.

      +

      anywhere rules

      + +

      The last type of rule introduced in this lesson is the anywhere rule. An
      +anywhere rule is specified by adding the anywhere attribute to a rule. Such a
      +rule is similar to a function rule in that it does not participate in cell
      +completion, and will apply anywhere that the left-hand-side matches in the
      +configuration, but distinct in that the symbol in question can still be matched
      +against in the left-hand side of other rules, even during concrete rewriting.
      +The reasoning behind this is that instead of the symbol in question being a
      +constructor, it is a constructor modulo the axioms defined with the
      +anywhere attribute. Essentially, the rules with the anywhere attribute will
      +apply as soon as they appear in the right-hand side of a rule being applied,
      +but the symbol in question will still be treated as a symbol that can be
      +matched on if it is not completely removed by those rules.

      +

      This can be useful in certain cases to allow you to define transformations over
      +particular pieces of syntax while still generally giving those pieces of syntax
      +another meaning when the anywhere rule does not apply. For example, the ISO C
      +standard defines the semantics of *&x as exactly equal to x, with no
      +reading or writing of memory taking place, and the K semantics of C implements
      +this functionality using an anywhere rule that is applied at compilation time.

      +

      NOTE: the anywhere attribute is only implemented on the LLVM backend
      +currently. Attempting to use it in a semantics that is compiled with the
      +Haskell backend will result in an error being reported by the compiler. This
      +should be remembered when using this attribute, as it may not be suitable for
      +a segment of a semantics which is intended to be symbolically executed.

      +

      Exercises

      + +
        +
      1. Write a version of the calculator from Lesson 1.14 Exercise 1, which uses
        +the same syntax for evaluating expressions, but defines its arithmetic logic
        +using anywhere rules rather than top-level rewrite rules.
        2. +
      +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.2: Fresh Constants

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.3: KLabels and Abstract Syntax

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.4: Overloaded Symbols

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.5: Matching Logic Connectives and #Or Patterns

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.6: Function Context

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.7: Record Productions and Named Nonterminals

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.8: #fun and #let

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.9: #as Patterns

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.10: The Matching Operators, :=K and :/=K

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.11: Uncommon Evaluation Order Concepts

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.12: IEEE 754 Floating Point and Fixed Width Integers

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.13: Alpha-renaming-aware Substitution

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.14: File I/O

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.15: String Buffers and Byte Sequences

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.16: The Intermediate Language of K, KORE

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      Lesson 2.17: Debugging Proofs using the Haskell Backend REPL

      + +

      Return to Top

      + +

      Click here to return to the Table of Contents for Section 2.

      +

      K User Manual

      +

      NOTE: The K User Manual is still under construction; some features of K
      +may have partial or missing documentation.

      +

      Introduction

      +

      Why K?

      + +

      The K Framework is a programming language and system design toolkit made for
      +practioners and researchers alike.

      +

      K For Practioners:
      +K is a framework for deriving programming languages tools from their semantic
      +specifications.

      +

      Typically, programming language tool development follows a similar pattern.
      +After a new programming language is designed, separate teams will develop
      +separate language tools (e.g. a compiler, interpreter, parser, symbolic
      +execution engine, etc). Code reuse is uncommon. The end result is that for each
      +new language, the same basic tools and patterns are re-implemented again and
      +again.

      +

      K approaches the problem differently -- it generates each of these tools from a single language specification.
      +The work of programming language design and tool implementation are made separate concerns.
      +The end result is that the exercise of
      +designing new languages and their associated tooling is now reduced to
      +developing a single language specification from which we derive our tooling for
      +free      .

      +

      K For Researchers:
      +K is a configuration- and rewrite-based executable semantic framework.

      +

      In more detail, K specifications are:

      +
        +
      1. Executable: compile into runnable and testable programs;
        2. +
      2. Semantic: correspond to a logical theory with a sound and relatively
        +complete proof system;
        3. +
      3. Configuration-based: organize system states into compositional,
        +hierarchical, labelled units called cells;
        4. +
      4. Rewrite-based: define system transitions using rewrite rules.
        5. +
      +

      K specifications are compiled into particular matching logic theories, giving
      +them a simple and expressive semantics. K semantic rules are implicitly defined
      +over the entire configuration structure, but omit unused cells, enabling a
      +highly modular definitional style. Furthermore, K has been used to develop
      +programming languages, type systems, and formal analysis tools.

      +

      Manual Objectives

      + +

      As mentioned in the Why K? section above, the K Framework is designed as a
      +collection of language-generic command-line interface (CLI) tools which revolve
      +around K specifications. These tools cover a broad range of uses, but they
      +typically fall into one of the following categories:

      +
        +
      1. Transforming K Specs (e.g. compilation)
        2. +
      2. Running K Specs (e.g. concrete and symbolic execution)
        3. +
      3. Analyzing K Specs (e.g. theorem proving)
        4. +
      +

      The main user-facing K tools include:

      +
        +
      • kompile - the K compiler driver
        • +
      • kparse - the stanadlone K parser and abstract syntax tree (AST)
        +transformation tool
        • +
      • krun - the K interpreter and symbolic execution engine driver
        • +
      • kprove - the K theorem prover
        • +
      +

      This user manual is designed to be a tool reference.
      +In particular, it is not desgined to be a tutorial on how to write K
      +specifications or to teach the logical foundations of K. New K users should
      +consult our dedicated
      +K tutorial,
      +or the more language-design oriented
      +PL tutorial.
      +Researchers seeking to learn more about the logic underlying K are encouraged
      +to peruse the
      +growing literature on K and matching logic.
      +We will consider the manual complete when it provides a complete description of
      +all user-facing K tools and features.

      +

      Introduction to K

      +

      Since K specifications are the primary input into the entire system, let us
      +take a moment to describe them. At the highest level, K specifications describe
      +a programming language or system using three different pieces:

      +
        +
      1. the system primitives, the base datatypes used during system operation,
        +e.g., numbers, lists, maps, etc;
        2. +
      2. the system state, a tuple or record over system primitives which gives a
        +complete snapshot of the system at any given moment;
        3. +
      3. the system behavior, a set of rules which defines possible system
        +evolutions.
        4. +
      +

      K specifications are then defined by a collection of sentences which
      +correspond to the three concepts above:

      +
        +
      1. syntax declarations encode the system primitives;
        2. +
      2. configuration declarations encode the system state;
        3. +
      3. context and rule declarations encode the system behavior.
        4. +
      +

      K sentences are then organized into one or modules which are stored in one or
      +more files. In this scheme, files may require other files and modules may
      +import other modules, giving rise to a hierarchy of files and modules. We
      +give an intuitive sketch of the two levels of grouping in the diagram below:

      +
         example.k file
+  +=======================+
+  | requires ".." --------|--> File_1
+  | ...                   |
+  | requires ".." --------|--> File_N
+  |                       |
+  |  +-----------------+  |
+  |  | module ..       |  |
+  |  |   imports .. ---|--|--> Module_1
+  |  |   ...           |  |
+  |  |   imports .. ---|--|--> Module_M
+  |  |                 |  |
+  |  |   sentence_1    |  |
+  |  |   ...           |  |
+  |  |   sentence_K    |  |
+  |  | endmodule       |  |
+  |  +-----------------+  |
+  |                       |
+  +=======================+
+

      where:

      +
        +
      • files and modules are denoted by double-bordered and single-borded boxes
        +respectively;
        • +
      • file or module identifiers are denoted by double dots (..);
        • +
      • potential repititions are denoted by triple dots (...).
        • +
      +

      In the end, we require that the file and module hierarchies both form a
      +directed acyclic graph (DAG). This is, no file may recursively require itself,
      +and likewise, no module may recursively import itself.

      +

      We now zoom in further to discuss the various kinds of sentences contained in K
      +specifications:

      +
        +
      1. +

        sentences that define our system's primitives, including:

        +
          +
        • sort declarations: define new categories of primitive datatypes
          • +
        • Backus-Naur Form (BNF) grammar declarations: define the
          +operators that inhabit our primitive datatypes
          • +
        • lexical syntax declarations: define lexemes/tokens for the
          +lexer/tokenizer
          • +
        • syntax associativity declarations: specify the
          +associativity/grouping of our declared operators
          • +
        • syntax priority declarations: specify the priority of
          +potential ambiguous operators
          • +
        +
        2. +
      2. +

        sentences that define our system's state, including:

        +
          +
        • configuration declarations: define labelled, hierarchical records
          +using an nested XML-like syntax
          • +
        +
        3. +
      3. +

        sentences that define our system's behavior, including:

        +
          +
        • context declarations: describe how primitives and configurations
          +can simplify
          • +
        • context alias declarations: define templates that can generate new
          +contexts
          • +
        • rule declarations: define how the system transitions from one state
          +to the next
          • +
        +
        4. +
      +

      K Process Overview

      + +

      We now examine how the K tools are generally used. The main input to all of the
      +K tools is a K specification. For effieciency reasons, this specification is
      +first compiled into an intermediate representation called Kore. Once we have
      +obtained this intermediate representation, we can use it to do:

      +
        +
      1. parsing/pretty-printing, i.e., converting a K term, whose syntax is defined
        +by a K specification, into a alternate representation
        2. +
      2. concrete and abstract execution of a K specification
        3. +
      3. theorem proving, i.e., verifying whether a set of claims about a K
        +specification hold
        4. +
      +

      We represent the overall process using the graphic below:

      +
       K Compilation Process
++============================================================+
+|                     +---------+                            |
+|  K Specification ---| kompile |--> Kore Specification --+  |
+|                     +---------+                         |  |
++=========================================================|==+
+                                                          |
+ K Execution Process                                      |
++=========================================================|==+
+|                                                         |  |
+|             +-------------------------------------------+  |
+|             |                                              |
+|             |       +---------+                            |
+|  K Term ----+-------| kparse  |--> K Term                  |
+|             |       +---------+                            |
+|             |                                              |
+|             |       +---------+                            |
+|  K Term ----+-------|  krun   |--> K Term                  |
+|             |       +---------+                            |
+|             |                                              |
+|             |       +---------+                            |
+|  K Claims --+-------| kprove  |--> K Claims                |
+|                     +---------+                            |
+|                                                            |
++============================================================+
+

      where:

      +
        +
      • process outlines are denoted by boxes with double-lined borders
        • +
      • executables are denoted by boxes with single-lined borders
        • +
      • inputs and outputs are denoted by words attached to lines
        • +
      • K terms typically correspond to programs defined in a particular
        +language's syntax (which are either parsed using kparse or executed using
        +krun)
        • +
      • K claims are a notation for describing how certain K programs should
        +execute (which are checked by our theorem prover kprove)
        • +
      +

      K Compilation Process:
      +Let us start with a description of the compilation process. According to the
      +above diagram, the compiler driver is called kompile. For our purposes, it is
      +enough to view the K compilation process as a black box that transforms a K
      +specification into a lower-level Kore specification that encodes the same
      +information, but that is easier to work with programmatically.

      +

      K Execution Process:
      +We now turn our attention to the K execution process. Abstractly, we can divide
      +the K execution process into the following stages:

      +
        +
      1. the kore specification is loaded (which defines a lexer, parser, and
        +unparser among other things)
        2. +
      2. the input string is lexed into a token stream
        3. +
      3. the token stream is parsed into K terms/claims
        4. +
      4. the K term/claims are transformed according the K tool being used (e.g.
        +kparse, krun, or kprove)
        5. +
      5. the K term/claims are unparsed into a string form and printed
        6. +
      +

      Note that all of the above steps performed in K execution process are fully
      +prescribed by the input K specification. Of course, there are entire languages
      +devoted to encoding these various stages proces individually, e.g., flex for
      +lexers, bison for parsers, etc. What K offers is a consistent language to
      +package the above concepts in a way that we believe is convenient and practical
      +for a wide range of uses.

      +

      Module Declaration

      +

      K modules are declared at the top level of a K file. They begin with the
      +module keyword and are followed by a module ID and an optional set of
      +attributes. They continue with zero or more imports and zero or more sentences
      +until the endmodule keyword is reached.

      +

      A module ID consists of an optional # at the beginning, followed by one or
      +more components separated by hyphens. Each component can contain letters,
      +numbers, or underscores.

      +

      After the module ID, attributes can be specified in square brackets. See below
      +for an (incomplete) list of allowed module attributes.

      +

      Following the attributes, a module can contain zero or more imports. An
      +import consists of the import or imports keywords followed by a module ID.
      +An import tells the compiler that this module should contain all the sentences
      +(recursively) contained by the module being imported.

      +

      Imports can be public or private. By default, they are public, which
      +means that all the imported syntax can be used by any module that imports the
      +module doing the import. However, you can explicitly override the visibility
      +of the import with the public or private keyword immediately prior to the
      +module name. A module imported privately does not export its syntax to modules
      +that import the module doing the import.

      +

      Following imports, a module can contain zero or more sentences. A sentence can
      +be a syntax declaration, a rule, a configuration declaration, a context, a
      +claim, or a context alias. Details on each of these can be found in subsequent
      +sections.

      +

      private attribute

      + +

      If the module is given the private attribute, all of its imports and syntax
      +are private by default. Individual pieces of syntax can be made public with
      +the public attribute, and individual imports can be made public with the
      +public keyword. See relevant sections on syntax and modules for more details
      +on what it means for syntax and imports to be public or private.

      +

      symbolic and concrete attribute

      + +

      These attributes may be placed on modules to indicate that they should only
      +be used by the Haskell and LLVM backends respectively. If the definition is
      +compiled on the opposite backend, they are implicitly removed from the
      +definition prior to parsing anywhere they are imported. This can be useful when
      +used in limited capacity in order to provide alternate semantics for certain
      +features on different backends. It should be used sparingly as it makes it more
      +difficult to trust the correctness of your semantics, even in the presence of
      +testing.

      +

      Syntax Declaration

      +

      Named Non-Terminals

      + +

      We have added a syntax to Productions which allows non-terminals to be given a
      +name in productions. This significantly improves the ability to document K, by
      +providing a way to explicitly explain what a field in a production corresponds
      +to instead of having to infer it from a comment or from the rule body.

      +

      The syntax is:

      +
      name: Sort
+

      This syntax can be used anywhere in a K definition that expects a non-terminal.

      +

      klabel(_) and symbol attributes

      + +

      By default K generates for each syntax definition a long and obfuscated klabel
      +string, which serves as a unique internal identifier and also is used in kast
      +format of that syntax. If we need to reference a certain syntax production
      +externally, we have to manually define the klabels using the klabel attribute.
      +One example of where you would want to do this is to be able to refer to a given
      +symbol via the syntax priorities attribute, or to enable overloading of a
      +given symbol.

      +

      If you only provide the klabel attribute, you can use the provided klabel to
      +refer to that symbol anywhere in the frontend K code. However, the internal
      +identifier seen by the backend for that symbol will still be the long obfuscated
      +generated string. Sometimes you want control over the internal identfier used as
      +well, in which case you use the symbol attribute. This tells the frontend to
      +use whatever the declared klabel is directly as the internal identfier.

      +

      For example:

      +
      module MYMODULE
+    syntax FooBarBaz ::= #Foo( Int, Int ) [klabel(#Foo), symbol] // symbol1
+                       | #Bar( Int, Int ) [klabel(#Bar)]         // symbol2
+                       | #Baz( Int, Int )                        // symbol3
+endmodule
+

      Here, we have that:

      +
        +
      • In frontend K, you can refer to "symbol1" as #Foo (from klabel(#Foo)),
        +and the backend will see 'Hash'Foo as the symbol name.
        • +
      • In frontend K, you can refer to "symbol2" as #Bar (from klabel(#Bar)),
        +and the backend will see
        +'Hash'Bar'LParUndsCommUndsRParUnds'MYMODULE'Unds'FooBarBaz'Unds'Int'Unds'Int
        +as the symbol name.
        • +
      • In frontend K, you can refer to "symbol3" as
        +#Baz(_,_)_MYMODULE_FooBarBaz_Int_Int (from auto-generated klabel), and
        +the backend will see
        +'Hash'Baz'LParUndsCommUndsRParUnds'MYMODULE'Unds'FooBarBaz'Unds'Int'Unds'Int
        +as the symbol name.
        • +
      +

      The symbol provided must be unique to this definition. This is enforced by K.
      +In general, it's recommended to use symbol attribute whenever you use klabel
      +unless you explicitely have a reason not to (eg. you want to overload symbols,
      +or you're using a deprecated backend). It can be very helpful use the symbol
      +attribute for debugging, as many debugging messages are printed in Kast format
      +which will be more readable with the symbol names you explicitely declare.
      +In addition, if you are programatically manipulating definitions via the JSON
      +Kast format, building terms using the user-provided pretty
      +symbol, klabel(...) is easier and less error-prone when the auto-generation
      +process for klabels changes.

      +

      Parametric productions and bracket attributes

      + +

      Some syntax productions, like the rewrite operator, the bracket operator, and
      +the #if #then #else #fi operator, cannot have their precise type system
      +expressed using only concrete sorts.

      +

      Prior versions of K solved this issue by using the K sort in this case, but
      +this introduces inexactness in which poorly typed terms can be created even
      +without having a cast operator present in the syntax, which is a design
      +consideration we would prefer to avoid.

      +

      It also introduces cases where terms cannot be placed in positions where they
      +ought to be well sorted unless their return sort is made to be KBott, which in
      +turn vastly complicates the grammar and makes parsing much slower.

      +

      In order to introduce this, we provide a new syntax for parametric productions
      +in K. This allows you to express syntax that has a sort signature based on
      +parametric polymorphism. We do this by means of an optional curly-brace-
      +enclosed list of parameters prior to the return sort of a production.

      +

      Some examples:

      +
      syntax {Sort} Sort ::= "(" Sort ")" [bracket]
+syntax {Sort} KItem ::= Sort
+syntax {Sort} Sort ::= KBott
+syntax {Sort} Sort ::= Sort "=>" Sort
+syntax {Sort} Sort ::= "#if" Bool "#then" Sort "#else" Sort "#fi"
+syntax {Sort1, Sort2} Sort1 ::= "#fun" "(" Sort2 "=>" Sort1 ")" "(" Sort2 ")"
+

      Here we have:

      +
        +
      1. Brackets, which can enclose any sort but should be of the same sort that was
        +enclosed.
        2. +
      2. Every sort is a KItem.
        3. +
      3. A KBott term can appear inside any sort.
        4. +
      4. Rewrites, which can rewrite a value of any sort to a value of the same sort.
        +Note that this allows the lhs or rhs to be a subsort of the other.
        5. +
      5. If then else, which can return any sort but which must contain that sort on
        +both the true and false branches.
        6. +
      6. lambda applications, in which the argument and parameter must be the same
        +sort, and the return value of the application must be the same sort as the
        +return value of the function.
        7. +
      +

      Note the last case, in which two different parameters are specified separated
      +by a comma. This indicates that we have multiple independent parameters which
      +must be the same each place they occur, but not the same as the other
      +parameters.

      +

      In practice, because every sort is a subsort of K, the Sort2
      +parameter in #6 above does nothing during parsing. It cannot
      +actually reject any parse, because it can always infer that the sort of the
      +argument and parameter are K, and it has no effect on the resulting sort of
      +the term. However, it will nevertheless affect the kore generated from the term
      +by introducing an additional parameter to the symbol generated for the term.

      +

      function and total attributes

      + +

      Many times it becomes easier to write a semantics if you have "helper"
      +functions written which can be used in the RHS of rules. The function
      +attribute tells K that a given symbol should be simplified immediately when it
      +appears anywhere in the configuration. Semantically, it means that evaluation
      +of that symbol will result in at most one return value (that is, the symbol is
      +a partial function).

      +

      The total attribute indicates that a symbol cannot be equal to matching logic
      +bottom; in other words, it has at least one value for every possible set of
      +arguments. It can be added to a production with the function attribute to
      +indicate to the symbolic reasoning engine that a given symbol is a
      +total function, that is it has exactly one return value for every possible
      +input. Other uses of the total attribute (i.e., on multi-valued symbols to
      +indicate they always have at least one value) are not yet implemented.

      +

      For example, here we define the _+Word_ total function and the _/Word_
      +partial function, which can be used to do addition/division modulo
      +2 ^Int 256. These functions can be used anywhere in the semantics where
      +integers should not grow larger than 2 ^Int 256. Notice how _/Word_ is
      +not defined when the denominator is 0.

      +
      syntax Int ::= Int "+Word" Int [function, total]
+             | Int "/Word" Int [function]
+
+rule I1 +Word I2 => (I1 +Int I2) modInt (2 ^Int 256)
+rule I1 /Word I2 => (I1 /Int I2) modInt (2 ^Int 256) requires I2 =/=Int 0
+

      freshGenerator attribute

      + +

      In K, you can access "fresh" values in a given domain using the syntax
      +!VARNAME:VarSort (with the !-prefixed variable name). This is supported for
      +builtin sorts Int and Id already. For example, you can generate fresh
      +memory locations for declared identifiers as such:

      +
      rule <k> new var x ; => . ... </k>
+     <env> ENV => ENV [ x <- !I:Int ] </env>
+     <mem> MEM => MEM [ !I <- 0     ] </mem>
+

      Each time a !-prefixed variable is encountered, a new integer will be used,
      +so each variable declared with new var _ ; will get a unique position in the
      +<mem>.

      +

      Sometimes you want to have generation of fresh constants in a user-defined
      +sort. For this, K will still generate a fresh Int, but can use a converter
      +function you supply to turn it into the correct sort. For example, here we can
      +generate fresh Foos using the freshFoo(_) function annotated with
      +freshGenerator.

      +
      syntax Foo ::= "a" | "b" | "c" | d ( Int )
+
+syntax Foo ::= freshFoo ( Int ) [freshGenerator, function, total]
+
+rule freshFoo(0) => a
+rule freshFoo(1) => b
+rule freshFoo(2) => c
+rule freshFoo(I) => d(I) [owise]
+
+rule <k> new var x ; => . ... </k>
+     <env> ENV => ENV [ x <- !I:Int  ] </env>
+     <mem> MEM => MEM [ !I <- !F:Foo ] </mem>
+

      Now each newly allocated memory slot will have a fresh Foo placed in it.

      +

      token attribute

      + +

      The token attribute signals to the Kore generator that the associated sort
      +will be inhabited by domain values. Sorts inhabited by domain values must not
      +have any constructors declared.

      +
      syntax Bytes [hook(BYTES.Bytes), token]
+

      Converting between [token] sorts

      + +

      You can convert between tokens of one sort via Strings by defining functions
      +implemented by builtin hooks.
      +The hook STRING.token2string allows conversion of any token to a string:

      +
      syntax String ::= FooToString(Foo)  [function, total, hook(STRING.token2string)]
+

      Similarly, the hook STRING.string2Token allows the inverse:

      +
      syntax Bar ::= StringToBar(String) [function, total, hook(STRING.string2token)]
+

      WARNING: This sort of conversion does NOT do any sort of parsing or validation.
      +Thus, we can create arbitary tokens of any sort:

      +
      StringToBar("The sun rises in the west.")
+

      Composing these two functions lets us convert from Foo to Bar

      +
      syntax Bar ::= FooToBar(Foo) [function]
+rule FooToBar(F) => StringToBar(FooToString(F))
+

      Parsing comments, and the #Layout sort

      + +

      Productions for the #Layout sort are used to describe tokens that are
      +considered "whitespace". The scanner removes tokens matching these productions
      +so they are not even seen by the parser. Below, we use it to define
      +lines begining with ; (semicolon) as comments.

      +
      syntax #Layout ::= r"(;[^\\n\\r]*)"    // Semi-colon comments
+                 | r"([\\ \\n\\r\\t])" // Whitespace
+

      prec attribute

      + +

      Consider the following naive attempt at creating a language what syntax that
      +allows two types of variables: names that contain underbars, and names that
      +contain sharps/hashes/pound-signs:

      +
      syntax NameWithUnderbar ::= r"[a-zA-Z][A-Za-z0-9_]*"  [token]
+syntax NameWithSharp    ::= r"[a-zA-Z][A-Za-z0-9_#]*" [token]
+syntax Pgm ::= underbar(NameWithUnderbar)
+             | sharp(NameWithSharp)
+

      Although, it seems that K has enough information to parse the programs
      +underbar(foo) and sharp(foo) with, the lexer does not take into account
      +whether a token is being parsed for the sharp or for the underbar
      +production. It chooses an arbitary sort for the token foo (perhaps
      +NameWithUnderbar). Thus, during paring it is unable to construct a valid term
      +for one of those programs (sharp(foo)) and produces the error message:
      +Inner Parser: Parse error: unexpected token 'foo'.

      +

      Since calculating inclusions and intersections between regular expressions is
      +tricky, we must provide this information to K. We do this via the prec(N)
      +attribute. The lexer will always prefer longer tokens to shorter tokens.
      +However, when it has to choose between two different tokens of equal length,
      +token productions with higher precedence are tried first. Note that the default
      +precedence value is zero when the prec attribute is not specified.

      +

      For example, the BUILTIN-ID-TOKENS module defines #UpperId and #LowerId with
      +the prec(2) attribute.

      +
        syntax #LowerId ::= r"[a-z][a-zA-Z0-9]*"                    [prec(2), token]
+  syntax #UpperId ::= r"[A-Z][a-zA-Z0-9]*"                    [prec(2), token]
+

      Furthermore, we also need to make sorts with more specific tokens subsorts of ones with more
      +general tokens. We add the token attribute to this production so that all
      +tokens of a particular sort are marked with the sort they are parsed as and not a
      +subsort thereof. e.g. we get underbar(#token("foo", "NameWithUnderbar"))
      +instead of underbar(#token("foo", "#LowerId"))

      +
      imports BUILTIN-ID-TOKENS
+syntax NameWithUnderbar ::= r"[a-zA-Z][A-Za-z0-9_]*" [prec(1), token]
+                          | #UpperId                [token]
+                          | #LowerId                [token]
+syntax NameWithSharp ::= r"[a-zA-Z][A-Za-z0-9_#]*" [prec(1), token]
+                       | #UpperId                 [token]
+                       | #LowerId                 [token]
+syntax Pgm ::= underbar(NameWithUnderbar)
+             | sharp(NameWithSharp)
+

      unused attribute

      + +

      K will warn you if you declare a symbol that is not used in any of the rules of
      +your definition. Sometimes this is intentional, however; in this case, you can
      +suppress the warning by adding the unused attribute to the production or
      +cell.

      +
      syntax Foo ::= foo() [unused]
+
+configuration <foo unused=""> .K </foo>
+

      Symbol priority and associativity

      + +

      Unlike most other parser generators, K combines the task of parsing with AST
      +generation. A production declared with the syntax keyword in K is both a
      +piece of syntax used when parsing, and a symbol that is used when rewriting.
      +As a result, it is generally convenient to describe expression grammars using
      +priority and associativity declarations rather than explicitly transforming
      +your grammar into a series of nonterminals, one for each level of operator
      +precedence. Thus, for example, a simple grammar for addition and multiplication
      +will look like this:

      +
      syntax Exp ::= Exp "*" Exp
+             | Exp "+" Exp
+

      However, this grammar is ambiguous. The term x+y*z might refer to x+(y*z)
      +or to (x+y)*z. In order to differentiate this, we introduce a partial
      +ordering between productions known as priority. A symbol "has tighter priority"
      +than another symbol if the first symbol can appear under the second, but the
      +second cannot appear under the first without a bracket. For example, in
      +traditional arithmetic, multiplication has tighter priority than addition,
      +which means that x+y*z cannot parse as (x+y)*z because the addition
      +operator would appear directly beneath the multiplication, which is forbidden
      +by the priority filter.

      +

      Priority is applied individually to each possible ambiguous parse of a term. It
      +then either accepts or rejects that parse. If there is only a single remaining
      +parse (after all the other disambiguation steps have happened), this is the
      +parse that is chosen. If all the parses were rejected, it is a parse error. If
      +multiple parses remain, they might be resolved by further disambiguation such
      +as via the prefer and avoid attributes, but if multiple parses remain after
      +disambiguation finishes, this is an ambiguous parse error, indicating there is
      +not a unique parse for that term. In the vast majority of cases, this is
      +an error and indicates that you ought to either change your grammar or add
      +brackets to the term in question.

      +

      Priority is specified in K grammars by means of one of two different
      +mechanisms. The first, and simplest, simply replaces the | operator in a
      +sequence of K productions with the > operator. This operator indicates that
      +everything prior to the > operator (including transitively) binds tighter
      +than what comes after. For example, a more complete grammar for simple
      +arithmetic might be:

      +
      syntax Exp ::= Exp "*" Exp
+             | Exp "/" Exp
+             > Exp "+" Exp
+             | Exp "-" Exp
+

      This indicates that multiplication and division bind tigher than addition
      +and subtraction, but that there is no relationship in priority between
      +multiplication and division.

      +

      As you may have noticed, this grammar is also ambiguous. x*y/z might refer to
      +x*(y/z) or to (x*y)/z. Indeed, if we removed division and subtraction
      +entirely, the grammar would still be ambiguous: x*y*z might parse as
      +x*(y*z), or as (x*y)*z. To resolve this, we introduce another feature:
      +associativity. Roughly, asssociativity tells us how symbols are allowed to nest
      +within other symbols with the same priority. If a set of symbols is left
      +associative, then symbols in that set cannot appear as the rightmost child
      +of other symbols in that set. If a set of symbols is right associative, then
      +symbols in that set cannot appear as the leftmost child of other symbols in
      +that set. Finally, if a set of symbols is non-associative, then symbols
      +in that set cannot appear as the rightmost or leftmost child of other symbols
      +in that set. For example, in the above example, if addition and subtraction
      +are left associative, then x+y+z will parse as (x+y)+z and x+y-z will
      +parse as (x+y)-z (because the other parse will have been rejected).

      +

      You might notice that this seems to apply only to binary infix operators. In
      +fact, the real behavior is slightly more complicated. Priority and
      +associativity (for technical reasons that go beyond the scope of this document)
      +really only apply when the rightmost or leftmost item in a production is a
      +nonterminal. If the rightmost nonterminal is followed by a terminal (or
      +respectively the leftmost preceded), priority and associativity do not apply.
      +Thus we can generalize these concepts to arbitrary context-free grammars.

      +

      Note that in some cases, this is not the behavior you want. You may actually
      +want to reject parses even though the leftmost and rightmost item in a
      +production are terminals. You can accomplish this by means of the
      +applyPriority attribute. When placed on a production, it tells the parser
      +which nonterminals of a production the priority filter ought to reject children
      +under, overriding the default behavior. For example, I might have a production
      +like syntax Exp ::= foo(Exp, Exp) [applyPriority(1)]. This tells the parser
      +to reject terms with looser priority binding under the first Exp, but not
      +the second. By default, with this production, neither position would apply
      +to the priority filter, because the first and last items of the production
      +are both terminals.

      +

      Associativity is specified in K grammars by means of one of two different
      +mechanisms. The first, and simplest, adds the associativity of a priority block
      +of symbols prior to that block. For example, we can remove the remaining
      +ambiguities in the above grammar like so:

      +
      syntax Exp ::= left:
+               Exp "*" Exp
+             | Exp "/" Exp
+             > right:
+               Exp "+" Exp
+             | Exp "-" Exp
+

      This indicates that multiplication and division are left-associative, ie, after
      +symbols with higher priority are parsed as innermost, symbols are nested with
      +the rightmost on top. Addition and subtraction are right associative, which
      +is the opposite and indicates that symbols are nested with the leftmost on top.
      +Note that this is similar but different from evaluation order, which also
      +concerns itself with the ordering of symbols, which is described in the next
      +section.

      +

      You may note we have not yet introduced the second syntax for priority
      +and associativity. In some cases, syntax for a grammar might be spread across
      +multiple modules, sometimes for very good reasons with respect to code
      +modularity. As a result, it becomes infeasible to declare priority and
      +associativity inline within a set of productions, because the productions
      +are not contiguous within a single file.

      +

      For this purpose, we introduce the equivalent syntax priorities,
      +syntax left, syntax right, and syntax non-assoc declarations. For
      +example, the above grammar can be written equivalently as:

      +
      syntax Exp ::= Exp "*" Exp [group(mult)]
+             | Exp "/" Exp [group(div)]
+             | Exp "+" Exp [group(add)]
+             | Exp "-" Exp [group(sub)]
+
+syntax priorities mult div > add sub
+syntax left mult div
+syntax right add sub
+

      Here, the group(_) attribute is used to create user-defined groups of
      +sentences. A particular group name collectively refers to the whole set of
      +sentences within that group. The sets are flattened together, so we could
      +equivalently have written:

      +
      syntax Exp ::= Exp "*" Exp [group(mult)]
+             | Exp "/" Exp [group(mult)]
+             | Exp "+" Exp [group(add)]
+             | Exp "-" Exp [group(add)]
+
+syntax priorities mult > add
+syntax left mult
+syntax right add
+

      Note that syntax [left|right|non-assoc] should not be used to group together
      +productions with different priorities. For example, this code would be invalid:

      +
      syntax priorities mult > add
+syntax left mult add
+

      Note that there is one other way to describe associativity, but it is
      +prone to a very common mistake. You can apply the attribute left, right,
      +or non-assoc directly to a production to indicate that it is, by itself,
      +left-, right-, or non-associative.

      +

      However, this often does not mean what users think it means. In particular:

      +
      syntax Exp ::= Exp "+" Exp [left]
+             | Exp "-" Exp [left]
+

      is not equivalent to:

      +
      syntax Exp ::= left:
+               Exp "+" Exp
+             | Exp "-" Exp
+

      Under the first, each production is associative with itself, but not each
      +other. Thus, x+y+z will parse unambiguously as (x+y)+z, but x+y-z will
      +be ambiguous. However, in the second, x+y-z will parse unambiguously as
      +(x+y)-z.

      +

      Think carefully about how you want your grammar to parse. In general, if you're
      +not sure, it's probably best to group associativity together into the same
      +blocks you use for priority, rather than using left, right, or non-assoc
      +attributes on the productions.

      +

      Lexical identifiers

      + +

      Sometimes it is convenient to be able to give a certain regular expression a
      +name and then refer to it in one or more regular expression terminals. This
      +can be done with a syntax lexical sentence in K:

      +
      syntax lexical Alphanum = r"[0-9a-zA-Z]"
+

      This defines a lexical identifier Alphanum which can be expanded in any
      +regular expression terminal to the above regular expression. For example, I
      +might choose to then implement the syntax of identifiers as follows:

      +
      syntax Id ::= r"[a-zA-Z]{Alphanum}*" [token]
+

      Here {Alphanum} expands to the above regular expression, making the sentence
      +equivalent to the following:

      +
      syntax Id ::= r"[a-zA-Z]([0-9a-zA-Z])*" [token]
+

      This feature can be used to more modularly construct the lexical syntax of your
      +language. Note that K does not currently check that lexical identifiers used
      +in regular expressions have been defined; this will generate an error when
      +creating the scanner, however, and the user ought to be able to debug what
      +happened.

      +

      assoc, comm, idem, and unit attributes

      + +

      These attributes are used to indicate whether a collection or a production
      +is associative, commutative, idempotent, and/or has a unit.
      +In general, you should not need to apply these attributes to productions
      +yourself, however, they do have certain special meaning to K. K will generate
      +axioms related to each of these concepts into your definition for you
      +automatically. It will also automatically sort associative-commutative
      +collections, and flatten the indentation of associative collections, when
      +unparsing.

      +

      public and private attribute

      + +

      K allows users to declare certain pieces of syntax as either public or private.
      +All syntax is public by default. Public syntax can be used from any module that
      +imports that piece of syntax. A piece of syntax can be declared private with
      +the private attribute. This means that that syntax can only be used in the
      +module in which it is declared; it is not visible from modules that import
      +that module.

      +

      You can also change the default visibility of a module with the private
      +attribute, when it is placed directly on a module. A module with the private
      +attribute has all syntax private by default; this can be overridden on
      +specific sentences with the public attribute.

      +

      Note that the private module attribute also changes the default visiblity
      +of imports; please refer to the appropriate section elsewhere in the manual
      +for more details.

      +

      Here is an example usage:

      +
      module WIDGET-SYNTAX
+
+  syntax Widget ::= foo()
+  syntax WidgetHelper ::= bar() [private] // this production is not visible
+                                          // outside this module
+endmodule
+
+module WIDGET [private]
+  imports WIDGET-SYNTAX
+
+  syntax Widget ::= fooImpl() // this production is not visible outside this
+                              // module
+
+  // this production is visible outside this module
+  syntax KItem ::= adjustWidget(Widget) [function, public]
+endmodule
+

      Configuration Declaration

      +

      exit attribute

      + +

      A single configuration cell containing an integer may have the "exit"
      +attribute. This integer will then be used as the return value on the console
      +when executing the program.

      +

      For example:

      +
      configuration <k> $PGM:Pgm </k>
+              <status-code exit=""> 1 </status-code>
+

      declares that the cell status-code should be used as the exit-code for
      +invocations of krun. Additionally, we state that the default exit-code is 1
      +(an error state). One use of this is for writing testing harnesses which assume
      +that the test fails until proven otherwise and only set the <status-code> cell
      +to 0 if the test succeeds.

      +

      Collection Cells: multiplicity and type attributes

      + +

      Sometimes a semantics needs to allow multiple copies of the same cell, for
      +example if you are making a concurrent multi-threading programming language.
      +For this purpose, K supports the multiplicity and type attributes on cells
      +declared in the configuration.

      +

      multiplicity can take on values * and ?. Declaring multiplicity="*"
      +indicates that the cell may appear any number of times in a runtime
      +configuration. Setting multiplicity="?" indicates that the cell may only
      +appear exactly 0 or 1 times in a runtime configuration. If there are no
      +configuration variables present in the cell collection, the initial
      +configuration will start with exactly 0 instances of the cell collection. If
      +there are configuration variables present in the cell collection, the initial
      +configuration will start with exactly 1 instance of the cell collection.

      +

      type can take on values Set, List, and Map. For example, here we declare
      +several collecion cells:

      +
      configuration <k> $PGM:Pgm </k>
+              <sets>  <set  multiplicity="?" type="Set">  0:Int </set>  </sets>
+              <lists> <list multiplicity="*" type="List"> 0:Int </list> </lists>
+              <maps>
+                <map multiplicity="*" type="Map">
+                  <map-key> 0:Int </map-key>
+                  <map-value-1> "":String </map-value-1>
+                  <map-value-2> 0:Int     </map-value-2>
+                </map>
+              </maps>
+

      Declaring type="Set" indicates that duplicate occurrences of the cell should
      +be de-duplicated, and accesses to instances of the cell will be nondeterministic
      +choices (constrained by any other parts of the match and side-conditions).
      +Similarly, declaring type="List" means that new instances of the cell can be
      +added at the front or back, and elements can be accessed from the front or back,
      +and the order of the cells will be maintained. The following are examples of
      +introduction and elimination rules for these collections:

      +
      rule <k> introduce-set(I:Int) => . ... </k>
+     <sets> .Bag => <set> I </set> </sets>
+
+rule <k> eliminate-set => I ... </k>
+     <sets> <set> I </set> => .Bag </sets>
+
+rule <k> introduce-list-start(I:Int) => . ... </k>
+     <lists> (.Bag => <list> I </list>) ... </lists>
+
+rule <k> introduce-list-end(I:Int) => . ... </k>
+     <lists> ... (.Bag => <list> I </list>) </lists>
+
+rule <k> eliminate-list-start => I ... </k>
+     <lists> (<list> I </list> => .Bag) ... </lists>
+
+rule <k> eliminate-list-end => I ... </k>
+     <lists> ... (<list> I </list> => .Bag) </lists>
+

      Notice that for multiplicity="?", we only admit a single <set> instance at
      +a time. For the type=List cell, we can add/eliminate cells from the from or
      +back of the <lists> cell. Also note that we use .Bag to indicate the empty
      +cell collection in all cases.

      +

      Declaring type="Map" indicates that the first sub-cell will be used as a
      +cell-key. This means that matching on those cells will be done as a map-lookup
      +operation if the cell-key is mentioned in the rule (for performance). If the
      +cell-key is not mentioned, it will fallback to normal nondeterministic
      +constrained by other parts of the match and any side-conditions. Note that there
      +is no special meaning to the name of the cells (in this case <map>,
      +<map-key>, <map-value-1>, and <map-value-2>). Additionally, any number of
      +sub-cells are allowed, and the entire instance of the cell collection is
      +considered part of the cell-value, including the cell-key (<map-key> in this
      +case) and the surrounding collection cell (<map> in this case).

      +

      For example, the following rules introduce, set, retrieve from, and eliminate
      +type="Map" cells:

      +
      rule <k> introduce-map(I:Int) => . ... </k>
+     <maps> ... (.Bag => <map> <map-key> I </map-key> ... </map>) ... </maps>
+
+rule <k> set-map-value-1(I:Int, S:String) => . ... </k>
+     <map> <map-key> I </map-key> <map-value-1> _ => S </map-value-1> ... </map>
+
+rule <k> set-map-value-2(I:Int, V:Int) => . ... </k>
+     <map> <map-key> I </map-key> <map-value-2> _ => V </map-value-2> ... </map>
+
+rule <k> retrieve-map-value-1(I:Int) => S ... </k>
+     <map> <map-key> I </map-key> <map-value-1> S </map-value-1> ... </map>
+
+rule <k> retrieve-map-value-2(I:Int) => V ... </k>
+     <map> <map-key> I </map-key> <map-value-2> V </map-value-2> ... </map>
+
+rule <k> eliminate-map(I:Int) => . ... </k>
+     <maps> ... (<map> <map-key> I </map-key> ... </map> => .Bag) ... </maps>
+

      Note how each rule makes sure that <map-key> cell is mentioned, and we
      +continue to use .Bag to indicate the empty collection. Also note that
      +when introducing new map elements, you may omit any of the sub-cells which are
      +not the cell-key. In case you do omit sub-cells, you must use structural
      +framing ... to indicate the missing cells, they will receive the default
      +value given in the configuration ... declaration.

      +

      Rule Declaration

      +

      Rule Structure

      + +

      Each K rule follows the same basic structure (given as an example here):

      +
      rule LHS => RHS requires REQ ensures ENS [ATTRS]
+

      The portion between rule and requires is referred to as the rule body,
      +and may contain one or more rewrites (though not nested). Here, the rule body is
      +LHS => RHS, where LHS and RHS are used as placeholders for the pre- and
      +post- states. Note that we lose no generality referring to the LHS or the
      +RHS, even in the presence of multiple rewrites, as the rewrites are pulled to
      +the top-level anyway.

      +

      Next is the requires clause, represented here as REQ. The requires clause is
      +an additional predicate (function-like term of sort Bool), which is to be
      +evaluated before applying the rule. If the requires clause does not evaluate to
      +true, then the rule does not apply.

      +

      Finally is the ensures clause, represented here as ENS. The ensures clause
      +is to be interpreted as a post-condition, and will be automatically added to the
      +path condition if the rule applies. It may cause the entire term to become
      +undefined, but the backend will not stop itself from applying the rule in this
      +case. Note that concrete backends (eg. the LLVM backend) are free to ignore the
      +ensures clause.

      +

      Overall, the transition represented by such a rule is from a state
      +LHS #And REQ ending in a state RHS #And ENS. When backends apply this rule
      +as a transition/rewrite, they should:

      +
        +
      • Check if pattern LHS matches (or unifies) with the current term, giving
        +substitution alpha.
        • +
      • Check if the instantiation alpha(REQ) is valid (or satisfiable).
        • +
      • Build the new term alpha(RHS #And ENS), and check if it's satisfiable.
        • +
      +

      Pattern Matching operator

      + +

      Sometimes when you want to express a side condition, you want to say that a
      +rule matches if a particular term matches a particular pattern, or if it
      +instead does /not/ match a particular pattern.

      +

      The syntax in K for this is :=K and :/=K. It has similar meaning to ==K and
      +=/=K, except that where ==K and =/=K express equality, :=K and =/=K express
      +model membership. That is to say, whether or not the rhs is a member of the set
      +of terms expressed by the lhs pattern. Because the lhs of these operators is a
      +pattern, the user can use variables in the lhs of the operator. However, due to
      +current limitations, these variables are NOT bound in the rest of the term.
      +The user is thus encouraged to use anonymous variables only, although this is
      +not required.

      +

      This is compiled by the K frontend down to an efficient pattern matching on a
      +fresh function symbol.

      +

      Anonymous function applications

      + +

      There are a number of cases in K where you would prefer to be able to take some
      +term on the RHS, bind it to a variable, and refer to it in multiple different
      +places in a rule.

      +

      You might also prefer to take a variable for which you know some of its
      +structure, and modify some of its internal structure without requiring you to
      +match on every single field contained inside that structure.

      +

      In order to do this, we introduce syntax to K that allows you to construct
      +anonymous functions in the RHS of a rule and apply them to a term.

      +

      The syntax for this is:

      +
      #fun(RuleBody)(Argument)
+

      Note the limitations currently imposed by the implementation. These functions
      +are not first-order: you cannot bind them to a variable and inject them like
      +you can with a regular klabel for a function. You also cannot express multiple
      +rules or multiple parameters, or side conditions. All of these are extensions
      +we would like to support in the future, however.

      +

      In the following, we use three examples to illustrate the behavior of #fun.
      +We point out that the support for #fun is provided by the frontend, not the
      +backends.

      +

      The three examples are real examples borrowed or modified from existing language
      +semantics.

      +

      Example 1 (A Simple Self-Explained Example).

      +
      #fun(V:Val => isFoo(V) andBool isBar(V))(someFunctionReturningVal())
+

      Example 2 (Nested #fun).

      +
         #fun(C
+=> #fun(R
+=> #fun(E
+=> foo1(E, R, C)
+  )(foo2(C))
+  )(foo3(0))
+  )(foo4(1))
+

      This example is from the beacon
      +semantics:https://github.com/runtimeverification/beacon-chain-spec/blob/master/b
      +eacon-chain.k at line 302, with some modification for simplicity. Note how
      +variables C, R, E are bound in the nested #fun.

      +

      Example 3 (Matching a structure).

      +
      rule foo(K, RECORD) =>
+  #fun(record(... field: _ => K))(RECORD)
+

      Unlike previous examples, the LHS of #fun in this example is no longer a
      +variable, but a structure. It has the same spirit as the first two examples,
      +but we match the RECORD with a structure record( DotVar, field: X), instead
      +of a standalone variable. We also use K's local rewrite syntax (i.e., the
      +rewriting symbol => does not occur at the top-level) to prevent writing
      +duplicate expressions on the LHS and RHS of the rewriting.

      +

      Macros and Aliases

      + +

      A production can be tagged with the macro, alias, macro-rec, or alias-rec
      +attributes. In all cases, what this signifies is that this is a macro production.
      +Macro rules are rules where the top symbol of the left-hand-side are macro
      +labels. Macro rules are applied statically during compilation on all terms that
      +they match, and statically before program execution on the initial configuration.
      +Currently, macro rules are required to not have side conditions, although they
      +can contain sort checks.

      +

      alias rules are also applied statically in reverse prior to unparsing on the
      +final configuration. Note that a macro rule can have unbound variables in the
      +right hand side. When such a macro exists, it should be used only on the left
      +hand side of rules, unless the user is performing symbolic execution and expects
      +to introduce symbolic terms into the subject being rewritten.

      +

      However, when used on the left hand side of a rule, it functions similarly to a
      +pattern alias, and allows the user to concisely express a reusable pattern that
      +they wish to match on in multiple places.

      +

      For example, consider the following semantics:

      +
      syntax KItem ::= "foo" [alias] | "foobar"
+syntax KItem ::= bar(KItem) [macro] | baz(Int, KItem)
+rule foo => foobar
+rule bar(I) => baz(?_, I)
+rule bar(I) => I
+

      This will rewrite baz(0, foo) to foo. First baz(0, foo) will be rewritten
      +statically to baz(0, foobar). Then the non-macro rule will apply (because
      +the rule will have been rewritten to rule baz(_, I) => I). Then foobar will
      +be rewritten statically after rewriting finishes to foo via the reverse form
      +of the alias.

      +

      Note that macros do not apply recursively within their own expansion. This is
      +done so as to ensure that macro expansion will always terminate. If the user
      +genuinely desires a recursive macro, the macro-rec and alias-rec attributes
      +can be used to provide this behavior.

      +

      For example, consider the following semantics:

      +
      syntax Exp ::= "int" Exp ";" | "int" Exps ";" [macro] | Exp Exp | Id
+syntax Exps ::= List{Exp,","}
+
+rule int X:Id, X':Id, Xs:Exps ; => int X ; int X', Xs ;
+

      This will expand int x, y, z; to int x; int y, z; because the macro does
      +not apply the second time after applying the substitution of the first
      +application. However, if the macro attribute were changed to the macro-rec
      +attribute, it would instead expand (as the user likely intended) to
      +int x; int y; int z;.

      +

      The alias-rec attribute behaves with respect to the alias attribute the
      +same way the macro-rec attribute behaves with respect to macro.

      +

      anywhere rules

      + +

      Some rules are not functional, but you want them to apply anywhere in the
      +configuration (similar to functional rules). You can use the anywhere
      +attribute on a rule to instruct the backends to make sure they apply anywhere
      +they match in the entire configuration.

      +

      For example, if you want to make sure that some associative operator is always
      +right-associated anywhere in the configuration, you can do:

      +
      syntax Stmt ::= Stmt ";" Stmt
+
+rule (S1 ; S2) ; S3 => S1 ; (S2 ; S3) [anywhere]
+

      Then after every step, all occurrences of _;_ will be re-associated. Note that
      +this allows the symbol _;_ to still be a constructor, even though it is
      +simplified similarly to a function.

      +

      trusted claims

      + +

      You may add the trusted attribute to a given claim for the K prover to
      +automatically add it to the list of proven circularities, instead of trying to
      +discharge it separately.

      +

      Projection and Predicate functions

      + +

      K automatically generates certain predicate and projection functions from the
      +syntax you declare. For example, if you write:

      +
      syntax Foo ::= foo(bar: Bar)
+

      It will automatically generate the following K code:

      +
      syntax Bool ::= isFoo(K) [function]
+syntax Foo ::= "{" K "}" ":>Foo" [function]
+syntax Bar ::= bar(Foo) [function]
+
+rule isFoo(F:Foo) => true
+rule isFoo(_) => false [owise]
+
+rule { F:Foo }:>Foo => F
+rule bar(foo(B:Bar)) => B
+

      The first two types of functions are generated automatically for every sort in
      +your K definition, and the third type of function is generated automatically
      +for each named nonterminal in your definition. Essentially, isFoo for some
      +sort Foo will tell you whether a particular term of sort K is a Foo,
      +{F}:>Foo will cast F to sort Foo if F is of sort Foo and will be
      +undefined (i.e., theoretically defined as #Bottom, the bottom symbol in
      +matching logic) otherwise. Finally, bar will project out the child of a foo
      +named bar in its production declaration.

      +

      Note that if another term of equal or smaller sort to Foo exists and has a
      +child named bar of equal or smaller sort to Bar, this will generate an
      +ambiguity during parsing, so care should be taken to ensure that named
      +nonterminals are sufficiently unique from one another to prevent such
      +ambiguities. Of course, the compiler will generate a warning in this case.

      +

      simplification attribute

      + +

      The simplification attribute identifies rules outside the main semantics that
      +are used to simplify function patterns.

      +

      Conditions: A simplification rule is applied by matching the function
      +arguments, instead of unification as when applying function definition
      +rules. This allows function symbols to appear nested as arguments to other
      +functions on the left-hand side of a simplification rule, which is forbidden in
      +function definition rules. For example, this rule would not be accepted as a
      +function definition rule:

      +
      rule (X +Int Y) +Int Z => X +Int (Y +Int Z) [simplification]
+

      A simplification rule is only applied when the current side condition implies
      +the requires clause of the rule, like function definition rules.

      +

      Order: The simplification attribute accepts an optional integer argument
      +which is the rule's simplification priority; if the optional argument is not
      +specified, it is equivalent to a simplification priority of 50. Backends
      +should attempt simplification rules in order of their simplification
      +priority      , but are not required to do so; in fact, the backend is free to apply
      +simplification rules at any time. Because of this, users must ensure that
      +simplification rules are sound regardless of their order of application. This
      +differs from the priority attribute in that rules with the priority
      +attribute must be applied in their priority order by the backend. It is an
      +error to have the priority attribute on a simplification rule.

      +

      For example, for the following definition:

      +
          syntax WordStack ::= Int ":" WordStack | ".WordStack"
+    syntax Int ::= sizeWordStack    ( WordStack       ) [function]
+                 | sizeWordStackAux ( WordStack , Int ) [function]
+ // --------------------------------------------------------------
+    rule sizeWordStack(WS) => sizeWordStackAux(WS, 0)
+
+    rule sizeWordStackAux(.WordStack, N) => N
+    rule sizeWordStackAux(W : WS    , N) => sizeWordStackAux(WS, N +Int 1)
+

      We might add the following simplification lemma:

      +
          rule sizeWordStackAux(WS, N) => N +Int sizeWordStackAux(WS, 0)
+      requires N =/=Int 0
+      [simplification]
+

      Then this simplification rule will only apply if the Haskell backend can prove
      +that notBool N =/=Int 0 is unsatisfiable. This avoids an infinite cycle of
      +applying this simplification lemma.

      +

      NOTE: The frontend and Haskell backend do not check that supplied
      +simplification rules are sound, this is the developer's responsibility. In
      +particular, rules with the simplification attribute must preserve definedness;
      +that is, if the left-hand side refers to any partial function then:

      +
        +
      • the right-hand side must be #Bottom when the left-hand side is #Bottom, or
        • +
      • the rule must have an ensures clause that is false when the left-hand
        +side is #Bottom, or
        • +
      • the rule must have a requires clause that is false when the left-hand
        +side is #Bottom.
        • +
      +

      These conditions are in order of decreasing preference: the best option is to
      +preserve #Bottom on the right-hand side, the next best option is to have an
      +ensures clause, and the least-preferred option is to have a requires clause.
      +The most preferred option is to write total functions and avoid the entire issue.

      +

      NOTE: The Haskell backend does not attempt to prove claims which right-hand
      +side is #Bottom. The reason for this is that the general case is undecidable,
      +and the backend might enter an infinite loop. Therefore, the backend emits a
      +warning if it encounters such a claim.

      +

      concrete and symbolic attributes (Haskell backend)

      + +

      Users can control the application of simplification rules using the concrete
      +and the symbolic attributes by specifying the type of patterns the rule's
      +arguments are to match.

      +

      A concrete pattern is a pattern which does not contain variables or unevaluated
      +functions, otherwise the pattern is symbolic.

      +

      The semantics of the two attributes is defined as follows:

      +
        +
      • If a simplification rule is marked concrete, then all arguments must be
        +concrete for the rule to match.
        • +
      • If a simplification rule is marked symbolic, then all arguments must be
        +symbolic for the rule to match.
        • +
      • The following syntax concrete(<variables>) (resp. symbolic(<variables>)),
        +where <variables> is a list of variable names separated by commas, can be used
        +to specify the exact arguments the user expects to match concrete (resp. symbolic)
        +patterns.
        • +
      +

      For example, the following will only match when all arguments
      +are concrete:

      +
      rule X +Int (Y +Int Z) => (X +Int Y) +Int Z [simplification, concrete]
+

      Conversely, the following will only match when all arguments
      +are symbolic:

      +
      rule X +Int (Y +Int Z) => (X +Int Y) +Int Z [simplification, symbolic]
+

      In practice, the following rules will re-associate and commute terms to combine
      +concrete arguments:

      +
      rule (A +Int Y) +Int Z => A +Int (Y +Int Z)
+  [concrete(Y, Z), symbolic(A), simplification]
+
+rule X +Int (B +Int Z) => B +Int (X +Int Z)
+  [concrete(X, Z), symbolic(B), simplification]
+

      The unboundVariables attribute

      + +

      Normally, K rules are not allowed to contain regular (i.e., not fresh, not
      +existential) variables in the RHS / requires / ensures clauses which are not
      +bound in the LHS.

      +

      However, in certain cases this behavior might be desired, like, for example,
      +when specifying a macro rule which is to be used in the LHS of other rules.
      +To allow for such cases, but still be useful and perform the unboundness checks
      +in regular cases, the unboundVariables attributes allows the user to specify
      +a comma-separated list of names of variables which can be unbound in the rule.

      +

      For example, in the macro declaration

      +
        rule cppEnumType => bar(_, scopedEnum() #Or unscopedEnum() ) [unboundVariables(_)]
+

      the declaration unboundVariables(_) allows the rule to pass the unbound
      +variable checks, and this in turn allows for cppEnumType to be used in
      +the LHS of a rule to mean the pattern above:

      +
        rule inverseConvertType(cppEnumType, foo((cppEnumType #as T::CPPType => underlyingType(T))))
+

      The memo attribute

      + +

      The memo attribute is a hint from the user to the backend to memoize a
      +function. Not all backends support memoization, but when the attribute is used
      +and the definition is compiled for a memo-supporting backend, then calls to
      +the function may be cached. At the time of writing, only the Haskell
      +backend supports memoization.

      +

      Limitations of memoization with the Haskell backend

      + +

      The Haskell backend will only cache a function call if all arguments are concrete.

      +

      It is recommended not to memoize recursive functions, as each recursive call
      +will be stored in the cache, but only the first iteration will be retrieved from
      +the cache; that is, the cache will be filled with many unreachable
      +entries. Instead, we recommend to perform a worker-wrapper transformation on
      +recursive functions, and apply the memo attribute to the wrapper.

      +

      Warning: A function declared with the memo attribute must not use
      +uninterpreted functions in the side-condition of any rule. Memoizing such an
      +impure function is unsound. To see why, consider the following rules:

      +
      syntax Bool ::= impure( Int ) [function]
+
+syntax Int ::= unsound( Int ) [function, memo]
+rule unsound(X:Int) => X +Int 1 requires impure(X)
+rule unsound(X:Int) => X        requires notBool impure(X)
+

      Because the function impure is not given rules to cover all inputs, unsound
      +can be memoized incoherently. For example,

      +
      {unsound(0) #And {impure(0) #Equals true}} #Equals 1
+

      but

      +
      {unsound(0) #And {impure(0) #Equals false}} #Equals 0
+

      The memoized value of unsound(0) would be incoherently determined by which
      +pattern the backend encounters first.

      +

      Variable Sort Inference

      + +

      In K, it is not required that users declare the sorts of variables in rules or
      +in the initial configuration. If the user does not explicitly declare the sort
      +of a variable somewhere via a cast (see below), the sort of the variable is
      +inferred from context based on the sort signature of every place the variable
      +appears in the rule.

      +

      As an example, consider the rule for addition in IMP:

      +
          syntax Exp ::= Exp "+" Exp | Int
+
+    rule I1 + I2 => I1 +Int I2
+

      Here +Int is defined in the INT module with the following signature:

      +
          syntax Int ::= Int "+Int" Int [function]
+

      In the rule above, the sort of both I1 and I2 is inferred as Int. This is because
      +a variable must have the same sort every place it appears within the same rule.
      +While a variable appearing only on the left-hand-side of the rule could have
      +sort Exp instead, the same variable appears as a child of +Int, which
      +constriants the sorts of I1 and I2 more tightly. Since the sort must be a
      +subsort of Int or equal to Int, and Int has no subsorts, we infer Int
      +as the sorts of I1 and I2. This means that the above rule will not match
      +until I1 and I2 become integers (i.e., have already been evaluated).

      +

      More complex examples are possible, however:

      +
          syntax Exp ::= Exp "+" Int | Int
+    rule _ + _ => 0
+

      Here we have two anonymous variables. They do not refer to the same variable
      +as one another, so they can have different sorts. The right side is constrained
      +by + to be of sort Int, but the left side could be either Exp or Int.
      +When this occurs, we have multiple solutions to the sorts of the variables in
      +the rule. K will only choose solutions which are maximal, however. To be
      +precise, if two different solutions exist, but the sorts of one solution are
      +all greater than or equal to the sorts of the other solution, K will discard
      +the smaller solution. Thus, in the case above, the variable on the left side
      +of the + is inferred of sort Exp, because the solution (Exp, Int) is
      +strictly greater than the solution (Int, Int).

      +

      It is possible, however, for terms to have multiple maximal solutions:

      +
          syntax Exp ::= Exp "+" Int | Int "+" Exp | Int
+    rule I1 + I2 => 0
+

      In this example, there is an ambiguous parse. This could parse as either
      +the first + or the second. In the first case, the maximal solution chosen is
      +(Exp, Int). In the second, it is (Int, Exp). Neither of these solutions is
      +greater than the other, so both are allowed by K. As a result, this program
      +will emit an error because the parse is ambiguous. To pick one solution over
      +the other, a cast or a prefer or avoid attribute can be used.

      +

      Casting

      + +

      There are three main types of casts in K: the semantic cast, the strict cast,
      +and the projection cast.

      +

      Semantic casts

      + +

      For every sort S declared in your grammar, K will define the following
      +production for you for use in rules:

      +
          syntax S ::= S ":S"
+

      The meaning of this cast is that the term inside the cast must be less than
      +or equal to Sort. This can be used to resolve ambiguities, but its principle
      +purpose is to guide execution by telling K what sort variables must match in
      +order for the rule to apply. When compiled, it will generate a pattern that
      +matches on an injection into Sort.

      +

      Strict casts

      + +

      K also introduces the strict cast:

      +
          syntax S ::= S "::S"
+

      The meaning at runtime is exactly the same as the semantic cast; however, it
      +restricts the sort of the term inside the cast to exactly Sort. That is
      +to say, if you use it on something that is a strictly smaller sort, it will
      +generate a type error. This is useful in certain circumstances to help
      +disambiguate terms, when a semantic cast would not have resolved the ambiguity.
      +As such, it is primarily used to solve ambiguities rather than to guide
      +execution.

      +

      Projection casts

      + +

      K also introduces the projection cast:

      +
          syntax {S2} S ::= "{" S2 "}" ":>S"
+

      The meaning of this cast at runtime is that if the term inside is of sort
      +Sort, it should have it injection stripped away and the value inside is
      +returned as a term of static sort Sort. However, if the term is of a
      +different sort, it is an error and execution will get stuck. Thus the primary
      +usefulness of this cast is to cast the return value of a function with a
      +greater sort down to a strictly smaller sort that you expect the return value
      +of the function to have. For example:

      +
          syntax Exp ::= foo(Exp) [function] | bar(Int) | Int
+    rule foo(I:Int) => I
+    rule bar(I) => bar({foo(I +Int 1)}:>Int)
+

      Here we know that foo(I +Int 1) will return an Int, but the return sort of
      +foo is Exp. So we project the result into the Int sort so that it can
      +be placed as the child of a bar.

      +

      owise and priority attributes.

      + +

      Sometimes, it is simply not convenient to explicitly describe every
      +single negative case under which a rule should not apply. Instead,
      +we simply wish to say that a rule should only apply after some other set of
      +rules have been tried. K introduces two different attributes that can be
      +added to rules which will automatically generate the necessary matching
      +conditions in a manner which is performant for concrete execution (indeed,
      +it generally outperforms during concrete execution code where the conditions
      +are written explicitly).

      +

      The first is the owise attribute. Very roughly, rules without an attribute
      +indicating their priority apply first, followed by rules with the owise
      +attribute only if all the other rules have been tried and failed. For example,
      +consider the following function:

      +
      syntax Int ::= foo(Int) [function]
+rule foo(0) => 0
+rule foo(_) => 1 [owise]
+

      Here foo(0) is defined explicitly as 0. Any other integer yields the
      +integer 1. In particular, the second rule above will only be tried after the
      +first rule has been shown not to apply.

      +

      This is because the first rule has a lower number assigned for its priority
      +than the second rule. In practice, each rule in your semantics is implicitly
      +or explicitly assigned a numerical priority. Rules are tried in increasing
      +order of priority, starting at zero and trying each increasing numerical value
      +successively.

      +

      You can specify the priority of a rule with the priority attribute. For
      +example, I could equivalently write the second rule above as:

      +
      rule foo(_) => 1 [priority(200)]
+

      The number 200 is not chosen at random. In fact, when you use the owise
      +attribute, what you are doing is implicitly setting the priority of the rule
      +to 200. This has a couple of implications:

      +
        +
      1. Multiple rules with the owise attribute all have the same priority and thus
        +can apply in any order.
        2. +
      2. Rules with priority higher than 200 apply after all rules with the
        +owise attribute have been tried.
        3. +
      +

      There is one more rule by which priorities are assigned: a rule with no
      +attributes indicating its priority is assigned the priority 50. Thus,
      +with each priority explicitly declared, the above example looks like:

      +
      syntax Int ::= foo(Int) [function]
+rule foo(0) => 0 [priority(50)]
+rule foo(_) => 1 [owise]
+

      One final note: the llvm backend reserves priorities between 50 and 150
      +inclusive for certain specific purposes. Because of this, explicit
      +priorities which are given within this region may not behave precisely as
      +described above. This is primarily in order that it be possible where necessary
      +to provide guidance to the pattern matching algorithm when it would otherwise
      +make bad choices about which rules to try first. You generally should not
      +give any rule a priority within this region unless you know exactly what the
      +implications are with respect to how the llvm backend orders matches.

      +

      Evaluation Strategy

      +

      strict and seqstrict attributes

      + +

      The strictness attributes allow defining evaluation strategies without having
      +to explicitly make rules which implement them. This is done by injecting
      +heating and cooling rules for the subterms. For this to work, you need to
      +define what a result is for K, by extending the KResult sort.

      +

      For example:

      +
      syntax AExp ::= Int
+              | AExp "+" AExp [strict, klabel(addExp)]
+

      This generates two heating rules (where the hole syntaxes "[]" "+" AExp and
      +AExp "+" "[]" is automatically added to create an evaluation context):

      +
      rule [addExp1-heat]: <k> HOLE:AExp +  AE2:AExp => HOLE ~>  [] + AE2 ... </k> [heat]
+rule [addExp2-heat]: <k>  AE1:AExp + HOLE:AExp => HOLE ~> AE1 +  [] ... </k> [heat]
+

      And two corresponding cooling rules:

      +
      rule [addExp1-cool]: <k> HOLE:AExp ~>  [] + AE2 => HOLE +  AE2 ... </k> [cool]
+rule [addExp2-cool]: <k> HOLE:AExp ~> AE1 +  [] =>  AE1 + HOLE ... </k> [cool]
+

      Note that the rules are given labels based on the klabel of the production, which
      +nonterminal is the hole, and whether it's the heating or the cooling rule.

      +

      You will note that these rules can apply one after another infinitely. In
      +practice, the KResult sort is used to break this cycle by ensuring that only
      +terms that are not part of the KResult sort will be heated. The heat and
      +cool attributes are used to tell the compiler that these are heating and
      +cooling rules and should be handled in the manner just described. Nothing stops
      +the user from writing such heating and cooling rules directly if they wish,
      +although we describe other more convenient syntax for most of the advanced
      +cases below.

      +

      One other thing to note is that in the above sentences, HOLE is just a
      +variable, but it has special meaning in the context of sentences with the
      +heat or cool attribute. In heating or cooling rules, the variable named
      +HOLE is considered to be the term being heated or cooled and the compiler
      +will generate isKResult(HOLE) and notBool isKResult(HOLE) side conditions
      +appropriately to ensure that the backend does not loop infinitely. The module
      +BOOL will also be automatically and privately included for semantic
      +purposes. The syntax for parsing programs will not be affected.

      +

      In order for this functionality to work, you need to define the KResult sort.
      +For instance, we tell K that a term is fully evaluated once it becomes an Int
      +here:

      +
      syntax KResult ::= Int
+

      Note that you can also say that a given expression is only strict only in
      +specific argument positions. Here we use this to define "short-circuiting"
      +boolean operators.

      +
      syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [strict(1)]
+              | BExp "&&" BExp [strict(1)]
+
+rule <k> true  || _    => true ... </k>
+rule <k> false || REST => REST ... </k>
+
+rule <k> true  && REST => REST  ... </k>
+rule <k> false && _    => false ... </k>
+

      If you want to force a specific evaluation order of the arguments, you can use
      +the variant seqstrict to do so. For example, this would make the boolean
      +operators short-circuit in their second argument first:

      +
      syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [seqstrict(2,1)]
+              | BExp "&&" BExp [seqstrict(2,1)]
+
+rule <k> _    || true  => true ... </k>
+rule <k> REST || false => REST ... </k>
+
+rule <k> REST && true  => REST  ... </k>
+rule <k> _    && false => false ... </k>
+

      This will generate rules like this in the case of _||_ (note that BE1 will
      +not be heated unless isKResult(BE2) is true, meaning that BE2 must be
      +evaluated first):

      +
      rule <k>  BE1:BExp || HOLE:BExp => HOLE ~> BE1 ||  [] ... </k> [heat]
+rule <k> HOLE:BExp ||  BE2:BExp => HOLE ~>  [] || BE2 ... </k> requires isKResult(BE2) [heat]
+
+rule <k> HOLE:BExp ~>  [] || BE2 => HOLE ||  BE2 ... </k> [cool]
+rule <k> HOLE:BExp ~> BE1 ||  [] =>  BE1 || HOLE ... </k> [cool]
+

      Context Declaration

      + +

      Sometimes more advanced evaluation strategies are needed. By default, the
      +strict and seqstrict attributes are limited in that they cannot describe
      +the context in which heating or cooling should occur. When this type of
      +control over the evaluation strategy is required, context sentences can be
      +used to simplify the process of declaring heating and cooling when it would be
      +unnecessarily verbose to write heating and cooling rules directly.

      +

      For example, if the user wants to heat a term if it exists under a foo
      +constructor if the term to be heated is of sort bar, one might write the
      +following context (with the optional label):

      +
      context [foo]: foo(HOLE:Bar)
+

      Once again, note that HOLE is just a variable, but one that has special
      +meaning to the compiler indicating the position in the context that should
      +be heated or cooled.

      +

      This will automatically generate the following sentences:

      +
      rule [foo-heat]: <k> foo(HOLE:Bar) => HOLE ~> foo([]) ... </k> [heat]
+rule [foo-cool]: <k> HOLE:Bar ~> foo([]) => foo(HOLE) ... </k> [cool]
+

      The user may also write the K cell explicitly in the context declaration
      +if they want to match on another cell as well, for example:

      +
      context <k> foo(HOLE:Bar) ... </k> <state> .Map </state>
+

      This context will now only heat or cool if the state cell is empty.

      +

      Side conditions in context declarations

      + +

      The user is allowed to write a side condition in a context declaration, like
      +so:

      +
      context foo(HOLE:Bar) requires baz(HOLE)
+

      This side condition will be appended verbatim to the heating rule that is
      +generated, however, it will not affect the cooling rule that is generated:

      +
      rule <k> foo(HOLE:Bar) => HOLE ~> foo([]) ... </k> requires baz(HOLE) [heat]
+rule <k> HOLE:Bar ~> foo([]) => foo(HOLE) ... </k> [cool]
+

      Rewrites in context declarations

      + +

      The user can also include exactly one rewrite operation in a context
      +declaration if that rule rewrites the variable HOLE on the left hand side
      +to a term containing HOLE on the right hand side. For exampl;e:

      +
      context foo(HOLE:Bar => bar(HOLE))
+

      In this case, the code generated will be as follows:

      +
      rule <k> foo(HOLE:Bar) => bar(HOLE) ~> foo([]) ... </k> [heat]
+rule <k> bar(HOLE:Bar) ~> foo([]) => foo(HOLE) ... </k> [cool]
+

      This can be useful if the user wishes to evaluate a term using a different
      +set of rules than normal.

      +

      result attribute

      + +

      Sometimes it is necessary to be able to evaluate a term to a different sort
      +than KResult. This is done by means of adding the result attribute to
      +a strict production, a context, or an explicit heating or cooling rule:

      +
      syntax BExp ::= Bool
+              | BExp "||" BExp [seqstrict(2,1), result(Bool)]
+

      In this case, the sort check used by seqstrict and by the heat and cool
      +attributes will be isBool instead of isKResult. This particular example
      +does not really require use of the result attribute, but if the user wishes
      +to evaluate a term of sort KResult further, the result attribute would be
      +required.

      +

      hybrid attribute

      + +

      In certain situations, it is desirable to treat a particular production which
      +has the strict attribute as a result if the term has had its arguments fully
      +evaluated. This can be accomplished by means of the hybrid attribute:

      +
      syntax KResult ::= Bool
+
+syntax BExp ::= Bool
+              | BExp "||" BExp [strict(1), hybrid]
+

      This attribute is equivalent in this case to the following additional axiom
      +being added to the definition of isKResult:

      +
      rule isKResult(BE1:BExp || BE2:BExp) => true requires isKResult(BE1)
+

      Sometimes you wish to declare a production hybrid with respect to a predicate
      +other than isKResult. You can do this by specifying a sort as the body of the
      +hybrid attribute, e.g.:

      +
      syntax BExp ::= BExp "||" BExp [strict(1), hybrid(Foo)]
+

      generates the rule:

      +
      rule isFoo(BE1:BExp || BE2:BExp) => true requires isFoo(BE1)
+

      Properly speaking, hybrid takes an optional comma-separated list of sort
      +names. If the list is empty, the attribute is equivalent to hybrid(KResult).
      +Otherwise, it generates hybrid predicates for exactly the sorts named.

      +

      Context aliases

      + +

      Sometimes it is necessary to define a fairly complicated evaluation strategy
      +for a lot of different operators. In this case, the user could simply write
      +a number of complex context declarations, however, this quickly becomes
      +tedious. For this purpose, K has a concept called a context alias. A context
      +alias is a bit like a template for describing contexts. The template can then
      +be instantiated against particular productions using the strict and
      +seqstrict attributes.

      +

      Here is a (simplified) example taken from the K semantics of C++:

      +
      context alias [c]: <k> HERE:K ... </k> <evaluate> false </evaluate>
+context alias [c]: <k> HERE:K ... </k> <evaluate> true </evaluate> [result(ExecResult)]
+
+syntax Expr ::= Expr "=" Init [strict(c; 1)]
+

      This defines the evaluation strategy during the translation phase of a C++
      +program for the assignment operator. It is equivalent to writing the following
      +context declarations:

      +
      context <k> HOLE:Expr = I:Init ... </k> <evaluate> false </evaluate>
+context <k> HOLE:Expr = I:Init ... </k> <evaluate> true </evaluate> [result(ExecResult)]
+

      What this is saying is, if the evaluate cell is false, evaluate the term
      +like normal to a KResult. But if the evaluate cell is true, instead
      +evaluate it to the ExecResult sort.

      +

      Essentially, we have given a name to this evaluation strategy in the form of
      +the rule label on the context alias sentences (in this case, c). We can
      +then say that we want to use this evaluation strategy to evaluate particular
      +arguments of particular productions by referring to it by name in a strict
      +attribute. For example, strict(c) will instantiate these contexts once for
      +each argument of the production, whereas strict(c; 1) will instantiate it
      +only for the first argument. The special variable HERE is used to tell the
      +compiler where you want to place the production that is to be heated or cooled.

      +

      You can also specify multiple context aliases for different parts of a production,
      +for example:

      +
      syntax Exp ::= foo(Exp, Exp) [strict(left; 1; right; 2)]
+

      This says that we can evaluate the left and right arguments in either order, but to evaluate
      +the left using the left context alias and the right using the right context alias.

      +

      We can also say seqstrict(left; 1; right; 2), in which case we additionally must evaluate
      +the left argument before the right argument. Note, all strict positions are considered collectively
      +when determining the evaluation order of seqstrict or the hybrid predicates.

      +

      A strict attribute with no rule label associated with it is equivalent to
      +a strict attribute given with the following context alias:

      +
      context alias [default]: <k> HERE:K ... </k>
+

      One syntactic convenience that is provided is that if you wish to declare the following context:

      +
      context foo(HOLE => bar(HOLE))
+

      you can simply write the following:

      +
      syntax Foo ::= foo(Bar) [strict(alias)]
+
+context alias [alias]: HERE [context(bar)]
+

      Pattern Matching

      +

      As Patterns

      + +

      New syntax has been added to K for matching a pattern and binding the resulting
      +match in its entirety to a variable.

      +

      The syntax is:

      +
      Pattern #as V::Var
+

      In this case, Pattern, including any variables, is matched and the resulting
      +variables are added to the substitution if matching succeeds. Furthermore, the
      +term matched by Pattern is added to the substitution as V.

      +

      This code can also be used outside of any rewrite, in which case matching
      +occurs as if it appeared on the left hand side, and the right hand side becomes
      +a variable corresponding to the alias.

      +

      It is an error to use an as pattern on the right hand side of a rule.

      +

      Record-like KApply Patterns

      + +

      We have added a syntax for matching on KApply terms which mimics the record
      +syntax in functional languages. This allows us to more easily express patterns
      +involving a KApply term in which we don't care about some or most of the
      +children, without introducing a dependency into the code on the number of
      +arguments which could be changed by a future refactoring.

      +

      The syntax is:

      +
      record(... field1: Pattern1, field2: Pattern2)
+

      Note that this only applies to productions that are prefix productions.
      +A prefix production is considered by the implementation to be any production
      +whose production items match the following regular expression:

      +
      (Terminal(_)*) Terminal("(")
+(NonTerminal (Terminal(",") NonTerminal)* )?
+Terminal(")")
+

      In other words, any sequence of terminals followed by an open parenthesis, an
      +optional comma separated list of non-terminals, and a close parenthesis.

      +

      If a prefix production has no named nonterminals, a record(...) syntax is
      +allowed, but in order to reference specific fields, it is necessary to give one
      +or more of the non-terminals in the production names.

      +

      Note: because the implementation currently creates one production per possible
      +set of fields to match on, and because all possible permutations of all
      +possible subsets of a list of n elements is a number that scales factorially
      +and reaches over 100 thousand productions at n=8, we currently do not allow
      +fields to be matched in any order like a true record, but only in the same
      +order as appears in the production itself.

      +

      Given that this only reduces the number of productions to the size of the power
      +set, this will still explode the parsing time if we create large productions of
      +10 or more fields that all have names. This is something that should probably
      +be improved, however, productions with that large of an arity are rare, and
      +thus it has not been viewed as a priority.

      +

      Or Patterns

      + +

      Sometimes you wish to express that a rule should match if one out of multiple
      +patterns should match the same subterm. We can now express this in K by means
      +of using the #Or ML connective on the left hand side of a rule.

      +

      For example:

      +
      rule foo #Or bar #Or baz => qux
+

      Here any of foo, bar, or baz will match this rule. Note that the behavior is
      +ill-defined if it is not the case that all the clauses of the or have the same
      +bound variables.

      +

      Matching global context in function rules

      + +

      On occasion it is highly desirable to be able to look up information from the
      +global configuration and match against it when evaluating a function. For this
      +purpose, we introduce a new syntax for function rules.

      +

      This syntax allows the user to match on function context from within a
      +function rule:

      +
      syntax Int ::= foo(Int) [function]
+
+rule [[ foo(0) => I ]]
+     <bar> I </bar>
+
+rule something => foo(0)
+

      This is completely desugared by the K frontend and does not require any special
      +support in the backend. It is an error to have a rewrite inside function
      +context, as we do not currently support propagating such changes back into the
      +global configuration. It is also an error if the context is not at the top
      +level of a rule body.

      +

      Desugared code:

      +
      syntax Int ::= foo(Int, GeneratedTopCell) [function]
+
+rule foo(0, <generatedTop>
+              <bar> I </bar>
+              ...
+            </generatedTop> #as Configuration) => I
+rule <generatedTop>
+       <k> something ... </k>
+       ...
+     </generatedTop> #as Configuration
+  => <generatedTop>
+       <k> foo(0, Configuration> ... </k>
+       ...
+     </generatedTop>
+

      Collection patterns

      + +

      It is allowed to write patterns on the left hand side of rules which refer to
      +complex terms of sort Map, List, and Set, despite these patterns ostensibly
      +breaking the rule that terms which are functions should not appear on the left
      +hand side of rules. Such terms are destructured into pattern matching
      +operations.

      +

      The following forms are allowed:

      +
      // 0 or more elements followed by 0 or 1 variables of sort List followed by
+// 0 or more elements
+ListItem(E1) ListItem(E2) L:List ListItem(E3) ListItem(E4)
+
+// the empty list
+.List
+
+// 0 or more elements in any order plus 0 or 1 variables of sort Set
+// in any order
+SetItem(K1) SetItem(K2) S::Set SetItem(K3) SetItem(K4)
+
+// the empty set
+.Set
+
+// 0 or more elements in any order plus by 0 or 1 variables of sort Map
+// in any order
+K1 |-> E1 K2 |-> E2 M::Map K3 |-> E3 K4 |-> E4
+
+// the empty map
+.Map
+

      Here K1, K2, K3, K4 etc can be any pattern except a pattern containing both
      +function symbols and unbound variables. An unbound variable is a variable whose
      +binding cannot be determined by means of decomposing non-set-or-map patterns or
      +map elements whose keys contain no unbound variables.

      +

      This is determined recursively, ie, the term K1 |-> E2 E2 |-> E3 E3 |-> E4 is
      +considered to contain no unbound variables.

      +

      Note that in the pattern K1 |-> E2 K3 |-> E4 E4 |-> E5, K1 and K3 are
      +unbound, but E4 is bound because it is bound by deconstructing the key E3, even
      +though E3 is itself unbound.

      +

      In the above examples, E1, E2, E3, and E4 can be any pattern that is normally
      +allowed on the lhs of a rule.

      +

      When a map or set key contains function symbols, we know that the variables in
      +that key are bound (because of the above restriction), so it is possible to
      +evaluate the function to a concrete term prior to performing the lookup.

      +

      Indeed, this is the precise semantics which occurs; the function is evaluated
      +and the result is looked up in the collection.

      +

      For example:

      +
      syntax Int ::= f(Int) [function]
+rule f(I:Int) => I +Int 1
+rule <k> I:Int => . ... </k> <state> ... SetItem(f(I)) ... </state>
+

      This will rewrite I to . if and only if the state cell contains
      +I +Int 1.

      +

      Note that in the case of Set and Map, one guarantee is that K1, K2, K3, and K4
      +represent /distinct/ elements. Pattern matching fails if the correct number of
      +distinct elements cannot be found.

      +

      Matching on cell fragments

      + +

      K allows matching fragments of the configuration and using them to construct
      +terms and use as function parameters.

      +
      configuration <t>
+                <k> #init ~> #collectOdd ~> $PGM </k>
+                <fs>
+                  <f multiplicity="*" type="Set"> 1 </f>
+                </fs>
+              </t>
+

      The #collectOdd construct grabs the entire content of the <fs> cell.
      +We may also match on only a portion of its content. Note that the fragment
      +must be wrapped in a <f> cell at the call site.

      +
      syntax KItem ::= "#collectOdd"
+rule <k> #collectOdd => collectOdd(<fs> Fs </fs>) ... </k>
+     <fs> Fs </fs>
+

      The collectOdd function collects the items it needs

      +
      syntax Set ::= collectOdd(FsCell) [function]
+rule collectOdd(<fs> <f> I </f> REST </fs>) => SetItem(I) collectOdd(<fs> REST </fs>) requires I %Int 2 ==Int 1
+rule collectOdd(<fs> <f> I </f> REST </fs>) =>            collectOdd(<fs> REST </fs>) requires I %Int 2 ==Int 0
+rule collectOdd(<fs> .Bag </fs>) => .Set
+

      all-path and one-path attributes to distinguish reachability claims

      + +

      As the Haskell backend can handle both one-path and all-path reachability
      +claims, but both these are encoded as rewrite rules in K, these attributes can
      +be used to clarify what kind of claim a rule is.

      +

      In addition of being able to annotate a rule with one of them
      +(if annotating with more at the same time, only one of them would be chosen),
      +one can also annotate whole modules, to give a default claim type for all rules
      +in that module.

      +

      Additionally, the Haskell backend introduces an extra command line option
      +for the K frontend, --default-claim-type, with possible values
      +all-path and one-path to allow choosing a default type for all
      +claims.

      +

      Set Variables

      + +

      Motivation

      + +

      Set variables were introduced as part of Matching Mu Logic, the mathematical
      +foundations for K. In Matching Mu Logic, terms evaluate to sets of values.
      +This is useful for both capturing partiality (as in 3/0) and capturing
      +non-determinism (as in 3 #Or 5). Consequently, symbol interpretation is
      +extended to have a collective interpretation over sets of input values.

      +

      Usually, K rules are given using regular variables, which expect that the term
      +they match is both defined and has a unique interpretation.

      +

      However, it is sometimes useful to have simplification rules which work over
      +any kind of pattern, be it undefined or non-deterministic. This behavior can be
      +achieved by using set variables to stand for any kind of pattern.

      +

      Syntax

      + +

      Any variable prefixed by @ will be considered a set variable.

      +

      Example

      + +

      Below is a simplification rule which motivated this extension:

      +
        rule #Ceil(@I1:Int /Int @I2:Int) =>
+    {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2)
+    [anywhere]
+

      This rule basically says that @I1:Int /Int @I2:Int is defined if @I1 and
      +@I2 are defined and @I2 is not 0. Using sets variables here is important as
      +it allows the simplification rule to apply any symbolic patterns, without
      +caring whether they are defined or not.

      +

      This allows simplifying the expression #Ceil((A:Int /Int B:Int) / C:Int) to:

      +
      {(C =/=Int 0) #Equals true} #And #Ceil(C) #And ({(B =/=Int 0) #Equals true}
+#And #Ceil(B) #And #Ceil(A)`
+

      See kframework/kore#729 for
      +more details.

      +

      SMT Translation

      + +

      K makes queries to an SMT solver (Z3) to discharge proof obligations when doing
      +symbolic execution. You can control how these queries are made using the
      +attributes smtlib, smt-hook, and smt-lemma on declared productions.
      +These attributes guide the prover when it tries to apply rules to discharge a
      +proof obligation.

      +
        +
      • smt-hook(...) allows you to specify a term in SMTLIB2 format which should
        +be used to encode that production, and assumes that all symbols appearing in
        +the term are already declared by the SMT solver.
        • +
      • smtlib(...) allows you to declare a new SMT symbol to be used when that
        +production is sent to Z3, and gives it uninterpreted function semantics.
        • +
      • smt-lemma can be applied to a rule to encode it as a conditional equality
        +when sending queries to Z3. A rule rule LHS => RHS requires REQ will be
        +encoded as the conditional equality (=> REQ (= (LHS RHS)). Every symbol
        +present in the rule must have an smt-hook(...) or smtlib(...) attribute.
        • +
      +
      syntax Int ::= "~Int" Int          [function, klabel(~Int_), symbol,
+                                    smtlib(notInt)]
+             | Int "^%Int" Int Int [function, klabel(_^%Int__), symbol,
+                                    smt-hook((mod (^ #1 #2) #3))]
+

      In the example above, we declare two productions ~Int_ and _^%Int__, and
      +tell the SMT solver to:

      +
        +
      • use uninterpreted function semantics for ~Int_ via SMTLIB2 symbol
        +notInt, and
        • +
      • use the SMTLIB2 term (mod (^ #1 #2) #3) (where #N marks the Nth
        +production non-terminal argument positions) for _^%Int__, where mod and
        +^ already are declared by the SMT solver.
        • +
      +

      Caution

      + +

      Set variables are currently only supported by the Haskell backend.
      +The use of rules with set variables should be sound for all other backends
      +which just execute by rewriting, however it might not be safe for backends
      +which want to guarantee coverage.

      +

      Variables occurring only in the RHS of a rule

      + +

      This section presents possible scenarios requiring variables to only appear in
      +the RHS of a rule.

      +

      Summary

      + +

      Except for ? variables and ! (fresh) variables, which are
      +required to only appear in the RHS of a rule, all other variables must
      +also appear in the LHS of a rule. This restriction also applies to anonymous
      +variables; in particular, for claims, ?_ (not _) should be used in the RHS
      +to indicate that something changes but we don't care to what value.

      +

      To support specifying random-like behavior, the above restriction can be relaxed
      +by annotating a rule with the unboundVariables attribute whenever the rule
      +intentionally contains regular variables only occurring in the RHS.

      +

      Introduction

      + +

      K uses question mark variables of the form ?X to refer to
      +existential variables, and uses ensures to specify logical constraints on
      +those variables.
      +These variables are only allowed to appear in the RHS of a K rule.

      +

      If the rules represent rewrite (semantic) steps or verification claims,
      +then the ? variables are existentially quantified at the top of the RHS;
      +otherwise, if they represent equations, the ? variables are quantified at the
      +top of the entire rule.

      +

      Note that when both ?-variables and regular variables are present,
      +regular variables are (implicitly) universally quantified on top of the rule
      +(already containing the existential quantifications).
      +This essentially makes all ? variables depend on all regular variables.

      +

      All examples below are intended more for program verification /
      +symbolic execution, and thus concrete implementations might choose to ignore
      +them altogether or to provide ad-hoc implementations for them.

      +

      Example: Verification claims

      + +

      Consider the following definition of a (transition) system:

      +
      module A
+  rule foo => true
+  rule bar => true
+  rule bar => false
+endmodule
+

      Consider also, the following specification of claims about the definition above:

      +
      module A-SPEC
+  rule [s1]: foo => ?X:Bool
+  rule [s2]: foo =>  X:Bool  [unboundVariables(X)]
+  rule [s3]: bar => ?X:Bool
+  rule [s4]: bar =>  X:Bool  [unboundVariables(X)]
+endmodule
+
      One-path interpretation
      + +
        +
      • (s1) says that there exists a path from foo to some boolean, which is
        +satisfied easily using the foo => true rule
        • +
      • (s3) says the same thing about bar and can be satisfied by either of
        +bar => true and bar => false rules
        • +
      • (s2) and (s4) can be better understood by replacing them with instances for
        +each element of type Bool, which can be interpreted that
        +both true and false are reachable from foo for (s2), or bar for (s4),
        +respectively. +
          +
        • (s2) cannot be verified as we cannot find a path from foo to false.
          • +
        • (s4) can be verified by using bar => true to show true is reachable and
          +bar => false to achieve the same thing for false
          • +
        +
        • +
      +
      All-path interpretation
      + +
        +
      • +

        (s1) says that all paths from foo will reach some boolean, which is
        +satisfied by the foo => true rule and the lack of other rules for foo

        +
        • +
      • +

        (s3) says the same thing about bar and can be satisfied by checking that
        +both bar => true and bar => false end in a boolean, and there are no
        +other rules for bar

        +
        • +
      • +

        (s2) and (s4) can be better understood by replacing them with instances for
        +each element of type Bool, which can be interpreted that
        +both true and false are reachable in all paths originating in
        +foo for (s2), or bar for (s4), respectively.
        +This is a very strong claim, requiring that all paths originating in
        +foo (bar) pass through both true and false,
        +so neither (s2) nor (s4) can be verified.

        +

        Interestingly enough, adding a rule like false => true would make both
        +(s2) and (s4) hold.

        +
        • +
      +

      Example: Random Number Construct rand()

      + +

      The random number construct rand() is a language construct which could be
      +easily conceived to be part of the syntax of a programming language:

      +
      Exp ::= "rand" "(" ")"
+

      The intended semantics of rand() is that it can rewrite to any integer in
      +a single step. This could be expressed as the following following infinitely
      +many rules.

      +
      rule  rand() => 0
+rule  rand() => 1
+rule  rand() => 2
+  ...    ...
+rule rand() => (-1)
+rule rand() => (-2)
+  ...    ...
+

      Since we need an instance of the rule for every integer, one could summarize
      +the above infinitely many rules with the rule

      +
      rule rand() => I:Int [unboundVariables(I)]
+

      Note that I occurs only in the RHS in the rule above, and thus the rule
      +needs the unboundVariables(I) attribute to signal that this is intentionally.

      +

      One can define variants of rand() by further constraining the output variable
      +as a precondition to the rule.

      +
      Rand-like examples
      + +
        +
      1. +

        randBounded(M,N) can rewrite to any integer between M and N

        +
        syntax Exp ::= randBounded(Int, Int)
+rule randBounded(M, N) => I
+  requires M <=Int I andBool I <=Int N
+  [unboundVariables(I)]
+
        2. +
      2. +

        randInList(Is) takes a list Is of items
        +and can rewrite in one step to any item in Is.

        +
        syntax Exp ::= randInList (List)
+rule randInList(Is) => I
+  requires I inList Is
+  [unboundVariables(I)]
+
        3. +
      3. +

        randNotInList(Is) takes a list Is of items
        +and can rewrite in one step to any item not in Is.

        +
        syntax Exp ::= randNotInList (List)
+rule randNotInList(Is) => I
+  requires notBool(I inList Is)
+  [unboundVariables(I)]
+
        4. +
      4. +

        randPrime(), can rewrite to any prime number.

        +
        syntax Exp ::= randPrime ()
+rule randPrime() => X:Int
+  requires isPrime(X)
+  [unboundVariables(X)]
+

        where isPrime(_) is a predicate that can be defined in the usual way.

        +
        5. +
      +

      Note 1: all above are not function symbols, but language constructs.

      +

      Note 2: Currently the frontend does not allow rules with universally quantified
      +variables in the RHS which are not bound in the LHS.

      +

      Note 3. Allowing these rules in a concrete execution engine would require an
      +algorithm for generating concrete instances for such variables, satisfying the
      +given constraints; thus the unboundVariables attribute serves two purposes:

      +
        +
      • to allow such rules to pass the variable checks, and
        • +
      • to signal (concrete execution) backends that specialized algorithm would be
        +needed to instantiate these variables.
        • +
      +

      Example: Fresh Integer Construct fresh(Is)

      + +

      The fresh integer construct fresh(Is) is a language construct.

      +
      Exp ::= ... | "fresh" "(" List{Int} ")"
+

      The intended semantics of fresh(Is) is that it can always rewrite to an
      +integer that in not in Is.

      +

      Note that fresh(Is) and randNotInList(Is) are different; the former
      +does not need to be able to rewrite to every integers not in Is,
      +while the latter requires so.

      +

      For example, it is correct to implement fresh(Is) so it always returns the
      +smallest positive integer that is not in Is, but same implementation for
      +randNotInList(Is) might be considered inadequate.
      +In other words, there exist multiple correct implementations of fresh(Is),
      +some of which may be deterministic, but there only exists a unique
      +implementation of randNotInList(Is).
      +Finally, note that randNotInList(Is) is a correct implementation
      +for fresh(Is); Hence, concrete execution engines can choose to handle
      +such rules accordingly.

      +

      We use the following K syntax to define fresh(Is)

      +
      syntax Exp ::= fresh (List{Int})
+rule fresh(Is:List{Int}) => ?I:Int
+  ensures notBool (?I inList{Int} Is)
+

      A variant of this would be a choiceInList(Is) language construct which would
      +choose some number from a list:

      +
      syntax Exp ::= choiceInList (List{Int})
+rule choiceInList(Is:List{Int}) => ?I:Int
+  ensures ?I inList{Int} Is
+

      Note: This definition is different from one using a ! variable to indicate
      +freshness because using ! is just syntactic sugar for generating globally
      +unique instances and relies on a special configuration cell, and cannot be
      +constrained, while the fresh described here is local and can be constrained.
      +While the first is more appropriate for concrete execution, this might be
      +better for symbolic execution / program verification.

      +

      Example: Arbitrary Number (Unspecific Function) arb()

      + +

      The function arb() is not a PL construct, but a mathematical function.
      +Therefore, its definition should not be interpreted as an execution step, but
      +rather as an equality.

      +

      The intended semantics of arb() is that it is an unspecified nullary function.
      +The exact return value of arb() is unspecified in the semantics but up to the
      +implementations.
      +However, being a mathematical function, arb() must return the same value in
      +any one implementation.

      +

      We do not need special frontend syntax to define arb().
      +We only need to define it in the usual way as a function
      +(instead of a language construct), and provide no axioms for it.
      +The total attribute ensures that the function is total, i.e.,
      +that it evaluates to precisely one value for each input.

      +
      Variants
      + +

      There are many variants of arb(). For example, arbInList(Is) is
      +an unspecified function whose return value must be an element from Is.

      +

      Note that arbInList(Is) is different from choiceInList(Is), because
      +choiceInList(Is) transitions to an integer in Is (could be a different one
      +each time it is used), while arbInList(Is) is equal to a (fixed)
      +integer not in Is.

      +

      W.r.t. the arb variants, we can use ? variables and the function
      +annotation to signal that we're defining a function and the value of the
      +function is fixed, but non-determinate.

      +
      syntax Int ::= arbInList(List{Int}) [function]
+rule arbInList(Is:List{Int}) => ?I:Int
+  ensures ?I inList{Int} Is
+

      If elimination of existentials in equational rules is needed, one possible
      +approach would be through Skolemization,
      +i.e., replacing the ? variable with a new uninterpreted function depending
      +on the regular variables present in the function.

      +

      Example: Interval (Non-function Symbols) interval()

      + +

      The symbol interval(M,N) is not a PL construct, nor a function in the
      +first-order sense, but a proper matching-logic symbol, whose interpretation is
      +in the powerset of its domain.
      +Its axioms will not use rewrites but equalities.

      +

      The intended semantics of interval(M,N) is that it equals the set of
      +integers that are larger than or equal to M and smaller than or equal to N.

      +

      Since expressing the axiom for interval requires an an existential
      +quantification on the right-hand-side, thus making it a non-total symbol
      +defined through an equation, using ? variables might be confusing since their
      +usage would be different from that presented in the previous sections.

      +

      Hence, the proposal to support this would be to write this as a proper ML rule.
      +A possible syntax for this purpose would be:

      +
      eq  interval(M,N)
+    ==
+    #Exists X:Int .
+        (X:Int #And { X >=Int M #Equals true } #And { X <=Int N #Equals true })
+

      Additionally, the symbol declaration would require a special attribute to
      +signal the fact that it is not a constructor but a defined symbol.

      +

      Since this feature is not clearly needed by K users at the moment, it is only
      +presented here as an example; its implementation will be postponed for such time
      +when its usefulness becomes apparent.

      +

      Parser Generation

      +

      In addition to on-the-fly parser generation using kast, K is capable of
      +ahead-of-time parser generation of LR(1) or GLR parsers using Flex and Bison.
      +This can be done one of two different ways.

      +
        +
      1. You can explicitly request for a particular parser to be generated by
        +invoking kast --gen-parser <outputFile> or
        +kast --gen-glr-parser <outputFile> respectively. kast will then create a
        +parser based on the same command line flags that govern on-the-fly parsing,
        +like -s to specify the starting sort, and -m to specify the module to
        +parse under. By default, this generates a parser for the sort of the $PGM
        +configuration variable in the main syntax module of the definition.
        2. +
      2. You can request that a specific set of parsers be generated for all the
        +configuration variables of your definition by passing the
        +--gen-bison-parser or --gen-glr-bison-parser flags to kompile.
        +kompile will decide the sorts to use as start symbols based on the sorts
        +in the configuration declaration for the configuration variables. The $PGM
        +configuration variable will be generated based on the main syntax module
        +of the definition. The user must explicitly annotate the configuration
        +declaration with the other modules to use to parse the other configuration
        +variables as attributes. For example, if I have the following cell in the
        +configuration declaration: <cell> foo($FOO:Foo, $BAR:Bar) </cell>,
        +One might annotate it with the attribute pair parser="FOO, TEST; BAR, TEST2"
        +to indicate that configuration variable $FOO should be parsed in the
        +TEST module, and configuration variable $BAR should be parsed in the
        +TEST2 module. If the user forgets to annotate the declaration with the
        +parser attribute, only the $PGM parser will be generated.
        3. +
      +

      Bison-generated parsers are extremely fast compared to kast, but they have
      +some important limitations:

      +
        +
      • Bison parsers will always output Kore. You can then pass the resulting AST
        +directly to llvm-krun or kore-exec and bypass the krun frontend, making
        +them very fast, but lower-level.
        • +
      • Bison parsers do not yet support macros. This may change in a future release.
        +Note that you can use anywhere rules instead of macros in most cases to get
        +around this limitation, although they will not benefit from unparsing via the
        +alias attribute.
        • +
      • Obligation falls on the user to ensure that the grammar they write is LR(1)
        +if they choose to use LR(1) parsing. If this does not happen, the parser
        +generated will have shift/reduce or reduce/reduce conflicts and the parser
        +may behave differently than kast would (kast is a GLL parser, ie, it
        +is based on LL parsers and parses all unambiguous context-free grammars).
        +K provides an attribute, not-lr1, which can be applied to modules known to
        +not be LR(1), and will trigger a warning if the user attempts to generate an
        +LR(1) parser which recursively imports that module.
        • +
      • If you are using LR(1) based parsing, the prefer and avoid attributes are
        +ignored. It is only possible to implement these attributes by means of
        +generalized LL or LR parsing and a postprocessing on the AST to remove the
        +undesirable ambiguity.
        • +
      • Obligation falls on the user to ensure that the grammar they write has as
        +few conflicts as possible if they are using GLR parsing. Bison's GLR support
        +is quite primitive, and in the worst case it can use exponential space and
        +time to parse a program, which generally leads the generated parser to report
        +"memory exhausted", indicating that the parse could not be completed within
        +the stack space allocated by Bison. It's best to ensure that the grammar is
        +as close to LR(1) as possible and only utilizes conflicts where absolutely
        +necessary. One tool that can be used to facilitate this is to pass
        +--bison-lists to kompile. This will disable support for the List{Sort}
        +syntax production, and it will make NeList{Sort} left associative, but the
        +resulting productions generated for NeList{Sort} will be LR(1) and use bounded
        +stack space.
        • +
      • If the grammar you are parsing is context-sensitive (for example, because
        +it requires a symbol table to parse), one thing you can do to make this
        +language parse in K is to implement the language as an ambiguous grammar.
        +Bison's GLR parser will generate an amb production that is parametric in
        +the sort of the ambiguity. You can then import the K-AMBIGUITIES module
        +and use rewriting to resolve the ambiguities using whatever preprocessing
        +mechanisms you prefer.
        • +
      +

      Location Information

      +

      K is able to insert file, line, and column metadata into the parse tree on a
      +per-sort basis when parsing using a bison-generated parser. To enable this,
      +mark the sort with the locations attribute.

      +
        syntax Exp [locations]
+  syntax Exp ::= Exp "/" Exp | Int
+

      K implicitly wraps productions of these sorts in a #location term (see the
      +K-LOCATIONS module in kast.md). The metadata can thus be accessed with
      +ordinary rewrite rules:

      +
        rule #location(_ / 0, File, StartLine, _StartColumn, _EndLine, _EndColumn) =>
+  "Error: Division by zero at " +String File +String ":" Int2String(StartLine)
+

      Sometimes it is desirable to allow code to be written in a file which
      +overwrites the current location information provided by the parser. This can be
      +done via a combination of the #LineMarker sort and the --bison-file flag to
      +the parser generator. If you declare a production of sort #LineMarker which
      +contains a regular expression terminal, this will be treated as a
      +line marker by the bison parser. The user will then be expected to provide
      +an implementation of the parser for the line marker in C. The function expected
      +by the parser has the signature void line_marker(char *, yyscan_t), where
      +yyscan_t is a
      +reentrant flex scanner.
      +The string value of the line marker token as specified by your regular
      +expression can be found in the first parameter of the function, and you can
      +set the line number used by the scanner using yyset_lineno(int, yyscan_t). If
      +you declare the variable extern char *filename, you can also set the current
      +file name by writing a malloc'd, zero-terminated string to that variable.

      +

      Unparsing

      +

      A number of factors go into how terms are unparsed in K. Here we describe some
      +of the features the user can use to control how unparsing happens.

      +

      Brackets

      + +

      One of the phases that the unparser goes through is to insert productions
      +tagged with the bracket attribute where it believes this is necessary
      +in order to create a correct string that will be parsed back into the original
      +AST. The most common case of this is in expression grammars. For example,
      +consider the following grammar:

      +
      syntax Exp ::= Int
+             | Exp "*" Exp
+             > Exp "+" Exp
+

      Here we have declared that expressions can contain integer addition and
      +multiplication, and that multiplication binds tighter than addition. As a
      +result, when writing a program, if we want to write an expression that first
      +applies addition, then multiplication, we must use brackets: (1 + 2) * 3.
      +Similarly, if we have such an AST, we must insert brackets into the AST
      +in order to faithfully unparse the term in a manner that will be parsed back
      +into the same ast, because if we do not, we end up unparsing the term as
      +1 + 2 * 3, which will be parsed back as 1 + (2 * 3) because of the priority
      +declaration in the grammar.

      +

      You can control how the unparser will insert such brackets by adding a
      +production with the bracket attribute and the correct sort. For example, if,
      +instead of parentheses, you want to use curly braces, you could write:

      +
      syntax Exp ::= "{" Exp "}" [bracket]
+

      This would signal to the unparser how brackets should look for terms of sort
      +Exp, and it will use this syntax when unparsing terms of sort Exp.

      +

      Commutative collections

      + +

      One thing that K will do (unless you pass the --no-sort-collections flag to
      +krun) is to sort associative, commutative collections (such as Set and Map)
      +alphanumerically. For example, if I have a collection whose keys are sort Id
      +and they have the values a, b, c, and d, then unparsing will always print
      +first the key a, then b, then c, then d, because this is the alphabetic order
      +of these keys when unparsed.

      +

      Furthermore, K will sort numeric keys numerically. For example, if I have a
      +collection whose keys are 1, 2, 5, 10, 30, it will first display 1, then 2,
      +then 5, then 10, then 30, because it will sort these keys numerically. Note
      +that this is different than an alphabetic sort, which would sort them as
      +1, 10, 2, 30, 5. We believe the former is more intuitive to users.

      +

      Substitution filtering

      + +

      K will remove substitution terms corresponding to anonymous variables when
      +using the --pattern flag if those anonymous variables provide no information
      +about the named variables in your serach pattern. You can disable this behavior
      +by passing --no-substitution-filtering to krun. When this flag is not passed,
      +and you are using the Haskell backend, any equality in a substitution (ie, an
      +#Equals under an #And under an #Or), will be hidden from the user if the
      +left hand side is a variable that was anonymous in the --pattern passed by
      +the user, unless that variable appears elsewhere in the substitution. If you
      +want to see that variable in the substitution, you can either disable this
      +filtering, or give that variable a name in the original search pattern.

      +

      Variable alpha renaming

      + +

      K will automatically rename variables that appear in the output configuration.
      +Similar to commutative collections, this is done to normalize the resulting
      +configuration so that equivalent configurations will be printed identically
      +regardless of how they happen to be reached. This pass can be disabled by
      +passing --no-alpha-renaming to krun.

      +

      Macro expansion

      + +

      K will apply macros in reverse on the output configuration if the macro was
      +created with the alias or alias-rec attribute. See the section on macro
      +expansion for more details.

      +

      Formatting

      + +

      format attribute

      + +

      K allows you to control how terms are unparsed using the format attribute.
      +By default, a domain value is unparsed by printing its string value verbatim,
      +and an application pattern is unparsed by printing its terminals and children
      +in the sequence implied by its concrete syntax, separated by spaces. However,
      +K gives you complete control over how you want to unparse the symbol.

      +

      A format attribute is a string containing zero or more escape sequences that
      +tell K how to unparse the symbol. Escape sequences begin with a '%' and are
      +followed by either an integer, or a single non-digit character. Below is a
      +list of escape sequences recognized by the formatter:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      Escape SequenceMeaning
      nInsert '\n' followed by the current indentation level
      iIncrease the current indentation level by 1
      dDecrease the current indentation level by 1
      cMove to the next color in the list of colors for this production
      rReset color to the default foreground color for the terminal (See below for more information on how colors work)
      an integerPrint a terminal or nonterminal from the production (See below for more information)
      any other charPrint that character verbatim
      +

      Using the integer escape sequence

      + +

      In the integer escape sequence %a, the integer a is treated as a 1-based
      +index into the terminals and nonterminals of the production.

      +
        +
      • +

        If the offset refers to a terminal, move to the next color in the list of
        +colors for this production, print the value of that terminal, then reset the
        +color to the default foreground color for the terminal.

        +
        • +
      • +

        If the offset refers to a regular expression terminal, it is an error.

        +
        • +
      • +

        If the offset refers to a nonterminal, print the unparsed representation of
        +the corresponding child of the current term.

        +
        • +
      +

      color and colors attributes

      + +

      K allows you to take advantage of ANSI terminal codes for foreground color
      +in order to colorize output pretty-printed by the unparser. This is controlled
      +via the color and colors attributes of productions. These attributes
      +combine with the format attribute to control how a term is colorized.

      +

      The first thing to understand about how colorization works is that the color
      +and colors attributes are used to construct a list of colors associated
      +with each production, and the format attribute then uses that list to choose
      +the color for each part of the production. For more information on how the
      +format attribute chooses a color from the list, see above, but essentially,
      +each terminal or %c in the format attribute advances the pointer in the list
      +by one element, and terminals and %r reset the current color to the default
      +foreground color of the terminal afterwards.

      +

      There are two ways you can construct a list of colors associated with a
      +production:

      +
        +
      • +

        The color attribute creates the entire list all with the same color, as
        +specified by the value of the attribute. When combined with the default format
        +attribute, this will color all the terminals in that production that color, but
        +more advanced techniques can be used as well.

        +
        • +
      • +

        The colors attribute creates the list from a manual, comma-separated list
        +of colors. The attribute is invalid if the length of the list is not equal to
        +the number of terminals in the production plus the number of %c substrings in
        +the format attribute.

        +
        • +
      +

      Attributes Reference

      +

      Attribute Syntax Overview

      + +

      In K, many different syntactic categories accept an optional trailing list of
      +keywords known as attributes. Attribute lists have two different syntaxes,
      +depending on where they occur. Each attribute also has a type which describes
      +where it may occur.

      +

      The first syntax is a square-bracketed ([]) list of words. This syntax is
      +available for following attribute types:

      +
        +
      1. module attributes - may appear immediately after the module keyword
        2. +
      2. sort attributes - may appear immediately after a sort declaration
        3. +
      3. production attributes - may appear immediately after a BNF production
        +alternative
        4. +
      4. rule attributes - may appear immediately after a rule
        5. +
      5. context attributes - may appear immediately after a context or context
        +alias
        6. +
      6. context alias attributes - may appear immediately after a context alias
        7. +
      7. claim attributes - may appear immediately after a claim
        8. +
      +

      The second syntax is the XML attribute syntax, i.e., a space delemited list of
      +key-and-quoted-value pairs appearing inside the start tag of an XML element:
      +<element key1="value" key2="value2" ... > </element>. This syntax is
      +available for the following attribute types:

      +
        +
      1. cell attributes - may appear inside of the cell start tag in
        +configuration declarations
        2. +
      +

      Unrecognized attributes are reported as an error. When we talk about
      +the type of an attribute, we mean a syntactic category to which an attribute
      +can be attached where the attribute has some semantic effect.

      +

      Attribute Index

      + +

      We now provide an index of available attributes organized alphabetically with a
      +brief description of each. Note that the same attribute may appear in the index
      +multiple times to indicate its effect in different contexts or with/without
      +arguments. A legend describing how to interpret the index follows.

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      NameTypeBackendReference
      alias-recprodallMacros and Aliases
      aliasprodallMacros and Aliases
      all-pathclaimhaskellall-path and one-path attributes to distinguish reachability claims
      anywhereruleallanywhere rules
      applyPriority(_)prodallSymbol priority and associativity
      avoidprodallSymbol priority and associativity
      binderprodallNo reference yet.
      bracketprodallParametric productions and bracket attributes
      color(_)prodallcolor and colors attributes
      colors(_)prodallcolor and colors attributes
      concretemodllvmsymbolic and concrete attribute
      concrete(_)rulehaskellconcrete and symbolic attributes (Haskell backend)
      concreterulehaskellconcrete and symbolic attributes (Haskell backend)
      context(_)aliasallContext aliases
      exit = ""cellallexit attribute
      formatprodallformat attribute
      freshGeneratorprodallfreshGenerator attribute
      functionprodallfunction and total attributes
      group(_)allallSymbol priority and associativity
      hook(_)prodallNo reference yet
      hybrid(_)prodallhybrid attribute
      hybridprodallhybrid attribute
      klabel(_)prodallklabel(_) and symbol attributes
      latex(_)prodallNo reference yet
      leftprodallSymbol priority and associativity
      locationssortallLocation Information
      macro-recprodallMacros and Aliases
      macroprodallMacros and Aliases
      memorulehaskellThe memo attribute
      multiplicity = "_"cellallCollection Cells: multiplicity and type attributes
      non-assocprodallSymbol priority and associativity
      one-pathclaimhaskellall-path and one-path attributes to distinguish reachability claims
      owiseruleallowise and priority attributes
      prec(_)tokenallprec attribute
      preferprodallSymbol priority and associativity
      priority(_)ruleallowise and priority attributes
      privatemodallprivate attribute
      privateprodallpublic and private attribute
      publicmodallNo reference yet.
      publicprodallpublic and private attribute
      result(_)ctxtallresult attribute
      result(_)ruleallresult attribute
      rightprodallSymbol priority and associativity
      seqstrict(_)prodallstrict and seqstrict attributes
      seqstrictprodallstrict and seqstrict attributes
      simplificationrulehaskellsimplification attribute (Haskell backend)
      simplification(_)rulehaskellsimplification attribute (Haskell backend)
      smt-hook(_)prodhaskellSMT Translation
      smtlib(_)prodhaskellSMT Translation
      smt-lemmarulehaskellSMT Translation
      strictprodallstrict and seqstrict attributes
      strict(_)prodallstrict and seqstrict attributes
      symbolicmodhaskellsymbolic and concrete attribute
      symbolicrulehaskellconcrete and symbolic attributes (Haskell backend)
      symbolic(_)rulehaskellconcrete and symbolic attributes (Haskell backend)
      symbolprodallklabel(_) and symbol attributes
      tokenprodalltoken attribute
      tokensortalltoken attribute
      totalprodallfunction and total attributes
      trustedclaimhaskelltrusted attribute
      type = "_"cellallCollection Cells: multiplicity and type attributes
      unboundVariables(_)ruleallThe unboundVariables attribute
      unusedprodallunused attribute
      kastmodallSpecify that this module should only be included in KAST backends (Java backend).
      koremodallSpecify that this module should only be included in Kore backends (Haskell/LLVM backend).
      concretemodallSpecify that this module should only be included in concrete backends (LLVM backend).
      symbolicmodallSpecify that this module should only be included in symbolic backends (Haskell/Java backend).
      stream = "_"cellallSpecify that this cell should be hooked up to a stream, either stdin, stdout, or stderr.
      +

      Internal Attribute Index

      + +

      Some attributes should not generally appear in user code, except in some
      +unusual or complex examples. Such attributes are typically generated by the
      +compiler and used internally. We list these attributes below as a reference for
      +interested readers:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      NameTypeBackendReference
      assocprodallassoc, comm, idem and unit attributes
      commprodallassoc, comm, idem and unit attributes
      digestmodallContains the hash of the textual contents of the module.
      idemprodallassoc, comm, idem and unit attributes
      unitprodallassoc, comm, idem and unit attributes
      userListprodallIdentifies the desugared form of Lst ::= List{Elm,"delim"}
      predicateprodallSpecifies the sort of a predicate label
      elementprodallSpecifies the label of the elements in a list
      bracketLabelprodallKeep track of the label of a bracket production since it can't have a klabel
      injectiveprodallLabel a given production as injective (unique output for each input)
      internalprodallProduction is reserved for internal use by the compiler
      coolruleallstrict and seqstrict attributes
      heatruleallstrict and seqstrict attributes
      +

      Index Legend

      + +
        +
      • +

        Name - the attribute's name (optionally followed by an underscore _ to indicate the attribute takes arguments)

        +
        • +
      • +

        Type - the syntactic categories where this attribute is not ignored;
        +the possible values are the types mentioned above or shorthands:

        +
          +
        1. all - short for any type except cell
          2. +
        2. mod - short for module
          3. +
        3. sort
          4. +
        4. prod - short for production
          5. +
        5. rule
          6. +
        6. ctxt - short for context or context alias
          7. +
        7. claim
          8. +
        8. cell
          9. +
        +
        • +
      • +

        Backend - the backends that do not ignore this attribute; possible values:

        +
          +
        1. all - all backends
          2. +
        2. llvm - the LLVM backend
          3. +
        3. haskell - the Haskell backend
          4. +
        +
        • +
      • +

        Effect - the attribute's effect (when it applies)

        +
        • +
      +

      Pending Documentation

      +

      Backend features not yet given documentation:

      +
        +
      • Parser of KORE terms and definitions
        • +
      • Term representation of K terms
        • +
      • Hooked sorts and symbols
        • +
      • Substituting a substitution into the RHS of a rule +
          +
        • domain values
          • +
        • functions
          • +
        • variables
          • +
        • symbols
          • +
        • polymorphism
          • +
        • hooks
          • +
        • injection compaction
          • +
        • overload compaction
          • +
        +
        • +
      • Pattern Matching / Unification of subject and LHS of rule +
          +
        • domain values
          • +
        • symbols
          • +
        • side conditions
          • +
        • and/or patterns
          • +
        • list patterns
          • +
        • nonlinear variables
          • +
        • map/set patterns +
            +
          • deterministic
            • +
          • nondeterministic
            • +
          +
          • +
        • modulo injections
          • +
        • modulo overloads
          • +
        +
        • +
      • Stepping +
          +
        • initialization
          • +
        • termination
          • +
        +
        • +
      • Print kore terms
        • +
      • Equality/comparison of terms
        • +
      • Owise rules
        • +
      • Strategy #STUCK axiom
        • +
      • User substitution +
          +
        • binders
          • +
        • kvar
          • +
        +
        • +
      +

      To get a complete list of hooks supported by K, you can run:

      +
      grep -P -R "(?<=[^-])hook\([^)]*\)" k-distribution/include/kframework/builtin/ \
+     --include "*.k" -ho | \
+sed 's/hook(//' | sed 's/)//' | sort | uniq | grep -v org.kframework
+

      All of these hooks will also eventually need documentation.

      +

      K Cheat Sheet

      +

      This is a quick reference of the most commonly used K tools.

      +
      kompile (--gen-bison-parser)? {file}                : generate parser, optionally with ahead of time
+krun {file}                                         : interpret file
+krun -cPGM='{string}'                               : interpret string
+kast --output (kore | kast) (-e|{file})             : parse expression or file
+kompile (--enable-search --backend haskell)? {file} : generate parser, enabling non-deterministic run
+krun (--search-all)? {file}                         : interpret file, evaluating non-deterministic runs as well
+foo-kompiled/parser_PGM {file}                      : ahead of time parse
+kompile (--main-module)? (--syntax-module)? {file}  : generate parser for {file}.k {file}-syntax.k, explicitly state main modules
+kparse <file> | kore-print -                        : parse and unparse a file
+kompile {file} --enable-llvm-debug                  : generate debuggable output for {file}.k
+krun {file} --debugger                              : debug K code
+kprove {file}                                       : Verify specs in {file}
+

      During GDB debugging session (see here for
      +LLDB breakpoint syntax):

      +
      break {file}:{linenum}                              : add a breakpoint to {file}'s {linenum} numbered line
+k match {module}.{label} subject                    : investigate matching
+

      K Tools

      +

      Here we document how to use some of the most commonly used K tools.

      +

      Minimizing Output

      +

      When one is working with kore-repl or the prover in general and looking at
      +specific configurations using config, sometimes the configurations can be huge.

      +

      One tool to help print configuration compactly is the pyk print utility:

      +
      pyk print
+

      We are going to use --minimize option (which is actually used automatically
      +when printing with pyk). This will filter out many uninteresting cells for the
      +current config and make the result more compact.

      +

      Then, when invoking the prover, you can minimize your output by piping it into
      +the pyk print ... facility with arguments for controlling the output:

      +
      kprove --output json --definition DEFN ... \
+    | jq .term                             \
+    | pyk print DEFN /dev/stdin --omit-labels ... --keep-labels ...
+

      You can also use this in the kore-repl more easily, by making a help script.
      +In your current directory, save a new script pykprint.sh:

      +
      #!/bin/bash
+
+kast --input kore --output json --definition $1 /dev/stdin \
+    | jq .term                                             \
+    | pyk print $1 /dev/stdin --omit-labels $2
+

      Now call config | bash pykprint.sh DEFN in Kore REPL to make the output
      +smaller.

      +

      The options you have to control the output are as follows:

      +
        +
      • --no-minimize: do not remove uninteresting cells.
        • +
      • --omit-cells: remove the selected cells from the output.
        • +
      • --keep-cells: keep only the selected cells in the output.
        • +
      +

      Note: Make sure that there is no whitespace around , in the omit list,
      +otherwise you'll get an error (, is a list separator, so this
      +requirement is strict).

      +

      Debugging

      +

      The LLVM Backend has support for integration with GDB. You can run the debugger
      +on a particular program by passing the --debugger flag to krun, or by
      +invoking the llvm backend interpreter directly. Below we provide a simple
      +tutorial to explain some of the basic commands supported by the LLVM backend.

      +

      LLDB Support

      + +

      GDB is not well-supported on macOS, particularly on newer OS versions and Apple
      +Silicon ARM hardware. Consequently, if the --debugger option is passed to krun
      +on macOS, LLDB[^1] is launched instead of GDB. However, the K-specific debugger
      +scripts that GDB uses have not been ported to LLDB yet, and so the instructions
      +in the rest of this section will not work.

      +

      The K Definition

      + +

      Here is a sample K definition we will use to demonstrate debugging
      +capabilities:

      +
      module TEST
+  imports INT
+
+  configuration <k> foo(5) </k>
+  rule [test]: I:Int => I +Int 1 requires I <Int 10
+
+  syntax Int ::= foo(Int) [function]
+  rule foo(I) => 0 -Int I
+
+endmodule
+

      You should compile this definition with --backend llvm --enable-llvm-debug to
      +use the debugger most effectively.

      +

      Stepping

      + +

      Important: When you first run krun with option --debugger, GDB / LLDB
      +will instruct you on how to modify ~/.gdbinit or ~/.lldbinit to enable
      +printing abstract syntax of K terms in the debugger. If you do not perform this
      +step, you can still use all the other features, but K terms will be printed as
      +their raw address in memory.

      +

      GDB will need the kompiled interpreter in its safe path in order to access the
      +pretty printing python script within it. A good way to do this would be to pick
      +a minimum top-level path that covers all of your kompiled semantics (ie. set auto-load safe-path ~/k-semantics). LLDB has slightly different security
      +policies that do not require fully-arbitrary code execution.

      +

      This section uses GDB syntax to demonstrate the debugging features. Please
      +refer to the GDB to LLDB command map on
      +macOS.

      +

      You can break before every step of execution is taken by setting a breakpoint
      +on the k_step function.

      +
      (gdb) break definition.kore:k_step
+Breakpoint 1 at 0x25e340
+(gdb) run
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("0", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb) continue
+Continuing.
+
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("1", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb) continue 2
+Will ignore next crossing of breakpoint 1.  Continuing.
+
+Breakpoint 1, 0x000000000025e340 in step (subject=`<generatedTop>{}`(`<k>{}`(`kseq{}`(`inj{Int{}, KItem{}}`(#token("3", "Int")),dotk{}(.KList))),`<generatedCounter>{}`(#token("0", "Int"))))
+(gdb)
+

      Breaking on a specific rule

      + +

      You can break when a rule is applied by giving the rule a rule label. If the
      +module name is TEST and the rule label is test, you can break when the rule
      +applies by setting a breakpoint on the TEST.test.rhs function:

      +
      (gdb) break TEST.test.rhs
+Breakpoint 1 at 0x25e250: file /home/dwightguth/test/./test.k, line 4.
+(gdb) run
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+

      Note that the substitution associated with that rule is visible in the
      +description of the frame.

      +

      You can also break when a side condition is applied using the TEST.test.sc
      +function:

      +
      (gdb) break TEST.test.sc
+Breakpoint 1 at 0x25e230: file /home/dwightguth/test/./test.k, line 4.
+(gdb) run
+Breakpoint 1, TEST.test.sc (VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+

      Note that every variable used in the side condition can have its value
      +inspected when stopped at this breakpoint, but other variables are not visible.

      +

      You can also break on a rule by its location:

      +
      (gdb) break test.k:4
+Breakpoint 1 at 0x25e230: test.k:4. (2 locations)
+(gdb) run
+Breakpoint 1, TEST.test.sc (VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("0", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.sc (VarI=#token("1", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+

      Note that this sets a breakpoint at two locations: one on the side condition
      +and one on the right hand side. If the rule had no side condition, the first
      +would not be set. You can also view the locations of the breakpoints and
      +disable them individually:

      +
      (gdb) info breakpoint
+Num     Type           Disp Enb Address            What
+1       breakpoint     keep y   <MULTIPLE>
+        breakpoint already hit 3 times
+1.1                         y     0x000000000025e230 in TEST.test.sc at /home/dwightguth/test/./test.k:4
+1.2                         y     0x000000000025e250 in TEST.test.rhs at /home/dwightguth/test/./test.k:4
+(gdb) disable 1.1
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("1", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb) continue
+Continuing.
+
+Breakpoint 1, TEST.test.rhs (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList), VarI=#token("2", "Int")) at /home/dwightguth/test/./test.k:4
+4         rule [test]: I:Int => I +Int 1 requires I <Int 10
+(gdb)
+

      Now only the breakpoint when the rule applies is enabled.

      +

      Breaking on a function

      + +

      You can also break when a particular function in your semantics is invoked:

      +
      (gdb) info functions foo
+All functions matching regular expression "foo":
+
+File /home/dwightguth/test/./test.k:
+struct __mpz_struct *Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int(struct __mpz_struct *);
+(gdb) break Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int
+Breakpoint 1 at 0x25e640: file /home/dwightguth/test/./test.k, line 6.
+(gdb) run
+Breakpoint 1, Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+6         syntax Int ::= foo(Int) [function]
+(gdb)
+

      In this case, the variables have numbers instead of names because the names of
      +arguments in functions in K come from rules, and we are stopped before any
      +specific rule has applied. For example, _1 is the first argument to the
      +function.

      +

      You can also set a breakpoint in this location by setting it on the line
      +associated with its production:

      +
      (gdb) break test.k:6
+Breakpoint 1 at 0x25e640: file /home/dwightguth/test/./test.k, line 6.
+(gdb) run
+Breakpoint 1, Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+6         syntax Int ::= foo(Int) [function]
+

      These two syntaxes are equivalent; use whichever is easier for you.

      +

      You can also view the stack of function applications:

      +
      (gdb) bt
+#0  Lblfoo'LParUndsRParUnds'TEST'UndsUnds'Int (_1=#token("1", "Int")) at /home/dwightguth/test/./test.k:6
+#1  0x000000000025e5f8 in apply_rule_111 (VarDotVar0=`<generatedCounter>{}`(#token("0", "Int")), VarDotVar1=dotk{}(.KList)) at /home/dwightguth/test/./test.k:9
+#2  0x0000000000268a52 in take_steps ()
+#3  0x000000000026b7b4 in main ()
+(gdb)
+

      Here we see that foo was invoked while applying the rule on line 9 of test.k,
      +and we also can see the substitution of that rule. If foo was evaluated while
      +evaluating another function, we would also be able to see the arguments of that
      +function as well, unless the function was tail recursive, in which case no
      +stack frame would exist once the tail call was performed.

      +

      Breaking on a set of rules or functions

      + +

      Using rbreak <regex> you can set breakpoints on multiple functions.

      +
        +
      • +

        rbreak Lbl - sets a breakpoint on all non hooked functions

        +
        • +
      • +

        rbreak Lbl.*TEST - sets a breakpoint on all functions from module TEST

        +
        • +
      • +

        rbreak hook_INT - sets a breakpoint on all hooks from module INT

        +
        • +
      +

      Other debugger issues

      + +
        +
      • <optimized out> try kompiling without -O1, -O2, or -O3.
        • +
      • (gdb) break definition.kore:break -> No source file named definition.kore.
        +send --enable-llvm-debug to kompile in order to generate debug info symbols.
        • +
      +

      Profiling your K semantics

      +

      The first thing to be aware of is in order to get meaningful data,
      +you need to build the semantics and all of its dependencies with
      +optimizations enabled but without the frame pointer elimination
      +optimization      . For example, for EVM, this means rebuilding GMP, MPFR,
      +JEMalloc, Crypto++, SECP256K1, etc with the following exports.

      +
      export CFLAGS="-DNDEBUG -O2 -fno-omit-frame-pointer"
+export CXXFLAGS="-DNDEBUG -O2 -fno-omit-frame-pointer"
+

      You can skip this step, but if you do, any samples within these
      +libraries will not have correct stack trace information, which means
      +you will likely not get a meaningful set of data that will tell you
      +where the majority of time is really being spent. Don't worry about
      +rebuilding literally every single dependency though. Just focus on the
      +ones that you expect to take a non-negligible amount of runtime. You
      +will be able to tell if you haven't done enough later, and you can go
      +back and rebuild more. Once this is done, you then build K with
      +optimizations and debug info enabled, like so:

      +
      mvn package -Dproject.build.type="FastBuild"
+

      Next, you build the semantics with optimizations and debug info
      +enabled (i.e., kompile -ccopt -O2 --iterated -ccopt -fno-omit-frame-pointer).

      +

      Once all this is done, you should be ready to profile your
      +application. Essentially, you should run whatever test suite you
      +usually run, but with perf record -g -- prefixed to the front. For
      +example, for KEVM it's the following command. (For best data, don't
      +run this step in parallel.)

      +
      perf record -g -- make test-conformance
+

      Finally, you want to filter out just the samples that landed within
      +the llvm backend and view the report. For this, you need to know the
      +name of the binary that was generated by your build system. Normally
      +it is interpreter, but e.g. if you are building the web3 client for
      +kevm, it would be kevm-client. You will want to run the following
      +command.

      +
      perf report -g -c $binary_name
+

      If all goes well, you should see a breakdown of where CPU time has
      +been spent executing the application. You will know that sufficient
      +time was spent rebuilding dependencies with the correct flags when the
      +total time reported by the main method is close to 100%. If it's not
      +close to 100%, this is probably because a decent amount of self time
      +was reported in stack traces that were not built with frame pointers
      +enabled, meaning that perf was unable to walk the stack. You will have
      +to go back, rebuild the appropriate libraries, and then record your
      +trace again.

      +

      Your ultimate goal is to identify the hotspots that take the most
      +time, and make them execute faster. Entries like step and
      +step_1234 like functions refer to the cost of matching. An entry
      +like side_condition_1234 is a side condition and apply_rule_1234
      +is constructing the rhs of a rule. You can convert from this rule
      +ordinal to a location using the llvm-kompile-compute-loc script in
      +the bin folder of the llvm backend repo. For example,

      +
      llvm-kompile-compute-loc 5868 evm-semantics/.build/defn/llvm/driver-kompiled
+

      spits out the following text.

      +
      Line: 18529
+/home/dwightguth/evm-semantics/./.build/defn/llvm/driver.k:493:10
+

      This is the line of definition.kore that the axiom appears on as
      +well as the original location of the rule in the K semantics. You can
      +use this information to figure out which rules and functions are
      +causing the most time and optimize them to be more efficient.

      +

      Running tests - kserver

      +

      The kserver is a front-end tool based on Nailgun
      +which helps to reduce the startup time of the JVM. Calling kserver in a terminal
      +window will wait for all kompile/kprove calls and force them to run in the same process
      +and share the same threads. This also reduces the thread contention significantly. kompile
      +uses all the threads available to do rule parsing. Another benefit is that it saves caches,
      +and each time you call kprove/kast, you can access those directly w/o extra disk usage.
      +Running the regression-new integration tests on a powerful machine (32 threads) takes 8m,
      +with the kserver active it takes 2m. You can start the kserver in two ways.

      +
        +
      • blocking: call kserver in the command line. Close it after you are done testing. Useful for quick testing.
        • +
      • non-blocking: call spawn-kserver <log.flie> and close it with stop-kserver - this is used for automation on CI
        • +
      +

      Because we reuse caches, you should stop and restart the server between runs.
      +The Nailgun implementation hasn't been updated in the last 3-5 years, and it's not compatible with Java 18 onwards.

      +

      K Builtins

      +

      The K Builtins (also referred to as the K Prelude or the K Standard Library)
      +consists of several files which contain definitions that make working with K
      +simpler. These files can be found under include/kframework/builtin in your K
      +installation directory, and can be imported with requires "FILENAME" (without
      +the path prefix).

      +
        +
      • domains: Basic datatypes which are universally useful.
        • +
      • kast: Representation of K internal data-structures (not to be
        +included in normal definitions).
        • +
      • prelude: Automatically included into every K definition.
        • +
      • ffi: FFI interface for calling out to native C code from K.
        • +
      • json: JSON datatype and parsers/unparsers for JSON strings.
        • +
      • rat: Rational number representation.
        • +
      • substitution: Hooked implementation of capture-aware
        +sustitution for K definitions.
        • +
      • unification: Hooked implementation of unification
        +exposed directly to K definitions.
        • +
      +

      Basic Builtin Types in K

      +

      A major piece of the K prelude consists of a series of modules that contain
      +implementations of basic data types and language features in K. You do not need
      +to require this file yourself; it is required automatically in every K
      +definition unless --no-prelude is passed to kompile. K may not work correctly
      +if some of these modules do not exist or do not declare certain functions.

      +

      Note that some functions in the K prelude functions are not total, that is,
      +they are not defined on all possible input values. When you invoke such a
      +function on an undefined input, the behavior is undefined. In particular, when
      +this happens, interpreters generated by the K LLVM backend may crash.

      +
      require "kast.md"
+

      Default Modules

      +

      K declares certain modules that contain most of the builtins you usually want
      +when defining a language in K. In particular, this includes integers, booleans,
      +strings, identifiers, I/O, lists, maps, and sets. The DOMAINS-SYNTAX module
      +is designed to be imported by the syntax module of the language and contains
      +only the program-level syntax of identifiers, integers, booleans, and strings.
      +The DOMAINS module contains the rest of the syntax, including builtin
      +functions over those and the remaining types.

      +

      Note that not all modules are included in DOMAINS. A few less-common modules
      +are not, including ARRAY, COLLECTIONS, FLOAT, STRING-BUFFER, BYTES,
      +K-REFLECTION, MINT, and STRATEGY.

      +
      module DOMAINS-SYNTAX
+  imports SORT-K
+  imports ID-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+  imports STRING-SYNTAX
+endmodule
+
+module DOMAINS
+  imports DOMAINS-SYNTAX
+  imports INT
+  imports BOOL
+  imports STRING
+  imports BASIC-K
+  imports LIST
+  imports K-IO
+  imports MAP
+  imports SET
+  imports ID
+  imports K-EQUAL
+endmodule
+

      Arrays

      +

      Provided here is an implementation for fixed-sized, contiguous maps from Int
      +to KItem. In some previous versions of K, the Array type was a builtin type
      +backed by mutable arrays of objects. However, in modern K, the Array type is
      +implemented by means of the List type; users should not access this interface
      +directly and should instead make only of the functions listed below. Users of
      +this module should import only the ARRAY module.

      +
      module ARRAY-SYNTAX
+  imports private LIST
+
+  syntax Array [hook(ARRAY.Array)]
+

      Array lookup

      + +

      You can look up an element in an Array by its index in O(log(N)) time. Note
      +that the base of the logarithm is a relatively high number and thus the time is
      +effectively constant.

      +
        syntax KItem ::= Array "[" Int "]" [function, hook(ARRAY.lookup)]
+

      Array update

      + +

      You can create a new Array with a new value for a key in O(log(N)) time, or
      +effectively constant.

      +
        syntax Array ::= Array "[" key: Int "<-" value: KItem "]" [function, hook(ARRAY.update), klabel(_[_<-_]), symbol]
+

      Array reset

      + +

      You can create a new Array where a particular key is reset to its default
      +value in O(log(N)) time, or effectively constant.

      +
        syntax Array ::= Array "[" Int "<-" "undef" "]" [function, hook(ARRAY.remove)]
+

      Multiple array update

      + +

      You can create a new Array from a List L of size N where the N
      +elements starting at index are replaced with the contents of L, in
      +O(N*log(K)) time (where K is the size of the array), or effectively linear.
      +Having index + N > K yields an exception.

      +
        syntax Array ::= updateArray(Array, index: Int, List) [function, hook(ARRAY.updateAll)]
+

      Array fill

      + +

      You can create a new Array where the length elements starting at index
      +are replaced with value, in O(length*log(N)) time, or effectively linear.

      +
        syntax Array ::= fillArray(Array, index: Int, length: Int, value: KItem) [function, hook(ARRAY.fill)]
+

      Array range check

      + +

      You can test whether an integer is within the bounds of an array in O(1) time.

      +
        syntax Bool ::= Int "in_keys" "(" Array ")" [function, total, hook(ARRAY.in_keys)]
+
      endmodule
+
+module ARRAY-IN-K [private]
+  imports public ARRAY-SYNTAX
+  imports private LIST
+  imports private K-EQUAL
+  imports private INT
+  imports private BOOL
+

      Array creation

      + +

      You can create an array with length elements where each element is
      +initialized to value in O(1) time. Note that the array is stored in a manner
      +where only the highest element that is actually modified is given a value
      +in its internal representation, which means that subsequent array operations
      +may incur a one-time O(N) resizing cost, possibly amortized across multiple
      +operations.

      +
        syntax Array ::= makeArray(length: Int, value: KItem) [function, hook(ARRAY.make), public]
+

      Implementation of Arrays

      + +

      The remainder of this section consists of an implementation in K of the
      +operations listed above. Users of the ARRAY module should not make use
      +of any of the syntax defined in any of these modules.

      +
        syntax Array ::= arr(List, Int, KItem)
+
+  rule makeArray(I::Int, D::KItem) => arr(.List, I, D)
+
+  rule arr(L::List, _, _       ) [ IDX::Int ] => L[IDX] requires 0 <=Int IDX andBool IDX  <Int size(L)
+  rule arr(_      , _, D::KItem) [ _        ] => D      [owise]
+
+  syntax List ::= ensureOffsetList(List, Int, KItem) [function]
+  rule ensureOffsetList(L::List, IDX::Int, D::KItem) => L makeList(IDX +Int 1 -Int size(L), D) requires         IDX >=Int size(L)
+  rule ensureOffsetList(L::List, IDX::Int, _::KItem) => L                                      requires notBool IDX >=Int size(L)
+
+  rule arr(L::List, I::Int, D::KItem) [ IDX::Int <- VAL::KItem ] => arr(ensureOffsetList(L, IDX, D) [ IDX <- VAL ], I, D)
+
+  rule arr(L::List, I::Int, D::KItem) [ IDX::Int <- undef ] => arr(L, I, D) [ IDX <- D ]
+
+  rule updateArray(arr(L::List, I::Int, D::KItem), IDX::Int, L2::List) => arr(updateList(ensureOffsetList(L, IDX +Int size(L2) -Int 1, D), IDX, L2), I, D)
+
+  rule fillArray(arr(L::List, I::Int, D::KItem), IDX::Int, LEN::Int, VAL::KItem) => arr(fillList(ensureOffsetList(L, IDX +Int LEN -Int 1, D), IDX, LEN, VAL), I, D)
+
+  rule IDX::Int in_keys(arr(_, I::Int, _)) => IDX >=Int 0 andBool IDX <Int I
+endmodule
+
+module ARRAY-SYMBOLIC [symbolic]
+  imports ARRAY-IN-K
+endmodule
+
+module ARRAY-KORE [kore]
+  imports ARRAY-IN-K
+endmodule
+
+module ARRAY
+  imports ARRAY-SYMBOLIC
+  imports ARRAY-KORE
+endmodule
+

      Maps

      +

      Provided here is the syntax of an implementation of immutable, associative,
      +commutative maps from KItem to KItem. This type is hooked to an
      +implementation of maps provided by the backend. For more information on
      +matching on maps and allowable patterns for doing so, refer to K's
      +user documentation.

      +
      module MAP
+  imports private BOOL-SYNTAX
+  imports private INT-SYNTAX
+  imports private LIST
+  imports private SET
+
+  syntax Map [hook(MAP.Map)]
+

      Map concatenation

      + +

      The Map sort represents a generalized associative array. Each key can be
      +paired with an arbitrary value, and can be used to reference its associated
      +value. Multiple bindings for the same key are not allowed.

      +

      You can construct a new Map consisting of key/value pairs of two Maps. The
      +result is #False if the maps have keys in common (in particular, this will
      +yield an exception during concrete execution). This operation is O(Nlog(M))
      +where N is the size of the smaller map, when it appears on the right hand side.
      +When it appears on the left hand side and all variables are bound, it is
      +O(N      log(M)) where M is the size of the map it is matching and N is the number
      +of elements being matched. When it appears on the left hand side containing
      +variables not bound elsewhere in the term, it is O(N^K) where N is the size of
      +the map it is matching and K is the number of unbound keys being matched. In
      +other words, one unbound variable is linear, two is quadratic, three is cubic,
      +etc.

      +
        syntax Map ::= Map Map                        [left, function, hook(MAP.concat), klabel(_Map_), symbol, assoc, comm, unit(.Map), element(_|->_), index(0), format(%1%n%2)]
+

      Map unit

      + +

      The map with zero elements is represented by .Map.

      +
        syntax Map ::= ".Map"                         [function, total, hook(MAP.unit), klabel(.Map), symbol, latex(\dotCt{Map})]
+

      Map elements

      + +

      An element of a Map is constructed via the |-> operator. The key is on the
      +left and the value is on the right.

      +
        syntax Map ::= KItem "|->" KItem                      [function, total, hook(MAP.element), klabel(_|->_), symbol, latex({#1}\mapsto{#2}), injective]
+
+  syntax priorities _|->_ > _Map_ .Map
+  syntax non-assoc _|->_
+

      Map lookup

      + +

      You can look up the value associated with the key of a map in O(log(N)) time.
      +Note that the base of the logarithm is a relatively high number and thus the
      +time is effectively constant. The value is #False if the key is not in the
      +map (in particular, this will yield an exception during concrete execution).

      +
        syntax KItem ::= Map "[" KItem "]"                    [function, hook(MAP.lookup), klabel(Map:lookup), symbol]
+

      Map lookup with default

      + +

      You can also look up the value associated with the key of a map using a
      +total function that assigns a specific default value if the key is not present
      +in the map. This operation is also O(log(N)), or effectively constant.

      +
        syntax KItem ::= Map "[" KItem "]" "orDefault" KItem      [function, total, hook(MAP.lookupOrDefault), klabel(Map:lookupOrDefault)]
+

      Map update

      + +

      You can insert a key/value pair into a map in O(log(N)) time, or effectively
      +constant.

      +
        syntax Map ::= Map "[" key: KItem "<-" value: KItem "]"           [function, total, klabel(Map:update), symbol, hook(MAP.update), prefer]
+

      Map delete

      + +

      You can remove a key/value pair from a map via its key in O(log(N)) time, or
      +effectively constant.

      +
        syntax Map ::= Map "[" KItem "<-" "undef" "]"     [function, total, hook(MAP.remove), klabel(_[_<-undef]), symbol]
+

      Map difference

      + +

      You can remove the key/value pairs in a map that are present in another map in
      +O(N*log(M)) time (where M is the size of the first map and N is the size of the
      +second), or effectively linear. Note that only keys whose value is the same
      +in both maps are removed. To remove all the keys in one map from another map,
      +you can say removeAll(M1, keys(M2)).

      +
        syntax Map ::= Map "-Map" Map                 [function, total, hook(MAP.difference), latex({#1}-_{\it Map}{#2})]
+

      Multiple map update

      + +

      You can update a map by adding all the key/value pairs in the second map in
      +O(N*log(M)) time (where M is the size of the first map and N is the size of the
      +second map), or effectively linear. If any keys are present in both maps, the
      +value from the second map overwrites the value in the first. This function is
      +total, which is distinct from map concatenation, a partial function only
      +defined on maps with disjoint keys.

      +
        syntax Map ::= updateMap(Map, Map)            [function, total, hook(MAP.updateAll)]
+

      Multiple map removal

      + +

      You can remove a Set of keys from a map in O(N*log(M)) time (where M is the
      +size of the Map and N is the size of the Set), or effectively linear.

      +
        syntax Map ::= removeAll(Map, Set)            [function, total, hook(MAP.removeAll)]
+

      Map keys (as Set)

      + +

      You can get a Set of all the keys in a Map in O(N) time.

      +
        syntax Set ::= keys(Map)                      [function, total, hook(MAP.keys)]
+

      Map keys (as List)

      + +

      You can get a List of all the keys in a Map in O(N) time.

      +
        syntax List ::= "keys_list" "(" Map ")"       [function, hook(MAP.keys_list)]
+

      Map key membership

      + +

      You can check whether a key is present in a map in O(1) time.

      +
        syntax Bool ::= KItem "in_keys" "(" Map ")"       [function, total, hook(MAP.in_keys)]
+

      Map values (as List)

      + +

      You can get a List of all the values in a map in O(N) time.

      +
        syntax List ::= values(Map)                   [function, hook(MAP.values)]
+

      Map size

      + +

      You can get the number of key/value pairs in a map in O(1) time.

      +
        syntax Int ::= size(Map)                      [function, total, hook(MAP.size), klabel(sizeMap)]
+

      Map inclusion

      + +

      You can determine whether a Map is a strict subset of another Map in O(N)
      +time (where N is the size of the first map). Only keys that are bound to the
      +same value are considered equal.

      +
        syntax Bool ::= Map "<=Map" Map               [function, total, hook(MAP.inclusion)]
+

      Map choice

      + +

      You can get an arbitrarily chosen key of a Map in O(1) time. The same key
      +will always be returned for the same map, but no guarantee is given that two
      +different maps will return the same element, even if they are similar.

      +
        syntax KItem ::= choice(Map)                      [function, hook(MAP.choice), klabel(Map:choice)]
+

      Implementation of Maps

      + +

      The remainder of this section contains lemmas used by the Java and Haskell
      +backend to simplify expressions of sort Map. They do not affect the semantics
      +of maps, merely describing additional rules that the backend can use to
      +simplify terms.

      +
      endmodule
+
+module MAP-KORE-SYMBOLIC [kore,symbolic,haskell]
+  imports MAP
+  imports private K-EQUAL
+  imports private BOOL
+
+  rule #Ceil(@M:Map [@K:KItem]) => {(@K in_keys(@M)) #Equals true} #And #Ceil(@M) #And #Ceil(@K) [simplification]
+
+  // Symbolic update
+
+  // Adding the definedness condition `notBool (K in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K |-> _ M:Map) [ K <- V ] => (K |-> V M) [simplification]
+  rule M:Map [ K <- V ] => (K |-> V M) requires notBool (K in_keys(M)) [simplification]
+  rule M:Map [ K <- _ ] [ K <- V ] => M [ K <- V ] [simplification]
+  // Adding the definedness condition `notBool (K1 in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K1 also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K1 |-> V1 M:Map) [ K2 <- V2 ] => (K1 |-> V1 (M [ K2 <- V2 ])) requires K1 =/=K K2 [simplification]
+
+  // Symbolic remove
+  rule (K |-> _ M:Map) [ K <- undef ] => M ensures notBool (K in_keys(M)) [simplification]
+  rule M:Map [ K <- undef ] => M requires notBool (K in_keys(M)) [simplification]
+  // Adding the definedness condition `notBool (K1 in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K1 also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K1 |-> V1 M:Map) [ K2 <- undef ] => (K1 |-> V1 (M [ K2 <- undef ])) requires K1 =/=K K2 [simplification]
+
+  // Symbolic lookup
+  rule (K  |->  V M:Map) [ K ]  => V ensures notBool (K in_keys(M)) [simplification]
+  rule (K1 |-> _V M:Map) [ K2 ] => M [K2] requires K1 =/=K K2 ensures notBool (K1 in_keys(M)) [simplification]
+  rule (_MAP:Map [ K  <-  V1 ]) [ K ]  => V1 [simplification]
+  rule ( MAP:Map [ K1 <- _V1 ]) [ K2 ] => MAP [ K2 ] requires K1 =/=K K2 [simplification]
+
+  rule (K  |->  V M:Map) [  K ] orDefault _ => V ensures notBool (K in_keys(M)) [simplification]
+  rule (K1 |-> _V M:Map) [ K2 ] orDefault D => M [K2] orDefault D requires K1 =/=K K2 ensures notBool (K1 in_keys(M)) [simplification]
+  rule (_MAP:Map [ K  <-  V1 ]) [ K ] orDefault _ => V1 [simplification]
+  rule ( MAP:Map [ K1 <- _V1 ]) [ K2 ] orDefault D => MAP [ K2 ] orDefault D requires K1 =/=K K2 [simplification]
+  rule .Map [ _ ] orDefault D => D [simplification]
+
+  // Symbolic in_keys
+  rule K in_keys(_M [ K <- undef ]) => false [simplification]
+  rule K in_keys(_M [ K <- _ ]) => true [simplification]
+  rule K1 in_keys(M [ K2 <- _ ]) => true requires K1 ==K K2 orBool K1 in_keys(M) [simplification]
+  rule K1 in_keys(M [ K2 <- _ ]) => K1 in_keys(M) requires K1 =/=K K2 [simplification]
+
+  rule {false #Equals @Key in_keys(.Map)} => #Ceil(@Key) [simplification]
+  rule {@Key in_keys(.Map) #Equals false} => #Ceil(@Key) [simplification]
+  rule {false #Equals @Key in_keys(Key' |-> Val @M)} => #Ceil(@Key) #And #Ceil(Key' |-> Val @M) #And #Not({@Key #Equals Key'}) #And {false #Equals @Key in_keys(@M)} [simplification]
+  rule {@Key in_keys(Key' |-> Val @M) #Equals false} => #Ceil(@Key) #And #Ceil(Key' |-> Val @M) #And #Not({@Key #Equals Key'}) #And {@Key in_keys(@M) #Equals false} [simplification]
+
+/*
+// The rule below is automatically generated by the frontend for every sort
+// hooked to MAP.Map. It is left here to serve as documentation.
+
+  rule #Ceil(@M:Map (@K:KItem |-> @V:KItem)) => {(@K in_keys(@M)) #Equals false} #And #Ceil(@M) #And #Ceil(@K) #And #Ceil(@V)
+    [simplification]
+*/
+endmodule
+
+module MAP-SYMBOLIC
+  imports MAP-KORE-SYMBOLIC
+endmodule
+

      Range Maps

      +

      Provided here is the syntax of an implementation of immutable, associative,
      +commutative range maps from Int to KItem. This type is hooked to an
      +implementation of range maps provided by the LLVM backend.
      +Currently, this type is not supported by other backends.
      +Although the underlying range map data structure supports any key sort, the
      +current implementation by the backend only supports Int keys due to
      +limitations of the underlying ordering function.

      +
      module RANGEMAP
+  imports private BOOL-SYNTAX
+  imports private INT-SYNTAX
+  imports private LIST
+  imports private SET
+
+

      Range, bounded inclusively below and exclusively above.

      + +
        syntax Range ::= "[" KItem "," KItem ")"    [klabel(Rangemap:Range), symbol]
+  
+  syntax RangeMap [hook(RANGEMAP.RangeMap)]
+

      Range map concatenation

      + +

      The RangeMap sort represents a map whose keys are stored as ranges, bounded
      +inclusively below and exclusively above. Contiguous or overlapping ranges that
      +map to the same value are merged into a single range.

      +

      You can construct a new RangeMap consisting of range/value pairs of two
      +RangeMaps. If the RangeMaps have overlapping ranges an exception will be
      +thrown during concrete execution. This operation is O(N*log(M)) (where N is
      +the size of the smaller map and M is the size of the larger map).

      +
        syntax RangeMap ::= RangeMap RangeMap                        [left, function, hook(RANGEMAP.concat), klabel(_RangeMap_), symbol, assoc, comm, unit(.RangeMap), element(_r|->_), index(0), format(%1%n%2)]
+

      Range map unit

      + +

      The RangeMap with zero elements is represented by .RangeMap.

      +
        syntax RangeMap ::= ".RangeMap"                         [function, total, hook(RANGEMAP.unit), klabel(.RangeMap), symbol, latex(\dotCt{RangeMap})]
+

      Range map elements

      + +

      An element of a RangeMap is constructed via the r|-> operator. The range
      +of keys is on the left, and the value is on the right.

      +
        syntax RangeMap ::= Range "r|->" KItem                      [function, hook(RANGEMAP.elementRng), klabel(_r|->_), symbol, latex({#1}\mapsto{#2}), injective]
+
+  syntax priorities _r|->_ > _RangeMap_ .RangeMap
+  syntax non-assoc _r|->_
+

      Range map lookup

      + +

      You can look up the value associated with a key of a RangeMap in O(log(N))
      +time (where N is the size of the RangeMap). This will yield an exception
      +during concrete execution if the key is not in the range map.

      +
        syntax KItem ::= RangeMap "[" KItem "]"                    [function, hook(RANGEMAP.lookup), klabel(RangeMap:lookup), symbol]
+

      Range map lookup with default

      + +

      You can also look up the value associated with a key of a RangeMap using a
      +total function that assigns a specific default value if the key is not present
      +in the RangeMap. This operation is also O(log(N)) (where N is the size of
      +the range map).

      +
        syntax KItem ::= RangeMap "[" KItem "]" "orDefault" KItem      [function, total, hook(RANGEMAP.lookupOrDefault), klabel(RangeMap:lookupOrDefault)]
+

      Range map lookup for range of key

      + +

      You can look up for the range that a key of a RangeMap is stored in in
      +O(log(N)) time (where N is the size of the RangeMap). This will yield an
      +exception during concrete execution if the key is not in the range map.

      +
        syntax Range ::= "find_range" "(" RangeMap "," KItem ")"                    [function, hook(RANGEMAP.find_range), klabel(RangeMap:find_range)]
+

      Range map update

      + +

      You can insert a range/value pair into a RangeMap in O(log(N)) time (where N
      +is the size of the RangeMap). Any ranges adjacent to or overlapping with the
      +range to be inserted will be updated accordingly.

      +
        syntax RangeMap ::= RangeMap "[" keyRange: Range "<-" value: KItem "]"           [function, klabel(RangeMap:update), symbol, hook(RANGEMAP.updateRng), prefer]
+

      Range map delete

      + +

      You can remove a range/value pair from a RangeMap in O(log(N)) time (where N
      +is the size of the RangeMap). If all or any part of the range is present in
      +the range map, it will be removed.

      +
        syntax RangeMap ::= RangeMap "[" Range "<-" "undef" "]"     [function, hook(RANGEMAP.removeRng), klabel(_r[_<-undef]), symbol]
+

      Range map difference

      + +

      You can remove the range/value pairs in a RangeMap that are also present in
      +another RangeMap in O(max{M,N}*log(M)) time (where M is the size of the
      +first RangeMap and N is the size of the second RangeMap). Note that only
      +the parts of overlapping ranges whose value is the same in both range maps
      +will be removed.

      +
        syntax RangeMap ::= RangeMap "-RangeMap" RangeMap                 [function, total, hook(RANGEMAP.difference), latex({#1}-_{\it RangeMap}{#2})]
+

      Multiple range map update

      + +

      You can update a RangeMap by adding all the range/value pairs in the second
      +RangeMap in O(N*log(M+N)) time (where M is the size of the first RangeMap
      +and N is the size of the second RangeMap). If any ranges are overlapping,
      +the value from the second range map overwrites the value in the first for the
      +parts where ranges are overlapping. This function is total, which is distinct
      +from range map concatenation, a partial function only defined on range maps
      +with non overlapping ranges.

      +
        syntax RangeMap ::= updateRangeMap(RangeMap, RangeMap)            [function, total, hook(RANGEMAP.updateAll)]
+

      Multiple range map removal

      + +

      You can remove a Set of ranges from a RangeMap in O(N*log(M)) time (where
      +M is the size of the RangeMap and N is the size of the Set). For every
      +range in the set, all or any part of it that is present in the range map will
      +be removed.

      +
        syntax RangeMap ::= removeAll(RangeMap, Set)            [function, hook(RANGEMAP.removeAll)]
+

      Range map keys (as Set)

      + +

      You can get a Set of all the ranges in a RangeMap in O(N) time (where N
      +is the size of the RangeMap).

      +
        syntax Set ::= keys(RangeMap)                      [function, total, hook(RANGEMAP.keys)]
+

      Range map keys (as List)

      + +

      You can get a List of all the ranges in a RangeMap in O(N) time (where N
      +is the size of the RangeMap).

      +
        syntax List ::= "keys_list" "(" RangeMap ")"       [function, hook(RANGEMAP.keys_list)]
+

      Range map key membership

      + +

      You can check whether a key is present in a RangeMap in O(log(N)) time (where
      +N is the size of the RangeMap).

      +
        syntax Bool ::= KItem "in_keys" "(" RangeMap ")"       [function, total, hook(RANGEMAP.in_keys)]
+

      Range map values (as List)

      + +

      You can get a List of all values in a RangeMap in O(N) time (where N is the
      +size of the RangeMap).

      +
        syntax List ::= values(RangeMap)                   [function, hook(RANGEMAP.values)]
+

      Range map size

      + +

      You can get the number of range/value pairs in a RangeMap in O(1) time.

      +
        syntax Int ::= size(RangeMap)                      [function, total, hook(RANGEMAP.size), klabel(sizeRangeMap)]
+

      Range map inclusion

      + +

      You can determine whether a RangeMap is a strict subset of another RangeMap
      +in O(M+N) time (where M is the size of the first RangeMap and N is the size
      +of the second RangeMap). Only keys within equal or overlapping ranges that
      +are bound to the same value are considered equal.

      +
        syntax Bool ::= RangeMap "<=RangeMap" RangeMap               [function, total, hook(RANGEMAP.inclusion)]
+

      Range map choice

      + +

      You can get an arbitrarily chosen key of a RangeMap in O(1) time. The same
      +key will always be returned for the same range map, but no guarantee is given
      +that two different range maps will return the same element, even if they are
      +similar.

      +
        syntax KItem ::= choice(RangeMap)                      [function, hook(RANGEMAP.choice), klabel(RangeMap:choice)]
+endmodule
+

      Sets

      +

      Provided here is the syntax of an implementation of immutable, associative,
      +commutative sets of KItem. This type is hooked to an implementation of sets
      +provided by the backend. For more information on matching on sets and allowable
      +patterns for doing so, refer to K's
      +user documentation.

      +
      module SET
+  imports private INT-SYNTAX
+  imports private BASIC-K
+
+  syntax Set [hook(SET.Set)]
+

      Set concatenation

      + +

      The Set sort represents a mathematical set (A collection of unique items).
      +The sets are nilpotent, i.e., the concatenation of two sets containing elements
      +in common is #False (note however, this may be silently allowed during
      +concrete execution). If you intend to add an element to a set that might
      +already be present in the set, use the |Set operator instead.

      +

      The concatenation operator is O(Nlog(M)) where N is the size of the smaller
      +set, when it appears on the right hand side. When it appears on the left hand
      +side and all variables are bound, it is O(N      log(M)) where M is the size of the
      +set it is matching and N is the number of elements being matched. When it
      +appears on the left hand side containing variables not bound elsewhere in the
      +term, it is O(N^K) where N is the size of the set it is matching and K is the
      +number of unbound keys being mached. In other words, one unbound variable is
      +linear, two is quadratic, three is cubic, etc.

      +
        syntax Set ::= Set Set                  [left, function, hook(SET.concat), klabel(_Set_), symbol, assoc, comm, unit(.Set), idem, element(SetItem), format(%1%n%2)]
+

      Set unit

      + +

      The set with zero elements is represented by .Set.

      +
        syntax Set ::= ".Set"                   [function, total, hook(SET.unit), klabel(.Set), symbol, latex(\dotCt{Set})]
+

      Set elements

      + +

      An element of a Set is constructed via the SetItem operator.

      +
        syntax Set ::= SetItem(KItem)               [function, total, hook(SET.element), klabel(SetItem), symbol, injective]
+

      Set union

      + +

      You can compute the union of two sets in O(N*log(M)) time (Where N is the size
      +of the smaller set). Note that the base of the logarithm is a relatively high
      +number and thus the time is effectively linear. The union consists of all the
      +elements present in either set.

      +
        syntax Set ::= Set "|Set" Set              [left, function, total, hook(SET.union), comm]
+  rule S1:Set |Set S2:Set => S1 (S2 -Set S1) [concrete]
+

      Set intersection

      + +

      You can compute the intersection of two sets in O(N*log(M)) time (where N
      +is the size of the smaller set), or effectively linear. The intersection
      +consists of all the elements present in both sets.

      +
        syntax Set ::= intersectSet(Set, Set)   [function, total, hook(SET.intersection), comm]
+

      Set complement

      + +

      You can compute the relative complement of two sets in O(N*log(M)) time (where
      +N is the size of the second set), or effectively linear. This is the set of
      +elements in the first set that are not present in the second set.

      +
        syntax Set ::= Set "-Set" Set           [function, total, hook(SET.difference), latex({#1}-_{\it Set}{#2}), klabel(Set:difference), symbol]
+

      Set membership

      + +

      You can compute whether an element is a member of a set in O(1) time.

      +
        syntax Bool ::= KItem "in" Set              [function, total, hook(SET.in), klabel(Set:in), symbol]
+

      Set inclusion

      + +

      You can determine whether a Set is a strict subset of another Set in O(N)
      +time (where N is the size of the first set).

      +
        syntax Bool ::= Set "<=Set" Set         [function, total, hook(SET.inclusion)]
+

      Set size

      + +

      You can get the number of elements (the cardinality) of a set in O(1) time.

      +
        syntax Int ::= size(Set)                [function, total, hook(SET.size)]
+

      Set choice

      + +

      You can get an arbitrarily chosen element of a Set in O(1) time. The same
      +element will always be returned for the same set, but no guarantee is given
      +that two different sets will return the same element, even if they are similar.

      +
        syntax KItem ::= choice(Set)                [function, hook(SET.choice), klabel(Set:choice)]
+
      endmodule
+

      Implementation of Sets

      + +

      The following lemmas are simplifications that the Haskell backend can
      +apply to simplify expressions of sort Set.

      +
      module SET-KORE-SYMBOLIC [kore,symbolic,haskell]
+  imports SET
+  imports private K-EQUAL
+  imports private BOOL
+
+  //Temporarly rule for #Ceil simplification, should be generated in front-end
+
+// Matching for this version not implemented.
+  // rule #Ceil(@S1:Set @S2:Set) =>
+  //        {intersectSet(@S1, @S2) #Equals .Set} #And #Ceil(@S1) #And #Ceil(@S2)
+  //   [simplification]
+
+//simpler version
+  rule #Ceil(@S:Set SetItem(@E:KItem)) =>
+         {(@E in @S) #Equals false} #And #Ceil(@S) #And #Ceil(@E)
+    [simplification]
+
+  // -Set simplifications
+  rule S              -Set .Set           => S          [simplification]
+  rule .Set           -Set  _             => .Set       [simplification]
+  rule SetItem(X)     -Set (S SetItem(X)) => .Set
+                               ensures notBool (X in S) [simplification]
+  rule S              -Set (S SetItem(X)) => .Set
+                               ensures notBool (X in S) [simplification]
+  rule (S SetItem(X)) -Set S              => SetItem(X)
+                               ensures notBool (X in S) [simplification]
+  rule (S SetItem(X)) -Set SetItem(X)     => S
+                               ensures notBool (X in S) [simplification]
+  // rule SetItem(X)     -Set S              => SetItem(X)
+  //                            requires notBool (X in S)  [simplification]
+  // rule (S1 SetItem(X)) -Set (S2 SetItem(X))  => S1 -Set S2
+  //                             ensures notBool (X in S1)
+  //                             andBool notBool (X in S2) [simplification]
+
+
+
+  // |Set simplifications
+  rule S    |Set .Set => S    [simplification, comm]
+  rule S    |Set S    => S    [simplification]
+
+  rule (S SetItem(X)) |Set SetItem(X) => S SetItem(X)
+                             ensures notBool (X in S) [simplification, comm]
+  // Currently disabled, see runtimeverification/haskell-backend#3301
+  // rule (S SetItem(X)) |Set S          => S SetItem(X)
+  //                            ensures notBool (X in S) [simplification, comm]
+
+  // intersectSet simplifications
+  rule intersectSet(.Set, _   ) => .Set    [simplification, comm]
+  rule intersectSet( S  , S   ) =>  S      [simplification]
+
+  rule intersectSet( S SetItem(X), SetItem(X))     => SetItem(X)
+                                                        ensures notBool (X in S)      [simplification, comm]
+  // Currently disabled, see runtimeverification/haskell-backend#3294
+  // rule intersectSet( S SetItem(X) , S)             => S ensures notBool (X in S)      [simplification, comm]
+  rule intersectSet( S1 SetItem(X), S2 SetItem(X)) => intersectSet(S1, S2) SetItem(X)
+                                                        ensures notBool (X in S1)
+                                                        andBool notBool (X in S2)     [simplification]
+
+  // membership simplifications
+  rule _E in .Set           => false   [simplification]
+  rule E  in (S SetItem(E)) => true
+              ensures notBool (E in S) [simplification]
+
+// These two rules would be sound but impose a giant overhead on `in` evaluation:
+  // rule E1 in (S SetItem(E2)) => true requires E1 in S
+  //                                 ensures notBool (E2 in S) [simplification]
+  // rule E1 in (S SetItem(E2)) => E1 in S requires E1 =/=K E2
+  //                                 ensures notBool (E2 in S) [simplification]
+
+  rule X in ((SetItem(X) S) |Set  _            ) => true
+                                    ensures notBool (X in S) [simplification]
+  rule X in ( _             |Set (SetItem(X) S)) => true
+                                    ensures notBool (X in S) [simplification]
+
+endmodule
+
+module SET-SYMBOLIC
+  imports SET-KORE-SYMBOLIC
+endmodule
+

      Lists

      +

      Provided here is the syntax of an implementation of immutable, associative
      +lists of KItem. This type is hooked to an implementation of lists provided
      +by the backend. For more information on matching on lists and allowable
      +patterns for doing so, refer to K's
      +user documentation.

      +
      module LIST
+  imports private INT-SYNTAX
+  imports private BASIC-K
+
+  syntax List [hook(LIST.List)]
+

      List concatenation

      + +

      The List sort is an ordered collection that may contain duplicate elements.
      +They are backed by relaxed radix balanced trees, which means that they support
      +efficiently adding elements to both sides of the list, concatenating two lists,
      +indexing, and updating elements.

      +

      The concatenation operator is O(log(N)) (where N is the size of the longer
      +list) when it appears on the right hand side. When it appears on the left hand
      +side, it is O(N), where N is the number of elements matched on the front and
      +back of the list.

      +
        syntax List ::= List List               [left, function, total, hook(LIST.concat), klabel(_List_), symbol, smtlib(smt_seq_concat), assoc, unit(.List), element(ListItem), format(%1%n%2)]
+

      List unit

      + +

      The list with zero elements is represented by .List.

      +
        syntax List ::= ".List"                 [function, total, hook(LIST.unit), klabel(.List), symbol, smtlib(smt_seq_nil), latex(\dotCt{List})]
+

      List elements

      + +

      An element of a List is constucted via the ListItem operator.

      +
        syntax List ::= ListItem(KItem)             [function, total, hook(LIST.element), klabel(ListItem), symbol, smtlib(smt_seq_elem)]
+

      List indexing

      + +

      You can get an element of a list by its integer offset in O(log(N)) time, or
      +effectively constant. Positive indices are 0-indexed from the beginning of the
      +list, and negative indices are -1-indexed from the end of the list. In other
      +words, 0 is the first element and -1 is the last element.

      +
        syntax KItem ::= List "[" Int "]"           [function, hook(LIST.get), klabel(List:get), symbol]
+

      List update

      + +

      You can create a new List with a new value at a particular index in
      +O(log(N)) time, or effectively constant.

      +
        syntax List ::= List "[" index: Int "<-" value: KItem "]" [function, hook(LIST.update), klabel(List:set)]
+

      List of identical elements

      + +

      You can create a list with length elements, each containing value, in O(N)
      +time.

      +
        syntax List ::= makeList(length: Int, value: KItem) [function, hook(LIST.make)]
+

      Multiple list update

      + +

      You can create a new List which is equal to dest except the N elements
      +starting at index are replaced with the contents of src in O(N*log(K)) time
      +(where K is the size of destand N is the size of src), or effectively linear. Having index + N > K yields an exception.

      +
        syntax List ::= updateList(dest: List, index: Int, src: List) [function, hook(LIST.updateAll)]
+

      List fill

      + +

      You can create a new List where the length elements starting at index
      +are replaced with value, in O(length*log(N)) time, or effectively linear.

      +
        syntax List ::= fillList(List, index: Int, length: Int, value: KItem) [function, hook(LIST.fill)]
+

      List slicing

      + +

      You can compute a new List by removing fromFront elements from the front
      +of the list and fromBack elements from the back of the list in
      +O((fromFront+fromBack)*log(N)) time, or effectively linear.

      +
        syntax List ::= range(List, fromFront: Int, fromBack: Int)   [function, hook(LIST.range), klabel(List:range), symbol]
+

      List membership

      + +

      You can compute whether an element is in a list in O(N) time. For repeated
      +comparisons, it is much better to first convert to a set using List2Set.

      +
        syntax Bool ::= KItem "in" List             [function, total, hook(LIST.in), klabel(_inList_)]
+

      List size

      + +

      You can get the number of elements of a list in O(1) time.

      +
        syntax Int ::= size(List)               [function, total, hook(LIST.size), klabel (sizeList), smtlib(smt_seq_len)]
+
      endmodule
+

      Collection Conversions

      +

      It is possible to convert from a List to a Set or from a Set to a list.
      +Converting from a List to a Set and back will not provide the same list;
      +duplicates will have been removed and the list may be reordered. Converting
      +from a Set to a List and back will generate the same set.

      +

      Note that because sets are unordered and lists are ordered, converting from a
      +Set to a List will generate some arbitrary ordering of elements, which may
      +be different from the natural ordering you might assume, or may not. Two
      +equal sets are guaranteed to generate the same ordering, but no guarantee is
      +otherwise provided about what the ordering will be. In particular, adding an
      +element to a set may completely reorder the elements already in the set, when
      +it is converted to a list.

      +
      module COLLECTIONS
+  imports LIST
+  imports SET
+  imports MAP
+
+  syntax List ::= Set2List(Set) [function, total, hook(SET.set2list)]
+  syntax Set ::= List2Set(List) [function, total, hook(SET.list2set)]
+
+endmodule
+

      Booleans

      +

      Provided here is the syntax of an implementation of boolean algebra in K.
      +This type is hooked to an implementation of booleans provided by the backend.
      +Note that this algebra is different from the builtin truth in matching logic.
      +You can, however, convert from the truth of the Bool sort to the truth in
      +matching logic via the expression {B #Equals true}.

      +

      The boolean values are true and false.

      +
      module SORT-BOOL
+  syntax Bool [hook(BOOL.Bool)]
+endmodule
+
+module BOOL-SYNTAX
+  imports SORT-BOOL
+  syntax Bool ::= "true"  [token]
+  syntax Bool ::= "false" [token]
+endmodule
+
+module BOOL-COMMON
+  imports private BASIC-K
+  imports BOOL-SYNTAX
+

      Basic boolean arithmetic

      + +

      You can:

      +
        +
      • Negate a boolean value.
        • +
      • AND two boolean values.
        • +
      • XOR two boolean values.
        • +
      • OR two boolean values.
        • +
      • IMPLIES two boolean values (i.e., P impliesBool Q is the same as
        +notBool P orBool Q)
        • +
      • Check equality of two boolean values.
        • +
      • Check inequality of two boolean values.
        • +
      +

      Note that only andThenBool and orElseBool are short-circuiting. andBool
      +and orBool may be short-circuited in concrete backends, but in symbolic
      +ackends, both arguments will be evaluated.

      +
        syntax Bool ::= "notBool" Bool          [function, total, klabel(notBool_), symbol, smt-hook(not), group(boolOperation), latex(\neg_{\scriptstyle\it Bool}{#1}), hook(BOOL.not)]
+                > Bool "andBool" Bool     [function, total, klabel(_andBool_), symbol, left, smt-hook(and), group(boolOperation), latex({#1}\wedge_{\scriptstyle\it Bool}{#2}), hook(BOOL.and)]
+                | Bool "andThenBool" Bool [function, total, klabel(_andThenBool_), symbol, left, smt-hook(and), group(boolOperation), hook(BOOL.andThen)]
+                | Bool "xorBool" Bool     [function, total, klabel(_xorBool_), symbol, left, smt-hook(xor), group(boolOperation), hook(BOOL.xor)]
+                | Bool "orBool" Bool      [function, total, klabel(_orBool_), symbol, left, smt-hook(or), group(boolOperation), latex({#1}\vee_{\scriptstyle\it Bool}{#2}), hook(BOOL.or)]
+                | Bool "orElseBool" Bool  [function, total, klabel(_orElseBool_), symbol, left, smt-hook(or), group(boolOperation), hook(BOOL.orElse)]
+                | Bool "impliesBool" Bool [function, total, klabel(_impliesBool_), symbol, left, smt-hook(=>), group(boolOperation), hook(BOOL.implies)]
+                > left:
+                  Bool "==Bool" Bool      [function, total, klabel(_==Bool_), symbol, left, comm, smt-hook(=), hook(BOOL.eq)]
+                | Bool "=/=Bool" Bool     [function, total, klabel(_=/=Bool_), symbol, left, comm, smt-hook(distinct), hook(BOOL.ne)]
+

      Implementation of Booleans

      + +

      The remainder of this section consists of an implementation in K of the
      +operations listed above.

      +
        rule notBool true => false
+  rule notBool false => true
+
+  rule true andBool B:Bool => B:Bool
+  rule B:Bool andBool true => B:Bool
+  rule false andBool _:Bool => false
+  rule _:Bool andBool false => false
+
+  rule true andThenBool K::Bool => K
+  rule K::Bool andThenBool true => K
+  rule false andThenBool _ => false
+  rule _ andThenBool false => false
+
+  rule false xorBool B:Bool => B:Bool
+  rule B:Bool xorBool false => B:Bool
+  rule B:Bool xorBool B:Bool => false
+
+  rule true orBool _:Bool => true
+  rule _:Bool orBool true => true
+  rule false orBool B:Bool => B
+  rule B:Bool orBool false => B
+
+  rule true orElseBool _ => true
+  rule _ orElseBool true => true
+  rule false orElseBool K::Bool => K
+  rule K::Bool orElseBool false => K
+
+  rule true impliesBool B:Bool => B
+  rule false impliesBool _:Bool => true
+  rule _:Bool impliesBool true => true
+  rule B:Bool impliesBool false => notBool B
+
+  rule B1:Bool =/=Bool B2:Bool => notBool (B1 ==Bool B2)
+endmodule
+
+module BOOL-KORE [kore, symbolic]
+  imports BOOL-COMMON
+
+  rule {true #Equals notBool @B} => {false #Equals @B} [simplification]
+  rule {notBool @B #Equals true} => {@B #Equals false} [simplification]
+  rule {false #Equals notBool @B} => {true #Equals @B} [simplification]
+  rule {notBool @B #Equals false} => {@B #Equals true} [simplification]
+
+  rule {true #Equals @B1 andBool @B2} => {true #Equals @B1} #And {true #Equals @B2} [simplification]
+  rule {@B1 andBool @B2 #Equals true} => {@B1 #Equals true} #And {@B2 #Equals true} [simplification]
+  rule {false #Equals @B1 orBool @B2} => {false #Equals @B1} #And {false #Equals @B2} [simplification]
+  rule {@B1 orBool @B2 #Equals false} => {@B1 #Equals false} #And {@B2 #Equals false} [simplification]
+endmodule
+
+module BOOL
+  imports BOOL-COMMON
+  imports BOOL-KORE
+endmodule
+

      Integers

      +

      Provided here is the syntax of an implementation of arbitrary-precision
      +integer arithmetic in K. This type is hooked to an implementation of integers
      +provided by the backend. For a fixed-width integer type, see the MINT module
      +below.

      +

      The UNSIGNED-INT-SYNTAX module provides a syntax of whole numbers in K.
      +This is useful because often programming languages implement the sign of an
      +integer as a unary operator rather than part of the lexical syntax of integers.
      +However, you can also directly reference integers with a sign using the
      +INT-SYNTAX module.

      +
      module UNSIGNED-INT-SYNTAX
+  syntax Int [hook(INT.Int)]
+  syntax Int ::= r"[0-9]+" [prefer, token, prec(2)]
+endmodule
+
+module INT-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  syntax Int ::= r"[\\+-]?[0-9]+" [prefer, token, prec(2)]
+endmodule
+
+module INT-COMMON
+  imports INT-SYNTAX
+  imports private BOOL
+

      Integer arithmetic

      + +

      You can:

      +
        +
      • Compute the bitwise complement ~Int of an integer value in twos-complement.
        • +
      • Compute the exponentiation ^Int of two integers.
        • +
      • Compute the exponentiation of two integers modulo another integer (^%Int).
        +A ^%Int B C is equal in value to (A ^Int B) %Int C, but has a better
        +asymptotic complexity.
        • +
      • Compute the product *Int of two integers.
        • +
      • Compute the quotient /Int or modulus %Int of two integers using
        +t-division, which rounds towards zero. Division by zero is #False.
        • +
      • Compute the quotient divInt or modulus modInt of two integers using
        +Euclidean division, in which the remainder is always non-negative. Division
        +by zero is #False.
        • +
      • Compute the sum +Int or difference -Int of two integers.
        • +
      • Compute the arithmetic right shift >>Int of two integers. Shifting by a
        +negative quantity is #False.
        • +
      • Compute the left shift of two integers. Shifting by a negative quantity is
        +#False.
        • +
      • Compute the bitwise and of two integers in twos-complement.
        • +
      • Compute the bitwise xor of two integers in twos-complement.
        • +
      • Compute the bitwise inclusive-or of two integers in twos-complement.
        • +
      +
        syntax Int ::= "~Int" Int                     [function, klabel(~Int_), symbol, total, latex(\mathop{\sim_{\scriptstyle\it Int}}{#1}), hook(INT.not), smtlib(notInt)]
+               > left:
+                 Int "^Int" Int                 [function, klabel(_^Int_), symbol, left, smt-hook(^), latex({#1}\mathrel{{\char`\^}_{\!\scriptstyle\it Int}}{#2}), hook(INT.pow)]
+               | Int "^%Int" Int Int            [function, klabel(_^%Int__), symbol, left, smt-hook((mod (^ #1 #2) #3)), hook(INT.powmod)]
+               > left:
+                 Int "*Int" Int                 [function, total, klabel(_*Int_), symbol, left, comm, smt-hook(*), latex({#1}\mathrel{\ast_{\scriptstyle\it Int}}{#2}), hook(INT.mul)]
+               /* FIXME: translate /Int and %Int into smtlib */
+               /* /Int and %Int implement t-division, which rounds towards 0 */
+               | Int "/Int" Int                 [function, klabel(_/Int_), symbol, left, smt-hook(div), latex({#1}\mathrel{\div_{\scriptstyle\it Int}}{#2}), hook(INT.tdiv)]
+               | Int "%Int" Int                 [function, klabel(_%Int_), symbol, left, smt-hook(mod), latex({#1}\mathrel{\%_{\scriptstyle\it Int}}{#2}), hook(INT.tmod)]
+               /* divInt and modInt implement e-division according to the Euclidean division theorem, therefore the remainder is always positive */
+               | Int "divInt" Int               [function, klabel(_divInt_), symbol, left, smt-hook(div), hook(INT.ediv)]
+               | Int "modInt" Int               [function, klabel(_modInt_), symbol, left, smt-hook(mod), hook(INT.emod)]
+               > left:
+                 Int "+Int" Int                 [function, total, klabel(_+Int_), symbol, left, comm, smt-hook(+), latex({#1}\mathrel{+_{\scriptstyle\it Int}}{#2}), hook(INT.add)]
+               | Int "-Int" Int                 [function, total, klabel(_-Int_), symbol, left, smt-hook(-), latex({#1}\mathrel{-_{\scriptstyle\it Int}}{#2}), hook(INT.sub)]
+               > left:
+                 Int ">>Int" Int                [function, klabel(_>>Int_), symbol, left, latex({#1}\mathrel{\gg_{\scriptstyle\it Int}}{#2}), hook(INT.shr), smtlib(shrInt)]
+               | Int "<<Int" Int                [function, klabel(_<<Int_), symbol, left, latex({#1}\mathrel{\ll_{\scriptstyle\it Int}}{#2}), hook(INT.shl), smtlib(shlInt)]
+               > left:
+                 Int "&Int" Int                 [function, total, klabel(_&Int_), symbol, left, comm, latex({#1}\mathrel{\&_{\scriptstyle\it Int}}{#2}), hook(INT.and), smtlib(andInt)]
+               > left:
+                 Int "xorInt" Int               [function, total, klabel(_xorInt_), symbol, left, comm, latex({#1}\mathrel{\oplus_{\scriptstyle\it Int}}{#2}), hook(INT.xor), smtlib(xorInt)]
+               > left:
+                 Int "|Int" Int                 [function, total, klabel(_|Int_), symbol, left, comm, latex({#1}\mathrel{|_{\scriptstyle\it Int}}{#2}), hook(INT.or), smtlib(orInt)]
+

      Integer minimum and maximum

      + +

      You can compute the minimum and maximum minInt and maxInt of two integers.

      +
        syntax Int ::= "minInt" "(" Int "," Int ")"   [function, total, smt-hook((ite (< #1 #2) #1 #2)), hook(INT.min)]
+               | "maxInt" "(" Int "," Int ")"   [function, total, smt-hook((ite (< #1 #2) #2 #1)), hook(INT.max)]
+

      Absolute value

      + +

      You can compute the absolute value absInt of an integer.

      +
        syntax Int ::= absInt ( Int )                 [function, total, smt-hook((ite (< #1 0) (- 0 #1) #1)), hook(INT.abs)]
+

      Log base 2

      + +

      You can compute the log base 2, rounded towards zero, of an integer. The log
      +base 2 of an integer is equal to the index of the highest bit set in the
      +representation of a positive integer. Log base 2 of zero or a negative number
      +is #False.

      +
        syntax Int ::= log2Int ( Int )                [function, hook(INT.log2)]
+

      Bit slicing

      + +

      You can compute the value of a range of bits in the twos-complement
      +representation of an integer, as interpeted either unsigned or signed, of an
      +integer. index is offset from 0 and length is the number of bits, starting
      +with index, that should be read. The number is assumed to be represented
      +in little endian notation with each byte going from least significant to
      +most significant. In other words, 0 is the least-significant bit, and each
      +successive bit is more significant than the last.

      +
        syntax Int ::= bitRangeInt           ( Int, index: Int, length: Int ) [function, hook(INT.bitRange)]
+               | signExtendBitRangeInt ( Int, index: Int, length: Int ) [function, hook(INT.signExtendBitRange)]
+

      Integer comparisons

      + +

      You can compute whether two integers are less than or equal to, less than,
      +greater than or equal to, greater than, equal, or unequal to another integer.

      +
        syntax Bool ::= Int "<=Int" Int         [function, total, klabel(_<=Int_), symbol, smt-hook(<=), latex({#1}\mathrel{\leq_{\scriptstyle\it Int}}{#2}), hook(INT.le)]
+                | Int "<Int" Int          [function, total, klabel(_<Int_), symbol, smt-hook(<), latex({#1}\mathrel{<_{\scriptstyle\it Int}}{#2}), hook(INT.lt)]
+                | Int ">=Int" Int         [function, total, klabel(_>=Int_), symbol, smt-hook(>=), latex({#1}\mathrel{\geq_{\scriptstyle\it Int}}{#2}), hook(INT.ge)]
+                | Int ">Int" Int          [function, total, klabel(_>Int_), symbol, smt-hook(>), latex({#1}\mathrel{>_{\scriptstyle\it Int}}{#2}), hook(INT.gt)]
+                | Int "==Int" Int         [function, total, klabel(_==Int_), symbol, comm, smt-hook(=), latex({#1}\mathrel{{=}{=}_{\scriptstyle\it Int}}{#2}), hook(INT.eq)]
+                | Int "=/=Int" Int        [function, total, klabel(_=/=Int_), symbol, comm, smt-hook(distinct), latex({#1}\mathrel{{=}{/}{=}_{\scriptstyle\it Int}}{#2}), hook(INT.ne)]
+

      Divides

      + +

      You can compute whether one integer evenly divides another. This is the
      +case when the second integer modulo the first integer is equal to zero.

      +
        syntax Bool ::= Int "dividesInt" Int    [function]
+

      Random integers

      + +

      You can, on concrete backends, compute a pseudorandom integer, or seed the
      +pseudorandom number generator. These operations are represented as
      +uninterpreted functions on symbolic backends.

      +
        syntax Int ::= randInt(Int) [function, hook(INT.rand), impure]
+  syntax K ::= srandInt(Int) [function, hook(INT.srand), impure]
+

      Implementation of Integers

      + +

      The remainder of this section consists of an implementation in K of some
      +of the operators above, as well as lemmas used by the Java and Haskell backend
      +to simplify expressions of sort Int. They do not affect the semantics of
      +integers, merely describing additional rules that the backend can use to
      +simplify terms.

      +
      endmodule
+
+module INT-SYMBOLIC [symbolic]
+  imports INT-COMMON
+  imports INT-SYMBOLIC-KORE
+  imports private BOOL
+
+  // Arithmetic Normalization
+  rule I +Int 0 => I [simplification]
+  rule I -Int 0 => I [simplification]
+
+  rule X modInt N => X requires 0 <=Int X andBool X <Int N [simplification]
+  rule X   %Int N => X requires 0 <=Int X andBool X <Int N [simplification]
+
+  // Bit-shifts
+  rule X <<Int 0 => X [simplification]
+  rule 0 <<Int _ => 0 [simplification]
+  rule X >>Int 0 => X [simplification]
+  rule 0 >>Int _ => 0 [simplification]
+endmodule
+
+module INT-SYMBOLIC-KORE [symbolic, kore, haskell]
+  imports INT-COMMON
+  imports ML-SYNTAX
+  imports private BOOL
+
+  // Definability Conditions
+  rule #Ceil(@I1:Int /Int   @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int %Int   @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int modInt @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int >>Int  @I2:Int) => {(@I2 >=Int 0)  #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int <<Int  @I2:Int) => {(@I2 >=Int 0)  #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+endmodule
+
+module INT-KORE [kore, symbolic]
+  imports private K-EQUAL
+  imports private BOOL
+  imports INT-COMMON
+
+  rule I1:Int ==K I2:Int => I1 ==Int I2 [simplification]
+  rule {K1 ==Int K2 #Equals true} => {K1 #Equals K2} [simplification]
+  rule {true #Equals K1 ==Int K2} => {K1 #Equals K2} [simplification]
+  rule {K1 ==Int K2 #Equals false} => #Not({K1 #Equals K2}) [simplification]
+  rule {false #Equals K1 ==Int K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=Int K2 #Equals true} => #Not({K1 #Equals K2}) [simplification]
+  rule {true #Equals K1 =/=Int K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=Int K2 #Equals false} => {K1 #Equals K2} [simplification]
+  rule {false #Equals K1 =/=Int K2} => {K1 #Equals K2} [simplification]
+
+  // Arithmetic Normalization
+  rule I +Int B => B +Int I          [concrete(I), symbolic(B), simplification(51)]
+  rule A -Int I => A +Int (0 -Int I) [concrete(I), symbolic(A), simplification(51)]
+
+  rule (A +Int I2) +Int I3 => A +Int (I2 +Int I3) [concrete(I2, I3), symbolic(A), simplification]
+  rule I1 +Int (B +Int I3) => B +Int (I1 +Int I3) [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 -Int (B +Int I3) => (I1 -Int I3) -Int B [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 +Int (I2 +Int C) => (I1 +Int I2) +Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule I1 +Int (I2 -Int C) => (I1 +Int I2) -Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule (I1 -Int B) +Int I3 => (I1 +Int I3) -Int B [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 -Int (I2 +Int C) => (I1 -Int I2) -Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule I1 -Int (I2 -Int C) => (I1 -Int I2) +Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule (C -Int I2) -Int I3 => C -Int (I2 +Int I3) [concrete(I2, I3), symbolic(C), simplification]
+
+  rule I1 &Int (I2 &Int C) => (I1 &Int I2) &Int C [concrete(I1, I2), symbolic(C), simplification]
+endmodule
+
+module INT
+  imports INT-COMMON
+  imports INT-SYMBOLIC
+  imports INT-KORE
+  imports private K-EQUAL
+  imports private BOOL
+
+  rule bitRangeInt(I::Int, IDX::Int, LEN::Int) => (I >>Int IDX) modInt (1 <<Int LEN)
+
+  rule signExtendBitRangeInt(I::Int, IDX::Int, LEN::Int) => (bitRangeInt(I, IDX, LEN) +Int (1 <<Int (LEN -Int 1))) modInt (1 <<Int LEN) -Int (1 <<Int (LEN -Int 1))
+
+  rule I1:Int divInt I2:Int => (I1 -Int (I1 modInt I2)) /Int I2
+  requires I2 =/=Int 0
+  rule
+    I1:Int modInt I2:Int
+  =>
+    ((I1 %Int absInt(I2)) +Int absInt(I2)) %Int absInt(I2)
+  requires I2 =/=Int 0    [concrete, simplification]
+
+  rule minInt(I1:Int, I2:Int) => I1 requires I1 <=Int I2
+  rule minInt(I1:Int, I2:Int) => I2 requires I1 >=Int I2
+
+  rule I1:Int =/=Int I2:Int => notBool (I1 ==Int I2)
+  rule (I1:Int dividesInt I2:Int) => (I2 %Int I1) ==Int 0
+
+  syntax Int ::= freshInt(Int)    [freshGenerator, function, total, private]
+  rule freshInt(I:Int) => I
+endmodule
+

      IEEE 754 Floating-point Numbers

      +

      Provided here is the syntax of an implementation of arbitrary-precision
      +floating-point arithmetic in K based on a generalization of the IEEE 754
      +standard. This type is hooked to an implementation of floats provided by the
      +backend.

      +

      The syntax of ordinary floating-point values in K consists of an optional sign
      +(+ or -) followed by an optional integer part, followed by a decimal point,
      +followed by an optional fractional part. Either the integer part or the
      +fractional part must be specified. The mantissa is followed by an optional
      +exponent part, which consists of an e or E, an optional sign (+ or -),
      +and an integer. The expoennt is followed by an optional suffix, which can be
      +either f, F, d, D, or pNxM where N and M are positive integers.
      +p and x can be either upper or lowercase.

      +

      The value of a floating-point literal is computed as follows: First the
      +mantissa is read as a rational number. Then it is multiplied by 10 to the
      +power of the exponent, which is interpreted as an integer, and defaults to
      +zero if it is not present. Finally, it is rounded to the nearest possible
      +value in a floating-point type represented like an IEEE754 floating-point type,
      +with the number of bits of precision and exponent specified by the suffix.
      +A suffix of f or f represents the IEEE binary32 format. A suffix of d
      +or D, or no suffix, represents the IEEE binary64 format. A suffix of
      +pNxM (either upper or lowercase) specifies exactly N bits of precision and
      +M bits of exponent. The number of bits of precision is assumed to include
      +any optional 1 that precedes the IEEE 754 mantissa. In other words, p24x8
      +is equal to the IEEE binary32 format, and p53x11 is equal to the IEEE
      +binary64 format.

      +
      module FLOAT-SYNTAX
+  syntax Float [hook(FLOAT.Float)]
+  syntax Float ::= r"([\\+-]?[0-9]+(\\.[0-9]*)?|\\.[0-9]+)([eE][\\+-]?[0-9]+)?([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(1)]
+  syntax Float ::= r"[\\+-]?Infinity([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(3)]
+  syntax Float ::= r"NaN([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(3)]
+endmodule
+
+module FLOAT
+  imports FLOAT-SYNTAX
+  imports private BOOL
+  imports private INT-SYNTAX
+

      Float precision

      + +

      You can retrieve the number of bits of precision in a Float.

      +
        syntax Int ::= precisionFloat(Float) [function, total, hook(FLOAT.precision)]
+

      Float exponent bits

      + +

      You can retrieve the number of bits of exponent range in a Float.

      +
        syntax Int ::= exponentBitsFloat(Float) [function, total, hook(FLOAT.exponentBits)]
+

      Float exponent

      + +

      You can retrieve the value of the exponent bits of a Float as an integer.

      +
        syntax Int ::= exponentFloat(Float) [function, total, hook(FLOAT.exponent)]
+

      Float sign

      + +

      You can retrieve the value of the sign bit of a Float as a boolean. True
      +means the sign bit is set.

      +
        syntax Bool ::= signFloat(Float)      [function, total, hook(FLOAT.sign)]
+

      Float special values

      + +

      You can check whether a Float value is infinite or Not-a-Number.

      +
        syntax Bool ::= isNaN(Float)          [function, total, smt-hook(fp.isNaN), hook(FLOAT.isNaN)]
+                | isInfinite(Float)     [function, total]
+

      Float arithmetic

      + +

      You can:

      +
        +
      • Compute the unary negation --Float of a float. --Float X is distinct
        +from 0.0 -Float X. For example, 0.0 -Float 0.0 is positive zero.
        +--Float 0.0 is negative zero.
        • +
      • Compute the exponentation ^Float of two floats.
        • +
      • Compute the product *Float, quotient /Float, or remainder %Float of two
        +floats. The remainder is computed based on rounding the quotient of the two
        +floats to the nearest integer.
        • +
      • Compute the sum +Float or difference -Float of two floats.
        • +
      +
        syntax Float ::= "--Float" Float             [function, total, smt-hook(fp.neg), hook(FLOAT.neg)]
+                 > Float "^Float" Float        [function, left, latex({#1}^{#2}), hook(FLOAT.pow)]
+                 > left:
+                   Float "*Float" Float        [function, left, smt-hook((fp.mul roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\ast_{\scriptstyle\it Float}}{#2}), hook(FLOAT.mul)]
+                 | Float "/Float" Float        [function, left, smt-hook((fp.div roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\div_{\scriptstyle\it Float}}{#2}), hook(FLOAT.div)]
+                 | Float "%Float" Float        [function, left, smt-hook((fp.rem roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\%_{\scriptstyle\it Float}}{#2}), hook(FLOAT.rem)]
+                 > left:
+                   Float "+Float" Float        [function, left, smt-hook((fp.add roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{+_{\scriptstyle\it Float}}{#2}), hook(FLOAT.add)]
+                 | Float "-Float" Float        [function, left, smt-hook((fp.sub roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{-_{\scriptstyle\it Float}}{#2}), hook(FLOAT.sub)]
+

      Floating-point mathematics

      + +

      You can:

      +
        +
      • Compute the Nth integer root rootFloat of a float.
        • +
      • Compute the absolute value absFloat of a float.
        • +
      • Round a floating-point number to a specified precision and exponent
        +range (roundFloat). The resulting Float will yield the specified values
        +when calling precisionFloat and exponentBitsFloat and when performing
        +further computation.
        • +
      • Round a float to the next lowest floating-point value which is an integer
        +(floorFloat).
        • +
      • Round a float to the next highest floating-point value which is an integer
        +(ceilFloat).
        • +
      • Round a float to the next closest floating-point value which is an integer, in
        +the direction of zero (truncFloat).
        • +
      • Compute the natural exponential expFloat of a float (i.e. e^x).
        • +
      • Compute the natural logarithm logFloat of a float.
        • +
      • Compute the sine sinFloat of a float.
        • +
      • Compute the cosine cosFloat of a float.
        • +
      • Compute the tangent tanFlooat of a float.
        • +
      • Compute the arcsine asinFloat of a float.
        • +
      • Compute the arccosine acosFloat of a float.
        • +
      • Compute the arctangent atanFloat of a float.
        • +
      • Compute the arctangent atan2Float of two floats.
        • +
      • Compute the maximum maxFloat of two floats.
        • +
      • Compute the minimum minFloat of two floats.
        • +
      • Compute the square root sqrtFloat of a float.
        • +
      • Compute the largest finite value expressible in a specified precision and
        +exponent range (maxValueFloat).
        • +
      • Compute the smallest positive finite value expressible in a specified
        +precision and exponent range (minValueFloat).
        • +
      +
        syntax Float ::= rootFloat(Float, Int)        [function, hook(FLOAT.root)]
+                 | absFloat(Float)              [function, total, smt-hook(fp.abs), hook(FLOAT.abs)]
+                 | roundFloat(Float, precision: Int, exponentBits: Int)  [function, hook(FLOAT.round)]
+                 | floorFloat(Float)            [function, total, hook(FLOAT.floor)]
+                 | ceilFloat(Float)             [function, total, hook(FLOAT.ceil)]
+                 | truncFloat(Float)            [function, total, hook(FLOAT.trunc)]
+                 | expFloat(Float)              [function, total, hook(FLOAT.exp)]
+                 | logFloat(Float)              [function, hook(FLOAT.log)]
+                 | sinFloat(Float)              [function, total, hook(FLOAT.sin)]
+                 | cosFloat(Float)              [function, total, hook(FLOAT.cos)]
+                 | tanFloat(Float)              [function, hook(FLOAT.tan)]
+                 | asinFloat(Float)             [function, hook(FLOAT.asin)]
+                 | acosFloat(Float)             [function, hook(FLOAT.acos)]
+                 | atanFloat(Float)             [function, total, hook(FLOAT.atan)]
+                 | atan2Float(Float, Float)     [function, hook(FLOAT.atan2)]
+                 | maxFloat(Float, Float)       [function, smt-hook(fp.max), hook(FLOAT.max)]
+                 | minFloat(Float, Float)       [function, smt-hook(fp.min), hook(FLOAT.min)]
+                 | sqrtFloat(Float)             [function]
+                 | maxValueFloat(precision: Int, exponentBits: Int)      [function, hook(FLOAT.maxValue)]
+                 | minValueFloat(precision: Int, exponentBits: Int)      [function, hook(FLOAT.minValue)]
+

      Floating-point comparisons

      + +

      Compute whether a float is less than or equasl to, less than, greater than or
      +equal to, greater than, equal, or unequal to another float. Note that
      +X ==Float Y and X ==K Y might yield different values. The latter should be
      +used in cases where you want to compare whether two values of sort Float
      +contain the same term. The former should be used when you want to implement
      +the == operator of a programming language. In particular, NaN =/=Float NaN
      +is true, because NaN compares unequal to all values, including itself, in
      +IEEE 754 arithmetic. 0.0 ==Float -0.0 is also true.

      +
        syntax Bool ::= Float "<=Float" Float       [function, smt-hook(fp.leq), latex({#1}\mathrel{\leq_{\scriptstyle\it Float}}{#2}), hook(FLOAT.le)]
+                | Float "<Float" Float        [function, smt-hook(fp.lt), latex({#1}\mathrel{<_{\scriptstyle\it Float}}{#2}), hook(FLOAT.lt)]
+                | Float ">=Float" Float       [function, smt-hook(fp.geq), latex({#1}\mathrel{\geq_{\scriptstyle\it Float}}{#2}), hook(FLOAT.ge)]
+                | Float ">Float" Float        [function, smt-hook(fg.gt), latex({#1}\mathrel{>_{\scriptstyle\it Float}}{#2}), hook(FLOAT.gt)]
+                | Float "==Float" Float       [function, comm, smt-hook(fp.eq), latex({#1}\mathrel{==_{\scriptstyle\it Float}}{#2}), hook(FLOAT.eq), klabel(_==Float_)]
+                | Float "=/=Float" Float      [function, comm, smt-hook((not (fp.eq #1 #2))), latex({#1}\mathrel{\neq_{\scriptstyle\it Float}}{#2})]
+
+  rule F1:Float =/=Float F2:Float => notBool (F1 ==Float F2)
+

      Conversion between integer and float

      + +

      You can convert an integer to a floating-point number with the specified
      +precision and exponent range. You can also convert a floating-point number
      +to the nearest integer. This operation rounds to the nearest integer, but it
      +also avoids the double-rounding that is present in ceilFloat and floorFloat
      +if the nearest integer is not representable in the specified floating-point
      +type.

      +
        syntax Float ::= Int2Float(Int, precision: Int, exponentBits: Int)    [function, latex({\\it{}Int2Float}), hook(FLOAT.int2float)]
+  syntax Int ::= Float2Int(Float)    [function, total, latex({\\it{}Float2Int}), hook(FLOAT.float2int)]
+

      Implementation of Floats

      + +

      The remainder of this section consists of an implementation in K of some of the
      +operators above.

      +
        rule sqrtFloat(F:Float) => rootFloat(F, 2)
+
+  rule isInfinite(F:Float) => F >Float maxValueFloat(precisionFloat(F), exponentBitsFloat(F)) orBool F <Float --Float maxValueFloat(precisionFloat(F), exponentBitsFloat(F))
+
+endmodule
+

      Strings

      +

      Provided here is the syntax of an implementation of Unicode strings in K. This
      +type is hooked to an implementation of strings provided by the backend. The
      +implementation is currently incomplete and does not fully support encodings
      +and code points beyond the initial 256 code points of the Basic Latin and
      +Latin-1 Supplement blocks. In the future, there may be breaking changes to
      +the semantics of this module in order to support this functionality.

      +

      The syntax of strings in K is delineated by double quotes. Inside the double
      +quotes, any character can appear verbatim except double quotes, backslash,
      +newline, and carriage return. K also supports the following escape sequences:

      +
        +
      • " - the " character
        • +
      • \ - the \ character
        • +
      • \n - newline character
        • +
      • \r - carriage return character
        • +
      • \t - tab character
        • +
      • \f - form feed character
        • +
      • \xFF - \x followed by two hexadecimal characters indicates a code point
        +between 0x00 and 0xff
        • +
      • \uFFFF - \u followed by four hexadecimal characters indicates a code point
        +between 0x0000 and 0xffff
        • +
      • \UFFFFFFFF - \U followed by eight hexadecimal characters indicates a code
        +point between 0x000000 and 0x10ffff
        • +
      +
      module STRING-SYNTAX
+  syntax String [hook(STRING.String)]
+  syntax String ::= r"[\\\"](([^\\\"\\n\\r\\\\])|([\\\\][nrtf\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2})|([\\\\][u][0-9a-fA-F]{4})|([\\\\][U][0-9a-fA-F]{8}))*[\\\"]"      [token]
+endmodule
+
+module STRING-COMMON
+  imports STRING-SYNTAX
+  imports private INT
+  imports private FLOAT-SYNTAX
+  imports private K-EQUAL
+  imports private BOOL
+

      String concatenation

      + +

      You can concatenate two strings in O(N) time. For successive concatenation
      +operations, it may be better to use the STRING-BUFFER module.

      +
        syntax String ::= String "+String" String    [function, total, left, latex({#1}+_{\scriptstyle\it String}{#2}), hook(STRING.concat)]
+

      String length

      + +

      You can get the length of a string in O(1) time.

      +
        syntax Int ::= lengthString ( String ) [function, total, hook(STRING.length)]
+

      Character and integer conversion

      + +

      You can convert between a character (as represented by a string containing
      +a single code point) and an integer in O(1) time.

      +
        syntax String ::= chrChar ( Int )      [function, hook(STRING.chr)]
+  syntax Int ::= ordChar ( String )      [function, hook(STRING.ord)]
+

      String substring

      + +

      You can compute a substring of a string in O(N) time (where N is the
      +length of the substring). There are two important facts to note:

      +
        +
      1. the range generated includes the character at startIndex but excludes the
        +character at endIndex, i.e., the range is [startIndex..endIndex).
        2. +
      2. this function is only defined on valid indices (i.e., it is defined when
        +startIndex < endIndex and endIndex is less than or equal to the string
        +length).
        3. +
      +
        syntax String ::= substrString ( String , startIndex: Int , endIndex: Int ) [function, total, hook(STRING.substr)]
+
      + +

      You can find the first (respectively, last) occurrence of a substring, starting
      +at a certain index, in another string in O(N*M) time.
      +Returns -1 if the substring is not found.

      +
        syntax Int ::= findString ( haystack: String , needle: String , index: Int )   [function, hook(STRING.find)]
+  syntax Int ::= rfindString ( haystack: String , needle: String , index: Int )  [function, hook(STRING.rfind)]
+
      + +

      You can find the first (respectively, last) occurrence of one of the characters
      +of the search string, starting at a certain index, in another string in
      +O(N*M) time.

      +
        syntax Int ::= findChar ( haystack: String , needles: String , index: Int )     [function, hook(STRING.findChar)]
+  syntax Int ::= rfindChar ( haystack: String , needles: String , index: Int )    [function, hook(STRING.rfindChar)]
+

      String and Bool conversion

      + +
        syntax String ::= Bool2String(Bool) [function, total]
+  rule Bool2String(true)  => "true"
+  rule Bool2String(false) => "false"
+
        syntax Bool ::= String2Bool(String) [function]
+  rule String2Bool("true")  => true
+  rule String2Bool("false") => false
+

      String and float conversion

      + +

      You can convert between a String and a Float. The String will be
      +represented in the syntax of the Float sort (see the section on the FLOAT
      +module above for details of that syntax). Which particular string is returned
      +by Float2String is determined by the backend, but the same Float is
      +guaranteed to return the same String, and converting that String back to a
      +Float is guaranteed to return the original Float.

      +

      You can also convert a Float to a string in a particular syntax using the
      +variant of Float2String with a format. In this case, the resulting string
      +is one which results directly from passing that format to mpfr_printf. This
      +functionality may not be supported on backends that do not use Gnu MPFR to
      +implement floating-point numbers.

      +
        syntax String ::= Float2String ( Float )              [function, total, hook(STRING.float2string)]
+  syntax String ::= Float2String ( Float , format: String )     [function, klabel(FloatFormat), hook(STRING.floatFormat)]
+  syntax Float  ::= String2Float ( String )             [function, hook(STRING.string2float)]
+

      String and integer conversions

      + +

      You can convert between a String and an Int. The String will be represented
      +in the syntax of the INT module (i.e., a nonempty sequence of digits
      +optionally prefixed by a sign). When converting from an Int to a String,
      +the sign will not be present unless the integer is negative.

      +

      You can also convert between a String and an Int in a particular radix.
      +This radix can be anywhere between 2 and 36. For a radix 2 <= N <= 10, the
      +digits 0 to N-1 will be used. For a radix 11 <= N <= 36, the digits 0 to 9
      +and the first N-10 letters of the Latin alphabet will be used. Both uppercase
      +and lowercase letters are supported by String2Base. Whether the letters
      +returned by Base2String are upper or lowercase is determined by the backend,
      +but the backend will consistently choose one or the other.

      +
        syntax Int    ::= String2Int   ( String )             [function, hook(STRING.string2int)]
+  syntax String ::= Int2String   ( Int )                [function, total, hook(STRING.int2string)]
+  syntax String ::= Base2String  ( Int , base: Int )          [function, hook(STRING.base2string)]
+  syntax Int    ::= String2Base  ( String , base: Int )       [function, hook(STRING.string2base)]
+

      String count and replace

      + +

      You can replace one, some, or all occurrences of a string within another
      +string in O(N*M) time. The replaceAll, replace, and replaceFirst methods
      +are identical, except replaceFirst replaces exactly one ocurrence of the
      +string, the first occurrence. replace replaces the first times occurrences.
      +And replaceAll replaces every occurrence.

      +

      You can also count the number of times a string occurs within another string
      +using countAllOccurrences.

      +
        syntax String ::= "replaceAll" "(" haystack: String "," needle: String "," replacement: String ")"      [function, total, hook(STRING.replaceAll)]
+  syntax String ::= "replace" "(" haystack: String "," needle: String "," replacement: String "," times: Int ")" [function, hook(STRING.replace)]
+  syntax String ::= "replaceFirst" "(" haystack: String "," needle: String "," replacement: String ")"    [function, total, hook(STRING.replaceFirst)]
+  syntax Int ::= "countAllOccurrences" "(" haystack: String "," needle: String ")"            [function, total, hook(STRING.countAllOccurrences)]
+

      String equality and lexicographic comparison

      + +

      You can compare whether two strings are equal or unequal, or whether one string
      +is less than, less than or equal to, greater than, or greater than or equal to
      +another according to the natural lexicographic ordering of strings.

      +
        syntax Bool ::= String "==String" String  [function, total, comm, hook(STRING.eq)]
+                | String "=/=String" String [function, total, comm, hook(STRING.ne)]
+  rule S1:String =/=String S2:String => notBool (S1 ==String S2)
+
+  syntax Bool ::= String  "<String" String [function, total, hook(STRING.lt)]
+                | String "<=String" String [function, total, hook(STRING.le)]
+                | String  ">String" String [function, total, hook(STRING.gt)]
+                | String ">=String" String [function, total, hook(STRING.ge)]
+

      Implementation of Strings

      + +

      What follows is a few String hooks which are deprecated and only are supported
      +on certain outdated backends of K, as well as an implementation of several
      +of the above operations in K.

      +
        syntax String ::= categoryChar(String)       [function, hook(STRING.category)]
+                  | directionalityChar(String) [function, hook(STRING.directionality)]
+
+  syntax String ::= "newUUID" [function, hook(STRING.uuid), impure]
+
+  rule S1:String <=String S2:String => notBool (S2 <String S1)
+  rule S1:String >String S2:String => S2 <String S1
+  rule S1:String >=String S2:String => notBool (S1 <String S2)
+
+  rule findChar(S1:String, S2:String, I:Int) => #if findString(S1, substrString(S2, 0, 1), I) ==Int -1 #then findChar(S1, substrString(S2, 1, lengthString(S2)), I) #else #if findChar(S1, substrString(S2, 1, lengthString(S2)), I) ==Int -1 #then findString(S1, substrString(S2, 0, 1), I) #else minInt(findString(S1, substrString(S2, 0, 1), I), findChar(S1, substrString(S2, 1, lengthString(S2)), I)) #fi #fi requires S2 =/=String ""
+  rule findChar(_, "", _) => -1
+  rule rfindChar(S1:String, S2:String, I:Int) => maxInt(rfindString(S1, substrString(S2, 0, 1), I), rfindChar(S1, substrString(S2, 1, lengthString(S2)), I)) requires S2 =/=String ""
+  rule rfindChar(_, "", _) => -1
+
+  rule countAllOccurrences(Source:String, ToCount:String) => 0
+            requires findString(Source, ToCount, 0) <Int 0
+  rule countAllOccurrences(Source:String, ToCount:String) => 1 +Int countAllOccurrences(substrString(Source, findString(Source, ToCount, 0) +Int lengthString(ToCount), lengthString(Source)), ToCount)
+            requires findString(Source, ToCount, 0) >=Int 0
+
+  rule replaceFirst(Source:String, ToReplace:String, Replacement:String) => substrString(Source, 0, findString(Source, ToReplace, 0))
+                +String Replacement +String substrString(Source, findString(Source, ToReplace, 0) +Int lengthString(ToReplace), lengthString(Source))
+                requires findString(Source, ToReplace, 0) >=Int 0
+  rule replaceFirst(Source:String, ToReplace:String, _:String) => Source
+        requires findString(Source, ToReplace, 0) <Int 0
+
+
+  rule replace(Source:String, ToReplace:String, Replacement:String, Count:Int) =>
+       substrString(Source, 0, findString(Source, ToReplace, 0)) +String Replacement +String
+       replace(substrString(Source, findString(Source, ToReplace, 0) +Int lengthString(ToReplace), lengthString(Source)), ToReplace, Replacement, Count -Int 1)
+        requires Count >Int 0
+  rule replace(Source:String, _, _, 0) => Source
+  rule replaceAll(Source:String, ToReplace:String, Replacement:String) => replace(Source, ToReplace, Replacement, countAllOccurrences(Source, ToReplace))
+
+endmodule
+
+module STRING-KORE [kore, symbolic]
+  imports private K-EQUAL
+  imports STRING-COMMON
+
+  rule S1:String ==K S2:String => S1 ==String S2 [simplification]
+
+endmodule
+
+module STRING
+  imports STRING-COMMON
+  imports STRING-KORE
+endmodule
+

      String Buffers

      +

      It is a well known fact that repeated string concatenations are quadratic
      +in performance whereas use of an efficient mutable representation of arrays
      +can yield linear performance. We thus provide such a sort, the StringBuffer
      +sort. Axiomatically, it is implemented below on symbolic backends using the
      +String module. However, on concrete backends it provides an efficient
      +implementation of string concatenation. There are three operations:

      +
        +
      • .StringBuffer creates a new StringBuffer with current content equal
        +to the empty string.
        • +
      • +String takes a StringBuffer and a String and appends the String to
        +the end of the StringBuffer
        • +
      • StringBuffer2String converts a StringBuffer to a String. This operation
        +copies the string so that subsequent modifications to the StringBuffer
        +will not change the value of the String returned by this function.
        • +
      +
      module STRING-BUFFER-IN-K [symbolic]
+  imports private BASIC-K
+  imports STRING
+
+  syntax StringBuffer ::= ".StringBuffer" [function, total]
+  syntax StringBuffer ::= StringBuffer "+String" String [function, total, avoid]
+  syntax StringBuffer ::= String
+  syntax String ::= StringBuffer2String ( StringBuffer ) [function, total]
+
+  rule {SB:String +String S:String}<:StringBuffer => (SB +String S)::String
+  rule .StringBuffer => ""
+  rule StringBuffer2String(S:String) => S
+endmodule
+
+module STRING-BUFFER-HOOKED [concrete]
+  imports private BASIC-K
+  imports STRING
+
+  syntax StringBuffer [hook(BUFFER.StringBuffer)]
+  syntax StringBuffer ::= ".StringBuffer" [function, total, hook(BUFFER.empty), impure]
+  syntax StringBuffer ::= StringBuffer "+String" String [function, total, hook(BUFFER.concat), avoid]
+  syntax String ::= StringBuffer2String ( StringBuffer ) [function, total, hook(BUFFER.toString)]
+endmodule
+
+module STRING-BUFFER
+  imports STRING-BUFFER-HOOKED
+  imports STRING-BUFFER-IN-K
+endmodule
+

      Byte Arrays

      +

      Provided here is the syntax of an implementation of fixed-width arrays of Bytes
      +in K. This type is hooked to an implementation of bytes provided by the
      +backend. In concrete backends, this representation is mutable and thus multiple
      +references can occur to the same Bytes object and when one is modified, the
      +others are also modified. Care should be taken not to rely on this fact however
      +as this is not the case in symbolic backends and thus you will experience
      +divergent behavior unless the Bytes type is used in a manner that preserves
      +consistency.

      +
      module BYTES-SYNTAX
+  imports private STRING-SYNTAX
+
+  syntax Bytes [hook(BYTES.Bytes)]
+  syntax Bytes ::= r"b[\\\"](([\\x20\\x21\\x23-\\x5B\\x5D-\\x7E])|([\\\\][tnfr\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2}))*[\\\"]"      [token]
+endmodule
+
      module BYTES-STRING-ENCODE [kore, symbolic]
+  imports BYTES-SYNTAX
+

      Encoding/decoding between Bytes and String

      + +

      You can encode/decode between Bytes and String using UTF-8, UTF-16LE, UTF-16BE, UTF-32LE, and UTF-32BE

      +
          syntax String ::= decodeBytes ( encoding: String , contents: Bytes ) [function, hook(BYTES.decodeBytes)]
+    syntax Bytes ::= encodeBytes ( encoding: String , contents: String ) [function, hook(BYTES.encodeBytes)]
+endmodule
+
      module BYTES-HOOKED
+  imports STRING-SYNTAX
+  imports BYTES-SYNTAX
+  imports BYTES-STRING-ENCODE
+

      Empty byte array

      + +

      The byte array of length zero is represented by .Bytes.

      +
        syntax Bytes ::= ".Bytes" [function, total, hook(BYTES.empty)]
+

      Endianness

      + +

      When converting to/from an integer, byte arrays can be treated as either little
      +endian (ie, least significant byte first) or big endian (ie, most significant
      +byte first).

      +
        syntax Endianness ::= "LE" [klabel(littleEndianBytes), symbol]
+                      | "BE" [klabel(bigEndianBytes), symbol]
+

      Signedness

      + +

      When converting to/from an integer, byte arrays can be treated as either signed
      +or unsigned.

      +
        syntax Signedness ::= "Signed" [klabel(signedBytes), symbol]
+                      | "Unsigned" [klabel(unsignedBytes), symbol]
+

      Integer and Bytes conversion

      + +

      You can convert from a Bytes to an Int. In order to do this, the endianness
      +and signedness of the Bytes must be provided. The resulting integer is
      +created by means of interpreting the Bytes as either a twos-complement
      +representation, or an unsigned representation, of an integer, in the specified
      +byte order.

      +

      You can also convert from an Int to a Bytes. This comes in two variants.
      +In the first, the length of the resulting Bytes in bytes is explicitly
      +specified. If the length is greater than the highest set bit in the magnitude
      +of the integer, the result is padded with 0 bits if the number is positive
      +and 1 bits if the number is negative. If the length is less than the highest
      +bit set in the magnitude of the integer, the most-significant bits of the
      +integer will be truncated. The endianness of the resulting Bytes object
      +is as specified.

      +

      In the second variant, both endianness and signedness are specified, and
      +the resulting Bytes object will be the smallest number of bytes necessary
      +for the resulting Bytes object to be convertible back to the original integer
      +via Bytes2Int. In other words, if the highest bit set in the magnitude of the
      +integer is N, then the byte array will be at least N+1 bits long, rounded up
      +to the nearest byte.

      +
        syntax Int ::= Bytes2Int(Bytes, Endianness, Signedness) [function, total, hook(BYTES.bytes2int)]
+  syntax Bytes ::= Int2Bytes(length: Int, Int, Endianness) [function, total, hook(BYTES.int2bytes)]
+                 | Int2Bytes(Int, Endianness, Signedness) [function, total, klabel(Int2BytesNoLen)]
+

      String and Bytes conversion

      + +

      You can convert between a Bytes and a String in O(N) time. The resulting
      +value is a copy of the original and will not be affected by subsequent
      +mutations of the input or output value.

      +
        syntax String ::= Bytes2String(Bytes) [function, total, hook(BYTES.bytes2string)]
+  syntax Bytes ::= String2Bytes(String) [function, total, hook(BYTES.string2bytes)]
+

      Bytes update

      + +

      You can set the value of a particular byte in a Bytes object in O(1) time.
      +The result is #False if value is not in the range [0..255] or if index
      +is not a valid index (ie, less than zero or greater than or equal to the length
      +of the Bytes term).

      +
        syntax Bytes ::= Bytes "[" index: Int "<-" value: Int "]" [function, hook(BYTES.update)]
+

      Bytes lookup

      + +

      You can get the value of a particular byte in a Bytes object in O(1) time.
      +The result is #False if index is not a valid index (see above).

      +
        syntax Int ::= Bytes "[" Int "]" [function, hook(BYTES.get)]
+

      Bytes substring

      + +

      You can get a new Bytes object containing a range of bytes from the input
      +Bytes in O(N) time (where N is the length of the substring). The range
      +of bytes included is [startIndex..endIndex). The resulting Bytes is
      +a copy and mutations to it do not affect mutations to the original Bytes.
      +The result is #False if startIndex or endIndex are not valid.

      +
        syntax Bytes ::= substrBytes(Bytes, startIndex: Int, endIndex: Int) [function, hook(BYTES.substr)]
+

      Multiple bytes update

      + +

      You can modify a Bytes to return a Bytes which is equal to dest except
      +the N elements starting at index are replaced with the contents of src in
      +O(N) time. This does not create a new Bytes object and will instead modify
      +the original on concrete backends. The result is #False if index + N
      +is not a valid index.

      +
        syntax Bytes ::= replaceAtBytes(dest: Bytes, index: Int, src: Bytes) [function, hook(BYTES.replaceAt)]
+

      Bytes padding

      + +

      You can create a new Bytes object which is at least length bytes long
      +by taking the input sequence and padding it on the right (respectively, on the
      +left) with the specified value. This does not create a new Bytes object
      +if the input is already at least length bytes long, and will instead
      +return the input unchanged. The result is #False if value is not in the
      +range [0..255], or if the length is negative.

      +
        syntax Bytes ::= padRightBytes(Bytes, length: Int, value: Int) [function, hook(BYTES.padRight)]
+                 | padLeftBytes(Bytes, length: Int, value: Int) [function, hook(BYTES.padLeft)]
+

      Bytes reverse

      + +

      You can reverse a Bytes object in O(N) time. This does not create a new
      +Bytes object and will instead modify the original on concrete backends.

      +
        syntax Bytes ::= reverseBytes(Bytes) [function, total, hook(BYTES.reverse)]
+

      Bytes length

      + +

      You can get the length of a Bytes term in O(1) time.

      +
        syntax Int ::= lengthBytes(Bytes) [function, total, hook(BYTES.length), smtlib(lengthBytes)]
+

      Bytes concatenation

      + +

      You can create a new Bytes object by concatenating two Bytes objects
      +together in O(N) time.

      +
        syntax Bytes ::= Bytes "+Bytes" Bytes [function, total, hook(BYTES.concat), right]
+
+endmodule
+

      Implementation of Bytes

      + +

      The remainder of this module consists of an implementation of some of the
      +operators listed above in K.

      +
      module BYTES-CONCRETE [concrete]
+  imports BYTES-HOOKED
+endmodule
+
+module BYTES-KORE [kore]
+  imports BYTES-HOOKED
+  imports BYTES-SYMBOLIC-CEIL
+endmodule
+
+module BYTES-SYMBOLIC-CEIL [symbolic, kore]
+  imports BYTES-HOOKED
+  imports private INT
+  imports private BOOL
+
+  rule #Ceil(padRightBytes(_, LEN, VAL)) => {(0 <=Int LEN andBool 0 <=Int VAL andBool VAL <Int 256) #Equals true} [simplification]
+  rule #Ceil(padLeftBytes(_, LEN, VAL))  => {(0 <=Int LEN andBool 0 <=Int VAL andBool VAL <Int 256) #Equals true} [simplification]
+endmodule
+
+module BYTES
+  imports BYTES-CONCRETE
+  imports BYTES-KORE
+  imports private INT
+
+  rule Int2Bytes(I::Int, E::Endianness, Unsigned) => Int2Bytes((log2Int(I) +Int 8) /Int 8, I, E)
+    requires I >Int 0
+  rule Int2Bytes(0, _::Endianness, _) => .Bytes
+  rule Int2Bytes(I::Int, E::Endianness, Signed) => Int2Bytes((log2Int(I) +Int 9) /Int 8, I, E)
+    requires I >Int 0
+  rule Int2Bytes(I::Int, E::Endianness, Signed) => Int2Bytes((log2Int(~Int I) +Int 9) /Int 8, I, E)
+    requires I <Int -1
+  rule Int2Bytes(-1, E::Endianness, Signed) => Int2Bytes(1, -1, E)
+endmodule
+

      Program identifiers

      +

      Provided here is an implementation for program identifiers in K. Developers
      +of semantics for a particular language may wish to use their own implementation
      +instead of the one provided here if their syntax differs from the syntax
      +defined below. However, this is provided for convenience for developers who
      +do not care about the lexical syntax of identifiers.

      +

      Provided are the following pieces of functionality:

      +
        +
      • Id2String - Convert an Id to a String containing its name
        • +
      • String2Id - Convert a String to an Id with the specified name
        • +
      • !X:Id - You can get a fresh identifier distinct from any previous identifier
        +generated by this syntax.
        • +
      +
      module ID-SYNTAX-PROGRAM-PARSING
+  imports BUILTIN-ID-TOKENS
+  syntax Id ::= r"(?<![A-Za-z0-9\\_])[A-Za-z\\_][A-Za-z0-9\\_]*"     [prec(1), token]
+              | #LowerId                                             [token]
+              | #UpperId                                             [token]
+endmodule
+
+module ID-SYNTAX
+  syntax Id [token]
+endmodule
+
+module ID-COMMON
+  imports ID-SYNTAX
+  imports private STRING
+
+  syntax String ::= Id2String ( Id )    [function, total, hook(STRING.token2string)]
+  syntax Id ::= String2Id (String) [function, total, hook(STRING.string2token)]
+  syntax Id ::= freshId(Int)    [freshGenerator, function, total, private]
+
+  rule freshId(I:Int) => String2Id("_" +String Int2String(I))
+endmodule
+
+module ID
+  imports ID-COMMON
+endmodule
+

      Equality and conditionals

      +

      Provided here are implementations of two important primitives in K:

      +
        +
      • ==K - the equality between two terms. Returns true if they are equal
        +and false if they are not equal.
        • +
      • #if #then #else #fi - polymorphic conditional function. If the first
        +argument evaluates to true, the second argument is returned. Otherwise,
        +the third argument is returned. Note that this does not short-circuit on
        +symbolic backends.
        • +
      +
      module K-EQUAL-SYNTAX
+  imports private BOOL
+  imports private BASIC-K
+
+  syntax Bool ::= left:
+                  K "==K" K           [function, total, comm, smt-hook(=), hook(KEQUAL.eq), klabel(_==K_), symbol, latex({#1}\mathrel{=_K}{#2}), group(equalEqualK)]
+                | K "=/=K" K          [function, total, comm, smt-hook(distinct), hook(KEQUAL.ne), klabel(_=/=K_), symbol, latex({#1}\mathrel{\neq_K}{#2}), group(notEqualEqualK)]
+
+  syntax priorities equalEqualK notEqualEqualK > boolOperation mlOp
+
+  syntax {Sort} Sort ::= "#if" Bool "#then" Sort "#else" Sort "#fi"   [function, total, smt-hook(ite), hook(KEQUAL.ite)]
+
+endmodule
+
+module K-EQUAL-KORE [kore, symbolic]
+  import private BOOL
+  import K-EQUAL-SYNTAX
+
+  rule K1:Bool ==K K2:Bool => K1 ==Bool K2 [simplification]
+  rule {K1 ==K K2 #Equals true} => {K1 #Equals K2} [simplification]
+  rule {true #Equals K1 ==K K2} => {K1 #Equals K2} [simplification]
+  rule {K1 ==K K2 #Equals false} => #Not({K1 #Equals K2}) [simplification]
+  rule {false #Equals K1 ==K K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=K K2 #Equals true} => #Not({K1 #Equals K2}) [simplification]
+  rule {true #Equals K1 =/=K K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=K K2 #Equals false} => {K1 #Equals K2} [simplification]
+  rule {false #Equals K1 =/=K K2} => {K1 #Equals K2} [simplification]
+
+endmodule
+
+module K-EQUAL
+  import private BOOL
+  import K-EQUAL-SYNTAX
+  import K-EQUAL-KORE
+
+  rule K1:K =/=K K2:K => notBool (K1 ==K K2)
+
+  rule #if C:Bool #then B1::K #else _ #fi => B1 requires C
+  rule #if C:Bool #then _ #else B2::K #fi => B2 requires notBool C
+
+endmodule
+

      Meta operations

      +

      Provided below are a few miscellaneous, mostly deprecated functions in K.
      +It is not recommended to use any of them directly as they are largely
      +unsupported in modern K. There are a few exceptions:

      +
        +
      • #getenv - Returns the value of an environment variable
        • +
      • #parseKORE - Takes a String containing a K intermediate representation of
        +a term such as is returned by kast -o kore and converts it to a term.
        +This is NOT type-safe. The responsibility is on the user to ensure that the
        +string they provide is a valid representation of a term of the sort exactly
        +equal to the sort where the function appears.
        • +
      • #kompiledDirectory - Returns the path to the current compiled K definition
        +directory.
        • +
      • #unparseKORE = Takes a K term and converts it to a string.
        • +
      +
      module K-REFLECTION
+  imports BASIC-K
+  imports STRING
+
+  syntax K ::= "#configuration" [function, impure, hook(KREFLECTION.configuration)]
+  syntax String ::= #sort(K) [function, hook(KREFLECTION.sort)]
+  syntax KItem ::= #fresh(String)   [function, hook(KREFLECTION.fresh), impure]
+  syntax KItem ::= getKLabel(K)  [function, hook(KREFLECTION.getKLabel)]
+
+  syntax K ::= #getenv(String) [function, impure, hook(KREFLECTION.getenv)]
+
+  syntax String ::= #kompiledDirectory() [function, hook(KREFLECTION.kompiledDir)]
+
+  // meaningful only for the purposes of compilation to a binary, otherwise
+  // undefined
+  syntax List ::= #argv() [function, hook(KREFLECTION.argv)]
+
+  // Takes as input a string and returns a K term
+  syntax {Sort} Sort ::= #parseKORE(String) [function, hook(KREFLECTION.parseKORE)]
+  syntax {Sort} String ::= #unparseKORE(Sort) [function, hook(KREFLECTION.printKORE)]
+  syntax IOError ::= "#noParse" "(" String ")" [klabel(#noParse), symbol]
+
+endmodule
+

      I/O in K

      +

      Concrete execution in K supports I/O operations. This functionality is not
      +supported during symbolic execution, because symbolic execution must exist
      +completely free of side-effects, and I/O is an irreducible type of side effect.
      +However, it is useful in many cases when defining concrete execution to be able
      +to make reference to I/O operations.

      +

      The design of these I/O operations is based on the POSIX standard, for the most
      +part. For example, the #read K function maps to the read POSIX function. We
      +do not at this time have a higher-level API for I/O, but this may be
      +implemented at some point in the future.

      +

      I/O operations generally return either their result, or an IOError term
      +corresponding to the errno returned by the underlying system call.

      +
      module K-IO
+  imports private LIST
+  imports private STRING
+  imports private INT
+

      I/O errors

      + +

      Aside from EOF, which is returned by #getc if the file is at end-of-file, all
      +of the below I/O errors correspond to possible values for errno after calling
      +a library function. If the errno returned is not one of the below errnos
      +known to K, #unknownIOError is returned along with the integer errno value.

      +
        syntax IOError ::= "#EOF" [klabel(#EOF), symbol] | #unknownIOError(errno: Int) [symbol]
+                   | "#E2BIG" [klabel(#E2BIG), symbol]
+                   | "#EACCES" [klabel(#EACCES), symbol]
+                   | "#EAGAIN" [klabel(#EAGAIN), symbol]
+                   | "#EBADF" [klabel(#EBADF), symbol]
+                   | "#EBUSY" [klabel(#EBUSY), symbol]
+                   | "#ECHILD" [klabel(#ECHILD), symbol]
+                   | "#EDEADLK" [klabel(#EDEADLK), symbol]
+                   | "#EDOM" [klabel(#EDOM), symbol]
+                   | "#EEXIST" [klabel(#EEXIST), symbol]
+                   | "#EFAULT" [klabel(#EFAULT), symbol]
+                   | "#EFBIG" [klabel(#EFBIG), symbol]
+                   | "#EINTR" [klabel(#EINTR), symbol]
+                   | "#EINVAL" [klabel(#EINVAL), symbol]
+                   | "#EIO" [klabel(#EIO), symbol]
+                   | "#EISDIR" [klabel(#EISDIR), symbol]
+                   | "#EMFILE" [klabel(#EMFILE), symbol]
+                   | "#EMLINK" [klabel(#EMLINK), symbol]
+                   | "#ENAMETOOLONG" [klabel(#ENAMETOOLONG), symbol]
+                   | "#ENFILE" [klabel(#ENFILE), symbol]
+                   | "#ENODEV" [klabel(#ENODEV), symbol]
+                   | "#ENOENT" [klabel(#ENOENT), symbol]
+                   | "#ENOEXEC" [klabel(#ENOEXEC), symbol]
+                   | "#ENOLCK" [klabel(#ENOLCK), symbol]
+                   | "#ENOMEM" [klabel(#ENOMEM), symbol]
+                   | "#ENOSPC" [klabel(#ENOSPC), symbol]
+                   | "#ENOSYS" [klabel(#ENOSYS), symbol]
+                   | "#ENOTDIR" [klabel(#ENOTDIR), symbol]
+                   | "#ENOTEMPTY" [klabel(#ENOTEMPTY), symbol]
+                   | "#ENOTTY" [klabel(#ENOTTY), symbol]
+                   | "#ENXIO" [klabel(#ENXIO), symbol]
+                   | "#EPERM" [klabel(#EPERM), symbol]
+                   | "#EPIPE" [klabel(#EPIPE), symbol]
+                   | "#ERANGE" [klabel(#ERANGE), symbol]
+                   | "#EROFS" [klabel(#EROFS), symbol]
+                   | "#ESPIPE" [klabel(#ESPIPE), symbol]
+                   | "#ESRCH" [klabel(#ESRCH), symbol]
+                   | "#EXDEV" [klabel(#EXDEV), symbol]
+                   | "#EWOULDBLOCK" [klabel(#EWOULDBLOCK), symbol]
+                   | "#EINPROGRESS" [klabel(#EINPROGRESS), symbol]
+                   | "#EALREADY" [klabel(#EALREADY), symbol]
+                   | "#ENOTSOCK" [klabel(#ENOTSOCK), symbol]
+                   | "#EDESTADDRREQ" [klabel(#EDESTADDRREQ), symbol]
+                   | "#EMSGSIZE" [klabel(#EMSGSIZE), symbol]
+                   | "#EPROTOTYPE" [klabel(#EPROTOTYPE), symbol]
+                   | "#ENOPROTOOPT" [klabel(#ENOPROTOOPT), symbol]
+                   | "#EPROTONOSUPPORT" [klabel(#EPROTONOSUPPORT), symbol]
+                   | "#ESOCKTNOSUPPORT" [klabel(#ESOCKTNOSUPPORT), symbol]
+                   | "#EOPNOTSUPP" [klabel(#EOPNOTSUPP), symbol]
+                   | "#EPFNOSUPPORT" [klabel(#EPFNOSUPPORT), symbol]
+                   | "#EAFNOSUPPORT" [klabel(#EAFNOSUPPORT), symbol]
+                   | "#EADDRINUSE" [klabel(#EADDRINUSE), symbol]
+                   | "#EADDRNOTAVAIL" [klabel(#EADDRNOTAVAIL), symbol]
+                   | "#ENETDOWN" [klabel(#ENETDOWN), symbol]
+                   | "#ENETUNREACH" [klabel(#ENETUNREACH), symbol]
+                   | "#ENETRESET" [klabel(#ENETRESET), symbol]
+                   | "#ECONNABORTED" [klabel(#ECONNABORTED), symbol]
+                   | "#ECONNRESET" [klabel(#ECONNRESET), symbol]
+                   | "#ENOBUFS" [klabel(#ENOBUFS), symbol]
+                   | "#EISCONN" [klabel(#EISCONN), symbol]
+                   | "#ENOTCONN" [klabel(#ENOTCONN), symbol]
+                   | "#ESHUTDOWN" [klabel(#ESHUTDOWN), symbol]
+                   | "#ETOOMANYREFS" [klabel(#ETOOMANYREFS), symbol]
+                   | "#ETIMEDOUT" [klabel(#ETIMEDOUT), symbol]
+                   | "#ECONNREFUSED" [klabel(#ECONNREFUSED), symbol]
+                   | "#EHOSTDOWN" [klabel(#EHOSTDOWN), symbol]
+                   | "#EHOSTUNREACH" [klabel(#EHOSTUNREACH), symbol]
+                   | "#ELOOP" [klabel(#ELOOP), symbol]
+                   | "#EOVERFLOW" [klabel(#EOVERFLOW), symbol]
+

      I/O result sorts

      + +

      Here we see sorts defined to contain either an Int or an IOError, or
      +either a String or an IOError. These sorts are used to implement the
      +return sort of functions that may succeed, in which case they return a value,
      +or may fail, in which case their return value indicates an error and the
      +error indicated is returned via errno.

      +
        syntax IOInt ::= Int | IOError
+  syntax IOString ::= String | IOError
+

      Opening a file

      + +

      You can open a file in K using #open. An optional mode indicates the file
      +open mode, which can have any value allowed by the fopen function in C.
      +The returned value is the file descriptor that was opened, or an error.

      +
        syntax IOInt ::= "#open" "(" path: String ")" [function]
+               | "#open" "(" path: String "," mode: String ")" [function, hook(IO.open), impure]
+
+  rule #open(S:String) => #open(S:String, "r+")
+

      Get/set position in file

      + +

      You can get the current offset in a file using #tell. You can also seek
      +to a particular offset using #seek or #seekEnd. #seek is implemented via
      +a call to lseek with the SEEK_SET whence. #seekEnd is implemented via a
      +call to lseek with the SEEK_END whence. You can emulate the SEEK_CUR
      +whence by means of #seek(FD, #tell(FD) +Int Offset).

      +
        syntax IOInt ::= "#tell" "(" fd: Int ")" [function, hook(IO.tell), impure]
+  syntax K ::= "#seek" "(" fd: Int "," index: Int ")" [function, hook(IO.seek), impure]
+             | "#seekEnd" "(" fd: Int "," fromEnd: Int ")" [function, hook(IO.seekEnd), impure]
+

      Read from file

      + +

      You can read a single character from a file using #getc. #EOF is returned
      +if you are at end-of-fie.

      +

      You can also read up to length characters in a file using #read. The
      +resulting read characters are returned, which may be fewer characters than
      +requested. A string of zero length being returned indicates end-of-file.

      +
        syntax IOInt ::= "#getc" "(" fd: Int ")"             [function, hook(IO.getc), impure]
+  syntax IOString ::= "#read" "(" fd: Int "," length: Int ")"    [function, hook(IO.read), impure]
+

      Write to file

      + +

      You can write a single character to a file using #putc. You can also write
      +a string to a file using #write. The returned value on success is .K.

      +
        syntax K ::= "#putc" "(" fd: Int "," value: Int ")"      [function, hook(IO.putc), impure]
+             | "#write" "(" fd: Int "," value: String ")" [function, hook(IO.write), impure]
+

      Closing a file

      + +

      You can close a file using #close. The returned value on success is .K.

      +
        syntax K ::= "#close" "(" fd: Int ")" [function, hook(IO.close), impure]
+

      Locking/unlocking a file

      + +

      You can lock or unlock parts of a file using the #lock and #unlock
      +functions. The lock starts at the beginning of the file and continues for
      +endIndex bytes. Note that Unix systems do not actually prevent locked files
      +from being read and modified; you will have to lock both sides of a concurrent
      +access to guarantee exclusivity.

      +
        syntax K ::= "#lock" "(" fd: Int "," endIndex: Int ")" [function, hook(IO.lock), impure]
+             | "#unlock" "(" fd: Int "," endIndex: Int ")" [function, hook(IO.unlock), impure]
+

      Networking

      + +

      You can accept a connection on a socket using #accept, or shut down the
      +write end of a socket with #shutdownWrite. Note that facility is not provided
      +for opening, binding, and listening on sockets. These functions are implemented
      +in order to support creating stateful request/response servers where the
      +request loop is implemented using rewriting in K, but the connection
      +initialization is written in native code and linked into the LLVM backend.

      +
        syntax IOInt ::= "#accept" "(" fd: Int ")" [function, hook(IO.accept), impure]
+  syntax K ::= "#shutdownWrite" "(" fd: Int ")" [function, hook(IO.shutdownWrite), impure]
+

      Time

      + +

      You can get the current time in seconds since midnight UTC on January 1, 1970
      +using #time.

      +
        syntax Int ::= "#time" "(" ")" [function, hook(IO.time), impure]
+

      Builtin file descriptors

      + +

      Provided here are functions that return the file descriptor for standard input,
      +standard output, and standard error.

      +
        syntax Int ::= "#stdin"   [function, total]
+                | "#stdout" [function, total]
+                | "#stderr" [function, total]
+
+  rule #stdin => 0
+  rule #stdout => 1
+  rule #stderr => 2
+

      Shell access

      + +

      You can execute a command using the shell using the #system operator. Care
      +must be taken to sanitize inputs to this function or security issues may
      +result. Note that K has no facility for reasoning about logic that happens
      +outside its process, so any functionality that you wish to be able to formally
      +reason about in K should not be implemented via the #system operator.

      +
        syntax KItem ::= #system ( String ) [function, hook(IO.system), impure]
+                 | "#systemResult" "(" Int /* exit code */ "," String /* stdout */ "," String /* stderr */ ")" [klabel(#systemResult), symbol]
+

      Temporary files

      + +

      You can get a temporary file and open it atomically using the #mkstemp
      +operator. The resulting file will be closed and deleted when K rewriting ends.
      +For more info on the argument to #mkstemp, see man mkstemp.

      +
        syntax IOFile ::= #mkstemp(template: String) [function, hook(IO.mkstemp), impure]
+  syntax IOFile ::= IOError
+                  | "#tempFile" "(" path: String "," fd: Int ")" [klabel(#tempFile), symbol]
+

      Deleting a file

      + +

      You can delete a file using its absolute or relative path using the #remove
      +operator. It returns .K on success or an IOError on failure.

      +
        syntax K ::= #remove(path: String) [function, total, hook(IO.remove), impure]
+

      Logging

      + +

      You can log information to disk using the #logToFile operator. Semantically,
      +this operator returns .K. However, it has a side effect that is not reasoned
      +about which is that value will be written to a uniquely-identified file
      +containing name in its name. The file is only flushed to disk when rewriting
      +finishes.

      +
        syntax K ::= #logToFile(name: String, value: String) [function, total, hook(IO.log), impure, returnsUnit, symbol]
+

      Strings can also be logged via the logging mechanisms available to the backend.
      +On the LLVM backend, this just means logging the text to standard error. On the
      +Haskell backend, a log message of type InfoUserLog is created with the
      +specified text.

      +
        syntax K ::= #log(value: String) [function, total, hook(IO.logString), impure, returnsUnit, symbol]
+

      Terms can also be logged to standard error in surface syntax, rather than as
      +KORE using #trace. This operator has similar semantics to #logToFile (i.e.
      +it returns .K, but prints as an impure side effect). Note that calling
      +#trace is equivalent to invoking the kprint tool for the first term that is
      +logged, which requires re-parsing the underlying K definition. Subsequent calls
      +do not incur this overhead again; the definition is cached.

      +
        syntax K ::= #trace(value: KItem) [function, total, hook(IO.traceTerm), impure, returnsUnit, symbol]
+             | #traceK(value: K)    [function, total, hook(IO.traceTerm), impure, returnsUnit, symbol]
+

      Implementation of high-level I/O streams in K

      + +

      Below is an implementation of the stream="stdin" and stream="stdout"
      +cell attributes in K. You should not refer to these symbols or modules directly
      +in your definition. It is provided only so that the K compiler can make use of
      +it. For more information on how to use this feature, refer to IMP++ in the K
      +tutorial.

      +
        syntax Stream ::= #buffer(K)
+                  | #istream(Int)
+                  | #parseInput(String, String)
+                  | #ostream(Int)
+
+endmodule
+
+// NOTE: DO NOT DIRECTLY IMPORT *-STREAM MODULES
+// These stream modules will be automatically instantiated and implicitly imported
+// into the main module when `stream` attributes appear in configuration cells.
+// Only `Stream` productions and `[stream]` rules will be imported.
+// The cell name will be replaced with the one of the main configuration.
+
+module STDIN-STREAM
+  imports K-IO
+  imports K-REFLECTION
+  imports LIST
+  imports INT
+  imports BOOL
+
+  configuration <stdin> ListItem(#buffer($STDIN:String)) ListItem($IO:String) ListItem(#istream(#stdin)) </stdin>
+
+  // read one character at a time until we read whitespace
+  rule [stdinGetc]:
+       <stdin>
+       ListItem(#parseInput(_:String, Delimiters:String))
+       ListItem(#buffer(S:String => S +String chrChar({#getc(N)}:>Int)))
+       ListItem("on")
+       ListItem(#istream(N:Int))
+       </stdin>
+    requires findChar(S, Delimiters, 0) ==Int -1 // [stdin]
+       [stream, priority(200)]
+
+  // when we reach whitespace, if it parses create a ListItem
+  rule [stdinParseString]:
+       <stdin>
+       (ListItem(#parseInput("String", Delimiters:String)) => ListItem(S))
+       ListItem(#buffer(S:String => ""))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1 // [stdin]
+       [stream]
+
+  // a hack: handle the case when we read integers without the help of the IO server
+  rule [stdinParseInt]:
+       <stdin>
+       (ListItem(#parseInput("Int", Delimiters:String))
+       => ListItem(String2Int(substrString(S, 0, findChar(S, Delimiters, 0)))))
+       ListItem(#buffer(S:String => substrString(S,findChar(S, Delimiters, 0) +Int 1, lengthString(S))))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1
+       andBool lengthString(S) >Int 1 // [stdin]
+       [stream]
+
+  rule [stdinTrim]:
+       <stdin>
+       ListItem(#parseInput(Sort:String, Delimiters:String))
+       ListItem(#buffer(S:String => substrString(S, 1, lengthString(S))))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1
+       andBool Sort =/=String "String"
+       andBool lengthString(S) <=Int 1 // [stdin]
+       [stream]
+
+  // NOTE: This unblocking rule will be instantiated and inserted carefully
+  // when necessary according to user-defined rules, since otherwise it will
+  // lead to a diverging (i.e., non-terminating) transition system definition.
+  // Currently, it supports only a simple pattern matching on the top of the
+  // input stream cell, e.g.,
+  //   rule <k> read() => V ... </k> <in> ListItem(V:Int) => .List ...  </in>
+  // Non-supported rules that refer to the input stream cell in a sophisticated
+  // way will get stuck in concrete execution mode with real IO enabled (i.e.,
+  // under `--io on` option), while they will still work in symbolic execution
+  // mode or concrete execution mode with real IO disabled (i.e., under `--io
+  // off`, `--search`, or `--debug` options).
+  //
+  // TODO: More patterns need to be supported as well. In that case, we need to
+  // have a way to specify such patterns.
+  rule [stdinUnblock]:
+       <stdin>
+         (.List => ListItem(#parseInput(?Sort:String, ?Delimiters:String)))
+         ListItem(#buffer(_:String))
+         ...
+       </stdin>
+
+  /*
+  syntax Stream ::= "#noIO"
+
+  rule ListItem(#buffer(_))
+       (ListItem(#noIO) ListItem(#istream(_:Int)) => .List) [stdin]
+  */
+
+endmodule
+
+module STDOUT-STREAM
+  imports K-IO
+  imports LIST
+  imports STRING
+
+  configuration <stdout> ListItem(#ostream(#stdout)) ListItem($IO:String) ListItem(#buffer("")) </stdout>
+//configuration <stderr> ListItem(#ostream(#stderr)) ListItem($IO:String) ListItem(#buffer("")) </stderr>
+
+  rule [stdoutBufferFloat]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String Float2String(F)))
+       (ListItem(F:Float) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+  rule [stdoutBufferInt]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String Int2String(I)))
+       (ListItem(I:Int) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+  rule [stdoutBufferString]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String S))
+       (ListItem(S:String) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+
+  // Send first char from the buffer to the server
+  rule [stdoutWrite]:
+       <stdout>
+       ListItem(#ostream(N:Int => {#write(N, S) ~> N:Int}:>Int))
+       ListItem("on")
+       ListItem(#buffer(S:String => ""))
+       _:List
+       </stdout>
+    requires S =/=String "" // [stdout, stderr]
+       [stream, priority(30)]
+
+  /*
+  syntax Stream ::= "#noIO"
+
+  rule ListItem(#buffer(Buffer:String => Buffer +String Float2String(F)))
+       (ListItem(F:Float) => .List)
+       _:List [stdout, stderr]
+  rule ListItem(#buffer(Buffer:String => Buffer +String Int2String(I)))
+       (ListItem(I:Int) => .List)
+       _:List [stdout, stderr]
+  rule ListItem(#buffer(Buffer:String => Buffer +String S))
+       (ListItem(S:String) => .List)
+       _:List [stdout, stderr]
+
+  rule (ListItem(#ostream(_:Int)) ListItem(#noIO) => .List)
+       ListItem(#buffer(_))
+       _:List [stdout, stderr]
+  */
+
+endmodule
+

      Machine Integers

      +

      Provided here is an implementation of arbitrarily large fixed-precision binary
      +integers in K. This type is hooked to an implementation of integers provided
      +by the backend, and in particular makes use of native machine integers for
      +certain sizes of integer. For arbitrary-precision integers, see the INT
      +module above.

      +

      The syntax of machine integers in K is the same as arbitrary-precision integers
      +(i.e., an optional sign followed by a sequence of digits) except that machine
      +integers always end in a suffix pN where N is an integer indicating the
      +width in bits of the integer. The MInt sort is parametric, and this is
      +reflected in the literals. For example, the sort of 0p8 is MInt{8}.

      +
      module MINT-SYNTAX
+  /*@\section{Description} The MInt implements machine integers of arbitrary
+   * bit width represented in 2's complement. */
+  syntax {Width} MInt{Width} [hook(MINT.MInt)]
+
+  /*@ Machine integer of bit width and value. */
+  syntax {Width} MInt{Width} ::= r"[\\+-]?[0-9]+[pP][0-9]+" [token, prec(2), hook(MINT.literal)]
+endmodule
+
+module MINT
+  imports MINT-SYNTAX
+  imports private INT
+  imports private BOOL
+

      Bitwidth of MInt

      + +

      You can get the number of bits of width in an MInt using bitwidthMInt.

      +
        syntax {Width} Int ::= bitwidthMInt(MInt{Width})   [function, total, hook(MINT.bitwidth)]
+

      Int and MInt conversions

      + +

      You can convert from an MInt to an Int using the MInt2Signed and
      +MInt2Unsigned functions. an MInt does not have a sign; its sign is instead
      +reflected in how operators interpret its value either as a signed integer or as
      +an unsigned integer. Thus, you can interpret a MInt as a signed integer witth
      +MInt2Signed, or as an unsigned integer respectively using MInt2Unsigned.

      +

      You can also convert from an Int to an MInt using Int2MInt. Care must
      +be given to ensure that the sort context where the Int2MInt operator appears
      +has the correct bitwidth, as this will influence the width of the resulting
      +MInt.

      +
        syntax {Width} Int ::= MInt2Signed(MInt{Width})     [function, total, hook(MINT.svalue)]
+                       | MInt2Unsigned(MInt{Width})     [function, total, hook(MINT.uvalue), smt-hook(bv2int)]
+
+  syntax {Width} MInt{Width} ::= Int2MInt(Int) [function, total, hook(MINT.integer), smt-hook(int2bv)]
+

      MInt min and max values

      + +

      You can get the minimum and maximum values of a signed or unsigned MInt
      +with az specified bit width using sminMInt, smaxMInt, uminMInt, and
      +umaxMInt.

      +
        syntax Int ::= sminMInt(Int)    [function]
+               | smaxMInt(Int)    [function]
+               | uminMInt(Int)    [function]
+               | umaxMInt(Int)    [function]
+  rule sminMInt(N:Int) => 0 -Int (1 <<Int (N -Int 1))
+  rule smaxMInt(N:Int) => (1 <<Int (N -Int 1)) -Int 1
+  rule uminMInt(_:Int) => 0
+  rule umaxMInt(N:Int) => (1 <<Int N) -Int 1
+

      MInt bounds checking

      + +

      You can check whether a specified Int will be represented in an MInt
      +with a specified width without any loss of precision when interpreted as
      +a signed or unsigned integer using soverflowMInt and uoverflowMInt.

      +
        syntax Bool ::= soverflowMInt(width: Int, Int)   [function]
+                | uoverflowMInt(width: Int, Int)   [function]
+  rule
+    soverflowMInt(N:Int, I:Int)
+  =>
+    I <Int sminMInt(N) orBool I >Int smaxMInt(N)
+  rule
+    uoverflowMInt(N:Int, I:Int)
+  =>
+    I <Int uminMInt(N) orBool I >Int umaxMInt(N)
+

      MInt arithmetic

      + +

      You can:

      +
        +
      • Compute the bitwise complement ~MInt of an MInt.
        • +
      • Compute the unary negation --MInt of an MInt.
        • +
      • Compute the product *MInt of two MInts.
        • +
      • Compute the quotient /sMInt of two MInts interpreted as signed integers.
        • +
      • Compute the modulus %sMInt of two MInts interpreted as signed integers.
        • +
      • Compute the quotient /uMInt of two MInts interpreted as unsigned
        +integers.
        • +
      • Compute the modulus %uMInt of two MInts interpreted as unsigned integers.
        • +
      • Compute the sum +MInt of two MInts.
        • +
      • Compute the difference -MInt of two MInts.
        • +
      • Compute the left shift <<MInt of two MInts. The second MInt is always
        +interpreted as positive.
        • +
      • Compute the arithmetic right shift >>aMInt of two MInts. The second
        +MInt is always interpreted as positve.
        • +
      • Compute the logical right shift >>lMInt of two MInts. The second MInt
        +is always interpreted as positive.
        • +
      • Compute the bitwise and &MInt of two MInts.
        • +
      • Compute the bitwise xor xorMInt of two MInts.
        • +
      • Compute the bitwise inclusive or |MInt of two MInts.
        • +
      +
        syntax {Width} MInt{Width} ::= "~MInt" MInt{Width} [function, total, hook(MINT.not), smt-hook(bvnot)]
+                               | "--MInt" MInt{Width} [function, total, hook(MINT.neg), smt-hook(bvuminus)]
+                               > left:
+                                 MInt{Width} "*MInt" MInt{Width} [function, total, hook(MINT.mul), smt-hook(bvmul)]
+                               | MInt{Width} "/sMInt" MInt{Width} [function, hook(MINT.sdiv), smt-hook(bvsdiv)]
+                               | MInt{Width} "%sMInt" MInt{Width} [function, hook(MINT.srem), smt-hook(bvsrem)]
+                               | MInt{Width} "/uMInt" MInt{Width} [function, hook(MINT.udiv), smt-hook(bvudiv)]
+                               | MInt{Width} "%uMInt" MInt{Width} [function, hook(MINT.urem), smt-hook(bvurem)]
+                               > left:
+                                 MInt{Width} "+MInt" MInt{Width} [function, total, hook(MINT.add), smt-hook(bvadd)]
+                               | MInt{Width} "-MInt" MInt{Width} [function, total, hook(MINT.sub), smt-hook(bvsub)]
+                               > left:
+                                 MInt{Width} "<<MInt" MInt{Width} [function, hook(MINT.shl), smt-hook(bvshl)]
+                               | MInt{Width} ">>aMInt" MInt{Width} [function, hook(MINT.ashr), smt-hook(bvashr)]
+                               | MInt{Width} ">>lMInt" MInt{Width} [function, hook(MINT.lshr), smt-hook(bvlshr)]
+                               > left:
+                                 MInt{Width} "&MInt" MInt{Width} [function, total, hook(MINT.and), smt-hook(bvand)]
+                               > left:
+                                 MInt{Width} "xorMInt" MInt{Width} [function, total, hook(MINT.xor), smt-hook(bvxor)]
+                               > left:
+                                 MInt{Width} "|MInt" MInt{Width} [function, total, hook(MINT.or), smt-hook(bvor)]
+

      MInt comparison

      + +

      You can compute whether one MInt is less than, less than or equal to, greater
      +than, or greater than or equal to another MInt when interpreted as signed
      +or unsigned integers. You can also compute whether one MInt is equal to or
      +unequal to another MInt.

      +
        syntax {Width} Bool ::= MInt{Width} "<sMInt" MInt{Width} [function, total, hook(MINT.slt), smt-hook(bvslt)]
+                        | MInt{Width} "<uMInt" MInt{Width} [function, total, hook(MINT.ult), smt-hook(bvult)]
+                        | MInt{Width} "<=sMInt" MInt{Width} [function, total, hook(MINT.sle), smt-hook(bvsle)]
+                        | MInt{Width} "<=uMInt" MInt{Width} [function, total, hook(MINT.ule), smt-hook(bvule)]
+                        | MInt{Width} ">sMInt" MInt{Width} [function, total, hook(MINT.sgt), smt-hook(bvsgt)]
+                        | MInt{Width} ">uMInt" MInt{Width} [function, total, hook(MINT.ugt), smt-hook(bvugt)]
+                        | MInt{Width} ">=sMInt" MInt{Width} [function, total, hook(MINT.sge), smt-hook(bvsge)]
+                        | MInt{Width} ">=uMInt" MInt{Width} [function, total, hook(MINT.uge), smt-hook(bvuge)]
+                        | MInt{Width} "==MInt" MInt{Width} [function, total, hook(MINT.eq), smt-hook(=)]
+                        | MInt{Width} "=/=MInt" MInt{Width} [function, total, hook(MINT.ne), smt-hook(distinct)]
+

      MInt min/max

      + +

      You can compute the signed minimum sMinMInt, the signed maximum sMaxMInt,
      +the unsigned minimum uMinMInt, and the unsigned maximum uMaxMInt of two
      +MInts.

      +
        syntax {Width} MInt{Width} ::= sMaxMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.smax), smt-hook((ite (bvslt #1 #2) #2 #1))]
+                               | sMinMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.smin), smt-hook((ite (bvslt #1 #2) #1 #2))]
+                               | uMaxMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.umax), smt-hook((ite (bvult #1 #2) #2 #1))]
+                               | uMinMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.umin), smt-hook((ite (bvult #1 #2) #1 #2))]
+

      MInt to MInt conversion

      + +

      You can convert an MInt of one width to another width with roundMInt.
      +The resulting MInt will be truncated starting from the most significant bit
      +if the resulting width is smaller than the input. The resulting MInt will be
      +zero-extended with the same low-order bits if the resulting width is larger
      +than the input.

      +
        syntax {Width1, Width2} MInt{Width1} ::= roundMInt(MInt{Width2}) [function, total, hook(MINT.round)]
+
      endmodule
+

      Strategies

      +

      K supports a built-in strategy language that allows you to control how rules
      +apply. In order to enable it, simply import the STRATEGY module in your
      +definition. This includes the following basic strategy constructs:

      +
        +
      • ^ Category:#RuleTag - This is a strategy that indicates that you should
        +apply a rule with the specified category exactly once. By default, all rules
        +get the regular tag. This can be changed on individual rules with the tag
        +attribute.
        • +
      • ~ Category:#RuleTag - This is the state the strategy cell will be in
        +after a rule has applied. In other words, all rules that do not mention the
        +strategy cell are automatically instrumented so that they rewrite the current
        +top of the <s> cell from ^ to ~
        • +
      • <s> - This is the strategy cell and contains the current strategy.
        • +
      • #STUCK() - By default, a rule is automatically inserted into the definition
        +which adds #STUCK() to the top of the <s> cell if no other rules apply
        +and if it is not already at the top of the <s> cell.
        • +
      +
      module STRATEGY
+    imports ML-SYNTAX
+    imports KVARIABLE-SYNTAX
+    imports private K-EQUAL
+
+    syntax #RuleTag ::= #KVariable
+
+    syntax Strategy ::= #STUCK()    [symbol]
+                      | "^" #RuleTag [symbol, klabel(#applyRule)]
+                      | "~" #RuleTag [symbol, klabel(#appliedRule)]
+
+    configuration <s> $STRATEGY:K </s>
+endmodule
+
+module RULE-TAG-SYNTAX
+    imports BUILTIN-ID-TOKENS
+
+    syntax #RuleTag ::= #LowerId [token]
+endmodule
+

      This is not a complete strategy language. However, it provides several basics.
      +The user can extend this strategy language into a complete strategy language
      +in one of three ways. First, they can import the DEFAULT-STRATEGY module,
      +which provides a very basic strategy that is essentially equivalent to
      +execution without a strategy. This can be useful if you wish to wholly
      +manipulate the strategy cell yourself within other rules.

      +
      module DEFAULT-STRATEGY-CONCRETE [concrete]
+    imports STRATEGY
+    imports RULE-TAG-SYNTAX
+    rule ~ regular => ^ regular [anywhere]
+endmodule
+
+module DEFAULT-STRATEGY-SYMBOLIC [symbolic]
+    imports STRATEGY
+    imports RULE-TAG-SYNTAX
+    rule <s> ~ regular => ^ regular ... </s>
+endmodule
+
+module DEFAULT-STRATEGY
+    imports STRATEGY
+    imports DEFAULT-STRATEGY-CONCRETE
+    imports DEFAULT-STRATEGY-SYMBOLIC
+
+    rule initSCell(_) => <s> ^ regular </s> [priority(25)]
+endmodule
+

      The second way you can extend the strategy language is with the
      +STRATEGY-ABSTRACT module. This provides a slightly more advanced set of
      +strategies which you can use to compose the basic strategies in the STRATEGY
      +module into more complex strategies. Note however that the functionality
      +provided is still relatively basic.

      +
      module STRATEGY-ABSTRACT
+    imports STRATEGY
+
+    syntax #RuleTag ::= "(" #RuleTag ")" [bracket]
+ // ----------------------------------------------
+
+    syntax Strategy ::= ".Strategy"
+                      | "(" Strategy ")" [bracket]
+ // ----------------------------------------------
+    rule <s> .Strategy => . ... </s>
+
+    syntax KItem ::= #catchSTUCK ( Strategy )
+ // -----------------------------------------
+    rule <s> #catchSTUCK(_) => . ... </s>
+    rule <s> #STUCK() ~> (_S:Strategy => .) ... </s>
+    rule <s> #STUCK() ~> #catchSTUCK(S) => S ... </s>
+
+    syntax Strategy ::= Strategy ";" Strategy [left]
+ // ------------------------------------------------
+    rule <s> S:Strategy ; S':Strategy => S ~> S' ... </s>
+
+    syntax #RuleTag ::= #RuleTag "|" #RuleTag [left, klabel(#alternateRule)]
+ // ------------------------------------------------------------------------
+    rule <s> ^ RT:#RuleTag | RT':#RuleTag => ^ RT ~> #catchSTUCK(^ RT') ... </s>
+
+    syntax #RuleTag ::= #RuleTag "*" [klabel(#repeatRule)]
+ // ------------------------------------------------------
+    rule <s> ^ RT:#RuleTag * => ^ RT ~> ^ RT * ~> #catchSTUCK(.Strategy) ... </s>
+endmodule
+

      The third mechanism for extending the strategy language is to define your own
      +strategy language. No special facility is required in K in order to compose
      +strategies together, so you are free to write whatever rules that manipulate
      +the strategy cell you want, and thus whatever more complex strategies you can
      +compose from the base builtin strategies.

      +

      K Language Features

      +

      Defined below is a series of modules used to parse inner syntax in K (ie, the
      +contents of rules, configuration declarations, and contexts).

      +

      Much of this file exists in tight correspondence with the K implementation, and
      +K will not work correctly if it is altered without corresponding changes to the
      +source code of the K tools.

      +

      Users should only import a few modules from this file. In particular, this
      +includes SORT-K, BASIC-K, ML-SYNTAX, DEFAULT-LAYOUT,
      +DEFAULT-CONFIGURATION, and K-AMBIGUITIES. The remaining modules should not
      +be imported by the user; they are used implicitly by the implementation of K.

      +

      Basic K Sorts

      +

      The SORT-K module declares the K sort, and nothing else.

      +
      module SORT-K
+  syntax K [hook(K.K)]
+endmodule
+

      The BASIC-K module declares the K, KItem, and KConfigVar sorts, and
      +imports the syntax of matching logic.

      +
      module BASIC-K
+  imports ML-SYNTAX
+  imports SORT-BOOL
+  syntax KItem [hook(K.KItem)]
+  syntax K     ::= KItem
+  syntax KConfigVar [token]
+  syntax KItem ::= KConfigVar
+endmodule
+

      KAST Abstract Syntax

      +

      Below is defined the abstract syntax of concrete terms in K, the KAST syntax.
      +Users should rarely if ever have to refer to this syntax; in general, it
      +suffices to use concrete syntax in rules, configuration declarations, contexts,
      +etc.

      +

      This syntax is used directly by the K implementation, and exists here as a
      +reference for the syntax of KAST, but it should not be imported directly by
      +the user.

      +
      module KSTRING
+  syntax KString ::= r"[\\\"](([^\\\"\\n\\r\\\\])|([\\\\][nrtf\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2})|([\\\\][u][0-9a-fA-F]{4})|([\\\\][U][0-9a-fA-F]{8}))*[\\\"]"      [token]
+  // optionally qualified strings, like in Scala "abc", i"abc", r"a*bc", etc.
+endmodule
+
+module BUILTIN-ID-TOKENS
+  syntax #LowerId ::= r"[a-z][a-zA-Z0-9]*"                    [prec(2), token]
+  syntax #UpperId ::= r"[A-Z][a-zA-Z0-9]*"                    [prec(2), token]
+endmodule
+
+module SORT-KBOTT
+  imports SORT-K
+  syntax KBott
+endmodule
+
+module KAST
+  imports BASIC-K
+  imports SORT-KBOTT
+  imports KSTRING
+  imports BUILTIN-ID-TOKENS
+
+  syntax KBott ::= "#token" "(" KString "," KString ")"  [klabel(#KToken), symbol]
+                 | "#klabel" "(" KLabel ")"              [klabel(#WrappedKLabel), symbol]
+                 | KLabel "(" KList ")"                  [klabel(#KApply), symbol]
+  syntax KItem ::= KBott
+
+  syntax KLabel ::= r"`(\\\\`|\\\\\\\\|[^`\\\\\\n\\r])+`" [token]
+                  | #LowerId                                   [token]
+                  | r"(?<![a-zA-Z0-9])[#a-z][a-zA-Z0-9]*"               [token, prec(1)]
+                       // something that doesn't collide with meta-variables
+
+  syntax KList ::= K
+                 | ".KList"          [klabel(#EmptyKList), symbol]
+                 | ".::KList"        [klabel(#EmptyKList), symbol]
+                 | KList "," KList   [klabel(#KList), left, assoc, unit(#EmptyKList), symbol, prefer]
+endmodule
+
+
+// To be used when parsing/pretty-printing ground configurations
+module KSEQ
+  imports KAST
+  imports K-TOP-SORT
+  syntax K ::= ".K"      [klabel(#EmptyK), symbol, unparseAvoid]
+             | "."       [klabel(#EmptyK), symbol]
+             | ".::K"    [klabel(#EmptyK), symbol, unparseAvoid]
+  syntax K ::= K "~>" K  [klabel(#KSequence), left, assoc, unit(#EmptyK), symbol]
+  syntax left #KSequence
+  syntax {Sort} Sort     ::= "(" Sort ")"    [bracket, group(defaultBracket), applyPriority(1)]
+endmodule
+

      Syntax of Matching Logic

      +

      K provides direct access to the symbols of Matching Logic, while giving them
      +their own concrete syntax distinct from the syntax of the KORE intermediate
      +representation. These symbols are primarily used during symbolic execution.
      +The LLVM Backend has relatively little understanding of Matching Logic directly
      +and use of these symbols directly in rules is likely to cause it to crash.
      +However, these symbols are necessary when providing lemmas and other types of
      +logical assistance to proofs and symbolic execution in the Haskell Backend.

      +

      The correspondance between K symbols and KORE symbols is as follows:

      +
        +
      • #Top - \top
        • +
      • #Bottom - \bottom
        • +
      • #Not - \not
        • +
      • #Ceil - \ceil
        • +
      • #Floor - \floor
        • +
      • #Equals - \equals
        • +
      • #And - \and
        • +
      • #Or - \or
        • +
      • #Implies - \implies
        • +
      • #Exists - \exists
        • +
      • #Forall - \forall
        • +
      • #AG - allPathGlobally
        • +
      • #wEF - weakExistsFinally
        • +
      • #wAF - weakAlwaysFinally
        • +
      +
      module ML-SYNTAX [not-lr1]
+  imports SORT-K
+
+  syntax {Sort} Sort ::= "#Top" [klabel(#Top), symbol, group(mlUnary)]
+                       | "#Bottom" [klabel(#Bottom), symbol, group(mlUnary)]
+                       | "#True" [klabel(#Top), symbol, group(mlUnary), unparseAvoid]
+                       | "#False" [klabel(#Bottom), symbol, group(mlUnary), unparseAvoid]
+                       | "#Not" "(" Sort ")" [klabel(#Not), symbol, mlOp, group(mlUnary)]
+
+  syntax {Sort1, Sort2} Sort2 ::= "#Ceil" "(" Sort1 ")" [klabel(#Ceil), symbol, mlOp, group(mlUnary)]
+                                | "#Floor" "(" Sort1 ")" [klabel(#Floor), symbol, mlOp, group(mlUnary)]
+                                | "{" Sort1 "#Equals" Sort1 "}" [klabel(#Equals), symbol, mlOp, group(mlEquals), comm, format(%1%i%n%2%d%n%3%i%n%4%d%n%5)]
+
+  syntax priorities mlUnary > mlEquals > mlAnd
+
+  syntax {Sort} Sort ::= Sort "#And" Sort [klabel(#And), symbol, assoc, left, comm, unit(#Top), mlOp, group(mlAnd), format(%i%1%d%n%2%n%i%3%d)]
+                       > Sort "#Or" Sort [klabel(#Or), symbol, assoc, left, comm, unit(#Bottom), mlOp, format(%i%1%d%n%2%n%i%3%d)]
+                       > Sort "#Implies" Sort [klabel(#Implies), symbol, mlOp, group(mlImplies), format(%i%1%d%n%2%n%i%3%d)]
+
+  syntax priorities mlImplies > mlQuantifier
+
+  syntax {Sort1, Sort2} Sort2 ::= "#Exists" Sort1 "." Sort2 [klabel(#Exists), symbol, mlOp, mlBinder, group(mlQuantifier)]
+                                | "#Forall" Sort1 "." Sort2 [klabel(#Forall), symbol, mlOp, mlBinder, group(mlQuantifier)]
+
+  syntax {Sort} Sort ::= "#AG" "(" Sort ")" [klabel(#AG), symbol, mlOp]
+                       | "#wEF" "(" Sort ")" [klabel(weakExistsFinally), symbol, mlOp]
+                       | "#wAF" "(" Sort ")" [klabel(weakAlwaysFinally), symbol, mlOp]
+endmodule
+

      Variables in K

      +

      Provided below is the syntax of variables in K. There are four types of
      +variables in K:

      +
        +
      1. Regular variables. These are denoted by variables that begin with an
        +underscore or a capital letter. These variables match exactly one value
        +and can be used to refer to it on the right-hand-side.
        2. +
      2. Fresh constants. These are denoted by variables that begin with an !. This
        +is a convenience syntax which can be used on the right-hand-side only, and
        +refer to a unique value of the specified sort which is distinct from any
        +other value that has been generated or will be generated by the !X syntax.
        +Note that this may not be distinct from values produced via other means.
        3. +
      3. Existential variables. This refers to variables that are existentially
        +quantified and begin with a ?. They are not required to appear on the
        +left-hand-side prior to appearing on the right-hand-side, and generally
        +refer to symbolic quantities that are introduced during rewriting. Refer to
        +K's documentation for more details.
        4. +
      4. Set variables. These are denoted by variables that begin with a @.
        +These variables refer to a set of values and are generally used when writing
        +simplification rules in the Haskell Backend. For more information, refer to
        +K's documentation.
        5. +
      +

      There is also a fifth type of "variable", although it is not technically a
      +variable. This refers to configuration variables, which are used to insert
      +values into the initial configuration that come from outside the semantics.
      +The most common of these is the $PGM variable, which conventionally contains
      +the program being executed and is placed in the <k> cell in the configuration
      +declaration. These "variables" begin with a $ and their values are populated
      +by the frontend prior to symbolic or concrete execution of a program.

      +
      module KVARIABLE-SYNTAX
+  syntax #KVariable
+endmodule
+
+// To be used when parsing/pretty-printing symbolic configurations
+module KSEQ-SYMBOLIC
+  imports KSEQ
+  imports ML-SYNTAX
+  imports KVARIABLE-SYNTAX
+
+  syntax #KVariable ::= r"(?<![A-Za-z0-9_\\$!\\?@])(\\!|\\?|@)?([A-Z][A-Za-z0-9'_]*|_|_[A-Z][A-Za-z0-9'_]*)"   [token, prec(1)]
+                      | #UpperId                                                          [token]
+  syntax KConfigVar ::= r"(?<![A-Za-z0-9_\\$!\\?@])(\\$)([A-Z][A-Za-z0-9'_]*)"            [token]
+  syntax KBott      ::= #KVariable
+  syntax KBott      ::= KConfigVar
+  syntax KLabel     ::= #KVariable
+endmodule
+

      Syntax of Cells

      +

      While the backend treats cells as regular productions like any other, the
      +frontend provides a significant amount of convenience notation for dealing with
      +groups of cells, in order to make writing modular definitions easier. As a
      +result, we need a syntax for groups of cells and for referring to cells within
      +rules, configuration declarations, and functions.

      +

      For historical reasons, the Bag sort is used to refer to groups of cells.
      +This may change in a future release. Users can combine cells in any order
      +by concatenating them together, and can refer to the absence of any cells with
      +the .Bag symbol. You can also refer to cells within a function by placing
      +the cell context symbol, [[ K ]] at the top of a rule, placing a function
      +symbol inside, and referring to cells afterwards. This implicitly inserts
      +a reference to the configuration at the time prior to the currently-applied
      +rule being applied which can be matched on within the function. Functions with
      +such context cannot be referred to in the initial configuration, because the
      +prior configuration does not yet exist.

      +
      module KCELLS
+  imports KAST
+
+  syntax Cell
+  syntax Bag ::= Bag Bag  [left, assoc, klabel(#cells), symbol, unit(#cells)]
+               | ".Bag"   [klabel(#cells), symbol]
+               | ".::Bag" [klabel(#cells), symbol]
+               | Cell
+  syntax Bag ::= "(" Bag ")" [bracket]
+  syntax KItem ::= Bag
+  syntax #RuleBody ::= "[" "[" K "]" "]" Bag    [klabel(#withConfig), symbol, avoid]
+  syntax non-assoc #withConfig
+  syntax Bag ::= KBott
+endmodule
+

      Users can also refer to cells in rules. When doing so, an optional ... can
      +be placed immediately after the start of the cell or immediately before the
      +end. In a cell whose contents are commutative, these are equivalent to one
      +another and are also equivalent to placing ... in both places. This means
      +that what is placed in the cell will be combined with the cell contents'
      +concatenation operator with an unnamed variable. In other words, you match on
      +some number of elements in the collection and do not care about the rest of
      +the collection.

      +

      In a cell whose contents are not commutative, the ... operators correspond
      +to a variable on the respective side of the contents of the cell that the
      +... appears. For example, <foo>... L </foo>, if L is a list, means
      +some number of elements followed by L. Note that not all combinations are
      +supported. Cells whose contents are sort K can only have ... appear at the
      +tail of the cell, and cells whose contents are sort List can only have ...
      +appear on at most one side in a single rule.

      +
      module RULE-CELLS
+  imports KCELLS
+  imports RULE-LISTS
+  // if this module is imported, the parser automatically
+  // generates, for all productions that have the attribute 'cell' or 'maincell',
+  // a production like below:
+  //syntax Cell ::= "<top>" #OptionalDots K #OptionalDots "</top>" [klabel(<top>)]
+
+  syntax #OptionalDots ::= "..." [klabel(#dots), symbol]
+                         | ""    [klabel(#noDots), symbol]
+
+  syntax Int
+  // this production will be added by the compiler to help handle bang variables,
+  // however, it is valuable to put it here because without this production, it
+  // will not exist at the point in time when rules and claims are parsed, and
+  // as a result it makes it very difficult to write proof claims over fragments
+  // of code that exercise rules containing bang variables. We put it here because
+  // this production will "vanish" after parsing finishes and not be picked up
+  // by the compiler, which is the behavior we want in this case since an actual
+  // production will be generated by the compiler later on.
+  syntax GeneratedCounterCell ::= "<generatedCounter>" Int "</generatedCounter>" [cell, klabel(<generatedCounter>), symbol, internal]
+endmodule
+

      Users can also declare cells in a configuration declaration. This generates a
      +specific set of productions that is used internally to implement the cell. The
      +most important of these is the cell itself, and attributes on this production
      +can be specified in an xml-attribute-like syntax.

      +

      You can also use an xml-short-tag-like syntax to compose configuration cells
      +together which were defined in different modules. However, it is a requirement
      +that any K definition have at most one fully-composed configuration; thus, all
      +other configuration declarations must appear composed within another
      +configuration declaration.

      +
      module CONFIG-CELLS
+  imports KCELLS
+  imports RULE-LISTS
+  syntax #CellName ::= r"[a-zA-Z][a-zA-Z0-9\\-]*"  [token, prec(1)]
+                     | #LowerId            [token]
+                     | #UpperId            [token]
+
+  syntax Cell ::= "<" #CellName #CellProperties ">" K "</" #CellName ">" [klabel(#configCell), symbol]
+  syntax Cell ::= "<" #CellName "/>" [klabel(#externalCell), symbol]
+
+  syntax #CellProperties ::= #CellProperty #CellProperties [klabel(#cellPropertyList), symbol]
+                           | ""                            [klabel(#cellPropertyListTerminator), symbol]
+  syntax #CellProperty ::= #CellName "=" KString           [klabel(#cellProperty), symbol]
+endmodule
+

      Syntax of Rules

      +

      Rules can have an optional requires clause or an ensures clause. For backwards-
      +compatibility, you can refer to the requires clause with both the requires
      +and when keywords; The latter, however, is deprecated and may be removed in
      +a future release.

      +

      The requires clause specifies the preconditions that must be true in order
      +for the rule to apply. The ensures clause specifies the information which
      +becomes true after the rule has applied. It is a requirement that information
      +present in the ensures clause refer to existential variables only.

      +

      When doing concrete execution, you can think of the requires clause as a
      +side-condition. In other words, even if the rule matches, it will not apply
      +unless the requires clause, which must be of sort Bool, evaluates to
      +true.

      +
      module REQUIRES-ENSURES
+  imports BASIC-K
+
+  syntax #RuleBody ::= K
+
+  syntax #RuleContent ::= #RuleBody                                 [klabel("#ruleNoConditions"), symbol]
+                        | #RuleBody "requires" Bool                 [klabel("#ruleRequires"), symbol]
+                        | #RuleBody "when" Bool                     [klabel("#ruleRequires"), symbol]
+                        | #RuleBody "ensures"  Bool                 [klabel("#ruleEnsures"), symbol]
+                        | #RuleBody "requires" Bool "ensures" Bool  [klabel("#ruleRequiresEnsures"), symbol]
+                        | #RuleBody "when" Bool "ensures" Bool      [klabel("#ruleRequiresEnsures"), symbol]
+endmodule
+

      Miscellaneous modules

      +

      The below modules are used in various ways as indicators to the implementation
      +that certain automatically generated syntax should be created by the parser.
      +These modules should not be imported directly by the user.

      +
      module K-TOP-SORT
+  imports SORT-KBOTT
+  syntax KItem ::= KBott
+  syntax {Sort} KItem ::= Sort
+endmodule
+
+module K-BOTTOM-SORT
+  imports SORT-KBOTT
+  syntax KItem ::= KBott
+  syntax {Sort} Sort ::= KBott
+endmodule
+
+module K-SORT-LATTICE
+  imports K-TOP-SORT
+  imports K-BOTTOM-SORT
+endmodule
+
+module AUTO-CASTS
+  // if this module is imported, the parser automatically
+  // generates, for all sorts, productions of the form:
+  // Sort  ::= Sort ":Sort"  // semantic cast - force the inner term to be `Sort` or a subsort
+  // Sort  ::= Sort "::Sort" // strict cast - force the inner term to be exactly `Sort`. Useful for disambiguation
+  // Sort ::= "{" Sort "}" "<:Sort" // synonym for strict cast
+  // Sort  ::= "{" K "}"    ":>Sort" // projection cast. Allows any term to be placed in a context that expects `Sort`
+  // this is part of the mechanism that allows concrete user syntax in K
+endmodule
+
+module AUTO-FOLLOW
+  // if this module is imported, the parser automatically
+  // generates a follow restriction for every terminal which is a prefix
+  // of another terminal. This is useful to prevent ambiguities such as:
+  // syntax K ::= "a"
+  // syntax K ::= "b"
+  // syntax K ::= "ab"
+  // syntax K ::= K K
+  // #parse("ab", "K")
+  // In the above example, the terminal "a" is not allowed to be followed by a "b"
+  // because it would turn the terminal into the terminal "ab".
+endmodule
+
+module PROGRAM-LISTS
+  imports SORT-K
+  // if this module is imported, the parser automatically
+  // replaces the default productions for lists:
+  // Es ::= E "," Es [userList("*"), klabel('_,_)]
+  //      | ".Es"    [userList("*"), klabel('.Es)]
+  // into a series of productions more suitable for programs:
+  // Es#Terminator ::= ""      [klabel('.Es)]
+  // Ne#Es ::= E "," Ne#Es     [klabel('_,_)]
+  //         | E Es#Terminator [klabel('_,_)]
+  // Es ::= Ne#Es
+  //      | Es#Terminator      // if the list is *
+endmodule
+
+module RULE-LISTS
+  // if this module is imported, the parser automatically
+  // adds the subsort production to the parsing module only:
+  // Es ::= E        [userList("*")]
+
+endmodule
+
+module RECORD-PRODUCTIONS
+  // if this module is imported, prefix productions of the form
+  // syntax Sort ::= name(Args)
+  // will be able to be parsed with don't-care variables according
+  // to their nonterminal's names
+endmodule
+
+module SORT-PREDICATES
+  // if this module is imported, the Bool sort will be annotated with
+  // syntax Bool ::= isSort(K) [function]
+  // and all sorts will be annotated with
+  // syntax Sort ::= project:Sort(K) [function]
+endmodule
+

      Additional Syntax for K Terms in Rules

      +

      Certain additional features are available when parsing the contents of rules
      +and contexts. For more information on each of these, refer to K's
      +documentation.

      +
      module KREWRITE
+  syntax {Sort} Sort ::= Sort "=>" Sort [klabel(#KRewrite), symbol]
+  syntax non-assoc #KRewrite
+  syntax priority #KRewrite > #withConfig
+endmodule
+
+// To be used to parse semantic rules
+module K
+  imports KSEQ-SYMBOLIC
+  imports REQUIRES-ENSURES
+  imports RECORD-PRODUCTIONS
+  imports SORT-PREDICATES
+  imports K-SORT-LATTICE
+  imports AUTO-CASTS
+  imports AUTO-FOLLOW
+  imports KREWRITE
+
+  syntax {Sort} Sort ::= Sort "#as" Sort [klabel(#KAs), symbol]
+  // functions that preserve sorts and can therefore have inner rewrites
+  syntax {Sort} Sort ::= "#fun" "(" Sort ")" "(" Sort ")" [klabel(#fun2), symbol, prefer]
+  // functions that do not preserve sort and therefore cannot have inner rewrites
+  syntax {Sort1, Sort2} Sort1 ::= "#fun" "(" Sort2 "=>" Sort1 ")" "(" Sort2 ")" [klabel(#fun3), symbol]
+
+  syntax {Sort1, Sort2} Sort1 ::= "#let" Sort2 "=" Sort2 "#in" Sort1 [klabel(#let), symbol]
+
+  /*@ Set membership over terms. In addition to equality over
+      concrete patterns, K also supports computing equality
+      between a concrete pattern and a symbolic pattern.
+      This is compiled efficiently down to pattern matching,
+      and can be used by putting a term with unbound variables
+      in the left child of :=K or =/=K. Note that this does not
+      bind variables used on the lhs however (although this may
+      change in the future).*/
+
+  syntax Bool ::= left:
+                  K ":=K" K           [function, total, klabel(_:=K_), symbol, group(equalEqualK)]
+                | K ":/=K" K          [function, total, klabel(_:/=K_), symbol, group(notEqualEqualK)]
+endmodule
+
+// To be used to parse terms in full K
+module K-TERM
+  imports KSEQ-SYMBOLIC
+  imports RECORD-PRODUCTIONS
+  imports SORT-PREDICATES
+  imports K-SORT-LATTICE
+  imports AUTO-CASTS
+  imports AUTO-FOLLOW
+  imports KREWRITE
+endmodule
+

      Layout Information

      +

      When constructing a scanner for use during parsing, often you wish to ignore
      +certain types of text, such as whitespace and comments. However, the specific
      +syntax which each language must ignore is a little different from language
      +to language, and thus you wish to specify it manually. You can do this by
      +defining productions of the #Layout sort. For more information, refer to
      +K's documentation. However, this module will be implicitly imported if no
      +productions are declared of sort #Layout. This module will also be used
      +for the purposes of parsing K rules. If you wish to declare a language with
      +no layout productions, simply create a sort declaration for the #Layout sort
      +in your code (e.g. syntax #Layout).

      +
      module DEFAULT-LAYOUT
+    syntax #Layout ::= r"(\\/\\*([^\\*]|(\\*+([^\\*\\/])))*\\*+\\/)" // C-style multi-line comments
+                     | r"(\\/\\/[^\\n\\r]*)"                         // C-style single-line comments
+                     | r"([\\ \\n\\r\\t])"                           // Whitespace
+endmodule
+

      Default Configuration

      +

      If the user has no configuration declaration in their seamantics, the below
      +configuration declaration will be implicitly imported.

      +
      module DEFAULT-CONFIGURATION
+  imports BASIC-K
+
+  configuration <k> $PGM:K </k>
+endmodule
+

      Parsing Ambiguous Languages

      +

      On occasion, it may be desirable to parse a language with an ambiguous grammar
      +when parsing a program, and perform additional semantic analysis at a later
      +time in order to resolve the ambiguities. A good example of this is as a
      +substitute for the lexer hack in parsers of the C programming language.

      +

      The following module contains a declaration for ambiguities in K. Usually,
      +an ambiguous parse is an error. However, when you use the --gen-glr-parser
      +flag to kast, or the --gen-glr-bison-parser flag to kompile, ambiguities
      +instead become instances of the below parametric production, which you can use
      +regular K rules to disambiguate as necessary.

      +
      module K-AMBIGUITIES
+
+  syntax {Sort} Sort ::= amb(Sort, Sort) [symbol]
+
+endmodule
+

      Annotating Parses with Locations

      +

      Another feature of K's Bison parser is the ability to annotate terms parsed
      +with location information about the file and line where they occurred. For
      +more information about how to use this, refer to K's documentation. However,
      +the below module exists to provide a user syntax for the annotations that
      +are generated by the parser.

      +
      module K-LOCATIONS
+  imports STRING-SYNTAX
+  imports INT-SYNTAX
+
+  // filename, startLine, startCol, endLine, endCol
+  syntax {Sort} Sort ::= #location(Sort, String, Int, Int, Int, Int) [symbol, format(%3)]
+
+endmodule
+

      K Prelude

      +

      The following files, integral to defining semantics in K, are automatically
      +required by every definition via this file. This behavior can be disabled
      +via kompile --no-prelude, however, semantics will likely break unless
      +they provide their own versions of these files, which are assumed to exist
      +by the compiler. There are, however, circumstances where passing this flag is
      +appropriate, such as if you are manually requiring these files in your
      +definition, if your definition was automatically condensed into a single file
      +with kompile -E, or if you wish to modify the inner syntax of K by providing
      +your own version of these files with different syntax.

      +
      require "kast.md"
+require "domains.md"
+

      K Foreign Function Interface

      +

      The K Foreign Function Interface (FFI) module provides a way to call native
      +functions directly from a K semantics using the C ABI. It also provides
      +utilities for allocating and deallocating byte buffers with static addresses
      +that are suitable for being passed to native code.

      +

      It is built off of the underlying libffi library
      +(https://sourceware.org/libffi/) and is subject to some of the same
      +limitations as that library. Bear in mind, because this library exposes
      +a number of unsafe C APIs directly, misuse of the library is likely to lead
      +to memory corruption in your interpreter and can cause segmentation faults or
      +corrupted term representations that lead to undefined behavior at runtime.

      +
      require "domains.md"
+
+module FFI-SYNTAX
+  imports private LIST
+

      The FFIType sort is used to declare the native C ABI types of operands passed
      +to the #ffiCall function. These types roughly correspond to the types
      +declared in ffi.h by libffi.

      +
        syntax FFIType ::= "#void" [klabel(#ffi_void), symbol]
+                  | "#uint8" [klabel(#ffi_uint8), symbol]
+                  | "#sint8" [klabel(#ffi_sint8), symbol]
+                  | "#uint16" [klabel(#ffi_uint16), symbol]
+                  | "#sint16" [klabel(#ffi_sint16), symbol]
+                  | "#uint32" [klabel(#ffi_uint32), symbol]
+                  | "#sint32" [klabel(#ffi_sint32), symbol]
+                  | "#uint64" [klabel(#ffi_uint64), symbol]
+                  | "#sint64" [klabel(#ffi_sint64), symbol]
+                  | "#float" [klabel(#ffi_float), symbol]
+                  | "#double" [klabel(#ffi_double), symbol]
+                  | "#uchar" [klabel(#ffi_uchar), symbol]
+                  | "#schar" [klabel(#ffi_schar), symbol]
+                  | "#ushort" [klabel(#ffi_ushort), symbol]
+                  | "#sshort" [klabel(#ffi_sshort), symbol]
+                  | "#uint" [klabel(#ffi_uint), symbol]
+                  | "#sint" [klabel(#ffi_sint), symbol]
+                  | "#ulong" [klabel(#ffi_ulong), symbol]
+                  | "#slong" [klabel(#ffi_slong), symbol]
+                  | "#longdouble" [klabel(#ffi_longdouble), symbol]
+                  | "#pointer" [klabel(#ffi_pointer), symbol]
+                  | "#complexfloat" [klabel(#ffi_complexfloat), symbol]
+                  | "#complexdouble" [klabel(#ffi_complexdouble), symbol]
+                  | "#complexlongdouble" [klabel(#ffi_complexlongdouble), symbol]
+                  | "#struct" "(" List ")" [klabel(#ffi_struct), symbol]
+endmodule
+
+module FFI
+  imports FFI-SYNTAX
+  imports private BYTES
+  imports private STRING
+  imports private BOOL
+  imports private LIST
+  imports private INT
+
+

      FFI Calls

      +

      The #ffiCall functions are designed to call a native C ABI function and
      +return a native result. They come in three variants:

      +

      Non-variadic

      + +

      In the first variant, #ffiCall(Address, Args, ArgTypes, ReturnType) takes
      +an integer address of a function (which can be obtained from
      +#functionAddress), a List of Bytes containing the arguments of the
      +function, a List of FFITypes containing the types of the parameters of the
      +function, and an FFIType containing the return type of the function, and
      +returns the return value of the function as a Bytes.

      +
        syntax Bytes ::= "#ffiCall" "(" Int "," List "," List "," FFIType ")" [function, hook(FFI.call)]
+

      Variadic

      + +

      In the second variant,
      +#ffiCall(Address, Args, FixedTypes, VariadicTypes, ReturnType takes an
      +integer address of a function, a List of Bytes containing the arguments
      +of the call, a List of FFITypes containing the types of the fixed
      +parameters of the function, a List of FFITypes containing the types of the
      +variadic parameters of the function, and an FFIType containing the return
      +type of the function, and returns the return value of the function as a
      +Bytes.

      +
        syntax Bytes ::= "#ffiCall" "(" Int "," List "," List "," List "," FFIType ")" [function, hook(FFI.call_variadic)]
+

      Generic

      + +

      In the third variant,
      +#ffiCall(IsVariadic, Address, Args, ArgTypes, NFixed, ReturnType takes
      +a boolean indicating whether the function is variadic or not, an integer
      +address of a function, a List of Bytes containing the arguments of the
      +call, a List of FFITypes containing the parameter typess of the call
      +followed by the types of the variadic arguments of the call, if any, an Int
      +containing how many of the arguments of the call are fixed or not, and an
      +FFIType containing the return type of the function, and returns the return
      +value of the function as a Bytes.

      +
        syntax Bytes ::= "#ffiCall" "(" Bool "," Int "," List "," List "," Int "," FFIType ")" [function]
+
+  rule #ffiCall(false, Addr::Int, Args::List, Types::List, _, Ret::FFIType) => #ffiCall(Addr, Args, Types, Ret)
+  rule #ffiCall(true, Addr::Int, Args::List, Types::List, NFixed::Int, Ret::FFIType) => #ffiCall(Addr, Args, range(Types, 0, size(Types) -Int NFixed), range(Types, NFixed, 0), Ret)
+

      Symbol Lookup

      +

      The FFI module provides a mechanism to look up any function symbol and return
      +that function's address.

      +
        syntax Int ::= "#functionAddress" "(" String ")" [function, hook(FFI.address)]
+

      Direct Memory Management

      +

      Most memory used by the LLVM backend to represent terms is managed
      +automatically via garbage collection. However, a consequence of this is that
      +a particular term does not have a fixed address across its entire lifetime
      +in most cases. Sometimes this is undesirable, especially if you intend for
      +the address of the memory to be taken by the semantics or if you intend
      +to pass this memory directly to native code. As a result, the FFI module
      +exposes the following unsafe APIs for memory management. Note that use of
      +these APIs leaves the burden of memory management completely on the user,
      +and thus misuse of these functions can lead to things like use-after-free
      +and other memory corruption bugs.

      +

      Allocation

      + +

      #alloc(Key, Size, Align) will allocate Size bytes with an alignment
      +requirement of Align (which must be a power of two), and return it as a
      +Bytes term. The memory is uniquely identified by its key and that key will
      +be used later to free the memory. The memory is not implicitly freed by garbage
      +collection; failure to call #free on the memory at a later date can lead to
      +memory leaks.

      +
        syntax Bytes ::= "#alloc" "(" KItem "," Int "," Int ")" [function, hook(FFI.alloc)]
+

      Addressing

      + +

      #addess(B) will return an Int representing the address of the first byte of
      +B, which must be a Bytes. Unless the Bytes term was allocated by #alloc,
      +the return value is unspecified and may not be the same across multipl
      +invocations on the same byte buffer. However, it is guaranteed that memory
      +allocated by #alloc will have the same address throughout its lifetime.

      +
        syntax Int ::= "#address" "(" Bytes ")" [function, hook(FFI.bytes_address)]
+

      Deallocation

      + +

      #free(Key) will free the memory of the Bytes object that was allocated
      +by a previous call to #alloc. If Key was not used in a previous call to
      +#alloc, or the memory was already freed, no action is taken. It will generate
      +undefined behavior if the Bytes term returned by the previous call to
      +#alloc is still referenced by any other term in the configuration or a
      +currently evaluating rule. The function returns .K.

      +
        syntax K ::= "#free" "(" KItem ")" [function, hook(FFI.free)]
+

      Reading

      + +

      #nativeRead(Addr, Mem) will read native memory at address Addr into Mem,
      +reading exactly lengthBytes(Mem) bytes. This will generate undefined behavior
      +if Addr does not point to a readable segment of memory at least
      +lengthBytes(Mem) bytes long.

      +
        syntax K ::= "#nativeRead" "(" Int "," Bytes ")" [function, hook(FFI.read)]
+

      Writing

      + +

      #nativeWrite(Addr, Mem) will write the contents of Mem to native memory at
      +address Addr. The memory will be read prior to being written, and a write
      +will only happen if the memory has a different value than the current value of
      +Mem. This will generate undefined behavior if Addr does not point to a
      +readable segment of memory at least lengthBytes(Mem) bytes long, or if the
      +memory at address Addr has a different value than currently contained in
      +Mem, and the memory in question is not writeable.

      +
        syntax K ::= "#nativeWrite" "(" Int "," Bytes ")" [function, hook(FFI.write)]
+endmodule
+

      Syntax of JSON

      +

      K provides builtin support for reading/writing to JSON. While the JSON-SYNTAX
      +module is not precisely the syntax of JSON (utilizing K's syntax for strings,
      +integers, and floating point numbers rather than the syntax used by JSON),
      +you can still convert directly to/from the actual syntax of JSON using
      +the JSON2String and String2JSON hooks.

      +
      module JSON-SYNTAX
+    imports INT-SYNTAX
+    imports STRING-SYNTAX
+    imports BOOL-SYNTAX
+    imports FLOAT-SYNTAX
+
+    syntax JSONs   ::= List{JSON,","}      [klabel(JSONs)      , symbol]
+    syntax JSONKey ::= String
+    syntax JSON    ::= "null"              [klabel(JSONnull)   , symbol]
+                     | String | Int | Float | Bool
+                     | JSONKey ":" JSON    [klabel(JSONEntry)  , symbol]
+                     | "{" JSONs "}"       [klabel(JSONObject) , symbol]
+                     | "[" JSONs "]"       [klabel(JSONList)   , symbol]
+endmodule
+

      Conversion between JSON and String

      +

      Given a string written in valid JSON, you can convert it to the JSON
      +sort with the String2JSON function. Assuming the user has not extended
      +the syntax of the JSON sort with their own constructors, any term of sort
      +JSON can also be converted to a String using the JSON2String function.

      +
      module JSON
+    imports JSON-SYNTAX
+
+    syntax String ::= JSON2String(JSON) [function, hook(JSON.json2string)]
+
+    syntax JSON ::= String2JSON(String) [function, hook(JSON.string2json)]
+endmodule
+

      Rational Numbers in K

      +

      K provides support for arbitrary-precision rational numbers represented as a
      +quotient between two integers. The sort representing these values is Rat.
      +Int is a subsort of Rat, and it is guaranteed that any integer will be
      +represented as an Int and can be matched as such on the left hand side
      +of rules. K also supports the usual arithmetic operators over rational numbers.

      +
      module RAT-SYNTAX
+  imports INT-SYNTAX
+  imports private BOOL
+
+  syntax Rat
+
+  syntax Rat ::= Int
+

      Arithmetic

      +

      You can:

      +
        +
      • Raise a rational number to any negative or nonnegative integer.
        • +
      • Multiply or divide two rational numbers to obtain a product or quotient.
        • +
      • Add or subtract two rational numbers to obtain a sum or difference.
        • +
      +
        syntax Rat ::= left:
+                 Rat "^Rat" Int [function, total, klabel(_^Rat_), symbol, smtlib(ratpow), hook(RAT.pow)]
+               > left:
+                 Rat "*Rat" Rat [function, total, klabel(_*Rat_), symbol, left, smtlib(ratmul), hook(RAT.mul)]
+               | Rat "/Rat" Rat [function,             klabel(_/Rat_), symbol, left, smtlib(ratdiv), hook(RAT.div)]
+               > left:
+                 Rat "+Rat" Rat [function, total, klabel(_+Rat_), symbol, left, smtlib(ratadd), hook(RAT.add)]
+               | Rat "-Rat" Rat [function, total, klabel(_-Rat_), symbol, left, smtlib(ratsub), hook(RAT.sub)]
+

      Comparison

      +

      You can determine whether two rational numbers are equal, unequal, or compare
      +one of less than, less than or equalto, greater than, or greater than or equal
      +to the other:

      +
        syntax Bool ::= Rat  "==Rat" Rat [function, total, klabel(_==Rat_),  symbol, smtlib(rateq), hook(RAT.eq)]
+                | Rat "=/=Rat" Rat [function, total, klabel(_=/=Rat_), symbol, smtlib(ratne), hook(RAT.ne)]
+                | Rat   ">Rat" Rat [function, total, klabel(_>Rat_),   symbol, smtlib(ratgt), hook(RAT.gt)]
+                | Rat  ">=Rat" Rat [function, total, klabel(_>=Rat_),  symbol, smtlib(ratge), hook(RAT.ge)]
+                | Rat   "<Rat" Rat [function, total, klabel(_<Rat_),   symbol, smtlib(ratlt), hook(RAT.lt)]
+                | Rat  "<=Rat" Rat [function, total, klabel(_<=Rat_),  symbol, smtlib(ratle), hook(RAT.le)]
+

      Min/Max

      +

      You can compute the minimum and maximum of two rational numbers:

      +
        syntax Rat ::= minRat(Rat, Rat) [function, total, klabel(minRat), symbol, smtlib(ratmin), hook(RAT.min)]
+               | maxRat(Rat, Rat) [function, total, klabel(maxRat), symbol, smtlib(ratmax), hook(RAT.max)]
+

      Conversion to Floating Point

      +

      You can convert a rational number to the nearest floating point number that
      +is representable in a Float of a specified number of precision and exponent
      +bits:

      +
        syntax Float ::= Rat2Float(Rat, precision: Int, exponentBits: Int) [function]
+endmodule
+

      Implementation of Rational Numbers

      +

      The remainder of this file consists of an implementation in K of the
      +operations listed above. Users of the RAT module should not use any of the
      +syntax defined in any of these modules.

      +

      As a point of reference for users, it is worth noting that rational numbers
      +are normalized to a canonical form by this module,. with the canonical form
      +bearing the property that it is either an Int, or a pair of integers
      +I /Rat J such that
      +I =/=Int 0 andBool J >=Int 2 andBool gcdInt(I, J) ==Int 1 is always true.

      +
      module RAT-COMMON
+  imports RAT-SYNTAX
+
+  // invariant of < I , J >Rat : I =/= 0, J >= 2, and I and J are coprime
+  syntax Rat ::= "<" Int "," Int ">Rat" [format(%2 /Rat %4)]
+endmodule
+
+module RAT-SYMBOLIC [symbolic, kore]
+  imports private RAT-COMMON
+  imports ML-SYNTAX
+  imports private BOOL
+
+  rule
+    #Ceil(@R1:Rat /Rat @R2:Rat)
+  =>
+    {(@R2 =/=Rat 0) #Equals true} #And #Ceil(@R1) #And #Ceil(@R2)
+  [simplification]
+endmodule
+
+module RAT-KORE [kore]
+  imports private RAT-COMMON
+  imports private K-EQUAL
+
+  /*
+   * equalities
+   */
+
+  // NOTE: the two rules below may not work correctly in non-kore backends
+
+  rule R ==Rat S => R ==K S
+
+  rule R =/=Rat S => R =/=K S
+endmodule
+
+module RAT [private]
+  imports private RAT-COMMON
+  imports public RAT-SYMBOLIC
+  imports public RAT-KORE
+  imports public RAT-SYNTAX
+  imports private INT
+  imports private BOOL
+
+  /*
+   * arithmetic
+   */
+
+  rule < I , I' >Rat +Rat < J , J' >Rat => ((I *Int J') +Int (I' *Int J)) /Rat (I' *Int J')
+  rule I:Int         +Rat < J , J' >Rat => ((I *Int J') +Int J) /Rat J'
+  rule < J , J' >Rat +Rat I:Int         => I +Rat < J , J' >Rat
+  rule I:Int         +Rat J:Int         => I +Int J
+
+  rule < I , I' >Rat *Rat < J , J' >Rat => (I *Int J) /Rat (I' *Int J')
+  rule I:Int         *Rat < J , J' >Rat => (I *Int J) /Rat J'
+  rule < J , J' >Rat *Rat I:Int         => I *Rat < J , J' >Rat
+  rule I:Int         *Rat J:Int         => I *Int J
+
+  rule < I , I' >Rat /Rat < J , J' >Rat => (I *Int J') /Rat (I' *Int J)
+  rule I:Int         /Rat < J , J' >Rat => (I *Int J') /Rat J
+  rule < I , I' >Rat /Rat J:Int         => I /Rat (I' *Int J) requires J =/=Int 0
+  rule I:Int         /Rat J:Int         => makeRat(I, J)      requires J =/=Int 0
+
+  // derived
+
+  rule R -Rat S => R +Rat (-1 *Rat S)
+
+  // normalize
+
+  syntax Rat ::= makeRat(Int, Int)      [function]
+               | makeRat(Int, Int, Int) [function]
+
+  rule makeRat(0, J) => 0 requires J =/=Int 0
+
+  rule makeRat(I, J) => makeRat(I, J, gcdInt(I,J)) requires I =/=Int 0 andBool J =/=Int 0
+
+  // makeRat(I, J, D) is defined when I =/= 0, J =/= 0, D > 0, and D = gcd(I,J)
+  rule makeRat(I, J, D) => I /Int D                       requires J ==Int D // implies J > 0 since D > 0
+  rule makeRat(I, J, D) => < I /Int D , J /Int D >Rat     requires J >Int 0 andBool J =/=Int D
+  rule makeRat(I, J, D) => makeRat(0 -Int I, 0 -Int J, D) requires J <Int 0
+
+  // gcdInt(a,b) computes the gcd of |a| and |b|, which is positive.
+  syntax Int ::= gcdInt(Int, Int) [function, public]
+
+  rule gcdInt(A, 0) => A        requires A >Int 0
+  rule gcdInt(A, 0) => 0 -Int A requires A <Int 0
+  rule gcdInt(A, B) => gcdInt(B, A %Int B) requires B =/=Int 0 // since |A %Int B| = |A| %Int |B|
+
+  /*
+   * exponentiation
+   */
+
+  rule _ ^Rat 0 => 1
+  rule 0 ^Rat N => 0 requires N =/=Int 0
+
+  rule < I , J >Rat ^Rat N => powRat(< I , J >Rat, N) requires N >Int 0
+  rule X:Int        ^Rat N => X ^Int N                requires N >Int 0
+
+  rule X ^Rat N => (1 /Rat X) ^Rat (0 -Int N) requires X =/=Rat 0 andBool N <Int 0
+
+  // exponentiation by squaring
+
+  syntax Rat ::= powRat(Rat, Int) [function]
+
+  // powRat(X, N) is defined when X =/= 0 and N > 0
+  rule powRat(X, 1) => X
+  rule powRat(X, N) => powRat(X *Rat X, N /Int 2) requires N >Int 1 andBool N %Int 2  ==Int 0
+  rule powRat(X, N) => powRat(X, N -Int 1) *Rat X requires N >Int 1 andBool N %Int 2 =/=Int 0
+
+  /*
+   * inequalities
+   */
+
+  rule R >Rat S => R -Rat S >Rat 0 requires S =/=Rat 0
+
+  rule < I , _ >Rat >Rat 0 => I >Int 0
+  rule I:Int        >Rat 0 => I >Int 0
+
+  // derived
+
+  rule R >=Rat S => notBool R <Rat S
+
+  rule R <Rat S => S >Rat R
+
+  rule R <=Rat S => S >=Rat R
+
+  rule minRat(R, S) => R requires R <=Rat S
+  rule minRat(R, S) => S requires S <=Rat R
+
+  rule maxRat(R, S) => R requires R >=Rat S
+  rule maxRat(R, S) => S requires S >=Rat R
+
+  syntax Float ::= #Rat2Float(Int, Int, Int, Int) [function, hook(FLOAT.rat2float)]
+  rule Rat2Float(Num:Int, Prec:Int, Exp:Int) => #Rat2Float(Num, 1, Prec, Exp)
+  rule Rat2Float(< Num, Dem >Rat, Prec, Exp) => #Rat2Float(Num, Dem, Prec, Exp)
+
+endmodule
+

      Capture-Aware Substitution in K

      +

      One of the traditional ways in which functional languages are given operational
      +semantics is via substitution. In particular, you can view a function as
      +declaring a particular bound variable, the parameter of the function, as well
      +as the body of the function, within which both bound and free variables can
      +occur, and implement the process of beta-reduction (one of the axioms of the
      +lambda calculus) by means of a substitution operator which is aware of the
      +difference between free variables and bound variables and prevents variable
      +capture.

      +

      In K this is implemented using two mechanisms: The KVar sort, and the
      +binder attribute.

      +

      The KVar Sort

      +

      K introduces a new hooked sort, KVar, which the substitution operator
      +(defined below) understands in a particular way. The syntax of KVar is the
      +same as for sort Id in DOMAINS, but with a different sort name. Similarly,
      +some of the same operators are defined over KVar which are defined for Id,
      +such as conversion from String to KVar and support for the !Var:KVar
      +syntax.

      +

      A KVar is simply an identifier with special meaning during substitution.
      +KVars must begin with a letter or underscore,
      +and can be followed by zero or more letters, numbers, or underscores.

      +
      module KVAR-SYNTAX-PROGRAM-PARSING
+  imports BUILTIN-ID-TOKENS
+
+  syntax KVar ::= r"(?<![A-Za-z0-9\\_])[A-Za-z\\_][A-Za-z0-9\\_]*"     [prec(1), token]
+                | #LowerId                                             [token]
+                | #UpperId                                             [token]
+endmodule
+
+module KVAR-SYNTAX
+  syntax KVar [token, hook(KVAR.KVar)]
+endmodule
+
+module KVAR-COMMON
+  imports KVAR-SYNTAX
+  imports private STRING
+
+  syntax KVar ::= String2KVar (String) [function, total, hook(STRING.string2token)]
+  syntax KVar ::= freshKVar(Int)    [freshGenerator, function, total, private]
+
+  rule freshKVar(I:Int) => String2KVar("_" +String Int2String(I))
+endmodule
+
+module KVAR
+  imports KVAR-COMMON
+endmodule
+

      The binder Attribute

      +

      A production can be given the attribute binder. Such a production must have
      +at least two nonterminals. The first nonterminal from left to right must be of
      +sort KVar, and contains the bound variable. The last nonterminal from left
      +to right contains the term that is bound. For example, I could describe lambdas
      +in the lambda calculus with the production
      +syntax Val ::= "lambda" KVar "." Exp [binder].

      +

      Substitution

      +

      K provides a hooked implementation of substitution, currently only implemented
      +on the Java and LLVM backends. Two variants exist: the first substitutes
      +a single KVar for a single KItem. The second takes a Map with KVar
      +keys and KItem values, and substitutes each element in the map atomically.

      +

      Internally, this is implemented in the LLVM backend by a combination of
      +de Bruijn indices for bound variables and names for free variables. Free
      +variables are also sometimes given a unique numeric identifier in order to
      +prevent capture, and the rewriter will automatically assign unique names to
      +such identifiers when rewriting finishes. The names assigned will always begin
      +with the original name of the variable and be followed by a unique integer
      +suffix. However, the names assigned after rewriting finishes might be different
      +from the names that would be assigned if rewriting were to halt prematurely,
      +for example due to krun --depth.

      +
      module SUBSTITUTION
+  imports private MAP
+  imports KVAR
+
+  syntax {Sort} Sort ::= Sort "[" KItem "/" KItem "]"  [function, hook(SUBSTITUTION.substOne), impure]
+  syntax {Sort} Sort ::= Sort "[" Map "]"      [function, hook(SUBSTITUTION.substMany), impure]
+endmodule
+

      K PL Tutorial

      + +

      Here you will learn how to use the K tool to define languages by means of a series of screencast movies. It is recommended to do these in the indicated order, because K features already discussed in a previous language definition will likely not be rediscussed in latter definitions. The screencasts follow quite closely the structure of the files under the tutorial folder in the K tool distribution. If you'd rather follow the instructions there and do the tutorial exercises yourself, then go back to https://kframework.org and download the K tool, if you have not done it already. Or, you can first watch the screencasts below and then do the exercises, or do them in parallel.

      +

      K Overview

      + +

      Make sure you watch the K overview video before you do the K tutorial:

      + +

      Learning K

      + +

      [34'46"] Part 1: Defining LAMBDA

      + +

      Here you will learn how to define a very simple functional language in K and the basics of how to use the K tool. The language is a call-by-value variant of lambda calculus with builtins and mu, and its definition is based on substitution.

      + +

      [37'07"] Part 2: Defining IMP

      + +

      Here you will learn how to define a very simple, prototypical textbook C-like imperative language, called IMP, and several new features of the K tool.

      + +

      [33'10"] Part 3: Defining LAMBDA++

      + +

      Here you will learn how to define constructs which abruptly change the execution control, as well as how to define functional languages using environments and closures. LAMBDA++ extends the LAMBDA language above with a callcc construct.

      + +

      [46'46"] Part 4: Defining IMP++

      + +

      Here you will learn how to refine configurations, how to generate fresh elements, how to tag syntactic constructs and rules, how to exhaustively search the space of non-deterministic or concurrent program executions, etc. IMP++ extends the IMP language above with increment, blocks and locals, dynamic threads, input/output, and abrupt termination.

      + +

      [17'03"] Part 5: Defining Type Systems

      + +

      Here you will learn how to define various kinds of type systems following various approaches or styles using K.

      + +

      [??'??"] Part 6: Miscellaneous Other K Features

      + +

      Here you will learn a few other K features, and better understand how features that you have already seen work.

      +
        +
      • [??'??"] ...
        • +
      +

      Learning Language Design and Semantics using K

      + +

      [??'??"] Part 7: SIMPLE: Designing Imperative Programming Languages

      + +

      Here you will learn how to design imperative programming languages using K. SIMPLE is an imperative language with functions, threads, pointers, exceptions, multi-dimensional arrays, etc. We first define an untyped version of SIMPLE, then a typed version. For the typed version, we define both a static and a dynamic semantics.

      + +

      [??'??"] Part 8: KOOL: Designing Object-Oriented Programming Languages

      + +

      Here woul will learn how to design object-oriented programming languages using K. KOOL is an object-oriented language that extends SIMPLE with classes and objects. We first define an untyped version of KOOL, then a typed version, with both a dynamic and a static semantics.

      + +

      [??'??"] Part 9: FUN: Designing Functional Programming Languages

      + +

      H
      +ere woul will learn how to design functional programming languages using K. FUN is a higher-order functional language with general let, letrec, pattern matching, references, lists, callcc, etc. We first define an untyped version of FUN, then a let-polymorphic type inferencer.

      + +

      [??'??"] Part 10: LOGIK: Designing Logic Programming Languages

      + +

      Here you will learn how to design a logic programming language using K.

      + +

      K overview

      + + +

      Go to Youtube mirror, if the above does not work.

      +

      Go back to https://kframework.org for further links, the K tool and contact information.

      +

      Learning K

      + +

      We start by introducing the basic features of K by means of a series
      +of very simple languages. The objective here is neither to learn those
      +languages nor to study their underlying paradigm, but simply to learn K.

      +
        +
      • LAMBDA: Lambda calculus defined.
        • +
      • IMP: A simple imperative language.
        • +
      • LAMBDA++: LAMBDA extended with control flow.
        • +
      • IMP++: IMP extended with threads and IO.
        • +
      • TYPES: LAMBDA type system.
        • +
      +

      Part 1: Defining LAMBDA

      + +

      Here you will learn how to define a very simple language in K and the basics
      +of how to use the K tool. The language is a variant of call-by-value lambda
      +calculus and its definition is based on substitution. Specifically, you will
      +learn the following:

      +
        +
      • How to define a module.
        • +
      • How to define a language syntax.
        • +
      • How to use the defined syntax to parse programs.
        • +
      • How to import predefined modules.
        • +
      • How to define evaluation strategies using strictness attributes.
        • +
      • How to define semantic rules.
        • +
      • How the predefined generic substitution works.
        • +
      • How to generate PDF and HTML documentation from ASCII definitions.
        • +
      • How to include builtins (integers and Booleans) into your language.
        • +
      • How to define derived language constructs.
        • +
      +

      This folder contains several lessons, each adding new features to LAMBDA.

      +

      Syntax Modules and Basic K Commands

      + +

      Here we define our first K module, which contains the initial syntax of the
      +LAMBDA language, and learn how to use the basic K commands.

      +

      Let us create an empty working folder, and open a terminal window
      +(to the left) and an editor window (to the right). We will edit our K
      +definition in the right window in a file called lambda.k, and will call
      +the K tool commands in the left window.

      +

      Let us start by defining a K module, containing the syntax of LAMBDA.

      +

      K modules are introduced with the keywords module ... endmodule.

      +

      The keyword syntax adds new productions to the syntax grammar, using a
      +BNF-like notation.

      +

      Terminals are enclosed in double-quotes, like strings.

      +

      You can define multiple productions for the same non-terminal in the same
      +syntax declaration using the | separator.

      +

      Productions can have attributes, which are enclosed in square brackets.

      +

      The attribute left tells the parser that we want the lambda application to be
      +left associative. For example, a b c d will then parse as (((a b) c) d).

      +

      The attribute bracket tells the parser to not generate a node for the
      +parenthesis production in the abstract syntax trees associated to programs.
      +In other words, we want to allow parentheses to be used for grouping, but we
      +do not want to bother to give them their obvious (ignore) semantics.

      +

      In our variant of lambda calculus defined here, identifiers and lambda
      +abstractions are meant to be irreducible, that is, are meant to be values.
      +However, so far Val is just another non-terminal, just like Exp,
      +without any semantic meaning. It will get a semantic meaning later.

      +

      After we are done typing our definition in the file lambda.k, we can kompile
      +it with the command:

      +
      kompile lambda.k
+
      +

      If we get no errors then a parser has been generated. This parser will be
      +called from now on by default by the krun tool. To see whether and how the
      +parser works, we are going to write some LAMBDA programs and store them in
      +files with the extension .lambda.

      +

      Let us create a file identity.lambda, which contains the identity lambda
      +abstraction:

      +
      lambda x . x
+
      +

      Now let us call krun on identity.lambda:

      +
      krun identity.lambda
+
      +

      Make sure you call the krun command from the folder containing your language
      +definition (otherwise type krun --help to learn how to pass a language
      +definition as a parameter to krun). The krun command produces the output:

      +
      <k>
+  lambda x . x
+</k>
+
      +

      If you see such an output it means that your program has been parsed (and then
      +pretty printed) correctly. If you want to see the internal abstract syntax
      +tree (AST) representation of the parsed program, which we call the K AST, then
      +type kast in the command instead of krun:

      +
      kast identity.lambda
+
      +

      You should normally never need to see this internal representation in your
      +K definitions, so do not get scared (yes, it is ugly for humans, but it is
      +very convenient for tools).

      +

      Note that krun placed the program in a <k> ... </k> cell. In K, computations
      +happen only in cells. If you do not define a configuration in your definition,
      +like we did here, then a configuration will be created automatically for you
      +which contains only one cell, the default k cell, which holds the program.

      +

      Next, let us create a file free-variable-capture.lambda, which contains an
      +expression which, in order to execute correctly in a substitution-based
      +semantics of LAMBDA, the substitution operation needs to avoid
      +variable-capture:

      +
      a (((lambda x.lambda y.x) y) z)
+
      +

      Next, file closed-variable-capture.lambda shows an expression which also
      +requires a capture-free substitution, but this expression is closed (that is,
      +it has no free variables) and all its bound variables are distinct (I believe
      +this is the smallest such expression):

      +
      (lambda z.(z z)) (lambda x.lambda y.(x y))
+
      +

      Finally, the file omega.lambda contains the classic omega combinator
      +(or closed expression), which is the smallest expression which loops forever
      +(not now, but after we define the semantics of LAMBDA):

      +
      (lambda x.(x x)) (lambda x.(x x))
+
      +

      Feel free to define and parse several other LAMBDA programs to get a feel for
      +how the parser works. Parse also some incorrect programs, to see how the
      +parser generates error messages.

      +

      In the next lesson we will see how to define semantic rules that iteratively
      +rewrite expressions over the defined syntax until they evaluate to a result.
      +This way, we obtain our first programming language defined using K.

      +

      Go to Lesson 2, LAMBDA: Module Importing, Rules, Variables

      +

      MOVIE (out of date) [4'07"]

      +

      Module Importing, Rules, Variables

      + +

      We here learn how to include a predefined module (SUBSTITUTION), how to
      +use it to define a K rule (the characteristic rule of lambda calculus),
      +and how to make proper use of variables in rules.

      +

      Let us continue our lambda.k definition started in the previous lesson.

      +

      The requires keyword takes a .k file containing language features that
      +are needed for the current definition, which can be found in the
      +k-distribution/include/kframework/builtin folder. Thus, the command

      +
      require "substitution.k"
+
      +

      says that the subsequent definition of LAMBDA needs the generic substitution,
      +which is predefined in file substitution.k under the folder
      +k-distribution/include/kframework/builtin. Note that substitution can be defined itself in K,
      +although it uses advanced features that we have not discussed yet in this
      +tutorial, so it may not be easy to understand now.

      +

      Using the imports keyword, we can now modify LAMBDA to import the module
      +SUBSTITUTION, which is defined in the required substitution.k file.

      +

      Now we have all the substitution machinery available for our definition.
      +However, since our substitution is generic, it cannot know which language
      +constructs bind variables, and what counts as a variable; however, this
      +information is critical in order to correctly solve the variable capture
      +problem. Thus, you have to tell the substitution that your lambda construct
      +is meant to be a binder, and that your Id terms should be treated as variables
      +for substitution. The former is done using the attribute binder.
      +By default, binder binds all the variables occurring anywhere in the first
      +argument of the corresponding syntactic construct within its other arguments;
      +you can configure which arguments are bound where, but that will be discussed
      +in subsequent lectures. To tell K which terms are meant to act as variables
      +for binding and substitution, we have to explicitly subsort the desired syntactic
      +categories to the builtin KVariable sort.

      +

      Now we are ready to define our first K rule. Rules are introduced with the
      +keyword rule and make use of the rewrite symbol, =>. In our case,
      +the rule defines the so-called lambda calculus beta-reduction, which
      +makes use of substitution in its right-hand side, as shown in lambda.k.

      +

      By convention, variables that appear in rules start with a capital letter
      +(the current implementation of the K tool may even enforce that).

      +

      Variables may be explicitly tagged with their syntactic category (also called
      +sort). If tagged, the matching term will be checked at run-time for
      +membership to the claimed sort. If not tagged, then no check will be made.
      +The former is safer, but involves the generation of a side condition to the
      +rule, so the resulting definition may execute slightly slower overall.

      +

      In our rule in lambda.k we tagged all variables with their sorts, so we chose
      +the safest path. Only the V variable really needs to be tagged there,
      +because we can prove (using other means, not the K tool, as the K tool is not
      +yet concerned with proving) that the first two variables will always have the
      +claimed sorts whenever we execute any expression that parses within our
      +original grammar.

      +

      Let us compile the definition and then run some programs. For example,

      +
      krun closed-variable-capture.lambda
+
      +

      yields the output

      +
      <k>
+  lambda y . ((lambda x . (lambda y . (x  y))) y)
+</k> 
+
      +

      Notice that only certain programs reduce (some even yield non-termination,
      +such as omega.lambda), while others do not. For example,
      +free-variable-capture.lambda does not reduce its second argument expression
      +to y, as we would expect. This is because the K rewrite rules between syntactic
      +terms do not apply anywhere they match. They only apply where they have been
      +given permission to apply by means of appropriate evaluation strategies of language
      +constructs, which is done using strictness attributes, evaluation contexts,
      +heating/cooling rules, etc., as discussed in the next lessons.

      +

      The next lesson will show how to add LAMBDA the desired evaluation strategies
      +using strictness attributes.

      +

      Go to Lesson 3, LAMBDA: Evaluation Strategies using Strictness

      +

      MOVIE (out of date) [4'03"]

      +

      Evaluation Strategies using Strictness

      + +

      Here we learn how to use the K strict attribute to define desired evaluation
      +strategies. We will also learn how to tell K which terms are already
      +evaluated, so it does not attempt to evaluate them anymore and treats them
      +internally as results of computations.

      +

      Recall from the previous lecture that the LAMBDA program
      +free-variable-capture.lambda was stuck, because K was not given permission
      +to evaluate the arguments of the lambda application construct.

      +

      You can use the attribute strict to tell K that the corresponding construct
      +has a strict evaluation strategy, that is, that its arguments need to be
      +evaluated before the semantics of the construct applies. The order of
      +argument evaluation is purposely unspecified when using strict, and indeed
      +the K tool allows us to detect all possible non-deterministic behaviors that
      +result from such intended underspecification of evaluation strategies. We will
      +learn how to do that when we define the IMP language later in this tutorial;
      +we will also learn how to enforce a particular order of evaluation.

      +

      In order for the above strictness declaration to work effectively and
      +efficiently, we need to tell the K tool which expressions are meant to be
      +results of computations, so that it will not attempt to evaluate them anymore.
      +One way to do it is to make Val a syntactic subcategory of the builtin
      +KResult syntactic category. Since we use the same K parser to also parse
      +the semantics, we use the same syntax keyword to define additional syntax
      +needed exclusively for the semantics (like KResults). See lambda.k.

      +

      Compile again and then run some programs. They should all work as expected.
      +In particular, free-variable-capture.lambda now evaluates to a y.

      +

      We now got a complete and working semantic definition of call-by-value
      +lambda-calculus. While theoretically correct, our definition is not
      +easy to use and disseminate. In the next lessons we will learn how to
      +generate formatted documentation for LAMBDA and how to extend LAMBDA
      +in order to write human readable and interesting programs.

      +

      Go to Lesson 4, LAMBDA: Generating Documentation; Latex Attributes.

      +

      MOVIE (out of date) [2'20"]

      +

      Generating Documentation; Latex Attributes

      + +

      In this lesson we learn how to generate formatted documentation from K
      +language definitions. We also learn how to use Latex attributes to control
      +the formatting of language constructs, particularly of ones which have a
      +mathematical flavor and we want to display accordingly.

      +

      To enhance readability, we may want to replace the keyword lambda by the
      +mathematical lambda symbol in the generated documentation. We can control
      +the way we display language constructs in the generated documentation
      +by associating them Latex attributes.

      +

      This is actually quite easy. All we have to do is to associate a latex
      +attribute to the production defining the construct in question, following
      +the Latex syntax for defining new commands (or macros).

      +

      In our case, we associate the attribute latex(\lambda{#1}.{#2}) to the
      +production declaring the lambda abstraction (recall that in Latex, #n refers
      +to the n-th argument of the defined new command).

      +

      We will later see, in Lesson 9, that we can add arbitrarily complex Latex
      +comments and headers to our language definitions, which give us maximum
      +flexibility in formatting our language definitions.

      +

      Now we have a simple programming language, with a nice documentation. However,
      +it is not easy to write interesting programs in this language. Almost all
      +programming languages build upon existing data-types and libraries. The K
      +tool provides a few of these (and you can add more).

      +

      In the next lesson we show how we can add builtin integers and Booleans to
      +LAMBDA, so we can start to evaluate meaningful expressions.

      +

      Go to Lesson 5, LAMBDA: Adding Builtins; Side Conditions.

      +

      MOVIE (out of date) [3'13"]

      +

      Adding Builtins; Side Conditions

      + +

      We have already added the builtin identifiers (sort Id) to LAMBDA expressions,
      +but those had no operations on them. In this lesson we add integers and
      +Booleans to LAMBDA, and extend the builtin operations on them into
      +corresponding operations on LAMBDA expressions. We will also learn how to add
      +side conditions to rules, to limit the number of instances where they can
      +apply.

      +

      The K tool provides several builtins, which are automatically included in all
      +definitions. These can be used in the languages that we define, typically by
      +including them in the desired syntactic categories. You can also define your
      +own builtins in case the provided ones are not suitable for your language
      +(e.g., the provided builtin integers and operations on them are arbitrary
      +precision).

      +

      For example, to add integers and Booleans as values to our LAMBDA, we have to
      +add the productions

      +
      syntax Val ::= Int | Bool
+
      +

      Int and Bool are the nonterminals that correspond to these builtins.

      +

      To make use of these builtins, we have to add some arithmetic operation
      +constructs to our language. We prefer to use the conventional infix notation
      +for these, and the usual precedences (i.e., multiplication and division bind
      +tighter than addition, which binds tighter than relational operators).
      +Inspired from SDF, we use > instead of
      +| to state that all the previous constructs bind tighter than all the
      +subsequent ones. See lambda.k.

      +

      The only thing left is to link the LAMBDA arithmetic operations to the
      +corresponding builtin operations, when their arguments are evaluated.
      +This can be easily done using trivial rewrite rules, as shown in lambda.k.
      +In general, the K tool attempts to uniformly add the corresponding builtin
      +name as a suffix to all the operations over builtins. For example, the
      +addition over integers is an infix operation named +Int.

      +

      Compile the new lambda.k definition and evaluate some simple arithmetic
      +expressions. For example, if arithmetic.lambda is (1+2*3)/4 <= 1, then

      +
      krun arithmetic.lambda
+
      +

      yields, as expected, true. Note that the parser took the desired operation
      +precedence into account.

      +

      Let us now try to evaluate an expression which performs a wrong computation,
      +namely a division by zero. Consider the expression arithmetic-div-zero.lambda
      +which is 1/(2/3). Since division is strict and 2/3 evaluates to 0, this
      +expression reduces to 1/0, which further reduces to 1 /Int 0 by the rule for
      +division, which is now stuck (with the current back-end to the K tool).

      +

      In fact, depending upon the back-end that we use to execute K definitions and
      +in particular to evaluate expressions over builtins, 1 /Int 0 can evaluate to
      +anything. It just happens that the current back-end keeps it as an
      +irreducible term. Other K back-ends may reduce it to an explicit error
      +element, or issue a segmentation fault followed by a core dump, or throw an
      +exception, etc.

      +

      To avoid requesting the back-end to perform an illegal operation, we may use a
      +side condition in the rule of division, to make sure it only applies when the
      +denominator is non-zero.

      +

      Like in other operational formalisms, the role of the K side
      +conditions is to filter the number of instances of the rule. The notion
      +of a side condition comes from logics, where a sharp distinction is made
      +between a side condition (cheap) and a premise (expensive). Premises are
      +usually resolved using further (expensive) logical derivations, while side
      +conditions are simple (cheap) conditions over the rule meta-variables within
      +the underlying mathematical domains (which in K can be extended by the user,
      +as we will see in future lessons). Regarded as a logic, K derives rewrite
      +rules from other rewrite rules; therefore, the K side conditions cannot
      +contain other rewrites in them (using =>). This contrasts other rewrite
      +engines, for example Maude, which
      +allow conditional rules with rewrites in conditions.
      +The rationale behind this deliberate restriction in K is twofold:

      +
        +
      • On the one hand, general conditional rules require a complex, and thus slower
        +rewrite engine, which starts recursive (sometimes exhaustive) rewrite sessions
        +to resolve the rewrites in conditions. In contrast, the side conditions in K
        +can be evaluated efficiently by back-ends, for example by evaluating builtin
        +expressions and/or by calling builtin functions.
        • +
      • On the other hand, the semantic definitional philosophy of K is that rule
        +premises are unnecessary, so there is no need to provide support for them.
        • +
      +

      Having builtin arithmetic is useful, but writing programs with just lambda
      +and arithmetic constructs is still a pain. In the next two lessons we will
      +add conditional (if_then_else) and binding (let and letrec) constructs,
      +which will allow us to write nicer programs.

      +

      Go to Lesson 6, LAMBDA: Selective Strictness; Anonymous Variables.

      +

      MOVIE (out of date) [4'52"]

      +

      Selective Strictness; Anonymous Variables

      + +

      We here show how to define selective strictness of language constructs,
      +that is, how to state that certain language constructs are strict only
      +in some arguments. We also show how to use anonymous variables.

      +

      We next define a conditional if construct, which takes three arguments,
      +evaluates only the first one, and then reduces to either the second or the
      +third, depending on whether the first one evaluated to true or to false.

      +

      K allows to define selective strictness using the same strict attribute,
      +but passing it a list of numbers. The numbers correspond to the arguments
      +in which we want the defined construct to be strict. In our case,

      +
      syntax Exp ::= "if" Exp "then" Exp "else" Exp   [strict(1)]
+
      +

      states that the conditional construct is strict in the first argument.

      +

      We can now assume that its first argument will eventually reduce to a value, so
      +we only write the following two semantic rules:

      +
      rule if true  then E else _ => E
+rule if false then _ else E => E
+
      +

      Thus, we assume that the first argument evaluates to either true or false.

      +

      Note the use of the anonymous variable _. We use such variables purely for
      +structural reasons, to state that something is there but we don't care what.
      +An anonymous variable is therefore completely equivalent to a normal variable
      +which is unsorted and different from all the other variables in the rule. If
      +you use _ multiple times in a rule, they will all be considered distinct.

      +

      Compile lambda.k and write and execute some interesting expressions making
      +use of the conditional construct. For example, the expression

      +
      if 2<=1 then 3/0 else 10
+
      +

      evaluates to 10 and will never evaluate 3/0, thus avoiding an unwanted
      +division-by-zero.

      +

      In the next lesson we will introduce two new language constructs, called
      +let and letrec and conventionally found in functional programming
      +languages, which will allow us to already write interesting LAMBDA programs.

      +

      Go to Lesson 7, LAMBDA: Derived Constructs; Extending Predefined Syntax.

      +

      MOVIE (out of date) [2'14"]

      +

      Derived Constructs, Extending Predefined Syntax

      + +

      In this lesson we will learn how to define derived language constructs, that
      +is, ones whose semantics is defined completely in terms of other language
      +constructs. We will also learn how to add new constructs to predefined
      +syntactic categories.

      +

      When defining a language, we often want certain language constructs to be
      +defined in terms of other constructs. For example, a let-binding construct
      +of the form

      +
      let x = e in e'
+
      +

      is nothing but syntactic sugar for

      +
      (lambda x . e') e
+
      +

      This can be easily achieved with a rule, as shown in lambda.k.

      +

      Compile lambda.k and write some programs using let binders.

      +

      For example, consider a lets.lambda program which takes arithmetic.lambda
      +and replaces each integer by a let-bound variable. It should evaluate to
      +true, just like the original arithmetic.lambda.

      +

      Let us now consider a more interesting program, namely one that calculates the
      +factorial of 10:

      +
      let f = lambda x . (
+        (lambda t . lambda x . (t t x))
+        (lambda f . lambda x . (if x <= 1 then 1 else (x * (f f (x + -1)))))
+        x
+      )
+in (f 10)
+
      +

      This program follows a common technique to define fixed points in untyped
      +lambda calculus, based on passing a function to itself.

      +

      We may not like to define fixed-points following the approach above, because
      +it requires global changes in the body of the function meant to be recursive,
      +basically to pass it to itself (f f in our case above). The approach below
      +isolates the fixed-point aspect of the function in a so-called fixed-point
      +combinator      , which we call fix below, and then apply it to the function
      +defining the body of the factorial, without any changes to it:

      +
      let fix = lambda f . (
+          (lambda x . (f (lambda y . (x x y))))
+          (lambda x . (f (lambda y . (x x y))))
+        )
+in let f = fix (lambda f . lambda x .
+                (if x <= 1 then 1 else (x * (f (x + -1)))))
+   in (f 10)
+
      +

      Although the above techniques are interesting and powerful (indeed, untyped
      +lambda calculus is in fact Turing complete), programmers will probably not
      +like to write programs this way.

      +

      We can easily define a more complex derived construct, called letrec and
      +conventionally encountered in functional programming languages, whose semantics
      +captures the fixed-point idea above. In order to keep its definition simple
      +and intuitive, we define a simplified variant of letrec, namely one which only
      +allows to define one recursive one-argument function. See lambda.k.

      +

      There are two interesting observations here.

      +

      First, note that we have already in-lined the definition of the fix
      +combinator in the definition of the factorial, to save one application of the
      +beta reduction rule (and the involved substitution steps). We could have
      +in-lined the definition of the remaining let, too, but we believe that the
      +current definition is easier to read.

      +

      Second, note that we extended the predefined Id syntactic category with two
      +new constants, $x and $y. The predefined identifiers cannot start with
      +$, so programs that will be executed with this semantics cannot possibly
      +contain the identifiers xandx andy. In other words, by adding them to Id they
      +become indirectly reserved for the semantics. This is indeed desirable,
      +because any possible uses of xinthebodyofthefunctiondefinedusing‘letrec‘wouldbecapturedbythe‘lambdax in the body of the function defined +using `letrec` would be captured by the `lambdaxdeclaration in the definition ofletrec`.

      +

      Using letrec, we can now write the factorial program as elegantly as it can
      +be written in a functional language:

      +
      letrec f x = if x <= 1 then 1 else (x * (f (x + -1)))
+in (f 10)
+
      +

      In the next lesson we will discuss an alternative definition of letrec, based
      +on another binder, mu, specifically designed to define fixed points.

      +

      Go to Lesson 8, LAMBDA: Multiple Binding Constructs.

      +

      MOVIE (out of date) [5'10"]

      +

      Multiple Binding Constructs

      + +

      Here we learn how multiple language constructs that bind variables can
      +coexist. We will also learn about or recall another famous binder besides
      +lambda, namely mu, which can be used to elegantly define all kinds of
      +interesting fixed-point constructs.

      +

      The mu binder has the same syntax as lambda, except that it replaces
      +lambda with mu.

      +

      Since mu is a binder, in order for substitution to know how to deal with
      +variable capture in the presence of mu, we have to tell it that mu is a
      +binding construct, same like lambda. We take advantage of being there and
      +also add mu its desired latex attribute.

      +

      The intuition for

      +
      mu x . e
+
      +

      is that it reduces to e, but each free occurrence of x in e behaves
      +like a pointer that points back to mu x . e.

      +

      With that in mind, let us postpone the definition of mu and instead redefine
      +letrec F X = E in E' as a derived construct, assuming mu available. The
      +idea is to simply regard F as a fixed-point of the function

      +
      lambda X . E
+
      +

      that is, to first calculate

      +
      mu F . lambda X . E
+
      +

      and then to evaluate E' where F is bound to this fixed-point:

      +
      let F = mu F . lambda X . E in E'
+
      +

      This new definition of letrec may still look a bit tricky, particularly
      +because F is bound twice, but it is much simpler and cleaner than our
      +previous definition. Moreover, now it is done in a type-safe manner
      +(this aspect goes beyond our objective in this tutorial).

      +

      Let us now define the semantic rule of mu.

      +

      The semantics of mu is actually disarmingly simple. We just have to
      +substitute mu X . E for each free occurrence of X in E:

      +
      mu X . E => E[(mu X . E) / X]
+
      +

      Compile lambda.k and execute some recursive programs. They should be now
      +several times faster. Write a few more recursive programs, for example ones
      +for calculating the Ackermann function, for calculating the number of moves
      +needed to solve the Hanoi tower problem, etc.

      +

      We have defined our first programming language in K, which allows us to
      +write interesting functional programs. In the next lesson we will learn how
      +to fully document our language definition, in order to disseminate it, to ship
      +it to colleagues or friends, to publish it, to teach it, and so on.

      +

      Go to Lesson 9, LAMBDA: A Complete and Commented Definition.

      +

      MOVIE (out of date) [2'40"]

      +

      A Complete and Documented K Definition

      + +

      In this lesson you will learn how to add formal comments to your K definition,
      +in order to nicely document it. The generated document can be then used for
      +various purposes: to ease understanding the K definition, to publish it,
      +to send it to others, etc.

      +

      The K tool allows a literate programming style, where the executable
      +language definition can be documented by means of annotations. One such
      +annotation is the latex(_) annotation, where you can specify how to format
      +the given production when producing Latex output via the --output latex
      +option to krun, kast, and kprove.

      +

      There are three types of comments, which we discuss next.

      +

      Ordinary comments

      + +

      These use // or /* ... */, like in various programming languages. These
      +comments are completely ignored.

      +

      Document annotations

      + +

      Use the @ symbol right after // or /* in order for the comment to be
      +considered an annotation and thus be processed by the K tool when it
      +generates documentation.

      +

      As an example, we can go ahead and add such an annotation at the beginning
      +of the LAMBDA module, explaining how we define the syntax of this language.

      +

      Header annotations

      + +

      Use the ! symbol right after // or /* if you want the comment to be
      +considered a header annotation, that is, one which goes before
      +\begin{document} in the generated Latex. You typically need header
      +annotations to include macros, or to define a title, etc.

      +

      As an example, let us set a Latex length and then add a title and an
      +author to this K definition.

      +

      Compile the documentation and take a look at the results. Notice the title.

      +

      Feel free to now add lots of annotations to lambda.k.

      +

      Then compile and check the result. Depending on your PDF viewer, you
      +may also see a nice click-able table of contents, with all the sections
      +of your document. This could be quite convenient when you define large
      +languages, because it helps you jump to any part of the semantics.

      +

      Tutorial 1 is now complete. The next tutorial will take us through the
      +definition of a simple imperative language and will expose us to more
      +feature of the K framework and the K tool.

      +

      MOVIE (out of date) [6'07"]

      +

      Part 2: Defining IMP

      + +

      Here you will learn how to define a very simple imperative language in K
      +and the basics of how to work with configurations, cells, and computations.
      +Specifically, you will learn the following:

      +
        +
      • How to define languages using multiple modules.
        • +
      • How to define sequentially strict syntactic constructs.
        • +
      • How to use K's syntactic lists.
        • +
      • How to define, initialize and configure configurations.
        • +
      • How the language syntax is swallowed by the builtin K syntactic category.
        • +
      • The additional syntax of the K syntactic category.
        • +
      • How the strictness annotations are automatically desugared into rules.
        • +
      • The first steps of the configuration abstraction mechanism.
        • +
      +

      Like in the previous tutorial, this folder contains several lessons, each
      +adding new features to IMP. Do them in order. Also, make sure you completed
      +and understood the previous tutorial.

      +

      Defining a More Complex Syntax

      + +

      Here we learn how to define a more complex language syntax than LAMBDA's,
      +namely the C-like syntax of IMP. Also, we will learn how to define languages
      +using multiple modules, because we are going to separate IMP's syntax from
      +its semantics using modules. Finally, we will also learn how to use K's
      +builtin support for syntactic lists.

      +

      The K tool provides modules for grouping language features. In general, we
      +can organize our languages in arbitrarily complex module structures.
      +While there are no rigid requirements or even guidelines for how to group
      +language features in modules, we often separate the language syntax from the
      +language semantics in different modules.

      +

      In our case here, we start by defining two modules, IMP-SYNTAX and IMP, and
      +import the first in the second, using the keyword imports. As their names
      +suggest, we will place all IMP's syntax definition in IMP-SYNTAX and all its
      +semantics in IMP.

      +

      Note, however, that K does no more than simply includes all the
      +contents of the imported module in the one which imports it (making sure
      +that everything is only kept once, even if you import it multiple times).
      +In other words, there is currently nothing fancy in K tool's module system.

      +

      IMP has six syntactic categories, as shown in imp.k: AExp for arithmetic
      +expressions, BExp for Boolean expressions, Block for blocks, Stmt for
      +statements, Pgm for programs and Ids for comma-separated lists of
      +identifiers. Blocks are special statements, whose role is to syntactically
      +constrain the conditional statement and the while loop statement to only
      +take blocks as branches and body, respectively.

      +

      There is nothing special about arithmetic and Boolean expressions. They
      +are given the expected strictness attributes, except for <= and &&,
      +for demonstration purposes.

      +

      The <= is defined to be seqstrict, which means that it evaluates its
      +arguments in order, from left-to-right (recall that the strict operators
      +can evaluate their arguments in any, fully interleaved, orders). Like
      +strict, the seqstrict annotation can also be configured; for example, one
      +can specify in which arguments and in what order. By default, seqstrict
      +refers to all the arguments, in their left-to-right order. In our case here,
      +it is equivalent with seqstrict(1 2).

      +

      The && is only strict in its first argument, because we will give it a
      +short-circuited semantics (its second argument will only be evaluated when
      +the first evaluates to true). Recall the K tool also allows us to associate
      +LaTex attributes to constructs, telling the document generator how to display
      +them. For example, we associate <= the attribute latex({#1}\leq{#2}),
      +which makes it be displayed ≤\leq everywhere in the generated LaTex
      +documentation.

      +

      In this tutorial we take the freedom to associate the various constructs
      +parsing precedences that we have already tested and we know work well, so that
      +we can focus on the semantics here instead of syntax. In practice, though,
      +you typically need to experiment with precedences until you obtain the desired
      +parser.

      +

      Blocks are defined using curly brackets, and they can either be empty or
      +hold a statement.

      +

      Nothing special about the IMP statements. Note that ; is an assignment
      +statement terminator, not a statement separator. Note also that blocks are
      +special statements.

      +

      An IMP program declares a comma-separated list of variables using the keyword
      +int like in C, followed by a semicolon ;, followed by a statement.
      +Syntactically, the idea here is that we can wrap any IMP program within a
      +main(){...} function and get a valid C program. IMP does not allow variable
      +declarations anywhere else except through this construct, at the top-level of
      +the program. Other languages provided with the K distribution (see, e.g., the
      +IMP++ language also discussed in this tutorial) remove this top-level program
      +construct of IMP and add instead variable declaration as a statement construct,
      +which can be used anywhere in the program, not only at the top level.

      +

      Note how we defined the comma-separated list of identifiers using
      +List{Id,","}. The K tool provides builtin support for generic syntactic
      +lists. In general,

      +
      syntax B ::= List{A,T}
+
      +

      declares a new non-terminal, B, corresponding to T-separated sequences of
      +elements of A, where A is a non-terminal and T is a terminal. These
      +lists can also be empty, that is, IMP programs declaring no variable are also
      +allowed (e.g., int; {} is a valid IMP program). To instantiate and use
      +the K builtin lists, you should alias each instance with a (typically fresh)
      +non-terminal in your syntax, like we do with the Ids nonterminal.

      +

      Like with other K features, there are ways to configure the syntactic lists,
      +but we do not discuss them here.

      +

      Recall from Tutorial 1 (LAMBDA) that in order for strictness to work well
      +we also need to tell K which computations are meant to be results. We do
      +this as well now, in the module IMP: integers and Booleans are K results.

      +

      Kompile imp.k and test the generated parser by running some programs.
      +Since IMP is a fragment of C, you may want to select the C mode in your
      +editor when writing these programs. This will also give your the feel that
      +you are writing programs in a real programming language.

      +

      For example, here is sum.imp, which sums in sum all numbers up to n:

      +
      int n, sum;
+n = 100;
+sum=0;
+while (!(n <= 0)) {
+  sum = sum + n;
+  n = n + -1;
+}
+
      +

      Now krun it and see how it looks parsed in the default k cell.

      +

      The program collatz.imp tests the Collatz conjecture for all numbers up to
      +m and accumulates the total number of steps in s:

      +
      int m, n, q, r, s;
+m = 10;
+while (!(m<=2)) {
+  n = m;
+  m = m + -1;
+  while (!(n<=1)) {
+    s = s+1;
+    q = n/2;
+    r = q+q+1;
+    if (r<=n) {
+      n = n+n+n+1;         // n becomes 3*n+1 if odd
+    } else {n=q;}          //        of   n/2 if even
+  }
+}
+
      +

      Finally, program primes.imp counts in s all the prime numbers up to m:

      +
      int i, m, n, q, r, s, t, x, y, z;
+m = 10;  n = 2;
+while (n <= m) {
+  // checking primality of n and writing t to 1 or 0
+  i = 2;  q = n/i;  t = 1;
+  while (i<=q && 1<=t) {
+    x = i;
+    y = q;
+    // fast multiplication (base 2) algorithm
+    z = 0;
+    while (!(x <= 0)) {
+      q = x/2;
+      r = q+q+1;
+      if (r <= x) { z = z+y; } else {}
+      x = q;
+      y = y+y;
+    } // end fast multiplication
+    if (n <= z) { t = 0; } else { i = i+1;  q = n/i; }
+  } // end checking primality
+  if (1 <= t) { s = s+1; } else {}
+  n = n+1;
+}
+
      +

      All the programs above will run once we define the semantics of IMP. If you
      +want to execute them now, wrap them in a main(){...} function and compile
      +them and run them with your favorite C compiler.

      +

      Before we move to the K semantics of IMP, we would like to make some
      +clarifications regarding the K builtin parser, kast. Although it is quite
      +powerful, you should not expect magic from it! While the K parser can parse
      +many non-trivial languages (see, for example, the KOOL language in
      +pl-tutorial/2_languages) in the K distribution), it was
      +never meant to be a substitute for real parsers. We often call the syntax
      +defined in K the syntax of the semantics, to highlight the fact that its
      +role is to serve as a convenient notation when writing the semantics, not
      +necessarily as a means to define concrete syntax of arbitrarily complex
      +programming languages. See the KERNELC language for an example on how to connect an external parser for concrete syntax to
      +the K tool.

      +

      The above being said, we strongly encourage you to strive to make the
      +builtin parser work with your desired language syntax! Do not give up
      +simply because you don't want to deal with syntactic problems. On the
      +contrary, fight for your syntax! If you really cannot define your desired
      +syntax because of tool limitations, we would like to know. Please tell us.

      +

      Until now we have only seen default configurations. In the next lesson we
      +will learn how to define a K custom configuration.

      +

      Go to Lesson 2, IMP: Defining a Configuration.

      +

      MOVIE (out of date) [09'15"]

      +

      Defining a Configuration

      + +

      Here we learn how to define a configuration in K. We also learn how to
      +initialize and how to display it.

      +

      As explained in the overview presentation on K, configurations are quite
      +important, because all semantic rules match and apply on them.
      +Moreover, they are the backbone of configuration abstraction, which allows
      +you to only mention the relevant cells in each semantic rule, the rest of
      +the configuration context being inferred automatically. The importance of
      +configuration abstraction will become clear when we define more complex
      +languages (even in IMP++). IMP does not really need it. K configurations
      +are constructed making use of cells, which are labeled and can be arbitrarily
      +nested.

      +

      Configurations are defined with the keyword configuration. Cells are
      +defined using an XML-ish notation stating clearly where the cell starts
      +and where it ends.

      +

      While not enforced by the tool, we typically like to put the entire
      +configuration in a top-level cell, called T. So let's define it:

      +
      configuration <T>...</T>
+
      +

      Cells can have other cells inside. In our case of IMP, we need a cell to
      +hold the remaining program, cell which we typically call k, and a cell to
      +hold the program state. Let us add them:

      +
      configuration <T> <k>...</k> <state>...</state> </T>
+
      +

      K allows us to also specify how to initialize a configuration at the same
      +time with declaring the configuration. All we have to do is to fill in
      +the contents of the cells with some terms. The syntactic categories of
      +those terms will also indirectly define the types of the corresponding
      +cells.

      +

      For example, we want the k cell to initially hold the program that is passed
      +to krun. K provides a builtin configuration variable, called $PGM, which
      +is specifically designed for this purpose: krun will place its program there
      +(after it parses it, or course). The K tool allows users to define their own
      +configuration variables, too, which can be used to develop custom
      +initializations of program configurations with the help of krun; this can be
      +quite useful when defining complex languages, but we do not discuss it in
      +this tutorial.

      +
      configuration <T> <k> $PGM </k> <state>...</state>  </T>
+
      +

      Moreover, we want the program to be a proper Pgm term (because we do not
      +want to allow krun to take fragments of programs, for example, statements).
      +Therefore, we tag $PGM with the desired syntactic category, Pgm:

      +
      configuration <T> <k> $PGM:Pgm </k> <state>...</state>  </T>
+
      +

      Like for other variable tags in K, a run-time check will be performed and the
      +semantics will get stuck if the passed term is not a well-formed program.

      +

      We next tell K that the state cell should be initialized with the empty map:

      +
      configuration <T> <k> $PGM:Pgm </k> <state> .Map </state>  </T>
+
      +

      Recall that in K . stands for nothing. However, since there are various
      +types of nothing, to avoid confusion we can suffix the . with its desired
      +type. K has several builtin data-types, including lists, sets, bags, and
      +maps. .Map is the empty map.

      +

      Kompile imp.k and run several programs to see how the configuration is
      +initialized as desired.

      +

      When configurations get large, and they do when defining large programming
      +languages, you may want to color the cells in order to more easily distinguish
      +them. This can be easily achieved using the color cell attribute, following
      +again an XML-ish style:

      +
      configuration <T color="yellow">
+                <k color="green"> $PGM:Pgm </k>
+                <state color="red"> .Map </state>
+              </T>
+
      +

      In the next lesson we will learn how to write rules that involve cells.

      +

      Go to Lesson 3, IMP: Computations, Results, Strictness; Rules Involving Cells.

      +

      MOVIE (out of date) [04'21"]

      +

      Computations, Results, Strictness; Rules Involving Cells

      + +

      In this lesson we will learn about the syntactic category K of computations,
      +about how strictness attributes are in fact syntactic sugar for rewrite rules
      +over computations, and why it is important to tell the tool which
      +computations are results. We will also see a K rule that involves cells.

      +

      K Computations

      + +

      Computation structures, or more simply computations, extend the abstract
      +syntax of your language with a list structure using ~> (read followed
      +by       or and then, and written ↷\curvearrowright in Latex) as a separator.
      +K provides a distinguished sort, K, for computations. The extension of the
      +abstract syntax of your language into computations is done automatically by
      +the K tool when you declare constructs using the syntax keyword, so the K
      +semantic rules can uniformly operate only on terms of sort K. The intuition
      +for computation structures of the form

      +
      t1 ~> t2 ~> ... ~> tn
+
      +

      is that the listed tasks are to be processed in order. The initial
      +computation typically contains the original program as its sole task, but
      +rules can then modify it into task sequences, as seen shortly.

      +

      Strictness in Theory

      + +

      The strictness attributes, used as annotations to language constructs,
      +actually correspond to rules over computations. For example, the
      +strict(2) attribute of the assignment statement corresponds to the
      +following two opposite rules (X ranges over Id and A over AExp):

      +
      X=A; => A ~> X=[];
+A ~> X=[]; => X=A;
+
      +

      The first rule pulls A from the syntactic context X=A; and schedules it
      +for processing. The second rule plugs A back into its context.
      +Inspired from the chemical abstract machine, we call rules of the first
      +type above heating rules and rules of the second type cooling rules.
      +Similar rules are generated for other arguments in which operations are
      +strict. Iterative applications of heating rules eventually bring to the
      +top of the computation atomic tasks, such as a variable lookup, or a
      +builtin operation, which then make computational progress by means of other
      +rules. Once progress is made, cooling rules can iteratively plug the result
      +back into context, so that heating rules can pick another candidate for
      +reduction, and so on and so forth.

      +

      When operations are strict only in some of their arguments, the corresponding
      +positions of the arguments in which they are strict are explicitly enumerated
      +in the argument of the strict attribute, e.g., strict(2) like above, or
      +strict(2 3) for an operation strict in its second and third arguments, etc.
      +If an operation is simply declared strict then it means that it is strict
      +in all its arguments. For example, the strictness of addition yields:

      +
      A1+A2 => A1 ~> []+A2
+A1 ~> []+A2 => A1+A2
+A1+A2 => A2 ~> A1+[]
+A2 ~> A1+[] => A1+A2
+
      +

      It can be seen that such heating/cooling rules can easily lead to
      +non-determinism, since the same term may be heated many different ways;
      +these different evaluation orders may lead to different behaviors in some
      +languages (not in IMP, because its expressions do not have side effects,
      +but we will experiment with non-determinism in its successor, IMP++).

      +

      A similar desugaring applies to sequential strictness, declared with the
      +keyword seqstrict. While the order of arguments of strict is irrelevant,
      +it matters in the case of seqstrict: they are to be evaluated in the
      +specified order; if no arguments are given, then they are assumed by default
      +to be evaluated from left-to-right. For example, the default heating/cooling
      +rules associated to the sequentially strict <= construct above are
      +(A1, A2 range over AExp and I1 over Int):

      +
      A1<=A2 => A1 ~> []<=A2
+A1 ~> []<=A2 => A1<=A2
+I1<=A2 => A2 ~> I1<=[]
+A2 ~> I1<=[] => I1<=A2
+
      +

      In other words, A2 is only heated/cooled after A1 is already evaluated.

      +

      While the heating/cooling rules give us a nice and uniform means to define
      +all the various allowable ways in which a program can evaluate, all based
      +on rewriting, the fact that they are reversible comes with a serious practical
      +problem: they make the K definitions unexecutable, because they lead to
      +non-termination.

      +

      Strictness in Practice; K Results

      + +

      To break the reversibility of the theoretical heating/cooling rules, and,
      +moreover, to efficiently execute K definitions, the current implementation of
      +the K tool relies on users giving explicit definitions of their languages'
      +results.

      +

      The K tool provides a predicate isKResult, which is automatically defined
      +as we add syntactic constructs to KResult (in fact the K tool defines such
      +predicates for all syntactic categories, which are used, for example, as
      +rule side conditions to check user-declared variable memberships, such as
      +V:Val stating that V belongs to Val).

      +

      The kompile tool, depending upon what it is requested to do, changes the
      +reversible heating/cooling rules corresponding to evaluation strategy
      +definitions (e.g., those corresponding to strictness attributes) to avoid
      +non-termination. For example, when one is interested in obtaining an
      +executable model of the language (which is the default compilation mode of
      +kompile), then heating is performed only when the to-be-pulled syntactic
      +fragment is not a result, and the corresponding cooling only when the
      +to-be-plugged fragment is a result. In this case, e.g., the heating/cooling
      +rules for assignment are modified as follows:

      +
      X=A; => A ~> X=[];  requires notBool isKResult(A)
+A ~> X=[]; => X=A;  requires isKResult(A)
+
      +

      Note that non-termination of heating/cooling is avoided now. The only thing
      +lost is the number of possible behaviors that a program can manifest, but
      +this is irrelevant when all we want is one behavior.

      +

      As will be discussed in the IMP++ tutorial, the heating/cooling rules are
      +modified differently by kompile when we are interested in other aspects
      +of the language definition, such us, for example, in a search-able model that
      +comprises all program behaviors. This latter model is obviously more general
      +from a theoretical perspective, but, in practice, it is also slower to execute.
      +The kompile tool strives to give you the best model of the language for the
      +task you are interested in.

      +

      Can't Results be Inferred Automatically?

      + +

      This is a long story, but the short answer is: No!. Maybe in some cases
      +it is possible, but we prefer to not attempt it in the K tool. For example,
      +you most likely do not want any stuck computation to count as a result,
      +since some of them can happen simply because you forgot a semantic rule that
      +could have further reduce it! Besides, in our experience with defining large
      +languages, it is quite useful to take your time and think of what the results
      +of your language's computations are. This fact in itself may help you improve
      +your overall language design. We typically do it at the same time with
      +defining the evaluation strategies of our languages. Although in theory K
      +could infer the results of your language as the stuck computations, based on
      +the above we have deliberately decided to not provide this feature, in spite
      +of requests from some users. So you currently do have to explicitly define
      +your K results if you want to effectively use the K tool. Note, however, that
      +theoretical definitions, not meant to be executed, need not worry about
      +defining results (that's because in theory semantic rules apply modulo the
      +reversible heating/cooling rules, so results are not necessary).

      +

      A K Rule Involving Cells

      + +

      All our K rules so far in the tutorial were of the form

      +
      rule left => right requires condition
+
      +

      where left and right were syntactic, or more generally computation, terms.

      +

      Here is our first K rule explicitly involving cells:

      +
      rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
+
      +

      Recall that the k cell holds computations, which are sequences of tasks
      +separated by ~>. Also, the state cell holds a map, which is a set of
      +bindings, each binding being a pair of computations (currently, the
      +K builtin data-structures, like maps, are untyped; or, said differently,
      +they are all over the type of computations, K).

      +

      Therefore, the two cells mentioned in the rule above hold collections
      +of things, ordered or not. The ...s, which we also call cell frames,
      +stand for more stuff there, which we do not care about.

      +

      The rewrite relation => is allowed in K to appear anywhere in a term, its
      +meaning being that the corresponding subterm is rewritten as indicated in the
      +shown context. We say that K's rewriting is local.

      +

      The rule above says that if the identifier X is the first task in the k
      +cell, and if X is bound to I somewhere in the state, then X rewrites
      +to I locally in the k cell. Therefore, IMP variables need to be already
      +declared when looked up.

      +

      Of course, the K rule above can be translated into an ordinary rewrite rule
      +of the form

      +
      rule <k> X ~> Rest </k> <state> Before (X |-> I) After </state>
+  => <k> I ~> Rest </k> <state> Before (X |-> I) After </state>
+
      +

      Besides being more verbose and thus tedious to write, this ordinary rule
      +is also more error-prone; for example, we may forget the Rest variable
      +in the right-hand-side, etc. Moreover, the concurrent semantics of K
      +allows for its rules to be interpreted as concurrent transactions, where
      +the context is the read-only component of the transaction, while the
      +subterms which are rewritten are read/write component of the transaction;
      +thus, K rule instances can apply concurrently if they only overlap
      +on read-only parts, while they cannot if regarded as ordinary rewrite logic
      +rules. Note: our current implementation of the K tool is not concurrent,
      +so K rules are in fact desugared as normal rewrite rules in the K tool.

      +

      Kompile imp.k using a documentation option and check out how the K rule
      +looks in the generated document. The ... frames are displayed as cell
      +tears, metaphorically implying that those parts of the cells that we
      +do not care about are torn away. The rewrite relation is replaced by a
      +horizontal line: specifically, the subterm which rewrites, X, is
      +underlined, and its replacement is written underneath the line.

      +

      In the next lesson we define the complete K semantics of IMP and
      +run the programs we parsed in the first lesson.

      +

      Go to Lesson 4, IMP: Configuration Abstraction, Part 1; Types of Rules.

      +

      MOVIE (out of date) [10'30"]

      +

      Configuration Abstraction, Part 1; Types of Rules

      + +

      Here we will complete the K definition of IMP and, while doing so, we will
      +learn the very first step of what we call configuration abstraction.

      +

      The IMP Semantic Rules

      + +

      Let us add the remaining rules, in the order in which the language constructs
      +were defined in IMP-SYNTAX.

      +

      The rules for the arithmetic and Boolean constructs are self-explanatory.
      +Note, however, that K will infer the correct sorts of all the variables in
      +these rules, because they appear as arguments of the builtin operations
      +(_+Int_, etc.). Moreover, the inferred sorts will be enforced dynamically.
      +Indeed, we do not want to apply the rule for addition, for example, when the
      +two arguments are not integers. In the rules for &&, although we prefer to
      +not do it here for simplicity, we could have eliminated the dynamic check by
      +replacing B (and similarly for _) with B:K. Indeed, it can be shown
      +that whenever any of these rules apply, B (or _) is a BExp anyway.
      +That's because there is no rule that can touch such a B (or _); this
      +will become clearer shortly, when we discuss the first step of configuration
      +abstraction. Therefore, since we know that B will be a BExp anyway, we
      +could save the time it takes to check its sort; such times may look minor,
      +but they accumulate, so some designers may prefer to avoid run-time checks
      +whenever possible.

      +

      The block rules are trivial. However, the rule for non-empty blocks is
      +semantically correct only because we do not have local variable declarations
      +in IMP. We will have to change this rule in IMP++.

      +

      The assignment rule has two =>: one in the k cell dissolving the
      +assignment statement, and the other in the state cell updating the value of
      +the assigned variable. Note that the one in the state is surrounded by
      +parentheses: (_ => I). That is because => is greedy: it matches as much
      +as it can to the left and to the right, until it reaches the cell boundaries
      +(closed or open). If you want to limit its scope, or for clarity, you can use
      +parentheses like here.

      +

      The rule for sequential composition simply desugars S1 S2 into S1 ~> S2.
      +Indeed, the two have exactly the same semantics. Note that statements
      +evaluate to nothing (.), so once S1 is processed in S1 ~> S2, then the
      +next task is automatically S2, without wasting any step for the transition.

      +

      The rules for the conditional and while statements are clear. One thing to
      +keep in mind now is that the while unrolling rule will not apply
      +indefinitely in the positive branch of the resulting conditional, because
      +of K's configuration abstraction, which will be discussed shortly.

      +

      An IMP program declares a set of variables and then executes a
      +statement in the state obtained after initializing all those variables
      +to 0. The rules for programs initialize the declared variables one by one,
      +checking also that there are no duplicates. We check for duplicates only for
      +demonstration purposes, to illustrate the keys predefined operation that
      +returns the set of keys of a map, and the set membership operation in.
      +In practice, we typically define a static type checker for our language,
      +which we execute before the semantics and reject inappropriate programs.

      +

      The use of the .Ids in the second rule is not necessary. We could have
      +written int; S instead of int .Ids; S and the K tool would parse it and
      +kompile the definition correctly, because it uses the same parser used for
      +parsing programs also to parse the semantics. However, we typically prefer to
      +explicitly write the nothing values in the semantics, for clarity;
      +the parser has been extended to accept these. Note that the first rule
      +matches the entire k cell, because int_;_ is the top-level program
      +construct in IMP, so there is nothing following it in the computation cell.
      +The anonymous variable stands for the second argument of this top-level program
      +construct, not for the rest of the computation. The second rule could have
      +also been put in a complete k cell, but we preferred not to, for simplicity.

      +

      Our IMP semantics is now complete, but there are a few more things that we
      +need to understand and do.

      +

      Configuration Abstraction, Part 1

      + +

      First, let us briefly discuss the very first step of configuration abstraction.
      +In K, all semantic rules are in fact rules between configurations. As soon
      +explained in the IMP++ tutorial, the declared configuration cell structure is
      +used to automatically complete the missing configuration parts in rules.
      +However, many rules do not involve any cells, being rules between syntactic
      +terms (of sort K); for example, we had only three rules involving cells in our
      +IMP semantics. In this case, the k cell will be added automatically and the
      +actual rewrite will happen on top of the enclosed computation. For example,
      +the rule for the while loop is automatically translated into the following:

      +
      rule <k> while (B) S => if (B) {S while (B) S} else {} ...</k>
+
      +

      Since the first task in computations is what needs to be done next, the
      +intuition for this rule completion is that the syntactic transition
      +only happens when the term to rewrite is ready for processing. This explains,
      +for example, why the while loop unrolling does not indefinitely apply in the
      +positive branch of the conditional: the inner while loop is not ready for
      +evaluation yet. We call this rule completion process, as well as other
      +similar ones, configuration abstraction. That is because the incomplete
      +rule abstracts away the configuration structure, thus being easier to read.
      +As seen soon when we define IMP++, configuration abstraction is not only a
      +user convenience; it actually significantly increases the modularity of our
      +definitions. The k-cell-completion is only the very first step, though.

      +

      If you really want certain rewrites over syntactic terms to apply
      +anywhere they match, then you should tag the rule with the attribute
      +anywhere, which was discussed in Tutorial 1, Lesson 2.5.

      +

      Kompile and then krun the programs that you only parsed in Lesson 1. They
      +should all execute as expected. The state cell shows the final state
      +of the program. The k cell shows the final code contents, which should be
      +empty whenever the IMP program executes correctly.

      +

      Kompile also with the documentation option and take a look at the generated
      +documentation. The assignment rule should particularly be of interest,
      +because it contains two local rewrites.

      +

      In the next lesson we comment the IMP definition and conclude this tutorial.

      +

      Go to Lesson 5, IMP: Completing and Documenting IMP.

      +

      MOVIE (out of date) [09'16"]

      +

      Completing and Documenting IMP

      + +

      We here learn no new concepts, but it is a good moment to take a break
      +and contemplate what we learned so far.

      +

      Let us add lots of formal annotations to imp.k.

      +

      Once we are done with the annotations, we kompile with the documentation
      +option and then take a look at the produced document. We often call these
      +documents language posters. Depending on how much information you add to
      +these language posters, they can serve as standalone, formal presentations
      +of your languages. For example, you can print them as large posters and
      +post them on the wall, or in poster sessions at conferences.

      +

      This completes our second tutorial. The next tutorials will teach us more
      +features of the K framework, such as how to define languages with complex
      +control constructs (like callcc), languages which are concurrent, and so on.

      +

      MOVIE (out of date) [03'45"]

      +

      Part 3: Defining LAMBDA++

      + +

      Here you will learn how to define language constructs which abruptly change
      +the execution control flow, and how to define language semantics following
      +and environment/store style. Specifically, you will learn the following:

      +
        +
      • How to define constructs like callcc, which allow you to take snapshots of
        +program executions and to go back in time at any moment.
        • +
      • How to define languages in an environment/store style.
        • +
      • Some basic notions about the use of closures and closure-like semantic
        +structures to save and restore execution environments.
        • +
      • Some basic intuitions about reusing existing semantics in new languages,
        +as well as some of the pitfalls in doing so.
        • +
      +

      Abrupt Changes of Control

      + +

      Here we add call-with-current-continuation (callcc) to the definition of
      +LAMBDA completed in Tutorial 1, and call the resulting language LAMBDA++.
      +While doing so, we will learn how to define language constructs that
      +abruptly change the execution control flow.

      +

      Take over the lambda.k definition from Lesson 8 in Part 1 of this Tutorial,
      +which is the complete definition of the LAMBDA language, but without the
      +comments.

      +

      callcc is a good example for studying the capabilities of a framework to
      +support abrupt changes of control, because it is one of the most
      +control-intensive language constructs known. Scheme is probably the first
      +programming language that incorporated the callcc construct, although
      +similar constructs have been recently included in many other languages in
      +one form or another.

      +

      Here is a quick description: callcc e passes the remaining computation
      +context, packaged as a function k, to e (which is expected to be a function);
      +if during its evaluation e passes any value to k, then the current
      +execution context is discarded and replaced by the one encoded by k and
      +the value is passed to it; if e evaluates normally to some value v and
      +passes nothing to k in the process, then v is returned as a result of
      +callcc e and the execution continues normally. For example, we want the
      +program callcc-jump.lambda:

      +
      (callcc (lambda k . ((k 5) + 2))) + 10
+
      +

      to evaluate to 15, not 17! Indeed, the computation context [] + 10 is
      +passed to callcc's argument, which then sends it a 5, so the computation
      +resumes to 5 + 10. On the other hand, the program callcc-not-jump.lambda

      +
      (callcc (lambda k . (5 + 2))) + 10
+
      +

      evaluates to 17.

      +

      If you like playing games, you can metaphorically think of callcc e as
      +saving your game state in a file and passing it to your friend e.
      +Then e can decide at some moment to drop everything she was doing, load
      +your game and continue to play it from where you were.

      +

      The behavior of many popular control-changing constructs can be obtained
      +using callcc. The program callcc-return.lambda shows, for example, how to
      +obtain the behavior of a return statement, which exits the current execution
      +context inside a function and returns a value to the caller's context:

      +
      letrec f x = callcc (lambda return . (
+  f (if (x <= 0) then ((return 1) / 0) else 2)
+))
+in (f -3)
+
      +

      This should evaluate to 1, in spite of the recursive call to f
      +and of the division by zero! Note that return is nothing but a variable
      +name, but one which is bound to the current continuation at the beginning of
      +the function execution. As soon as 1 is passed to return, the computation
      +jumps back in time to where callcc was defined! Change -3 to 3 and the
      +program will loop forever.

      +

      callcc is quite a powerful and beautiful language construct, although one
      +which is admittedly hard to give semantics to in some frameworks.
      +But not in K 😃 Here is the entire K syntax and semantics of callcc:

      +
      syntax Exp ::= "callcc" Exp  [strict]
+syntax Val ::= cc(K)
+rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+rule <k> cc(K) V ~> _ =>  V ~> K </k>
+
      +

      Let us first discuss the annotated syntax. We declared callcc strict,
      +because its argument may not necessarily be a function yet, so it may need
      +to be evaluated. As explained above, we need to encode the remaining
      +computation somehow and pass it to callcc's argument. More specifically,
      +since LAMBDA is call-by-value, we have to encode the remaining computation as
      +a value. We do not want to simply subsort computations to Val, because there
      +are computations which we do not want to be values. A simple solution to
      +achieve our goal here is to introduce a new value construct, say cc (from
      +current-continuation), which holds any computation.

      +

      Note that, inspired from SDF,
      +K allows you to define the syntax of helping semantic operations, like cc,
      +more compactly. Typically, we do not need a fancy syntax for such operators;
      +all we need is a name, followed by open parenthesis, followed by a
      +comma-separated list of arguments, followed by closed parenthesis. If this
      +is the syntax that you want for a particular construct, then K allows you to
      +drop all the quotes surrounding the terminals, as we did above for cc.

      +

      The semantic rules do exactly what the English semantics of callcc says.
      +Note that here, unlike in our definition of LAMBDA in Tutorial 1, we had
      +to mention the cell <k/> in our rules. This is because we need to make sure
      +that we match the entire remaining computation, not only a fragment of it!
      +For example, if we replace the two rules above with

      +
      rule (callcc V:Val => V cc(K)) ~> K
+rule cc(K) V ~> _ =>  V ~> K
+
      +

      then we get a callcc which is allowed to non-deterministically pick a
      +prefix of the remaining computation and pass it to its argument, and then
      +when invoked within its argument, a non-deterministic prefix of the new
      +computation is discarded and replaced by the saved one. Wow, that would
      +be quite a language! Would you like to write programs in it? 😃

      +

      Consequently, in K we can abruptly change the execution control flow of a
      +program by simply changing the contents of the <k/> cell. This is one of
      +the advantages of having an explicit representation of the execution context,
      +like in K or in reduction semantics with evaluation contexts. Constructs like
      +callcc are very hard and non-elegant to define in frameworks such as SOS,
      +because those implicitly represent the execution context as proof context,
      +and the latter cannot be easily changed.

      +

      Now that we know how to handle cells in configurations and use them in rules,
      +in the next lesson we take a fresh look at LAMBDA and define it using
      +an environment-based style, which avoids the complexity of substitution
      +(e.g., having to deal with variable capture) and is closer in spirit to how
      +functional languages are implemented.

      +

      Go to Lesson 2, LAMBDA++: Semantic (Non-Syntactic) Computation Items.

      +

      MOVIE (out of date) [6'28"]

      +

      Semantic (Non-Syntactic) Computation Items

      + +

      In this lesson we start another semantic definition of LAMBDA++, which
      +follows a style based on environments instead of substitution. In terms of
      +K, we will learn how easy it is to add new items to the syntactic category
      +of computations K, even ones which do not have a syntactic nature.

      +

      An environment binds variable names of interest to locations where their
      +values are stored. The idea of environment-based definitions is to maintain
      +a global store mapping locations to values, and then have environments
      +available when we evaluate expressions telling where the variables are
      +located in the store. Since LAMBDA++ is a relatively simple language, we
      +only need to maintain one global environment. Following a similar style
      +like in IMP, we place all cells into a top cell T:

      +
      configuration <T>
+                <k> $PGM:Exp </k>
+                <env> .Map </env>
+                <store> .Map </store>
+              </T>
+
      +

      Recall that $PGM is where the program is placed by krun after parsing. So
      +the program execution starts with an empty environment and an empty store.

      +

      In environment-based definitions of lambda-calculi, lambda abstractions
      +evaluate to so-called closures:

      +
      rule <k> lambda X:Id . E => closure(Rho,X,E) ...</k>
+     <env> Rho </env>
+
      +

      A closure is like a lambda abstraction, but it also holds the environment
      +in which it was declared. This way, when invoked, a closure knows where to
      +find in the store the values of all the variables that its body expression
      +refers to. We will define the lookup rule shortly.

      +

      Therefore, unlike in the substitution-based definitions of LAMBDA and
      +LAMBDA++, neither the lambda abstractions nor the identifiers are values
      +anymore here, because they both evaluate further: lambda abstractions to
      +closures and identifiers to their values in the store. In fact, the only
      +values at this moment are the closures, and they are purely semantic entities,
      +which cannot be used explicitly in programs. That's why we modified the
      +original syntax of the language to include no Val syntactic category
      +anymore, and that's why we need to add closures as values now; same like
      +before, we add a Val syntactic category which is subsorted
      +to KResult. In general, whenever you have any strictness attributes,
      +your should also define some K results.

      +

      Invoking a closure is a bit more involved than the substitution-based
      +beta-reduction: we need to switch to the closure's environment, then create a
      +new, or fresh, binding for the closure's parameter to the value passed to the
      +closure, then evaluate the closure's body, and then switch back to the
      +caller's environment, which needs to be stored somewhere in the meanwhile.
      +We can do all these with one rule:

      +
      rule <k> closure(Rho,X,E) V:Val => E ~> Rho' ...</k>
+     <env> Rho' => Rho[X <- !N] </env>
+     <store>... .Map => (!N:Int |-> V) ...</store>
+
      +

      Therefore, we atomically do all the following:

      +
        +
      • switch the computation to the closure's body, E, followed by a
        +caller-environment-recovery task Rho' (note that Rho' is the
        +current environment),
        • +
      • generate a fresh location !N (the ! is important, we discuss it below),
        +bind X to !N in closure's environment and switch the current environment
        +Rho' to that one,
        • +
      • write the value passed to the closure, V, at location !N.
        • +
      +

      This was the most complex K rule we've seen so far in the tutorial. Note,
      +however, that this one rule achieves a lot. It is, in fact, quite compact
      +considering how much it does. Note also that everything that this K rule
      +mentions is needed also conceptually in order to achieve this task, so it
      +is minimal from that point of view. That would not be the case if we
      +used, instead, a conventional rewrite rule, because we would have had to
      +mention the remaining store, say Sigma, in both sides of the rule, to say
      +it stays unchanged. Here we just use ....

      +

      The declaration of the fresh variable above, !N, is new and needs
      +some explanation. First, note that !N appears only in the right-hand-side
      +terms in the rule, that is, it is not matched when the rule is applied.
      +Instead, a fresh Nat element is generated each time the rule is applied.
      +In K, we can define syntactic categories which have the capability to
      +generate fresh elements like above, using unbound variables whose name starts
      +with a !. The details of how to do that are beyond the scope of this
      +tutorial (see Tutorial 6). All we need to know here is that an arbitrary
      +fresh element of that syntactic category is generated each time the rule
      +is applied. We cannot rely on the particular name or value of the generated
      +element, because that can change with the next version of the K tool, or
      +even from execution to execution with the same version. All you can rely
      +on is that each newly generated element is distinct from the previously
      +generated elements for the same syntactic category.

      +

      Unlike in the substitution-based definition, we now also need a lookup rule:

      +
      rule <k> X => V ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> V ...</store>
+
      +

      This rule speaks for itself: replace X by the value V located in the store
      +at X's location N in the current environment.

      +

      The only thing left to define is the auxiliary environment-recovery operation:

      +

      rule _:Val ~> (Rho => .) ... _ => Rho

      +

      When the item preceding the environment recovery task Rho in the
      +computation becomes a value, replace the current environment with Rho
      +and dissolve Rho from the computation.

      +

      Let us kompile and ... fail:

      +
      kompile lambda
+
      +

      gives a parsing error saying that V:Val does not fit there in the closure
      +invocation rule. That's because Val and Exp are currently completely
      +disconnected, so K rightfully complains that we want to apply a value to
      +another one, because application was defined to work with expressions, not
      +values. What we forgot here was to state that Exp includes Val:

      +
      syntax Exp ::= Val
+
      +

      Now everything works, but it is a good time to reflect a bit.

      +

      So we added closures, which are inherently semantic entities, to the syntax
      +of expressions. Does that mean that we can now write LAMBDA programs with
      +closures in them? Interestingly, with our current definition of LAMBDA,
      +which purposely did not follow the nice organization of IMP into syntax and
      +semantic modules, and with K's default parser, kast, you can. But you are
      +not supposed to speculate this! In fact, if you use an external parser, that
      +parser will reject programs with explicit closures. Also, if we split the
      +LAMBDA definition into two modules, one called LAMBDA-SYNTAX containing
      +exclusively the desired program syntax and one called LAMBDA importing the
      +former and defining the syntax of the auxiliary operations and the semantics,
      +then even K's default parser will reject programs using auxiliary syntactic
      +constructs.

      +

      Indeed, when you kompile a language, say lang.k, the tool will by default
      +attempt to find a module LANG-SYNTAX and generate the program parser from
      +that. If it cannot find it, then it will use the module LANG instead. There
      +are also ways to tell kompile precisely which syntax module you want to use
      +for the program parser if you don't like the default convention.
      +See kompile --help.

      +

      Another insightful thought to reflect upon, is the relationship between your
      +language's values and other syntactic categories. It is often the case that
      +values form a subset of the original language syntax, like in IMP (Part 2 of
      +the tutorial), but sometimes that is not true, like in our case here. When
      +that happens, in order for the semantics to be given smoothly and uniformly
      +using the original syntax, you need to extend your language's original
      +syntactic categories with the new values. The same holds true in other
      +semantic approaches, not only in K, even in ones which are considered purely
      +syntactic. As it should be clear by now, K does not enforce you to use a
      +purely syntactic style in your definitions; nevertheless, K does allow you to
      +develop purely syntactic definitions, like LAMBDA in Part 1 of the tutorial,
      +if you prefer those.

      +

      krun some programs, such as those provided in Lesson 1 of the LAMBDA
      +tutorial (Part 1). Note the closures, both as results in the <k/> cell,
      +and as values in the store. Also, since variables are not values anymore,
      +expressions that contain free variables may get stuck with one of those on
      +top of their computation. See, for example, free-variable-capture.lambda,
      +which gets stuck on z, because z is free, so it cannot evaluate it.
      +If you want, you can go ahead and manually provide a configuration with
      +z mapped to some location in the environment and that location mapped to
      +some value in the store, and then you can also execute this program. The
      +program omega.lambda should still loop.

      +

      Although we completely changed the definitional style of LAMBDA, the semantics
      +of the other constructs do not need to change, as seen in the next lesson.

      +

      Go to Lesson 3, LAMBDA++: Reusing Existing Semantics.

      +

      MOVIE (out of date) [8'02"]

      +

      Reusing Existing Semantics

      + +

      In this lesson we will learn that, in some cases, we can reuse existing
      +semantics of language features without having to make any change!

      +

      Although the definitional style of the basic LAMBDA language changed quite
      +radically in our previous lesson, compared to its original definition in
      +Part 1 of the tutorial, we fortunately can reuse a large portion of the
      +previous definition. For example, let us just cut-and-paste the rest of the
      +definition from Lesson 7 in Part 1 of the tutorial.

      +

      Let us kompile and krun all the remaining programs from Part 1 of the
      +tutorial. Everything should work fine, although the store contains lots of
      +garbage. Garbage collection is an interesting topic, but we do not do it
      +here. Nevertheless, much of this garbage is caused by the intricate use of
      +the fixed-point combinator to define recursion. In a future lesson in this
      +tutorial we will see that a different, environment-based definition of
      +fixed-points will allocate much less memory.

      +

      One interesting question at this stage is: how do we know when we can reuse
      +an existing semantics of a language feature? Well, I'm afraid the answer is:
      +we don't. In the next lesson we will learn how reuse can fail for quite subtle
      +reasons, which are impossible to detect statically (and some non-experts may
      +fail to even detect them at all).

      +

      Go to Lesson 4, LAMBDA++: Do Not Reuse Blindly!.

      +

      MOVIE (out of date) [3'21"]

      +

      Do Not Reuse Blindly!

      + +

      It may be tempting to base your decision to reuse an existing semantics of
      +a language feature solely on syntactic considerations; for example, to reuse
      +whenever the parser does not complain. As seen in this lesson, this could
      +be quite risky.

      +

      Let's try (and fail) to reuse the definition of callcc from Lesson 1:

      +
      syntax Exp ::= "callcc" Exp  [strict]
+syntax Val ::= cc(K)
+rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+rule <k> cc(K) V ~> _ =>  V ~> K </k>
+
      +

      The callcc examples that we tried in Lesson 1 work, so it may look it works.

      +

      However, the problem is that cc(K) should also include an environment,
      +and that environment should also be restored when cc(K) is invoked.
      +Let's try to illustrate this bug with callcc-env1.lambda

      +
      let x = 1 in
+  ((callcc lambda k . (let x = 2 in (k x))) + x)
+
      +

      where the second argument of +, x, should be bound to the top x, which
      +is 1. However, since callcc does not restore the environment, that x
      +should be looked up in the wrong, callcc-inner environment, so we should see
      +the overall result 4.

      +

      Hm, we get the right result, 3 ... (Note: you may get 4, depending on
      +your version of K and platform; but both 3 and 4 are possible results, as
      +explained below and seen in the tests). How can we get 3? Well, recall that
      ++ is strict, which means that it can evaluate its arguments in any order.
      +It just happened that in the execution that took place above its second
      +argument was evaluated first, to 1, and then the callcc was evaluated, but
      +its cc value K had already included the 1 instead of x ... In Part 4 of
      +the tutorial we will see how to explore all the non-deterministic behaviors of
      +a program; we could use that feature of K to debug semantics, too.
      +For example, in this case, we could search for all behaviors of this program
      +and we would indeed get two possible value results: 3 and 4.

      +

      One may think that the problem is the non-deterministic evaluation order
      +of +, and thus that all we need to do is to enforce a deterministic order
      +in which the arguments of + are evaluated. Let us follow this path to
      +see what happens. There are two simple ways to make the evaluation order
      +of +'s arguments deterministic. One is to make + seqstrict in the
      +semantics, to enforce its evaluation from left-to-right. Do it and then
      +run the program above again; you should get only one behavior for the
      +program above, 4, which therefore shows that copying-and-pasting our old
      +definition of callcc was incorrect. However, as seen shortly, that only
      +fixed the problem for the particular example above, but not in general.
      +Another conventional approach to enforce the desired evaluation order is to
      +modify the program to enforce the left-to-right evaluation order using let
      +binders, as we do in callcc-env2.lambda:

      +
      let x = 1 in
+  let a = callcc lambda k . (let x = 2 in (k x)) in
+    let b = x in
+      (a + b)
+
      +

      With your installation of K you may get the "expected" result 4 when you
      +execute this program, so it may look like our non-deterministic problem is
      +fixed. Unfortunately, it is not. Using the K tool to search for all the
      +behaviors in the program above reveals that the final result 3 is still
      +possible. Moreover, both the 3 and the 4 behaviors are possible regardless
      +of whether + is declared to be seqstrict or just strict. How is that
      +possible? The problem is now the non-deterministic evaluation strategy of
      +the function application construct. Indeed, recall that the semantics of
      +the let-in construct is defined by desugaring to lambda application:

      +
      rule let X = E in E' => (lambda X . E') E
+
      +

      With this, the program above eventually reduces to

      +
      (lambda a . ((lambda b . a + b) x))
+(callcc lambda k . (let x = 2 in (k x)))
+
      +

      in an environment where x is 1. If the first expression evaluates first,
      +then it does so to a closure in which x is bound to a location holding 1,
      +so when applied later on to the x inside the argument of callcc (which is
      +2), it will correctly lookup x in its enclosed environment and thus the
      +program will evaluate to 3. On the other hand, if the second expression
      +evaluates first, then the cc value will freeze the first expression as is,
      +breaking the relationship between its x and the current environment in which
      +it is bound to 1, being inadvertently captured by the environment of the
      +let-in construct inside the callcc and thus making the entire expression
      +evaluate to 4.

      +

      So the morale is: Do not reuse blindly. Think!

      +

      In the next lesson we fix the environment-based semantics of callcc by having
      +cc also wrap an environment, besides a computation. We will also give a more
      +direct semantics to recursion, based on environments instead of fixed-point
      +combinators.

      +

      Go to Lesson 5, LAMBDA++: More Semantic Computation Items.

      +

      MOVIE (out of date) [3'37"]

      +

      More Semantic Computation Items

      + +

      In this lesson we see more examples of semantic (i.e., non-syntactic)
      +computational items, and how useful they can be. Specifically, we fix the
      +environment-based definition of callcc and give an environment-based
      +definition of the mu construct for recursion.

      +

      Let us first fix callcc. As discussed in Lesson 4, the problem that we
      +noticed there was that we only recovered the computation, but not the
      +environment, when a value was passed to the current continuation. This is
      +quite easy to fix: we modify cc to take both an environment and a
      +computation, and its rules to take a snapshot of the current environment with
      +it, and to recover it at invocation time:

      +
      syntax Val ::= cc(Map,K)
+rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k> <env> Rho </env>
+rule <k> cc(Rho,K) V:Val ~> _ =>  V ~> K </k> <env> _ => Rho </env>
+
      +

      Let us kompile and make sure it works with the callcc-env2.lambda program,
      +which should evaluate to 3, not to 4.

      +

      Note that the cc value, which can be used as a computation item in the <k/>
      +cell, is now quite semantic in nature, pretty much the same as the closures.

      +

      Let us next add one more closure-like semantic computational item, for mu.
      +But before that, let us reuse the semantics of letrec in terms of mu that
      +was defined in Lesson 8 of Part 1 of the tutorial on LAMBDA:

      +
      syntax Exp ::= "letrec" Id Id "=" Exp "in" Exp [macro]
+             | "mu" Id "." Exp                 [latex(\mu{#1}.{#2})]
+rule letrec F:Id X = E in E' => let F = mu F . lambda X . E in E'
+
      +

      We removed the binder annotation of mu, because it is not necessary
      +anymore (since we do not work with substitutions anymore).

      +

      To save the number of locations needed to evaluate mu X . E, let us replace
      +it with a special closure which already binds X to a fresh location holding
      +the closure itself:

      +
      syntax Exp ::= muclosure(Map,Exp)
+
+rule <k> mu X . E => muclosure(Rho[X <- !N], E) ...</k>
+     <env> Rho </env>
+     <store>... .Map => (!N:Int |-> muclosure(Rho[X <- !N], E)) ...</store>
+
      +

      Since each time mu X . E is encountered during the evaluation it needs to
      +evaluate E, we conclude that muclosure cannot be a value. We can declare
      +it as either an expression or as a computation. Let's go with the former.

      +

      Finally, here is the rule unrolling the muclosure:

      +

      rule muclosure(Rho,E) => E ~> Rho' ...
      + Rho' => Rho

      +

      Note that the current environment Rho' needs to be saved before and
      +restored after E is executed, because the fixed point may be invoked
      +from a context with a completely different environment from the one
      +in which mu X . E was declared.

      +

      We are done. Let us now kompile and krun factorial-letrec.lambda from
      +Lesson 7 in Part 1 of the tutorial on LAMBDA. Recall that in the previous
      +lesson this program generated a lot of garbage into the store, due to the
      +need to allocate space for the arguments of all those lambda abstractions
      +needed to run the fixed-point combinator. Now we need much fewer locations,
      +essentially only locations for the argument of the factorial function, one at
      +each recursive call. Anyway, much better than before.

      +

      In the next lesson we wrap up the environment definition of LAMBDA++ and
      +generate its documentation.

      +

      Go to Lesson 6, LAMBDA++: Wrapping Up and Documenting LAMBDA++.

      +

      MOVIE (out of date) [5'19"]

      +

      Wrapping Up and Documenting LAMBDA++

      + +

      In this lesson we wrap up and nicely document LAMBDA++. In doing so, we also
      +take the freedom to reorganize the semantics a bit, to make it look better.

      +

      See the lambda.k file, which is self-explanatory.

      +

      Part 3 of the tutorial is now complete. Part 4 will teach you more features
      +of the K framework, in particular how to exhaustively explore the behaviors
      +of non-deterministic or concurrent programs.

      +

      MOVIE (out of date) [6'23"]

      +

      Part 4: Defining IMP++

      + +

      IMP++ extends IMP, which was discussed in Part 2 of this tutorial, with several
      +new syntactic constructs. Also, some existing syntax is generalized, which
      +requires non-modular changes of the existing IMP semantics. For example,
      +global variable declarations become local declarations and can occur
      +anywhere a statement can occur. In this tutorial we will learn the following:

      +
        +
      • That (and how) existing syntax/semantics may change as a language evolves.
        • +
      • How to refine configurations as a language evolves.
        • +
      • How to define and use fresh elements of desired sorts.
        • +
      • How to tag syntactic constructs and rules, and how to use such tags
        +with the superheat/supercool/transition options of kompile.
        • +
      • How the search option of krun works.
        • +
      • How to stream cells holding semantic lists to the standard input/output,
        +and thus obtain interactive interpreters for the defined languages.
        • +
      • How to delete, save and restore cell contents.
        • +
      • How to add/delete cells dynamically.
        • +
      • More details on how the configuration abstraction mechanism works.
        • +
      +

      Like in the previous tutorials, this folder contains several lessons, each
      +adding new features to IMP++. Do them in order and make sure you completed
      +and understood the previous tutorials.

      +

      Extending/Changing an Existing Language Syntax

      + +

      Here we learn how to extend the syntax of an existing language, both with
      +new syntactic constructs and with more general uses of existing constructs.
      +The latter, in particular, requires changes of the existing semantics.

      +

      Consider the IMP language, as defined in Lesson 4 of Part 2 of the tutorial.

      +

      Let us first add the new syntactic constructs, with their precedences:

      +
        +
      • variable increment, ++, which increments an integer variable and
        +evaluates to the new value;
        • +
      • read, which reads and evaluates to a new integer from the input buffer;
        • +
      • print, which takes a comma-separated list of arithmetic expressions and
        +evaluates and prints each of them in order, from left to right, to the
        +output buffer; we therefore define a new list syntactic category, AExps,
        +which we pass as an argument to print; note we do not want to declare
        +print to be strict, because we do not want to first evaluate the
        +arguments and then print them (for example, if the second argument performs
        +an illegal operation, say division by zero, we still want to print the first
        +argument); we also go ahead and add strings as arithmetic expressions,
        +because we intend print to also take strings, in order to print nice
        +messages to the user;
        • +
      • halt, which abruptly terminates the program; and
        • +
      • spawn, which takes a statement and creates a new concurrent thread
        +executing it and sharing its environment with the parent thread.
        • +
      +

      Also, we want to allow local variable declarations, which can appear anywhere
      +a statement can appear. Their scope ranges from the place they are defined
      +until the end of the current block, and they can shadow previous declarations,
      +both inside and outside the current block. The simplest way to define the
      +syntax of the new variable declarations is as ordinary statements, at the same
      +time removing the previous Pgm syntactic category and its construct.
      +Programs are now just statements.

      +

      We are now done with adding the new syntax and modifying the old one.
      +Note that the old syntax was modified in a way which makes the previous IMP
      +programs still parse, but this time as statements. Let us then modify
      +the configuration variable $PGM to have the sort Stmt instead of Pgm,
      +and let us try to run the old IMP programs, for example sum.imp.

      +

      Note that they actually get stuck with the global declaration on the top
      +of their computations. This is because variable declarations are now treated
      +like any statements, in particular, the sequential composition rule applies.
      +This makes the old IMP rule for global variable declarations not match anymore.
      +We can easily fix it by replacing the anonymous variable _, which matched
      +the program's statement that now turned into the remaining computation in
      +the <k/> cell, with the cell frame variable ..., which matches the
      +remaining computation. Similarly, we have to change the rule for the case
      +where there are no variables left to declare into one that dissolves itself.

      +

      We can now run all the previous IMP programs, in spite of the fact that
      +our IMP++ semantics is incomplete and, more interestingly, in spite of the
      +fact that our current semantics of blocks is incorrect in what regards the
      +semantics of local variable declarations (note that the old IMP programs do
      +not declare block-local variables, which is why they still run correctly).

      +

      Let us also write some proper IMP++ programs, which we would like to execute
      +once we give semantics to the new constructs.

      +

      div.imp is a program manifesting non-deterministic behaviors due to the
      +desired non-deterministic evaluation strategy of division and the fact that
      +expressions will have side effects once we add variable increment. We will
      +be able to see all the different behaviors of this program. Challenge: can
      +you identify the behavior where the program performs a division-by-zero?

      +

      If we run div.imp now, it will get stuck with the variable increment
      +construct on top of the computation cell. Once we give it a semantics,
      +div.imp will execute completely (all the other constructs in div.imp
      +already have their semantics defined as part of IMP).

      +

      Note that some people prefer to define all their semantics in a by need
      +style, that is, they first write and parse lots of programs, and then they
      +add semantics to each language construct on which any of the programs gets
      +stuck, and so on and so forth until they can run all the programs.

      +

      io.imp is a program which exercises the input/output capabilities of the
      +language: reads two integers and prints three strings and an integer.
      +Note that the variable declaration is not the first statement anymore.

      +

      sum-io.imp is an interactive variant of the sum program.

      +

      spawn.imp is a program which dynamically creates two threads that interact
      +with the main thread via the shared variable x. Lots of behaviors will be
      +seen here once we give spawn the right semantics.

      +

      Finally, locals.imp tests whether variable shadowing/unshadowing works well.

      +

      In the next lesson we will prepare the configuration for the new constructs,
      +and will see what it takes to adapt the semantics to the new configuration.
      +Specifically, we will split the state cell into an environment cell and a
      +store cell, like in LAMBDA++ in Part 3 of the tutorial.

      +

      Go to Lesson 2, IMP++: Configuration Refinement; Freshness.

      +

      MOVIE (out of date) [07'47"]

      +

      Configuration Refinement; Freshness

      + +

      To prepare for the semantics of threads and local variables, in this lesson we
      +split the state cell into an environment and a store. The environment and
      +the store will be similar to those in the definition of LAMBDA++ in Part
      +3 of the Tutorial. This configuration refinement will require us to change
      +some of IMP's rules, namely those that used the state.

      +

      To split the state map, which binds program variables to values, into an
      +environment mapping program variables to locations and a store mapping
      +locations to values, we replace in the configuration declaration the cell

      +
      <state color="red"> .Map </state>
+
      +

      with two cells

      +
      <env color="LightSkyBlue"> .Map </env>
+<store color="red"> .Map </store>
+
      +

      Structurally speaking, this split of a cell into other cells is a major
      +semantic change, which, unfortunately, requires us to revisit the existing
      +rules that used the state cell. One could, of course, argue that we could
      +have avoided this problem if we had followed from the very beginning the
      +good-practice style to work with an environment and a store, instead of a
      +monolithic state. While that is a valid argument, highlighting the fact that
      +modularity is not only a feature of the framework alone, but one should also
      +follow good practices to achieve it, it is also true that if all we wanted
      +in Part 2 of the tutorial was to define IMP as is, then the split of the state
      +in an environment and a store is unnecessary and not really justified.

      +

      The first rule which used a state cell is the lookup rule:

      +
      rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
+
      +

      We modify it as follows:

      +
      rule <k> X:Id => I ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> I ...</store>
+
      +

      So we first match the location N of X in the environment, then the value
      +I at location N in the store, and finally we rewrite X to I into the
      +computation. This rule also shows an instance of a more complex
      +multiset matching, where two variables (X and N) are matched each twice.

      +

      The assignment rule is modified quite similarly.

      +

      The variable declaration rule is trickier, though, because we need to allocate
      +a fresh location in the store and bind the newly declared variable to it.
      +This is quite similar to the way we allocated space for variables in
      +the environment-based definition of LAMBDA++ in Part 3 of the tutorial.

      +
      rule <k> int (X,Xs => Xs); ...</k>
+     <env> Rho => Rho[X <- !N:Int] </env>
+     <store>... .Map => !N |-> 0 ...</store>
+
      +

      Note the use of the fresh (!N) variable notation above. Recall from
      +the LAMBDA++ tutorial that each time the rule with fresh (!) variables is
      +applied, fresh elements of corresponding sorts are generated for the fresh
      +variables, distinct from all the previously generated elements; also, we
      +cannot and should not assume anything about the particular element that is
      +being generated, except that it is different from the previous ones.

      +

      kompile and krun sum.imp to see how the fresh locations have been
      +generated and used. There were two fresh locations needed, for the two
      +variables. Note also that a cell holding the counter has been added to the
      +configuration.

      +

      In the next lesson we will add the semantics of variable increment, and see
      +how that yields non-deterministic behaviors in programs and how to explore
      +those behaviors using the K tool.

      +

      Go to Lesson 3, IMP++: Tagging; Superheat/Supercool Kompilation Options.

      +

      MOVIE (out of date) [04'06"]

      +

      Tagging; Transition Kompilation Option

      + +

      In this lesson we add the semantics of variable increment. In doing so, we
      +learn how to tag syntactic constructs and rules and then use such tags to
      +instruct the kompile tool to generate the desired language model that is
      +amenable for exhaustive analysis.

      +

      The variable increment rule is self-explanatory:

      +
      rule <k> ++X => I +Int 1 ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> (I => I +Int 1) ...</store>
+
      +

      We can now run programs like our div.imp program introduced in Lesson 1.
      +Do it.

      +

      The addition of increment makes the evaluation of expressions have side
      +effects. That, in combination with the non-determinism allowed by the
      +strictness attributes in how expression constructs evaluate their
      +arguments, makes expressions in particular and programs in general have
      +non-deterministic behaviors. One possible execution of the div.imp program
      +assigns 1 to y's location, for example, but this program manifests several
      +other behaviors, too.

      +

      To see all the (final-state) behaviors that a program can have, you can call
      +the krun tool with the option --search. For example:

      +
      krun div.imp --search
+
      +

      Oops, we see only one solution, the same as when we ran it without search.

      +

      Here is what happens. krun can only explore as much of the transition
      +system associated to a program as kompile allowed the generated language
      +model to yield. Since most of the K users are interested in language models
      +that execute efficiently, that is, in faster interpreters for the defined
      +languages, by default kompile optimizes the generated language model for
      +execution. In particular, it inserts no backtracking markers, which krun
      +uses when called with the --search option in order to systematically generate
      +the entire transition system associated to a program. This is why krun
      +showed us only one solution when run with the --search option on div.imp.

      +

      We next explain how to tell kompile what kind of language model we are
      +interested in for analysis purposes. When you experiment with non-determinism
      +in a language semantics, you should keep it in mind that the --transition
      +option of kompile allows you to configure what counts as a transition in
      +your language model. We here only discuss transitions due to the
      +non-deterministic evaluation strategies of language constructs, but we will
      +see in future lectures (see Lesson 6 of IMP++, where we add concurrency) that
      +we can also have transitions due to non-deterministic applications of rewrite
      +rules.

      +

      If you want to explore the entire behavior space due to non-deterministic
      +evaluation strategies, then you should include all the language constructs
      +in the --transition option. This may sound like the obvious thing to
      +always do, but as soon as you do it you soon realize that it is way too much
      +in practice when you deal with large languages or programs. There are simply
      +too many program behaviors to consider, and krun will likely hang
      +on you or crush. For example, a small ten-statement program where each
      +statement uses one strict expression construct already has 1000+ behaviors for
      +krun to explore! Driven by practical needs of its users, the K tool
      +therefore allows you to finely tune the generated language models using the
      +--transition option.

      +

      To state which constructs are to be considered to generate transitions in the
      +generated language model, and for other reasons, too, the K tool allows you to
      +tag any production and any rule by putting the tag in brackets. You can associate
      +multiple tags to the same construct or rule, and more than one construct or
      +rule can have the same tag. As an example, let us tag the division construct
      +with division, the lookup rule with lookup and the increment rule with
      +increment. The tags of the rules are not needed in this lesson, we do it only
      +to demonstrate that rules can also be tagged.

      +

      The least intrusive way to enforce our current language to explore the
      +entire space of behaviors due to the strictness of division is to kompile it
      +with the following option:

      +
      kompile imp.k --transition "division"
+
      +

      It is interesting to note that the lookup and increment rules are the only
      +two rules which can trigger non-deterministic behaviors for division, because
      +no other rule but these two can ever apply while a division operation is
      +heated. Previous versions of K allowed you to also specify which rules could
      +trigger non-deterministic behaviors of operator evaluation strategies,
      +but that option was rarely used and is not available anymore.

      +

      Note that it is highly non-trivial to say precisely whether a strict language
      +construct may yield non-deterministic behaviors. For example, division's
      +strictness would yield no non-determinism if the language had no side effects.
      +It is even harder to say so for a particular program. Consequently, our K
      +implementation makes no attempt to automatically detect which operations
      +should be tagged as transitions. Instead, it provides the functionality to
      +let you decide it.

      +

      Now the command

      +
      krun div.imp --search
+
      +

      shows us all five behaviors of this program. Interestingly, one
      +of the five behaviors yields a division by zero!

      +

      The --transition option can be quite useful when you experiment with your
      +language designs or when you formally analyze programs for certain kinds of
      +errors. Please let us know if you ever need more finer-grained control over
      +the non-determinism of your language models.

      +

      Before we conclude this lesson, we'd like to let you know one trick, which
      +you will hopefully not overuse: you can tag elements in your K definition with
      +kompile option names, and those elements will be automatically included in
      +their corresponding options. For example, if you tag the division production
      +with transition then the command

      +
      kompile imp
+
      +

      is completely equivalent to the previous kompile command.

      +

      Please use this default behavior with caution, or even better, try to avoid
      +using it! You may be tempted to add the transition tag to lots of elements
      +and then forget about them; your language models will then be increasingly slower
      +when you execute them and you may wonder why ... This convention is typically
      +convenient when you want to quickly experiment with non-determinism and do not
      +want to bother inventing tag names and calling kompile with options.

      +

      In the next lesson we add input/output to our language and learn how to
      +generate a model of it which behaves like an interactive interpreter!

      +

      Go to Lesson 4, IMP++: Semantic Lists; Input/Output Streaming.

      +

      MOVIE (out of date) [06'56"]

      +

      Semantic Lists; Input/Output Streaming

      + +

      In this lesson we add semantics to the read and print IMP++ constructs.
      +In doing so, we also learn how to use semantic lists and how to connect
      +cells holding semantic lists to the standard input and standard output.
      +This allows us to turn the K semantics into an interactive interpreter.

      +

      We start by adding two new cells to the configuration,

      +
      <in color="magenta"> .List </in>
+<out color="Orchid"> .List </out>
+
      +

      each holding a semantic list, initially empty. Semantic lists are
      +space-separated sequences of items, each item being a term of the form
      +ListItem(t), where t is a term of sort K. Recall that the semantic maps,
      +which we use for states, environments, stores, etc., are sets of pairs
      +t1 |-> t2, where t1 and t2 are terms of sort K. The ListItem wrapper
      +is currently needed, to avoid parsing ambiguities.

      +

      Since we want the print statement to also print strings, we need to tell
      +K that strings are results. To make it more interesting, let us also overload
      +the + symbol on arithmetic expressions to also take strings and, as a
      +result, to concatenate them. Since + is already strict, we only need to add
      +a rule reducing the IMP addition of strings to the builtin operation +String
      +which concatenates two strings.

      +

      The semantics of read is immediate: read and consumes the first integer item
      +from the <in/> cell; note that our read only reads integer values (it gets
      +stuck if the first item in the <in/> cell is not an integer).

      +

      The semantics of print is a bit trickier. Recall that print takes an
      +arbitrary number of arithmetic expression arguments, and evaluates and outputs
      +each of them in order, from left to right. For example,
      +print("Hello", 3/0, "Bye"); outputs "Hello" and then gets stuck on the
      +illegal division by zero operation. In other words, we do not want it to
      +first evaluate all its arguments and then print them, because that would miss
      +outputting potentially valuable information. So the first step is to evaluate
      +the first argument of print. In some sense, what we'd like to say is that
      +print has the evaluation strategy strict(1). However, strictness
      +attributes only work with individual language constructs, while what we need
      +is an evaluation strategy that involves two constructs: print and the list
      +(comma) construct of AExps. If we naively associate print the strict(1)
      +evaluation strategy then its first and unique argument, an AExps list, will
      +be scheduled for evaluation and the execution will get stuck because we have
      +no rules for evaluating AExps terms. If we make the list construct of
      +AExps strict then we get the wrong semantics for print which first
      +evaluates all its arguments and then outputs them. The correct way to
      +tell K that print should evaluate only its first argument is by using a
      +context declaration:

      +
      context print(HOLE:AExp, _);
+
      +

      Note the HOLE of sort AExp above. Contexts allow us to define finer-grain
      +evaluation strategies than the strictness attributes, involving potentially
      +more than one language construct, like above. The HOLE indicates the
      +argument which is requested to be evaluated. For example, the strict
      +attribute of division corresponds to two contexts:

      +
      context HOLE / _
+context _ / HOLE
+
      +

      In their full generality, contexts can be any terms with precisely one
      +occurrence of a HOLE, and with arbitrary side conditions on any variables
      +occurring in the context term as well as on the HOLE. See Part 6 of the
      +tutorial for more examples.

      +

      Once evaluated, the first argument of print is expected to become either an
      +integer or a string. Since we want to print both integers and string values,
      +to avoid writing two rules, one for each type of value, we instead add a new
      +syntactic category, Printable, which is the union of integers and strings.

      +

      Let us kompile and krun the io.imp program discussed in Lesson 1. As
      +expected, it gets stuck with a read construct on top of the computation and
      +with an empty <in/> cell. To run it, we need to provide some items in the
      +<in/> cell, so that the rule of read can match. Let us add

      +
      <in> ListItem(3) ListItem(5) ListItem(7) </in>
+
      +

      Now, if we krun io.imp, we can see that its execution completes normally
      +(the <k/> cell is empty), that the first two items have been removed by the
      +two read constructs from the <in/> cell, and that the desired strings and
      +numbers have been placed into the <out/> cell.

      +

      Cells holding semantic lists can be connected to the standard input and
      +standard output buffers, and krun knows how to handle these appropriately.
      +Let us connect the <in/> cell to the standard input using the cell attribute
      +stream="stdin" and the <out/> cell to the standard output with the
      +attribute stream="sdtout". A cell connected to the standard input will
      +take its items from the standard input and block the rewriting process when
      +an input is needed until an item is available in the standard input buffer.
      +A cell connected to the standard output buffer will send all its items, in
      +order, to the standard output.

      +

      Let us kompile and krun io.imp again. It prints the message and then
      +waits for your input numbers. Type in two numbers, then press <Enter>.
      +A message with their sum is then printed, followed by the final configuration.
      +If you do not want to see the final configuration, and thus obtain a realistic
      +interpreter for our language, then call krun with the option --output none:

      +
      krun io.imp --output none
+
      +

      Let us now krun our interactive sum program, which continuously reads numbers
      +from the console and prints the sum of numbers up to them:

      +
      krun sum-io.imp
+
      +

      Try a few numbers, then 0. Note that the program terminated, but with junk
      +in the <k/> cell, essentially with a halt statement on its top. Of course,
      +because halt has been reached and it has no semantics yet.

      +

      In the next lesson we give the semantics of halt and also fix the semantics
      +of blocks with local variable declarations.

      +

      Go to Lesson 5, IMP++: Deleting, Saving and Restoring Cell Contents.

      +

      MOVIE (out of date) [05'21"]

      +

      Deleting, Saving and Restoring Cell Contents

      + +

      In this lesson we will see how easily we can delete, save and/or restore
      +contents of cells in order to achieve the desired semantics of language
      +constructs that involve abrupt changes of control or environments. We have
      +seen similar or related K features in the LAMBDA++ language in Part 3 of the
      +tutorial.

      +

      Let us start by adding semantics to the halt statement. As its name says,
      +what we want is to abruptly terminate the execution of the program. Moreover,
      +we want the program configuration to look as if the program terminated
      +normally, with an empty computation cell. The simplest way to achieve that is
      +to simply empty the computation cell when halt is encountered:

      +
      rule <k> halt; ~> _ => . </k>
+
      +

      It is important to mention the entire <k/> cell here, with both its membranes
      +closed, to make sure that its entire contents is discarded. Note the
      +anonymous variable, which matches the rest of the computation.

      +

      kompile and krun sum-io.imp. Note that unlike in Lesson 4, the program
      +terminates with an empty computation cell now.

      +

      As mentioned earlier, the semantics of blocks that was inherited from IMP is
      +wrong. Program locals.imp shows it very clearly: the environments are not
      +correctly restored at block exits. One way to fix the problem is to take
      +a snapshot of the current environment when a block is entered and save it
      +somewhere, and then to restore it when the block is left. There are many
      +ways to do this, which you can explore on your own: for example you can add
      +a new list cell for this task where to push/pop the environment snapshots in
      +a stack style; or you can use the existing environment cell for this purpose,
      +but then you need to change the variable access rules to search through the
      +stacked environments for the variable.

      +

      My preferred solution is to follow a style similar to how we saved/restored
      +LAMBDA++ environments in Part 3 of the Tutorial, namely to use the already
      +existing <k/> cell for such operations. More specifically, we place a
      +reminder item in the computation whenever we need to take a snapshot of
      +some cell contents; the item simply consists of the entire contents of the cell.
      +Then, when the reminder item is reached, we restore the contents of the cell:

      +
      rule <k> {S} => S ~> Rho ...</k> <env> Rho </env>
+
      +

      The only thing left now is to give the definition of environment restore:

      +
      rule <k> Rho => . ...</k> <env> _ => Rho </env>
+
      +

      Done. kompile and krun locals.imp. Everything should work correctly now.
      +Note that the rule above is different from the one we had for LAMBDA++ in
      +Part 3 of the tutorial, in that here there is no value preceding the environment
      +restoration item in the computation; that's because IMP++ statements,
      +unlike LAMBDA++'s expressions, evaluate to nothing (.).

      +

      In the next lesson we will give semantics to the spawn S construct, which
      +dynamically creates a concurrent shared-memory thread executing statement S.

      +

      Go to Lesson 6, IMP++: Adding/Deleting Cells Dynamically; Configuration Abstraction, Part 2.

      +

      MOVIE (out of date) [04'30"]

      +

      Adding/Deleting Cells Dynamically; Configuration Abstraction, Part 2

      + +

      In this lesson we add dynamic thread creation and termination to IMP, and
      +while doing so we learn how to define and use configurations whose structure
      +can evolve dynamically.

      +

      Recall that the intended semantics of spawn S is to spawn a new concurrent
      +thread that executes S. The new thread is being passed at creation time
      +its parent's environment, so it can share with its parent the memory
      +locations that its parent had access to at creation time. No other locations
      +can be shared, and no other memory sharing mechanism is available.
      +The parent and the child threads can evolve unrestricted, in particular they
      +can change their environments by declaring new variables or shadowing existing
      +ones, can create other threads, and so on.

      +

      The above suggests that each thread should have its own computation and its
      +own environment. This can be elegantly achieved if we group the <k/> and
      +<env/> cells in a <thread/> cell in the configuration. Since at any given
      +moment during the execution of a program there could be zero, one or more
      +instances of such a <thread/> cell in the configuration, it is a good idea
      +to declare the <thread/> cell with multiplicity * (i.e., zero, one or more):

      +
      <thread multiplicity="*" color="blue">
+  <k color="green"> $PGM:Stmt </k>
+  <env color="LightSkyBlue"> .Map </env>
+</thread>
+
      +

      This multiplicity declaration is not necessary, but it is a good idea to do
      +it for several reasons:

      +
        +
      1. it may help the configuration abstraction process,
        +which may in turn significantly increase the compactness and modularity of
        +your subsequent rules;
        2. +
      2. it may help various analysis and execution tools,
        +for example static analyzers to give you error messages when you create cells
        +where you should not, or K compilers to improve performance by starting
        +actual concurrent hardware threads or processes corresponding to each cell
        +instance; and
        3. +
      3. it may help you better understand and control the dynamics
        +of your configuration, and thus your overall semantics.
        4. +
      +

      For good encapsulation, I also prefer to put all thread cells into one cell,
      +<threads/>. This is technically unnecessary, though; to convince yourself
      +that this is indeed the case, you can remove this cell once we are done with
      +the semantics and everything will work without having to make any changes.

      +

      Before we continue, let us kompile an krun some programs that used to
      +work, say sum-io.imp. In spite of the relatively radical configuration
      +reorganization, those programs execute just fine! How is that possible?
      +In particular, why do rules like the lookup and assignment still work,
      +unchanged, in spite of the fact that the <k/> and <env/> cells are not at
      +the same level with the <store/> cell in the configuration anymore?

      +

      Welcome to configuration abstraction, part 2. Recall that the role of
      +configuration abstraction is to allow you to only write the relevant
      +information in each rule, and have the compiler fill-in the obvious and boring
      +details. According to the configuration that we declared for our new
      +language, there is only one reasonable way to complete rules like the lookup,
      +namely to place the <k/> and </env> cells inside a <thread/> cell,
      +inside a <threads/> cell:

      +
      rule <threads>...
+       <thread>...
+         <k> X:Id => I ...</k>
+         <env>... X |-> N ...</env>
+       ...</thread>
+     ...<threads/>
+     <store>... N |-> I ...</store>  [lookup]
+
      +

      This is the most direct, compact and local way to complete the configuration
      +context of the lookup rule. If for some reason you wanted here to match the
      +<k/> cell of one thread and the <env/> cell of another thread, then you
      +would need to explicitly tell K so, by mentioning the two thread cells,
      +for example:

      +
      rule <thread>...
+         <k> X:Id => I ...</k>
+     ...</thread>
+     <thread>...
+         <env>... X |-> N ...</env>
+     ...</thread>
+     <store>... N |-> I ...</store>  [lookup]
+
      +

      By default, K completes rules in a greedy style. Think this way: what is the
      +minimal number of changes to my rule to make it fit the declared
      +configuration? That's what the K tool will do.

      +

      Configuration abstraction is technically unnecessary, but once you start
      +using it and get a feel for how it works, it will become your best friend.
      +It allows you to focus on the essentials of your semantics, and at the same
      +time gives you flexibility in changing the configuration later on without
      +having to touch the rules. For example, it allows you to remove the
      +<threads/> cell from the configuration, if you don't like it, without
      +having to touch any rule.

      +

      We are now ready to give the semantics of spawn:

      +
      rule <k> spawn S => . ...</k> <env> Rho </env>
+     (. => <thread>... <k> S </k> <env> Rho </env> ...</thread>)
+
      +

      Note configuration abstraction at work, again. Taking into account
      +the declared configuration, and in particular the multiplicity information
      +* in the <thread/> cell, the only reasonable way to complete the rule
      +above is to wrap the <k/> and <env/> cells on the first line within a
      +<thread/> cell, and to fill-in the ...s in the child thread with the
      +default contents of the other subcells in <thread/>. In this case there
      +are no other cells, so we can get rid of those ...s, but that would
      +decrease the modularity of this rule: indeed, we may later on add other
      +cells within <thread/> as the language evolves, for example a function
      +or an exception stack, etc.

      +

      In theory, we should be able to write the rule above even more compactly
      +and modularly, namely as

      +
      rule <k> spawn S => . ...</k> <env> Rho </env>
+     (. => <k> S </k> <env> Rho </env>)
+
      +

      Unfortunately, this currently does not work in the K tool, due to some
      +known limitations of our current configuration abstraction algorithm.
      +This latter rule would be more modular, because it would not even depend
      +on the cell name thread. For example, we may later decide to change
      +thread into agent, and we would not have to touch this rule.
      +We hope this current limitation will be eliminated soon.

      +

      Once a thread terminates, its computation cell becomes empty. When that
      +happens, we can go ahead and remove the useless thread cell:

      +
      rule <thread>... <k> . </k> ...</thread> => .
+
      +

      Let's see what we've got. kompile and krun spawn.imp.
      +Note the following:

      +
        +
      • The <threads/> cell is empty, so all threads terminated normally;
        • +
      • The value printed is different from the value in the store; the store value
        +is not even the one obtained if the threads executed sequentially.
        • +
      +

      Therefore, interesting behaviors may happen; we would like to see them all!

      +

      Based on prior experience with krun's search option, we would hope that

      +
      krun spawn.imp --search
+
      +

      shows all the behaviors. However, the above does not work, for two reasons.

      +

      First, spawn.imp is an interactive program, which reads a number from the
      +standard input. When analyzing programs exhaustively using the search option,
      +krun has to disable the streaming capabilities (just think about it and you
      +will realize why). The best you can do in terms of interactivity with search
      +is to pipe some input to krun: krun will flush the standard input buffer
      +into the cells connected to it when creating the initial configuration (will
      +do that no matter whether you run it with or without the --search option).
      +For example:

      +
      echo 23 | krun spawn.imp --search
+
      +

      puts 23 in the standard input buffer, which is then transferred in the
      +<in/> cell as a list item, and then the exhaustive search procedure is
      +invoked.

      +

      Second, even after piping some input, the spawn.imp program still manifests
      +only one behavior, which does not seem right. There should be many more.

      +

      As explained in Lesson 3, by default kompile optimizes the generated
      +language model for execution. In particular, it does not insert any
      +backtracking markers where transition attempts should be made, so krun
      +lacks the information it needs to exhaustively search the generated language
      +model. Like we did in Lesson 3 with the language constructs, we also have
      +to explicitly tell kompile which rules should be considered as actual
      +transitions. A theoretically correct but practically unfeasible approach
      +to search all possible behaviors is to consider all rules as transitions.
      +Even more than with the non-deterministic strictness of language constructs
      +in Lesson 3, such a naive solution would make the number of behaviors, and
      +thus krun, explode. Remember that a two-thread program with 150 statements
      +each manifests more behaviors than particles in the known universe!
      +Consequently, unless your multi-threaded programs are very small, you will
      +most likely want to control which rules should be considered transitions and
      +which should not.

      +

      A good rule of thumb is to include as transitions only those rules which
      +compete for behaviors. That is, those rules which may yield a different
      +behavior if we choose to apply them when other rules match as well.
      +The rule for addition, for example, is a clear example of a rule which
      +should not be a transition: indeed, 3+7 will rewrite to 10 now and also
      +later. On the other hand, the lookup rule should be a transition. Indeed,
      +if we delay the lookup of variable x, then other threads may write x in the
      +meanwhile (with an increment or an assignment rule) and thus yield a
      +different behavior.

      +

      Let us discuss and tag those rules which should be transitions: lookup and
      +increment need to be transitions and we already tagged them in Lesson 3;
      +the read rule needs to also be a transition, because it may complete with
      +other instances of itself in other threads; assignment needs to also be a
      +transition, and so should be the first rule for print.

      +

      Let us now kompile with the transition option set as desired:

      +
      kompile imp --transition "lookup increment assignment read print"
+
      +

      Now echo 23 | krun spawn.imp --search gives us all 12 behaviors of the
      +spawn.imp program.

      +

      Like for non-deterministically strict operations which can be tagged as
      +transitions, it is highly non-trivial to say precisely which rules need
      +to be transitions. So krun makes no attempt to automatically detect it.
      +Instead, it provides the functionality to let you decide it.

      +

      We currently have no mechanism for thread synchronization. In the next lesson
      +we add a join statement, which allows a thread to wait until another completes.

      +

      Go to Lesson 7, IMP++: Everything Changes: Syntax, Configuration, Semantics.

      +

      MOVIE (out of date) [11'40"]

      +

      Everything Changes: Syntax, Configuration, Semantics

      + +

      In this lesson we add thread joining, one of the simplest thread
      +synchronization mechanisms. In doing so, we need to add unique ids
      +to threads in the configuration, and to modify the syntax to allow spawn
      +to return the id of the newly created thread. This gives us an opportunity
      +to make several other small syntactic and semantics changes to the language,
      +which make it more powerful or more compact at a rather low cost.

      +

      Before we start, let us first copy and modify the previous spawn.imp program
      +from Lesson 1 to make use of thread joining. Recall from Lesson 6 that in some
      +runs of this program the main thread completed before the child threads,
      +printing a possibly undesired value of x. What we want now is to assign
      +unique ids to the two spawned threads, and then to modify the main thread to
      +join the two child threads before printing. To avoid adding a new type to
      +the language, let's assume that thread ids are integer numbers. So we declare
      +two integers, t1 and t2, and assign them the two spawn commands. In order
      +for this to parse, we will have to change the syntax of spawn to be an
      +arithmetic expression construct instead of a statement. Once we do that,
      +we have a slight syntactic annoyance: we need to put two consecutive ;
      +after the spawn assignment, one for the assignment statement inside the spawn,
      +and another for the outer assignment. To avoid the two consecutive semicolons,
      +we can syntactically enforce spawn to take a block as argument, instead of a
      +statement. Now it looks better. The new spawn.imp program is still
      +non-deterministic, because the two threads can execute in any order and even
      +continue to have a data-race on the shared variable x, but we should see fewer
      +behaviors when we use the join statements. If we want to fully synchronize
      +this program, we can have the second thread start with a join(t1) statement.
      +Then we should only see one behavior for this program.

      +

      Let us now modify the language semantics. First, we move the spawn
      +construct from statements to expressions, and make it take a block.
      +Second, we add one more sub-cell to the thread cell in the configuration,
      +<id/>, to hold the unique identifier of the thread. We want the main
      +thread to have id 0, so we initialize this cell with 0. Third, we modify
      +the spawn rule to generate a fresh integer identifier, which is put in the
      +<id/> cell of the child thread and returned as a result of spawn in the
      +parent thread. Fourth, let us add the join statement to the language,
      +both syntactically and semantically. So in order for the join(T) statement
      +to execute, thread T must have its computation empty. However, in order
      +for this to work we have to get rid of the thread termination cleanup rule.
      +Indeed, we need to store somewhere the information that thread T terminated;
      +the simplest way to do it is to not remove the terminated threads. Feel free
      +to experiment with other possibilities, too, here. For example, you may add
      +another cell, <done/>, in which you can store all the thread ids of the
      +terminated and garbage-collected threads.

      +

      Let us now kompile imp.k and convince ourselves that the new spawn.imp
      +with join statements indeed has fewer behaviors than its variant without
      +join statements. Also, let us convince ourselves that the fully synchronized
      +variant of it indeed has only one behavior.

      +

      Note that now spawn, like variable increment, makes the evaluation of
      +expressions to have side effects. Many programming languages in fact allow
      +expressions to be evaluated only for their side effects, and not for their
      +value. This is typically done by simply adding a ; after the expression
      +and thus turning it into a statement. For example, ++x;. Let as also
      +allow arithmetic expressions in our language to be used as statements, by
      +simply adding the production AExp ";" to Stmt, with evaluation strategy
      +strict and with the expected semantics discarding the value of the AExp.

      +

      Another simple change in syntax and semantics which gives our language more
      +power, is to remove the ; from the syntax of variable assignments and to make
      +them expression instead of statement constructs. This change, combined with
      +the previous one, will still allow us to parse all the programs that we could
      +parse before, but will also allow us to parse more programs. For example, we
      +can now do sequence assignments like in C: x = y = z = 0. The semantics
      +of assignment now has to return the assigned value also to the computation,
      +because we want the assignment expression to evaluate to the assigned value.

      +

      Let us also make another change, but this time one which only makes the
      +definition more compact. Instead of defining statement sequential
      +composition as a binary construct for statements, let us define a new
      +syntactic construct, Stmts, as whitespace-separated lists of Stmt. This
      +allows us to get rid of the empty blocks, because we can change the syntax of
      +blocks to {Stmts} and Stmts also allows the empty sequence of statements.
      +However, we do have to make sure that .Stmts dissolves.

      +

      In general, unless you are defining a well-established programming language,
      +it is quite likely that your definitions will suffer lots of changes like the
      +ones seen in this lecture. You add a new construct, which suggests changes
      +in the existing syntax making in fact your language parse more programs,
      +which then requires corresponding changes in the semantics, and so on.
      +Also, compact definitions are desirable in general, because they are easier
      +to read and easier to change if needed later.

      +

      In the next lesson we wrap up and document the definition of IMP++.

      +

      Go to Lesson 8, IMP++: Wrapping up Larger Languages.

      +

      Wrapping up Larger Languages

      + +

      In this lesson we wrap up IMP++'s semantics and also generate its poster.
      +While doing so, we also learn how to display larger configurations in order
      +to make them easier to read and print.

      +

      Note that we rearrange a bit the semantics, to group the semantics of old
      +IMP's constructs together, and separate it from the new IMP++'s semantics.

      +

      There is a detailed discussion at the end of the document about the
      +--transition option of kompile, because that is important and we want
      +the poster to include everything we learned in this part of the tutorial.

      +

      You can go even further and manually edit the generated Latex document.
      +You typically want to do that when you want to publish your language
      +definition, or parts of it, and you need to finely tune it to fit the
      +editing requirements. For example, you may want to insert some negative
      +spaces, etc.

      +

      Part 4 of the tutorial is now complete. At this moment you should know most
      +of K framework's features and how to use the K tool. You can now define or
      +design your own programming languages, and then execute and analyze programs.

      +

      MOVIE (out of date) [06'26"]

      +

      Part 5: Defining Type Systems

      + +

      In this part of the tutorial we will show that defining type systems for
      +languages is essentially no different from defining semantics. The major
      +difference is that programs and fragments of programs now rewrite to their
      +types, instead of to concrete values. In terms of K, we will learn how
      +to use it for a certain particular but important kind of applications.

      +

      Imperative, Environment-Based Type Systems

      + +

      In this lesson you learn how to define a type system for an imperative
      +language (the IMP++ language defined in Part 4 of the tutorial), using a style
      +based on type environments.

      +

      Let us copy the imp.k file from Part 4 of the tutorial, Lesson 7, which holds
      +the semantics of IMP++, and modify it into a type system. The resulting type
      +system, when executed, yields a type checker.

      +

      We start by defining the new strictness attributes of the IMP++ syntax.
      +While doing so, remember that programs and fragments of programs now reduce
      +to their types. So types will be the new results of our new (type) semantics.
      +We also clean up the semantics by removing the unnecessary tags, and also
      +use strict instead of seqstrict wherever possible, because strict gives
      +implementations more freedom. Interestingly, note that spawn is strict now,
      +because the code of the child thread should type in the current parent's type
      +environment. Note that this is not always the case for threads, see for example
      +SIMPLE in the languages tutorial, but it works here for our simpler IMP++.

      +

      From a typing perspective, the && construct is strict in both its arguments;
      +its short-circuit (concrete) semantics is irrelevant for its (static) type
      +system. Similarly, both the conditional and the while loop are strict
      +constructs when regarded through the typing lenses.

      +

      Finally, the sequential composition is now sequentially strict! Indeed,
      +statements are now going to reduce to their type, stmt, and it is critical
      +for sequential composition to type its argument statements left-to-right;
      +for example, imagine that the second argument is a variable declaration (whose
      +type semantics will modify the type environment).

      +

      We continue by defining the new results of computations, that is, the actual
      +types. In this simple imperative language, we only have a few constant types:
      +int, bool, string, block and stmt.

      +

      We next define the new configuration, which is actually quite simple. Besides
      +the <k/> cell, all we need is a type environment cell, <tenv/>, which will
      +hold a map from identifiers to their types. A type environment is therefore
      +like a state in the abstract domain of type values.

      +

      Let us next modify the semantic rules, turning them into a type system. In
      +short, the idea is to reduce the basic values to their types, and then have a
      +rule for each language construct reducing it to its result type whenever its
      +arguments have the expected types.

      +

      We write the rules in the order given by the syntax declarations, to make
      +sure we do not forget any construct.

      +

      Integers reduce to their type, int.

      +

      So do the strings.

      +

      Variables are now looked up in the type environment and reduced to their type
      +there. Since we only declare integer variables in IMP++, their type in tenv
      +will always be int. Nevertheless, we write the rule generically, so that we
      +would not have to change it later if we add other type declarations to IMP++.
      +Note that we reject programs which lookup undeclared variables. Rejection,
      +in this case, means rewriting getting stuck.

      +

      Variable increment types to int, provided the variable has type int.

      +

      Read types to int, because we only allow integer input.

      +

      Division is only allowed on integers, so it rewrites to int provided that its
      +arguments rewrite to int. Note, however, that in order to write int / int,
      +we have to explicitly add int to the syntax of arithmetic expressions.
      +Otherwise, the K parser rightfully complains, because / was declared on
      +arithmetic expressions, not on types. One simple and generic way to allow
      +types to appear anywhere, is to define Type as a syntactic subcategory of all
      +the other syntactic categories. Let's do it on a by-need basis, though.

      +

      Addition is overloaded, so we add two typing rules for it: one for integers
      +and another for strings.

      +

      As discussed, spawn types to stmt provided that its argument types to
      +block.

      +

      The assignment construct was strict(2); its typing policy is that the declared
      +type of X should be identical to the type of the assigned value. Like for
      +lookup, we define this rule more generically than needed for IMP++, for any
      +type, not only for int.

      +

      The typing rules for Boolean expression constructs are in the same spirit.
      +Note that we need only one rule for &&.

      +

      The typing of blocks is a bit trickier. First, note that we still need to
      +recover the environment after the block is typed, because we do not want the
      +block-local variables to be visible in the outer type environment. We recover
      +the type environment only after the block-enclosed statements type; moreover,
      +we also opportunistically yield a block type on the computation when we
      +discard the type environment recovery item. To account for the fact that the
      +block-enclosed statement can itself be a block (e.g., {{S}}), we would need an
      +additional rule. Since we do not like repetition, we instead group the types
      +block and stmt into one syntactic category, BlockOrStmtType, and now we
      +can have only one rule. We also include BlockOrStmtType in Type, as a
      +replacement for the two basic types.

      +

      The expression statement types as expected. Recall that we only allow
      +arithmetic expressions, which type to int, to be used as statements in IMP++.

      +

      The conditional was declared strict in all its arguments. Its typing policy
      +is that its first argument types to bool and its two branches to block.
      +If that is the case, then it yields a stmt type.

      +

      For while, its first argument should type to bool and its second to block.

      +

      Variable declarations add new bindings to the type environment. Recall that
      +we can only declare variables of integer type in IMP++.

      +

      The typing policy of print is that it can only print integer or string values,
      +and in that case it types to stmt. Like for BlockOrStmtType, to avoid
      +having two similar rules, one for int and another for string, we prefer to
      +introduce an additional syntactic category, PrintableType, which includes both
      +int and string types.

      +

      halt types to stmt; so its subsequent code is also typed.

      +

      join types to stmt, provided that its argument types to int.

      +

      Sequential composition was declared as a whitespace-separated sequentially
      +strict list. Its typing policy is that all the statements in the list should
      +type to stmt or block in order for the list to type to stmt. Since
      +lists are maintained internally as cons-lists, this is probably the simplest
      +way to do it:

      +
      rule .Stmts => stmt
+rule _:BlockOrStmtType Ss => Ss
+
      +

      Note that the first rule, which types the empty sequence of statements to stmt,
      +is needed anyway, to type empty blocks {} (together with the block rule).

      +

      kompile imp.k and krun all the programs in Part 4 of the tutorial. They
      +should all type to stmt.

      +

      In the next lesson we will define a substitution-based type system for LAMBDA.

      +

      Go to Lesson 2, Type Systems: Substitution-Based Higher-Order Type Systems.

      +

      MOVIE (out of date) [10'11"]

      +

      Substitution-Based Higher-Order Type Systems

      + +

      In this lesson you learn how to define a substitution-based type system for
      +a higher-order language, namely the LAMBDA language defined in Part 1 of the
      +tutorial.

      +

      Let us copy the definition of LAMBDA from Part 1 of the tutorial, Lesson 8.
      +We are going to modify it into a type systems for LAMBDA.

      +

      Before we start, it is important to clarify an important detail, namely that
      +our type system will yield a type checker when executed, not a type
      +inferencer. In particular, we are going to change the LAMBDA syntax
      +to allow us to associate a type to each declared variable. The
      +constructs which declare variables are lambda, let, letrec and mu.
      +The syntax of all these will therefore change.

      +

      Since here we are not interested in a LAMBDA semantics anymore, we take the
      +freedom to eliminate the Val syntactic category, our previous results.
      +Our new results are going to be the types, because programs will now reduce
      +to their types.

      +

      As explained, the syntax of the lambda construct needs to change, to also
      +declare the type of the variable that it binds. We add the new syntactic
      +category Type, with the following constructs: int, bool, the function
      +type (which gives it its higher-order status), and parentheses as bracket.
      +Also, we make types our K results.

      +

      We are now ready to define the typing rules.

      +

      Let us start with the typing rule for lambda abstraction: lambda X : T . E
      +types to the function type T -> T', where T' is the type obtained by further
      +typing E[T/X]. This can be elegantly achieved by reducing the lambda
      +abstraction to T -> E[T/X], provided that we extend the function type construct
      +to take expressions, not only types, as arguments, and to be strict.
      +This can be easily achieved by redeclaring it as a strict expression construct
      +(strictness in the second argument would suffice in this example, but it is
      +more uniform to define it strict overall).

      +

      The typing rule for application is as simple as it can get: (T1->T2) T1 => T2.

      +

      Let us now give the typing rules of arithmetic and Boolean expression
      +constructs. First, let us get rid of Val. Second, rewrite each value to its
      +type, similarly to the type system for IMP++ in the previous lesson. Third,
      +replace each semantic rule by its typing rule. Fourth, make sure you
      +do not forget to subsort Type to Exp, so your rules above will parse.

      +

      The typing policy of the conditional statement is that its first argument
      +should type to bool and its other two arguments should type to the same type
      +T, which will also be the result type of the conditional. So we make the
      +conditional construct strict in all its three arguments and we write the
      +obvious rule: if bool then T:Type else T => T. We want a runtime check that
      +the latter arguments are actually typed, so we write T:Type.

      +

      There is nothing special about let, except that we have to make sure we
      +change its syntax to account for the type of the variable that it binds.
      +This rule is a macro, so the let is desugared statically.

      +

      Similarly, the syntax of letrec and mu needs to change to account for the
      +type of the variable that they bind. The typing of letrec remains based on
      +its desugaring to mu; we have to make sure the types are also included now.

      +

      The typing policy of mu is that its body should type to the same type T of
      +its variable, which is also the type of the entire mu expression. This can
      +be elegantly achieved by rewriting it to (T -> T) E[T/X]. Recall that
      +application is strict, so E[T/X] will be eventually reduced to its type.
      +Then the application types correctly only if that type is also T, and in
      +that case the result type will also be T.

      +

      kompile and krun some programs. You can, for example, take the LAMBDA
      +programs from the first tutorial, modify them by adding types to their
      +variable declarations, and then type check them using krun.

      +

      In the next lesson we will discuss an environment-based type system
      +for LAMBDA.

      +

      Go to Lesson 3, Type Systems: Environment-Based Higher-Order Type Systems.

      +

      MOVIE (out of date) [6'52"]

      +

      Environment-Based Higher-Order Type Systems

      + +

      In this lesson you learn how to define an environment-based type system for
      +a higher-order language, namely the LAMBDA language defined in Part 1 of the
      +tutorial.

      +

      The simplest and fastest way to proceed is to copy the substitution-based
      +type system of LAMBDA from the previous lesson and modify it into an
      +environment-based one. A large portion of the substitution-based definition
      +will remain unchanged. We only have to modify the rules that use
      +substitution.

      +

      We do not need the substitution anymore, so we can remove the require and
      +import statements. The syntax of types and expressions stays unchanged, but
      +we can now remove the binder tag of lambda.

      +

      Like in the type system of IMP++ in Lesson 1, we need a configuration that
      +contains, besides the <k/> cell, a <tenv/> cell that will hold the type
      +environment.

      +

      In an environment-based definition, unlike in a substitution-based one, we
      +need to lookup variables in the environment. So let us start with the
      +type lookup rule:

      +
      rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</k>
+
      +

      The type environment is populated by the semantic rule of lambda:

      +
      rule <k> lambda X : T . E => (T -> E) ~> Rho ...</k>
+     <tenv> Rho => Rho[X <- T] </tenv>
+
      +

      So X is bound to its type T in the type environment, and then T -> E
      +is scheduled for processing. Recall that the arrow type construct has been
      +extended into a strict expression construct, so E will be eventually reduced
      +to its type. Like in other environment-based definitions, we need to make
      +sure that we recover the type environment after the computation in the scope
      +of the declared variable terminates.

      +

      The typing rule of application does not change, so it stays as elegant as it
      +was in the substitution-based definition:

      +
      rule (T1 -> T2) T1 => T2
+
      +

      So do the rules for arithmetic and Boolean constructs, and those for the
      +if, and let, and letrec.

      +

      The mu rule needs to change, because it was previously defined using
      +substitution. We modify it in the same spirit as we modified the lambda
      +rule: bind X to its type in the environment, schedule its body for typing
      +in its right context, and then recover the type environment.

      +

      Finally, we give the semantics of environment recovery, making sure
      +the environment is recovered only after the preceding computation is
      +reduced to a type:

      +

      rule _:Type ~> (Rho => .) ... _ => Rho

      +

      The changes that we applied to the substitution-based definition were
      +therefore quite systematic: each substitution invocation was replaced with
      +an appropriate type environment update/recovery.

      +

      Go to Lesson 4, Type Systems: A Naive Substitution-Based Type Inferencer.

      +

      A Naive Substitution-Based Type Inferencer

      + +

      In this lesson you learn how to define a naive substitution-based type
      +inferencer for a higher-order language, namely the LAMBDA language
      +defined in Part 1 of the tutorial.

      +

      Unlike in the type checker defined in Lessons 2 and 3, where we had to
      +associate a type with each declared variable, a type inferencer
      +attempts to infer the types of all the variables from the way those
      +variables are used. Let us take a look at this program, say plus.lambda:

      +
      lambda x . lambda y . x + y
+
      +

      Since x and y are used in an integer addition context, we can infer
      +that they must have the type int and the result of the addition is
      +also an int, so the type of the entire expression is int -> int -> int.
      +Similarly, the program if.lambda

      +
      lambda x . lambda y . lambda z .
+  if x then y else z
+
      +

      can only make sense when x has type bool and y and z have the same
      +type, say t, in which case the type of the entire expression is
      +bool -> t -> t -> t. Since the type t can be anything, we say that
      +the type of this expression is polymorphic. That means that the code
      +above can be used in different contexts, where t can be an int, a
      +bool, a function type int -> int, and so on.

      +

      In the identity.lambda program

      +
      let f = lambda x . x
+in f 1
+
      +

      f has such a polymorphic type, which is then applied to an integer,
      +so this program is type-safe and its type is int.

      +

      A typical polymorphic expression is the composition

      +
      lambda f . lambda g . lambda x .
+  g (f x)
+
      +

      which has the type (t1 -> t2) -> (t2 -> t3) -> (t1 -> t3), polymorphic
      +in 3 types.

      +

      Let us now define our naive type inferencer and then we discuss more
      +examples. The idea is quite simple: we conceptually do the same
      +operations like we did within the type checker defined in Lesson 2,
      +with two important differences:

      +
        +
      1. instead of declaring a type with each declared variable, we assume
        +a fresh type for that variable; and
        2. +
      2. instead of checking that the types of expressions satisfy the
        +type properties of the context in which they are used, we impose
        +those properties as type equality constraints. A general-purpose
        +unification-based constraint solving mechanism is then used to solve
        +the generated type constraints.
        3. +
      +

      Let us start with the syntax, which is essentially identical to that
      +of the type checker in Lesson 2, except that bound variables are not
      +declared a type anymore. Also, to keep things more compact, we put
      +all the Exp syntax declarations in one syntax declaration this time.

      +

      Before we modify the rules, let us first define our machinery for
      +adding and solving constraints. First, we require and import the
      +unification procedure. We do not discuss unification here, but if you
      +are interested you can consult the unification.k files under
      +k-distribution/include/kframework/builtin, which contains our current generic
      +definition of unification, which is written also in K. The generic unification
      +provides a sort, Mgu, for most-general-unifier, an operation
      +updateMgu(Mgu,T1,T2) which updates Mgu with additional constraints
      +generated by forcing the terms T1 and T2 to be equal, and an operation
      +applyMgu(Mgu,T) which applies Mgu to term T. For our use
      +of unification here, we do not even need to know how Mgu terms are
      +represented internally.

      +

      We define a K item construct, =, which takes two Type terms and
      +enforces them to be equal by means of updating the current Mgu.
      +Once the constraints are added to the Mgu, the equality dissolves
      +itself. With this semantics of = in mind, we can now go ahead and
      +modify the rules of the type checker systematically into rules
      +for a type inferencer. The changes are self-explanatory and
      +mechanical: for example, the rule

      +
      rule int * int => int
+
      +

      changes into rule

      +
      rule T1:Type  * T2:Type => T1 = int ~> T2 = int ~> int
+
      +

      generating the constraints that the two arguments of multiplication
      +have the type int, and the result type is int. Recall that each type
      +equality on the <k/> cell updates the current Mgu appropriately and
      +then dissolves itself; thus, the above says that after imposing the
      +constraints T1=int and T2=int, multiplication yields a type int.

      +

      As mentioned above, since types of variables are not declared anymore,
      +but inferred, we have to generate a fresh type for each variable at its
      +declaration time, and then generate appropriately constraints for it.
      +For example, the type semantics of lambda and mu become:

      +
      rule lambda X . E => T -> E[T/X]  when fresh(T:Type)
+rule mu X . E => (T -> T) E[T/X]  when fresh(T:Type)
+
      +

      that is, we add a condition stating that the previously declared type
      +is now a fresh one. This type will be further constrained by how the
      +variable X is being used within E.

      +

      Interestingly, the previous typing rule for lambda application is not
      +powerful enough anymore. Indeed, since types are not given anymore,
      +it may very well be the case that the inferred type of the first
      +argument of the application construct is not yet a function type
      +(remember, for example, the program composition.lambda above). What
      +we have to do is to enforce it to be a function type, by means of
      +fresh types and constraints. We can introduce a fresh type for the
      +result of the application, and then write the expected rule as
      +follows:

      +
      rule T1:Type T2:Type => T1 = (T2 -> T) ~> T  when fresh(T:Type)
+
      +

      The conditional requires that its first argument is a bool and its
      +second and third arguments have the same type, which is also the
      +result type.

      +

      The macros do not change, in particular let is desugared into lambda
      +application. We will next see that this is a significant restriction,
      +because it limits the polymorphism of our type system.

      +

      We are done. We have a working type inferencer for LAMBDA.

      +

      Let's kompile it and krun the programs above. They all work as
      +expected. Let us also try some additional programs, to push it to its
      +limits.

      +

      First, let us test mu by means of a letrec example:

      +
      letrec f x = 3
+in f
+
      +

      We can also try all the programs that we had in our first tutorial, on
      +lambda, for example the factorial.imp program:

      +
      letrec f x = if x <= 1 then 1 else (x * (f (x + -1)))
+in (f 10)
+
      +

      Those programs are simple enough that they should all work as
      +expected with our naive type inferencer here.

      +

      Let us next try to type some tricky programs, which involve more
      +complex and indirect type constraints.

      +

      tricky-1.lambda:

      +
      lambda f . lambda x . lambda y . (
+  (f x y) + x + (let x = y in x)
+)
+
      +

      tricky-2.lambda:

      +
      lambda x .
+  let f = lambda y . if true then y else x
+  in (lambda x . f 0)
+
      +

      tricky-3.lambda:

      +
      lambda x . let f = lambda y . if true then x 7 else x y
+           in f
+
      +

      tricky-4.lambda:

      +
      lambda x . let f = lambda x . x
+           in let d = (f x) + 1
+              in x
+
      +

      tricky-5.lambda:

      +
      lambda x . let f = lambda y . x y
+           in let z = x 0 in f
+
      +

      It is now time to see the limitations of this naive type inferencer.
      +Consider the program

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      Our type inferencer fails graciously with a clash in the <mgu/> cell
      +between int and bool. Indeed, the desugaring macro of let turns it
      +into a lambda and an application, which further enforce id to have a
      +type of the form t -> t for some fresh type t. The first use of id
      +in the condition of if will then constrain t to be bool, while the
      +other uses in the two branches will enforce t to be int. Thus the
      +clash in the <mgu/> cell.

      +

      Similarly, the program

      +
      let id = lambda x . x
+in id id
+
      +

      yields a different kind of conflict: if id has type t -> t, in order
      +to apply id to itself it must be the case that its argument, t, equals
      +t -> t. These two type terms cannot be unified because there is a
      +circular dependence on t, so we get a cycle in the <mgu/> cell.

      +

      Both limitations above will be solved when we change the semantics of
      +let later on, to account for the desired polymorphism.

      +

      Before we conclude this lesson, let us see one more interesting
      +example, where the lack of let-polymorphism leads not to a type error,
      +but to a less generic type:

      +
      let f1 = lambda x . x in
+  let f2 = f1 in
+    let f3 = f2 in
+      let f4 = f3 in
+        let f5 = f4 in
+          if (f5 true) then f2 else f3
+
      +

      Our current type inferencer will infer the type bool -> bool for the
      +program above. Nevertheless, since all functions f1, f2, f3, f4, f5
      +are the identity function, which is polymorphic, we would expect the
      +entire program to type to the same polymorphic identity function type.

      +

      This limitation will be also addressed when we define our
      +let-polymorphic type inferencer.

      +

      Before that, in the next lesson we will show how easily we can turn
      +the naive substitution-based type inferencer discussed in this lesson
      +into a similarly naive, but environment-based type inferencer.

      +

      Go to Lesson 5, Type Systems: A Naive Environment-Based Type Inferencer.

      +

      A Naive Environment-Based Type Inferencer

      + +

      In this lesson you learn how to define a naive environment-based type
      +inferencer for a higher-order language. Specifically, we take the
      +substitution-based type inferencer for LAMBDA defined in Lesson 4 and
      +turn it into an environment-based one.

      +

      Recall from Lesson 3, where we defined an environment-based type
      +checker for LAMBDA based on the substitution-based one in Lesson 2,
      +that the transition from a substitution-based definition to an
      +environment-based one was quite systematic and mechanical: each
      +substitution occurrence E[T/X] is replaced by E, but at the same time
      +the variable X is bound to type T in the type environment. One benefit
      +of using type environments instead of substitution is that we replace
      +a linear complexity operation (the substitution) with a constant
      +complexity one (the variable lookup).

      +

      There is not much left to say which has not been already said in
      +Lesson 3: we remove the unnecessary binder annotations for the
      +variable binding operations, then add a <tenv/> cell to the
      +configuration to hold the type environment, then add a new rule for
      +variable lookup, and finally apply the transformation of substitutions
      +E[T/X] into E as explained above.

      +

      The resulting type inferencer should now work exactly the same way as
      +the substitution-based one, except, of course, that the resulting
      +configurations will contain a <tenv/> cell now.

      +

      As sanity check, let us consider two more LAMBDA programs that test
      +the static scoping nature of the inferencer. We do that because
      +faulty environment-based definitions often have this problem. The
      +program

      +
      let x = 1
+in let f = lambda a . x
+   in let x = true
+      in f 3
+
      +

      should type to int, not to bool, and so it does. Similarly, the
      +program

      +
      let y = 0
+in letrec f x = if x <= 0
+                then y
+                else let y = true
+                     in f (x + 1)
+   in f 1
+
      +

      should also type to int, not bool, and so it does, too.

      +

      The type inferencer defined in this lesson has the same limitations,
      +in terms of polymorphism, as the one in Lesson 4. In the next
      +lesson we will see how it can be parallelized, and in further lessons
      +how to make it polymorphic.

      +

      Go to Lesson 6, Type Systems: Parallel Type Checkers/Inferencers.

      +

      Parallel Type Checkers/Inferencers

      + +

      In this lesson you learn how to define parallel type checkers or
      +inferencers. For the sake of a choice, we will parallelize the one in
      +the previous lesson, but the ideas are general. We are using the same
      +idea to define type checkers for other languages in the K tool
      +distribution, such as SIMPLE and KOOL.

      +

      The idea is in fact quite simple. Instead of one monolithic typing
      +task, we generate many smaller tasks, which can be processed in
      +parallel. We use the same approach to define parallel semantics as we
      +used for threads in IMP++ in Part 4 of the tutorial, that is, we add a
      +cell holding all the parallel tasks, making sure we declare the cell
      +holding a task with multiplicity *. For the particular type
      +inferencer that we chose here, the one in Lesson 5, each task will
      +hold an expression to type together with a type environment (so it
      +knows where to lookup its free variables). We have the following
      +configuration then:

      +
      configuration <tasks color="yellow">
+                <task color="orange" multiplicity="*">
+                  <k color="green"> $PGM:Exp </k>
+                  <tenv color="red"> .Map </tenv>
+                </task>
+              </tasks>
+              <mgu color="blue"> .Mgu </mgu>
+
      +

      Now we have to take each typing rule we had before and change it to
      +yield parallel typing. For example, our rule for typing
      +multiplication was the following in Lesson 5:

      +
      rule T1:Type * T2:Type => T1 = int ~> T2 = int ~> int
+
      +

      Since * was strict, its two arguments eventually type, and once that
      +happens the rule above fires. Unfortunately, the strictness of
      +multiplication makes the typing of the two expressions sequential in
      +our previous definition. To avoid typing the two expressions
      +sequentially and instead generating two parallel tasks, we remove the
      +strict attribute of multiplication and replace the rule above with the
      +following:

      +
      rule <k> E1 * E2 => int ...</k> <tenv> Rho </tenv>
+     (. => <task> <k> E1 = int </k> <tenv> Rho </tenv> </task>
+           <task> <k> E2 = int </k> <tenv> Rho </tenv> </task>)
+
      +

      Therefore, we generate two tasks for typing E1 and E2 in the same type
      +environment as the current task, and let the current task continue by
      +simply optimistically reducing E1*E2 to its expected result type, int.
      +If E1 or E2 will not type to int, then either their corresponding
      +tasks will get stuck or the <mgu/> cell will result into a clash or cycle,
      +so the program will not type overall in spite of the fact that we
      +allowed the task containing the multiplication to continue. This is
      +how we get maximum of parallelism in this case.

      +

      Before we continue, note that the new tasks hold equalities in them,
      +where one of its arguments is an expression, while previously the
      +equality construct was declared to take types. What we want now is
      +for the equality construct to possibly take any expressions, and first
      +type them and then generate the type constraint like before. This can
      +be done very easily by just extending the equality construct to
      +expressions and declaring it strict:

      +
      syntax KItem ::= Exp "=" Exp  [strict]
+
      +

      Unlike before, where we only passed types to the equality construct,
      +we now need a runtime check that its arguments are indeed types before
      +we can generate the updateMgu command:

      +
      rule <k> T:Type = T':Type => . ...</k>
+     <mgu> Theta:Mgu => updateMgu(Theta,T,T') </mgu>
+
      +

      Like before, an equality will therefore update the <mgu/> cell and then
      +it dissolves itself, letting the <k/> cell in the corresponding task
      +empty. Such empty tasks are unnecessary, so they can be erased:

      +
      rule <task>... <k> . </k> ...</task> => .
+
      +

      We can now follow the same style as for multiplication to write the
      +parallel typing rules of the other arithmetic constructs, and even for
      +the conditional.

      +

      To parallelize the typing of lambda we generate two fresh types, one
      +for the variable and one for the body, and make sure that we generate
      +the correct type constraint and environment in the body task:

      +
      rule <k> lambda X . E => Tx -> Te ...</k> <tenv> TEnv </tenv>
+     (. => <task> <k> E = Te </k> <tenv> TEnv[Tx/X] </tenv> </task>)
+  when fresh(Tx:Type) andBool fresh(Te:Type)
+
      +

      Note that the above also allows us to not need to change and then
      +recover the environment of the current cell.

      +

      For function application we also need to generate two fresh types:

      +
      rule <k> E1 E2 => T ...</k> <tenv> Rho </tenv>
+     (. => <task> <k> E1 = T2 -> T </k> <tenv> Rho </tenv> </task>
+           <task> <k> E2 = T2 </k> <tenv> Rho </tenv> </task>)
+  when fresh(T2:Type) andBool fresh(T:Type)
+
      +

      The only rule left is that of mu X . E. In this case we only need one
      +fresh type, because X, E and mu X . E have all the same type:

      +
      rule <k> mu X . E => T ...</k>  <tenv> TEnv </tenv>
+     (. => <task> <k> E = T </k> <tenv> TEnv[T/X] </tenv> </task>)
+  when fresh(T:Type)
+
      +

      We do not need the type environment recovery operation, so we delete it.

      +

      We can now kompile and krun all the programs that we typed in Lesson 5.
      +Everything should work.

      +

      In this lesson we only aimed at parallelizing the type inferencer in
      +Lesson 5, not to improve its expressiveness; it still has the same
      +limitations in terms of polymorphism. The next lessons are dedicated
      +to polymorphic type inferencers.

      +

      Go to Lesson 7, Type Systems: A Naive Substitution-based Polymorphic Type Inferencer.

      +

      A Naive Substitution-based Polymorphic Type Inferencer

      + +

      In this lesson you learn how little it takes to turn a naive monomorphic
      +type inferencer into a naive polymorphic one, basically only changing
      +a few characters. In terms of the K framework, you will learn that
      +you can have complex combinations of substitutions in K, both over
      +expressions and over types.

      +

      Let us start directly with the change. All we have to do is to take
      +the LAMBDA type inferencer in Lesson 4 and only change the macro

      +
      rule let X = E in E' => (lambda X . E') E  [macro]
+
      +

      as follows:

      +
      rule let X = E in E' => E'[E/X]  [macro]
+
      +

      In other words, we are inlining the beta-reduction rule of
      +lambda-calculus within the original rule. In terms of typing,
      +the above forces the type inferencer to type E in place for each
      +occurrence of X in E'. Unlike in the first rule, where X had to get
      +one type only which satisfied the constrains of all X's occurrences in
      +E', we now never associate any type to X anymore.

      +

      Let us kompile and krun some examples. Everything that worked with
      +the type inferencer in Lesson 4 should still work here, although the
      +types of some programs can now be more general. For example, reconsider
      +the nested-lets.lambda program

      +
      let f1 = lambda x . x in
+  let f2 = f1 in
+    let f3 = f2 in
+      let f4 = f3 in
+        let f5 = f4 in
+          if (f5 true) then f2 else f3
+
      +

      which was previously typed to bool -> bool. With the new rule above,
      +the sequence of lets is iteratively eliminated and we end up with the
      +program

      +
      if (lambda x . x) true then (lambda x . x) else (lambda x . x)
+
      +

      which now types (with both type inferencers) to a type of the form
      +t -> t, for some type variable t, which is more general than the
      +previous bool -> bool type that the program typed to in Lesson 4.

      +

      We can also now type programs that were not typable before, such as

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      and

      +
      let id = lambda x . x
+in id id
+
      +

      Let us also test it on some trickier programs, also not typable
      +before, such as

      +
      let f = lambda x . x
+in let g = lambda y . f y
+   in g g
+
      +

      which gives us a type of the form t -> t for some type variable t,
      +and as

      +
      let f = let g = lambda x . x
+        in let h = lambda x . lambda x . (g g g g)
+           in h
+in f
+
      +

      which types to t1 -> t2 -> t3 -> t3 for some type variables t1, t2, t3.

      +

      Here is another program which was not typable before, which is
      +trickier than the others above in that a lambda-bound variable appears
      +free in a let-bound expression:

      +
      lambda x . (
+  let y = lambda z . x
+  in if (y true) then (y 1) else (y (lambda x . x))
+)
+
      +

      The above presents no problem now, because once lambda z . x gets
      +substituted for y we get a well-typed expression which yields that x
      +has the type bool, so the entire expression types to bool -> bool.

      +

      The cheap type inferencer that we obtained above therefore works as
      +expected. However, it has two problems which justify a more advanced
      +solution. First, substitution is typically considered an elegant
      +mathematical instrument which is not too practical in implementations,
      +so an implementation of this type inferencer will likely be based on
      +type environments anyway. Additionally, we mix two kinds of
      +substitutions in this definition, one where we substitute types and
      +another where we substitute expressions, which can only make things
      +harder to implement efficiently. Second, our naive substitution of E
      +for X in E' can yield an exponential explosion in size of the original
      +program. Consider, for example, the following classic example which
      +is known to generate a type whose size is exponential in the size of
      +the program (and is thus used as an argument for why let-polymorphic
      +type inference is exponential in the worst-case):

      +
      let f00 = lambda x . lambda y . x in
+  let f01 = lambda x . f00 (f00 x) in
+    let f02 = lambda x . f01 (f01 x) in
+      let f03 = lambda x . f02 (f02 x) in
+        let f04 = lambda x . f03 (f03 x) in
+          // ... you can add more nested lets here
+          f04
+
      +

      The particular instance of the pattern above generates a type which
      +has 17 type variables! The desugaring of each let doubles the size of
      +the program and of its resulting type. While such programs are little
      +likely to appear in practice, it is often the case that functions can
      +be quite complex and large while their type can be quite simple in the
      +end, so we should simply avoid retyping each function each time it is
      +used.

      +

      This is precisely what we will do next. Before we present the classic
      +let-polymorphic type inferencer in Lesson 9, which is based on
      +environments, we first quickly discuss in Lesson 8 an intermediate
      +step, namely a naive environment-based variant of the inferencer
      +defined here.

      +

      Go to Lesson 8, Type Systems: A Naive Environment-based Polymorphic Type Inferencer.

      +

      A Naive Environment-based Polymorphic Type Inferencer

      + +

      In this short lesson we discuss how to quickly turn a naive
      +environment-based monomorphic type inferencer into a naive let-polymorphic
      +one. Like in the previous lesson, we only need to change a few
      +characters. In terms of the K framework, you will learn how to have
      +both environments and substitution in the same definition.

      +

      Like in the previous lesson, all we have to do is to take the LAMBDA
      +type inferencer in Lesson 5 and only change the rule

      +
      rule let X = E in E' => (lambda X . E') E
+
      +

      as follows:

      +
      rule let X = E in E' => E'[E/X]
+
      +

      The reasons why this works have already been explained in the previous
      +lesson, so we do not repeat them here.

      +

      Since our new let rule uses substitution, we have to require the
      +substitution module at the top and also import SUBSTITUTION in the
      +current module, besides the already existing UNIFICATION.

      +

      Everything which worked with the type inferencer in Lesson 7 should
      +also work now. Let us only try the exponential type example,

      +
      let f00 = lambda x . lambda y . x in
+  let f01 = lambda x . f00 (f00 x) in
+    let f02 = lambda x . f01 (f01 x) in
+      let f03 = lambda x . f02 (f02 x) in
+        let f04 = lambda x . f03 (f03 x) in
+          f04
+
      +

      As expected, this gives us precisely the same type as in Lesson 7.

      +

      So the only difference between this type inferencer and the one in
      +Lesson 7 is that substitution is only used for LAMBDA-to-LAMBDA
      +transformations, but not for infusing types within LAMBDA programs.
      +Thus, the syntax of LAMBDA programs is preserved intact, which some
      +may prefer. Nevertheless, this type inferencer is still expensive and
      +wasteful, because the let-bound expression is typed over and over
      +again in each place where the let-bound variable occurs.

      +

      In the next lesson we will discuss a type inferencer based on the
      +classic Damas-Hindley-Milner type system, which maximizes the reuse of
      +typing work by means of parametric types.

      +

      Go to Lesson 9, Type Systems: Let-Polymorphic Type Inferencer (Damas-Hindley-Milner).

      +

      Let-Polymorphic Type Inferencer (Damas-Hindley-Milner)

      + +

      In this lesson we discuss a type inferencer based on what we call today
      +the Damas-Hindley-Milner type system, which is at the core of many
      +modern functional programming languages. The first variant of it was
      +proposed by Hindley in 1969, then, interestingly, Milner rediscovered
      +it in 1978 in the context of the ML language. Damas formalized it as
      +a type system in his PhD thesis in 1985. More specifically, our type
      +inferencer here, like many others as well as many implementations of
      +it, follows more closely the syntax-driven variant proposed by Clement
      +in 1987.

      +

      In terms of K, we will see how easily we can turn one definition which
      +is considered naive (our previous type inferencer in Lesson 8) into a
      +definition which is considered advanced. All we have to do is to
      +change one existing rule (the rule of the let binder) and to add a new
      +one. We will also learn some new predefined features of K, which make
      +the above possible.

      +

      The main idea is to replace the rule

      +
      rule let X = E in E' => E'[E/X]
+
      +

      which creates potentially many copies of E within E' with a rule
      +which types E once and then reuses that type in each place where X
      +occurs free in E'. The simplest K way to type E is to declare the
      +let construct strict(2). Now we cannot simply bind X to the type
      +of E, because we would obtain a variant of the naive type inferencer
      +we already discussed, together with its limitations, in Lesson 5 of this
      +tutorial. The trick here is to parameterize the type of E in all its
      +unconstrained fresh types, and then create fresh copies of those
      +parameters in each free occurrence of X in E'.

      +

      Let us discuss some examples, before we go into the technical details.
      +Consider the first let-polymorphic example which failed to be typed
      +with our first naive type-inferencer:

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      When typing lambda x . x, we get a type of the form t -> t, for some
      +fresh type t. Instead of assigning this type to id as we did in the
      +naive type inferencers, we now first parametrize this type in its
      +fresh variable t, written

      +
      (forall t) t -> t
+
      +

      and then bind id to this parametric type. The intuition for the
      +parameter is that it can be instantiated with any other type, so this
      +parametric type stands, in fact, for infinitely many non-parametric
      +types. This is similar to what happens in formal logic proof systems,
      +where rule schemas stand for infinitely many concrete instances of
      +them. For this reason, parametric types are also called type schemas.

      +

      Now each time id is looked up within the let-body, we create a fresh
      +copy of the parameter t, which can this way be independently
      +constrained by each local context. Let's suppose that the three id
      +lookups yield the types t1 -> t1, t2 -> t2, and respectively t3 -> t3.
      +Then t1 will be constrained to be bool, and t2 and t3 to be int,
      +so we can now safely type the program above to int.

      +

      Therefore, a type schema comprises a summary of all the typing work
      +that has been done for typing the corresponding expression, and an
      +instantiation of its parameters with fresh copies represents an
      +elegant way to reuse all that typing work.

      +

      There are some subtleties regarding what fresh types can be made
      +parameters. Let us consider another example, discussed as part of
      +Lesson 7 on naive let-polymorphism:

      +
      lambda x . (
+  let y = lambda z . x
+  in if (y true) then (y 1) else (y (lambda x . x))
+)
+
      +

      This program should type to bool -> bool, as explained in Lesson 7.
      +The lambda construct will bind x to some fresh type tx. Then the
      +let-bound expression lambda z . x types to tz -> tx for some
      +additional fresh type tz. The question now is what should the
      +parameters of this type be when we generate the type schema? If we
      +naively parameterize in all fresh variables, that is in both tz and
      +tx obtaining the type schema (forall tz,tx) tz -> tx, then there will
      +be no way to infer that the type of x, tx, must be a bool! The
      +inferred type of this expression would then wrongly be tx -> t for
      +some fresh types tx and t. That's because the parameters are replaced
      +with fresh copies in each occurrence of y, and thus their relationship
      +to the original x is completely lost. This tells us that we cannot
      +parameterize in all fresh types that appear in the type of the
      +let-bound expression. In particular, we cannot parameterize in those
      +which some variables are already bound to in the current type
      +environment (like x is bound to tx in our example above).
      +In our example, the correct type schema is (forall tz) tz -> tx,
      +which now allows us to correctly infer that tx is bool.

      +

      Let us now discuss another example, which should fail to type:

      +
      lambda x .
+  let f = lambda y . x y
+  in if (f true) then (f 1) else (f 2)
+
      +

      This should fail to type because lambda y . x y is equivalent to x,
      +so the conditional imposes the conflicting constraints that x should be
      +a function whose argument is either a bool or an int. Let us try to
      +type it using our currently informal procedure. Like in the previous
      +example, x will be bound to a fresh type tx. Then the let-bound
      +expression types to ty -> tz with ty and tz fresh types, adding also
      +the constraint tx = ty -> tz. What should the parameters of this type
      +be? If we ignore the type constraint and simply make both ty and tz
      +parameters because no variable is bound to them in the type
      +environment (indeed, the only variable x in the type environment is
      +bound to tx), then we can wrongly type this program to tx -> tz
      +following a reasoning similar to the one in the example above.
      +In fact, in this example, none of ty and tz can be parameters, because
      +they are constrained by tx.

      +

      The examples above tell us two things: first, that we have to take the
      +type constraints into account when deciding the parameters of the
      +schema; second, that after applying the most-general-unifier solution
      +given by the type constraints everywhere, the remaining fresh types
      +appearing anywhere in the type environment are consequently constrained
      +and cannot be turned into parameters. Since the type environment can in
      +fact also hold type schemas, which already bind some types, we only need
      +to ensure that none of the fresh types appearing free anywhere in the
      +type environment are turned into parameters of type schemas.

      +

      Thanks to generic support offered by the K tool, we can easily achieve
      +all the above as follows.

      +

      First, add syntax for type schemas:

      +
      syntax TypeSchema ::= "(" "forall" Set ")" Type  [binder]
+
      +

      The definition below will be given in such a way that the Set argument
      +of a type schema will always be a set of fresh types. We also declare
      +this construct to be a binder, so that we can make use of the generic
      +free variable function provided by the K tool.

      +

      We now replace the old rule for let

      +
      rule let X = E in E' => E'[E/X]
+
      +

      with the following rule:

      +
      rule <k> let X = T:Type in E => E ~> tenv(TEnv) ...</k>
+     <mgu> Theta:Mgu </mgu>
+     <tenv> TEnv
+      => TEnv[(forall freeVariables(applyMgu(Theta, T)) -Set
+                      freeVariables(applyMgu(Theta, values TEnv))
+              ) applyMgu(Theta, T) / X]
+     </tenv>
+
      +

      So the type T of E is being parameterized and then bound to X in the
      +type environment. The current mgu Theta, which comprises all the type
      +constraints accumulated so far, is applied to both T and the types in
      +the type environment. The remaining fresh types in T which do not
      +appear free in the type environment are then turned into type parameters.
      +The function freeVariables returns, as expected, the free variables of
      +its argument as a Set; this is why we declared the type schema to be a
      +binder above.

      +

      Now a LAMBDA variable in the type environment can be bound to either a
      +type or a type schema. In the first case, the previous rule we had
      +for variable lookup can be reused, but we have to make sure we check
      +that T there is of sort Type (adding a sort membership, for example).
      +In the second case, as explained above, we have to create fresh copies
      +of the parameters. This can be easily achieved with another
      +predefined K function, as follows:

      +
      rule <k> X:Id => freshVariables(Tvs,T) ...</k>
+     <tenv>... X |-> (forall Tvs) T ...</tenv>
+
      +

      Indeed, freshVariables takes a set of variables and a term, and returns the
      +same term but with each of the given variables replaced by a fresh copy.

      +

      The operations freeVariables and freshVariables are useful in many K
      +definitions, so they are predefined in module substitution.k.

      +

      Our definition of this let-polymorphic type inferencer is now
      +complete. To test it, kompile it and then krun all the LAMBDA
      +programs discussed since Lesson 4. They should all work as expected.

      +

      K Languages

      + +

      Here we present several "real-world" language examples. These languages
      +demonstrate many of the features you would expect to find in a full-fledged
      +programming language.

      +
        +
      • SIMPLE: Imperative programming language with threads.
        • +
      • KOOL: SIMPLE extended with object-oriented features.
        • +
      • FUN: A functional language with algebraic data-types and pattern-matching.
        • +
      • LOGIK: A logical programming language based on clause unification.
        • +
      +

      SIMPLE — Untyped

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped SIMPLE language.
      +SIMPLE is intended to be a pedagogical and research language that captures
      +the essence of the imperative programming paradigm, extended with several
      +features often encountered in imperative programming languages.
      +A program consists of a set of global variable declarations and
      +function definitions. Like in C, function definitions cannot be
      +nested and each program must have one function called main,
      +which is invoked when the program is executed. To make it more
      +interesting and to highlight some of K's strengths, SIMPLE includes
      +the following features in addition to the conventional imperative
      +expression and statement constructs:

      +
        +
      • +

        Multidimensional arrays and array references. An array evaluates
        +to an array reference, which is a special value holding a location (where
        +the elements of the array start) together with the size of the array;
        +the elements of the array can be array references themselves (particularly
        +when the array is multi-dimensional). Array references are ordinary values,
        +so they can be assigned to variables and passed/received by functions.

        +
        • +
      • +

        Functions and function values. Functions can have zero or
        +more parameters and can return abruptly using a return statement.
        +SIMPLE follows a call-by-value parameter passing style, with static scoping.
        +Function names evaluate to function abstractions, which hereby become ordinary
        +values in the language, same like the array references.

        +
        • +
      • +

        Blocks with locals. SIMPLE variables can be declared
        +anywhere, their scope being from the place where they are declared
        +until the end of the most nested enclosing block.

        +
        • +
      • +

        Input/Output. The expression read() evaluates to the
        +next value in the input buffer, and the statement write(e)
        +evaluates e and outputs its value to the output buffer. The
        +input and output buffers are lists of values.

        +
        • +
      • +

        Exceptions. SIMPLE has parametric exceptions (the value thrown as
        +an exception can be caught and bound).

        +
        • +
      • +

        Concurrency via dynamic thread creation/termination and
        +synchronization. One can spawn a thread to execute any statement.
        +The spawned thread shares with its parent its environment at creation time.
        +Threads can be synchronized via a join command which blocks the current thread
        +until the joined thread completes, via re-entrant locks which can be acquired
        +and released, as well as through rendezvous commands.

        +
        • +
      +

      Like in many other languages, some of SIMPLE's constructs can be
      +desugared into a smaller set of basic constructs. We do that at the end
      +of the syntax module, and then we only give semantics to the core constructs.

      +

      Note: This definition is commented slightly more than others, because it is
      +intended to be one of the first non-trivial definitions that the new
      +user of K sees. We recommend the beginner user to first check the
      +language definitions discussed in the K tutorial.

      +
      module SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      We start by defining the SIMPLE syntax. The language constructs discussed
      +above have the expected syntax and evaluation strategies. Recall that in K
      +we annotate the syntax with appropriate strictness attributes, thus giving
      +each language construct the desired evaluation strategy.

      +

      Identifiers

      + +

      Recall from the K tutorial that identifiers are builtin and come under the
      +syntactic category Id. The special identifier for the function
      +main belongs to all programs, and plays a special role in the semantics,
      +so we declare it explicitly. This would not be necessary if the identifiers
      +were all included automatically in semantic definitions, but that is not
      +possible because of parsing reasons (e.g., K variables used to match
      +concrete identifiers would then be ambiguously parsed as identifiers). They
      +are only included in the parser generated to parse programs (and used by the
      +kast tool). Consequently, we have to explicitly declare all the
      +concrete identifiers that play a special role in the semantics, like
      +main below.

      +
        syntax Id ::= "main" [token]
+

      Declarations

      + +

      There are two types of declarations: for variables (including arrays) and
      +for functions. We are going to allow declarations of the form
      +var x=10, a[10,10], y=23;, which is why we allow the var
      +keyword to take a list of expressions. The non-terminals used in the two
      +productions below are defined shortly.

      +
        syntax Stmt ::= "var" Exps ";"
+                | "function" Id "(" Ids ")" Block
+

      Expressions

      + +

      The expression constructs below are standard. Increment (++) takes
      +an expression rather than a variable because it can also increment an array
      +element. Recall that the syntax we define in K is what we call the syntax
      +of the semantics      : while powerful enough to define non-trivial syntaxes
      +(thanks to the underlying SDF technology that we use), we typically refrain
      +from defining precise syntaxes, that is, ones which accept precisely the
      +well-formed programs (that would not be possible anyway in general). That job
      +is deferred to type systems, which can also be defined in K. In other words,
      +we are not making any effort to guarantee syntactically that only variables
      +or array elements are passed to the increment construct, we allow any
      +expression. Nevertheless, we will only give semantics to those, so expressions
      +of the form ++5, which parse (but which will be rejected by our type
      +system in the typed version of SIMPLE later), will get stuck when executed.
      +Arrays can be multidimensional and can hold other arrays, so their
      +lookup operation takes a list of expressions as argument and applies to an
      +expression (which can in particular be another array lookup), respectively.
      +The construct sizeOf gives the size of an array in number of elements
      +of its first dimension. Note that almost all constructs are strict. The only
      +constructs which are not strict are the increment (since its first argument
      +gets updated, so it cannot be evaluated), the input read which takes no
      +arguments so strictness is irrelevant for it, the logical and and or constructs
      +which are short-circuited, the thread spawning construct which creates a new
      +thread executing the argument expression and return its unique identifier to
      +the creating thread (so it cannot just evaluate its argument in place), and the
      +assignment which is only strict in its second argument (for the same reason as
      +the increment).

      +
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+

      We also need comma-separated lists of identifiers and of expressions.
      +Moreover, we want them to be strict, that is, to evaluate to lists of results
      +whenever requested (e.g., when they appear as strict arguments of
      +the constructs above).

      +
        syntax Ids  ::= List{Id,","}           [klabel(Exps)]
+  syntax Exps ::= List{Exp,","}          [klabel(Exps), strict]  // automatically hybrid now
+  syntax Exps ::= Ids
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(Exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}    [klabel(Exps)]
+  syntax Ids ::= Bottoms
+

      Statements

      + +

      Most of the statement constructs are standard for imperative languages.
      +We syntactically distinguish between empty and non-empty blocks, because we
      +chose Stmts not to be a (;-separated) list of
      +Stmt. Variables can be declared anywhere inside a block, their scope
      +ending with the block. Expressions are allowed to be used for their side
      +effects only (followed by a semicolon ;). Functions are allowed
      +to abruptly return. The exceptions are parametric, i.e., one can throw a value
      +which is bound to the variable declared by catch. Threads can be
      +dynamically created and terminated, and can synchronize with join,
      +acquire, release and rendezvous. Note that the
      +strictness attributes obey the intended evaluation strategy of the various
      +constructs. In particular, the if-then-else construct is strict only in its
      +first argument (the if-then construct will be desugared into if-then-else),
      +while the loop constructs are not strict in any arguments. The print
      +statement construct is variadic, that is, it takes an arbitrary number of
      +arguments.

      +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+// NOTE: print strict allows non-deterministic evaluation of its arguments
+// Either keep like this but document, or otherwise make Exps seqstrict.
+// Of define and use a different expression list here, which is seqstrict.
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+

      The reason we allow Stmts as the first argument of for
      +instead of Stmt is because we want to allow more than one statement
      +to be executed when the loop is initialized. Also, as seens shorly, macros
      +may expand one statement into more statements; for example, an initialized
      +variable declaration statement var x=0; desugars into two statements,
      +namely var x; x=0;, so if we use Stmt instead of Stmts
      +in the production of for above then we risk that the macro expansion
      +of statement var x=0; happens before the macro expansion of for,
      +also shown below, in which case the latter would not apply anymore because
      +of syntactic mismatch.

      +
        syntax Stmt ::= Stmt Stmt                          [right]
+
+// I wish I were able to write the following instead, but confuses the parser.
+//
+// syntax Stmts ::= List{Stmt,""}
+// syntax Top ::= Stmt | "function" Id "(" Ids ")" Block
+// syntax Pgm ::= List{Top,""}
+//
+// With that, I could have also eliminated the empty block
+

      Desugared Syntax

      + +

      This part desugars some of SIMPLE's language constructs into core ones.
      +We only want to give semantics to core constructs, so we get rid of the
      +derived ones before we start the semantics. All desugaring macros below are
      +straightforward.

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule for(Start Cond; Step) {} => {Start while (Cond) {Step;}}
+  rule var E1:Exp, E2:Exp, Es:Exps; => var E1; var E2, Es;
+  rule var X:Id = E; => var X; X = E;
+

      For the semantics, we can therefore assume from now on that each
      +conditional has both branches, that there are only while loops, and
      +that each variable is declared alone and without any initialization as part of
      +the declaration.

      +
      endmodule
+
+
+module SIMPLE-UNTYPED
+  imports SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS
+

      Basic Semantic Infrastructure

      + +

      Before one starts adding semantic rules to a K definition, one needs to
      +define the basic semantic infrastructure consisting of definitions for
      +values and configuration. As discussed in the definitions
      +in the K tutorial, the values are needed to know when to stop applying
      +the heating rules and when to start applying the cooling rules corresponding
      +to strictness or context declarations. The configuration serves as a backbone
      +for the process of configuration abstraction which allows users to only
      +mention the relevant cells in each semantic rule, the rest of the configuration
      +context being inferred automatically. Although in some cases the configuration
      +could be automatically inferred from the rules, we believe that it is very
      +useful for language designers/semanticists to actually think of and design
      +their configuration explicitly, so the current implementation of K requires
      +one to define it.

      +

      Values

      + +

      We here define the values of the language that the various fragments of
      +programs evaluate to. First, integers and Booleans are values. As discussed,
      +arrays evaluate to special array reference values holding (1) a location from
      +where the array's elements are contiguously allocated in the store, and
      +(2) the size of the array. Functions evaluate to function values as
      +λ-abstractions (we do not need to evaluate functions to closures
      +because each function is executed in the fixed global environment and
      +function definitions cannot be nested). Like in IMP and other
      +languages, we finally tell the tool that values are K results.

      +
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+               | lambda(Ids,Stmt)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+                   | Vals  // TODO: should not need this
+

      The inclusion of values in expressions follows the methodology of
      +syntactic definitions (like, e.g., in SOS): extend the syntax of the language
      +to encompass all values and additional constructs needed to give semantics.
      +In addition to that, it allows us to write the semantic rules using the
      +original syntax of the language, and to parse them with the same (now extended
      +with additional values) parser. If writing the semantics directly on the K
      +AST, using the associated labels instead of the syntactic constructs, then one
      +would not need to include values in expressions.

      +

      Configuration

      + +

      The K configuration of SIMPLE consists of a top level cell, T,
      +holding a threads cell, a global environment map cell genv
      +mapping the global variables and function names to their locations, a shared
      +store map cell store mapping each location to some value, a set cell
      +busy holding the locks which have been acquired but not yet released
      +by threads, a set cell terminated holding the unique identifiers of
      +the threads which already terminated (needed for join), input
      +and output list cells, and a nextLoc cell holding a natural
      +number indicating the next available location. Unlike in the small languages
      +in the K tutorial, where we used the fresh predicate to generate fresh
      +locations, in larger languages, like SIMPLE, we prefer to explicitly manage
      +memory. The location counter in nextLoc models an actual physical
      +location in the store; for simplicity, we assume arbitrarily large memory and
      +no garbage collection. The threads cell contains one thread
      +cell for each existing thread in the program. Note that the thread cell has
      +multiplicity *, which means that at any given moment there could be zero,
      +one or more thread cells. Each thread cell contains a
      +computation cell k, a control cell holding the various
      +control structures needed to jump to certain points of interest in the program
      +execution, a local environment map cell env mapping the thread local
      +variables to locations in the store, and finally a holds map cell
      +indicating what locks have been acquired by the thread and not released so far
      +and how many times (SIMPLE's locks are re-entrant). The control cell
      +currently contains only two subcells, a function stack fstack which
      +is a list and an exception stack xstack which is also a list.
      +One can add more control structures in the control cell, such as a
      +stack for break/continue of loops, etc., if the language is extended with more
      +control-changing constructs. Note that all cells except for k are
      +also initialized, in that they contain a ground term of their corresponding
      +sort. The k cell is initialized with the program that will be passed
      +to the K tool, as indicated by the $PGM variable, followed by the
      +execute task (defined shortly).

      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+
+  syntax ControlCell
+  syntax ControlCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Map" color="yellow">
+                      <id color="pink"> -1 </id>
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <genv color="pink"> .Map </genv>
+                  <store color="white"> .Map </store>
+                  <busy color="cyan"> .Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                </T>
+

      Declarations and Initialization

      + +

      We start by defining the semantics of declarations (for variables,
      +arrays and functions).

      +

      Variable Declaration

      + +

      The SIMPLE syntax was desugared above so that each variable is
      +declared alone and its initialization is done as a separate statement.
      +The semantic rule below matches resulting variable declarations of the
      +form var X; on top of the k cell
      +(indeed, note that the k cell is complete, or round, to the
      +left, and is torn, or ruptured, to the right), allocates a fresh
      +location L in the store which is initialized with a special value
      +⊥ (indeed, the unit ., or nothing, is matched anywhere
      +in the map ‒note the tears at both sides‒ and replaced with the
      +mapping L ↦ ⊥), and binds X to L in the local
      +environment shadowing previous declarations of X, if any.
      +This possible shadowing of X requires us to therefore update the
      +entire environment map, which is expensive and can significantly slow
      +down the execution of larger programs. On the other hand, since we know
      +that L is not already bound in the store, we simply add the binding
      +L ↦ ⊥ to the store, thus avoiding a potentially complete
      +traversal of the the store map in order to update it. We prefer the approach
      +used for updating the store whenever possible, because, in addition to being
      +faster, it offers more true concurrency than the latter; indeed, according
      +to the concurrent semantics of K, the store is not frozen while
      +L ↦ ⊥ is added to it, while the environment is frozen during the
      +update operation Env[L/X]. The variable declaration command is
      +also removed from the top of the computation cell and the fresh location
      +counter is incremented. The undefined symbol ⊥ added in the store
      +is of sort KItem, instead of Val, on purpose; this way, the
      +store lookup rules will get stuck when one attempts to lookup an
      +uninitialized location. All the above happen in one transactional step,
      +with the rule below. Note also how configuration abstraction allows us to
      +only mention the needed cells; indeed, as the configuration above states,
      +the k and env cells are actually located within a
      +thread cell within the threads cell, but one needs
      +not mention these: the configuration context of the rule is
      +automatically transformed to match the declared configuration
      +structure.

      +
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      Array Declaration

      + +

      The K semantics of the uni-dimensional array declaration is somehow similar
      +to the above declaration of ordinary variables. First, note the
      +context declaration below, which requests the evaluation of the array
      +dimension. Once evaluated, say to a natural number N, then
      +N +Int 1 locations are allocated in the store for
      +an array of size N, the additional location (chosen to be the first
      +one allocated) holding the array reference value. The array reference
      +value array(L,N) states that the array has size N and its
      +elements are located contiguously in the store starting with location
      +L. The operation L … L' ↦ V, defined at the end of this
      +file in the auxiliary operation section, initializes each location in
      +the list L … L' to V. Note that, since the dimensions of
      +array declarations can be arbitrary expressions, this virtually means
      +that we can dynamically allocate memory in SIMPLE by means of array
      +declarations.

      +
        context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 +Int N </nextLoc>
+    requires N >=Int 0
+

      SIMPLE allows multi-dimensional arrays. For semantic simplicity, we
      +desugar them all into uni-dimensional arrays by code transformation.
      +This way, we only need to give semantics to uni-dimensional arrays.
      +First, note that the context rule above actually evaluates all the array
      +dimensions (that's why we defined the expression lists strict!):
      +Upon evaluating the array dimensions, the code generation rule below
      +desugars multi-dimensional array declaration to uni-dimensional declarations.
      +To this aim, we introduce two special unique variable identifiers,
      +$1 and $2. The first variable, $1, iterates
      +through and initializes each element of the first dimension with an array
      +of the remaining dimensions, declared as variable $2:

      +
        syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         for(var $1 = 0; $1 <= N1 - 1; ++$1) {
+           var $2[N2, Vs];
+           X[$1] = $2;
+         }
+       }
+

      Ideally, one would like to perform syntactic desugarings like the one
      +above before the actual semantics. Unfortunately, that was not possible in
      +this case because the dimension expressions of the multi-dimensional array need
      +to be evaluated first. Indeed, the desugaring rule above does not work if the
      +dimensions of the declared array are arbitrary expressions, because they can
      +have side effects (e.g., a[++x,++x]) and those side effects would be
      +propagated each time the expression is evaluated in the desugaring code (note
      +that both the loop condition and the nested multi-dimensional declaration
      +would need to evaluate the expressions given as array dimensions).

      +

      Function declaration

      + +

      Functions are evaluated to λ-abstractions and stored like any other
      +values in the store. A binding is added into the environment for the function
      +name to the location holding its body. Similarly to the C language, SIMPLE
      +only allows function declarations at the top level of the program. More
      +precisely, the subsequent semantics of SIMPLE only works well when one
      +respects this requirement. Indeed, the simplistic context-free parser
      +generated by the grammar above is more generous than we may want, in that it
      +allows function declarations anywhere any declaration is allowed, including
      +inside arbitrary blocks. However, as the rule below shows, we are not
      +storing the declaration environment with the λ-abstraction value as
      +closures do. Instead, as seen shortly, we switch to the global environment
      +whenever functions are invoked, which is consistent with our requirement that
      +functions should only be declared at the top. Thus, if one declares local
      +functions, then one may see unexpected behaviors (e.g., when one shadows a
      +global variable before declaring a local function). The type checker of
      +SIMPLE, also defined in K (see examples/simple/typed/static),
      +discards programs which do not respect this requirement.

      +
        rule <k> function F(Xs) S => . ...</k>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> lambda(Xs, S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      When we are done with the first pass (pre-processing), the computation
      +cell k contains only the token execute (see the configuration
      +declaration above, where the computation item execute was placed
      +right after the program in the k cell of the initial configuration)
      +and the cell genv is empty. In this case, we have to call
      +main() and to initialize the global environment by transferring the
      +contents of the local environment into it. We prefer to do it this way, as
      +opposed to processing all the top level declarations directly within the global
      +environment, because we want to avoid duplication of semantics: the syntax of
      +the global declarations is identical to that of their corresponding local
      +declarations, so the semantics of the latter suffices provided that we copy
      +the local environment into the global one once we are done with the
      +pre-processing. We want this separate pre-processing step precisely because
      +we want to create the global environment. All (top-level) functions end up
      +having their names bound in the global environment and, as seen below, they
      +are executed in that same global environment; all these mean, in particular,
      +that the functions "see" each other, allowing for mutual recursion, etc.

      +
        syntax KItem ::= "execute"
+  rule <k> execute => main(.Exps); </k>
+       <env> Env </env>
+       <genv> .Map => Env </genv>
+

      Expressions

      + +

      We next define the K semantics of all the expression constructs.

      +

      Variable lookup

      + +

      When a variable X is the first computational task, and X is bound to some
      +location L in the environment, and L is mapped to some value V in the
      +store, then we rewrite X into V:

      +
        rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Note that the rule above excludes reading ⊥, because ⊥ is not
      +a value and V is checked at runtime to be a value.

      +

      Variable/Array increment

      + +

      This is tricky, because we want to allow both ++x and ++a[5].
      +Therefore, we need to extract the lvalue of the expression to increment.
      +To do that, we state that the expression to increment should be wrapped
      +by the auxiliary lvalue operation and then evaluated. The semantics
      +of this auxiliary operation is defined at the end of this file. For now, all
      +we need to know is that it takes an expression and evaluates to a location
      +value. Location values, also defined at the end of the file, are integers
      +wrapped with the operation loc, to distinguish them from ordinary
      +integers.

      +
        context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I => I +Int 1) ...</store>  [group(increment)]
+

      Arithmetic operators

      + +

      There is nothing special about the following rules. They rewrite the
      +language constructs to their library counterparts when their arguments
      +become values of expected sorts:

      +
        rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 requires I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 requires I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+

      The equality and inequality constructs reduce to syntactic comparison
      +of the two argument values (which is what the equality on K terms does).

      +
        rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+

      The logical negation is clear, but the logical conjunction and disjunction
      +are short-circuited:

      +
        rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+

      Array lookup

      + +

      Untyped SIMPLE does not check array bounds (the dynamically typed version of
      +it, in examples/simple/typed/dynamic, does check for array out of
      +bounds). The first rule below desugars the multi-dimensional array access to
      +uni-dimensional array access; recall that the array access operation was
      +declared strict, so all sub-expressions involved are already values at this
      +stage. The second rule rewrites the array access to a lookup operation at a
      +precise location; we prefer to do it this way to avoid locking the store.
      +The semantics of the auxiliary lookup operation is straightforward,
      +and is defined at the end of the file.

      +
      // The [anywhere] feature is underused, because it would only be used
+// at the top of the computation or inside the lvalue wrapper. So it
+// may not be worth, or we may need to come up with a special notation
+// allowing us to enumerate contexts for [anywhere] rules.
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+

      Size of an array

      + +

      The size of the array is stored in the array reference value, and the
      +sizeOf construct was declared strict, so:

      +
        rule sizeOf(array(_,N)) => N
+

      Function call

      + +

      Function application was strict in both its arguments, so we can
      +assume that both the function and its arguments are evaluated to
      +values (the former expected to be a λ-abstraction). The first
      +rule below matches a well-formed function application on top of the
      +computation and performs the following steps atomically: it switches
      +to the function body followed by return; (for the case in
      +which the function does not use an explicit return statement); it
      +pushes the remaining computation, the current environment, and the
      +current control data onto the function stack (the remaining
      +computation can thus also be discarded from the computation cell,
      +because an unavoidable subsequent return statement ‒see
      +above‒ will always recover it from the stack); it switches the
      +current environment (which is being pushed on the function stack) to
      +the global environment, which is where the free variables in the
      +function body should be looked up; it binds the formal parameters to
      +fresh locations in the new environment, and stores the actual
      +arguments to those locations in the store (this latter step is easily
      +done by reducing the problem to variable declarations, whose semantics
      +we have already defined; the auxiliary operation mkDecls is
      +defined at the end of the file). The second rule pops the
      +computation, the environment and the control data from the function
      +stack when a return statement is encountered as the next
      +computational task, passing the returned value to the popped
      +computation (the popped computation was the context in which the
      +returning function was called). Note that the pushing/popping of the
      +control data is crucial. Without it, one may have a function that
      +contains an exception block with a return statement inside, which
      +would put the xstack cell in an inconsistent state (since the
      +exception block modifies it, but that modification should be
      +irrelevant once the function returns). We add an artificial
      +nothing value to the language, which is returned by the
      +nulary return; statements.

      +
        syntax KItem ::=  (Map,K,ControlCellFragment)
+
+  rule <k> lambda(Xs,S)(Vs:Vals) ~> K => mkDecls(Xs,Vs) S return; </k>
+       <control>
+         <fstack> .List => ListItem((Env,K,C)) ...</fstack>
+         C
+       </control>
+       <env> Env => GEnv </env>
+       <genv> GEnv </genv>
+
+  rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem((Env,K,C)) => .List ...</fstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+

      Like for division-by-zero, it is left unspecified what happens
      +when the nothing value is used in domain calculations. For
      +example, from the the perspective of the language semantics,
      +7 +Int nothing can evaluate to anything, or
      +may not evaluate at all (be undefined). If one wants to make sure that
      +such artificial values are never misused, then one needs to define a static
      +checker (also using K, like our the type checker in
      +examples/simple/typed/static) and reject programs that do.
      +Note that, unlike the undefined symbol ⊥ which had the sort K
      +instead of Val, we defined nothing to be a value. That
      +is because, as explained above, we do not want the program to get
      +stuck when nothing is returned by a function. Instead, we want the
      +behavior to be unspecified; in particular, if one is careful to never
      +use the returned value in domain computation, like it happens when we
      +call a function for its side effects (e.g., with a statement of the
      +form f(x);), then the program does not get stuck.

      +

      Read

      + +

      The read() expression construct simply evaluates to the next
      +input value, at the same time discarding the input value from the
      +in cell.

      +
        rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+

      Assignment

      + +

      In SIMPLE, like in C, assignments are expression constructs and not statement
      +constructs. To make it a statement all one needs to do is to follow it by a
      +semi-colon ; (see the semantics for expression statements below).
      +Like for the increment, we want to allow assignments not only to variables but
      +also to array elements, e.g., e1[e2] = e3 where e1 evaluates
      +to an array reference, e2 to a natural number, and e3 to any
      +value. Thus, we first compute the lvalue of the left-hand-side expression
      +that appears in an assignment, and then we do the actual assignment to the
      +resulting location:

      +
        context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+

      Statements

      + +

      We next define the K semantics of statements.

      +

      Blocks

      + +

      Empty blocks are simply discarded, as shown in the first rule below.
      +For non-empty blocks, we schedule the enclosed statement but we have to
      +make sure the environment is recovered after the enclosed statement executes.
      +Recall that we allow local variable declarations, whose scope is the block
      +enclosing them. That is the reason for which we have to recover the
      +environment after the block. This allows us to have a very simple semantics
      +for variable declarations, as we did above. One can make the two rules below
      +computational if one wants them to count as computational steps.

      +
        rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+

      The basic definition of environment recovery is straightforward and
      +given in the section on auxiliary constructs at the end of the file.

      +

      There are two common alternatives to the above semantics of blocks.
      +One is to keep track of the variables which are declared in the block and only
      +recover those at the end of the block. This way one does more work for
      +variable declarations but conceptually less work for environment recovery; we
      +say conceptually because it is not clear that it is indeed the case that
      +one does less work when AC matching is involved. The other alternative is to
      +work with a stack of environments instead of a flat environment, and push the
      +current environment when entering a block and pop it when exiting it. This
      +way, one does more work when accessing variables (since one has to search the
      +variable in the environment stack in a top-down manner), but on the other hand
      +uses smaller environments and the definition gets closer to an implementation.
      +Based on experience with dozens of language semantics and other K definitions,
      +we have found that our approach above is the best trade-off between elegance
      +and efficiency (especially since rewrite engines have built-in techniques to
      +lazily copy terms, by need, thus not creating unnecessary copies),
      +so it is the one that we follow in general.

      +

      Sequential composition

      + +

      Sequential composition is desugared into K's builtin sequentialization
      +operation (recall that, like in C, the semi-colon ; is not a
      +statement separator in SIMPLE — it is either a statement terminator or a
      +construct for a statement from an expression). Note that K allows
      +to define the semantics of SIMPLE in such a way that statements eventually
      +dissolve from the top of the computation when they are completed; this is in
      +sharp contrast to (artificially) evaluating them to a special
      +skip statement value and then getting rid of that special value, as
      +it is the case in other semantic approaches (where everything must evaluate
      +to something). This means that once S₁ completes in the rule below, S₂
      +becomes automatically the next computation item without any additional
      +(explicit or implicit) rules.

      +
        rule S1:Stmt S2:Stmt => S1 ~> S2
+

      A subtle aspect of the rule above is that S₁ is declared to have sort
      +Stmts and not Stmt. That is because desugaring macros can indeed
      +produce left associative sequential composition of statements. For example,
      +the code var x=0; x=1; is desugared to
      +(var x; x=0;) x=1;, so although originally the first term of
      +the sequential composition had sort Stmt, after desugaring it became
      +of sort Stmts. Note that the attribute [right] associated
      +to the sequential compositon production is an attribute of the syntax, and not
      +of the semantics: e.g., it tells the parser to parse
      +var x; x=0; x=1; as var x; (x=0; x=1;), but it
      +does not tell the rewrite engine to rewrite (var x; x=0;) x=1; to
      +var x; (x=0; x=1;).

      +

      Expression statements

      + +

      Expression statements are only used for their side effects, so their result
      +value is simply discarded. Common examples of expression statements are ones
      +of the form ++x;, x=e;, e1[e2]=e3;, etc.

      +
        rule _:Val; => .
+

      Conditional

      + +

      Since the conditional was declared with the strict(1) attribute, we
      +can assume that its first argument will eventually be evaluated. The rules
      +below cover the only two possibilities in which the conditional is allowed to
      +proceed (otherwise the rewriting process gets stuck).

      +
        rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+

      While loop

      + +

      The simplest way to give the semantics of the while loop is by unrolling.
      +Note, however, that its unrolling is only allowed when the while loop reaches
      +the top of the computation (to avoid non-termination of unrolling). The
      +simple while loop semantics below works because our while loops in SIMPLE are
      +indeed very basic. If we allowed break/continue of loops then we would need
      +a completely different semantics, which would also involve the control cell.

      +
        rule while (E) S => if (E) {S while(E)S}
+

      Print

      + +

      The print statement was strict, so all its arguments are now
      +evaluated (recall that print is variadic). We append each of
      +its evaluated arguments to the output buffer, and discard the residual
      +print statement with an empty list of arguments.

      +
        rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+

      Exceptions

      + +

      SIMPLE allows parametric exceptions, in that one can throw and catch a
      +particular value. The statement try S₁ catch(X) S₂
      +proceeds with the evaluation of S₁. If S₁ evaluates normally, i.e.,
      +without any exception thrown, then S₂ is discarded and the execution
      +continues normally. If S₁ throws an exception with a statement of the
      +form throw E, then E is first evaluated to some value V
      +(throw was declared to be strict), then V is bound to X, then
      +S₂ is evaluated in the new environment while the reminder of S₁ is
      +discarded, then the environment is recovered and the execution continues
      +normally with the statement following the try S₁ catch(X) S₂ statement.
      +Exceptions can be nested and the statements in the
      +catch part (S₂ in our case) can throw exceptions to the
      +upper level. One should be careful with how one handles the control data
      +structures here, so that the abrupt changes of control due to exception
      +throwing and to function returns interact correctly with each other.
      +For example, we want to allow function calls inside the statement S₁ in
      +a try S₁ catch(X) S₂ block which can throw an exception
      +that is not caught by the function but instead is propagated to the
      +try S₁ catch(X) S₂ block that called the function.
      +Therefore, we have to make sure that the function stack as well as other
      +potential control structures are also properly modified when the exception
      +is thrown to correctly recover the execution context. This can be easily
      +achieved by pushing/popping the entire current control context onto the
      +exception stack. The three rules below modularly do precisely the above.

      +
        syntax KItem ::= (Id,Stmt,K,Map,ControlCellFragment)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem((X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem((X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      The catch statement S₂ needs to be executed in the original environment,
      +but where the thrown value V is bound to the catch variable X. We here
      +chose to rely on two previously defined constructs when giving semantics to
      +the catch part of the statement: (1) the variable declaration with
      +initialization, for binding X to V; and (2) the block construct for
      +preventing X from shadowing variables in the original environment upon the
      +completion of S₂.

      +

      Threads

      + +

      SIMPLE's threads can be created and terminated dynamically, and can
      +synchronize by acquiring and releasing re-entrant locks and by rendezvous.
      +We discuss the seven rules giving the semantics of these operations below.

      +

      Thread creation

      + +

      Threads can be created by any other threads using the spawn S
      +construct. The spawn expression construct evaluates to the unique identifier
      +of the newly created thread and, at the same time, a new thread cell is added
      +into the configuration, initialized with the S statement and sharing the
      +same environment with the parent thread. Note that the newly created
      +thread cell is torn. That means that the remaining cells are added
      +and initialized automatically as described in the definition of SIMPLE's
      +configuration. This is part of K's configuration abstraction mechanism.

      +
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+             ...</thread>)
+

      Thread termination

      + +

      Dually to the above, when a thread terminates its assigned computation (the
      +contents of its k cell) is empty, so the thread can be dissolved.
      +However, since no discipline is imposed on how locks are acquired and released,
      +it can be the case that a terminating thread still holds locks. Those locks
      +must be released, so other threads attempting to acquire them do not deadlock.
      +We achieve that by removing all the locks held by the terminating thread in its
      +holds cell from the set of busy locks in the busy cell
      +(keys(H) returns the domain of the map H as a set, that is, only
      +the locks themselves ignoring their multiplicity). As seen below, a lock is
      +added to the busy cell as soon as it is acquired for the first time
      +by a thread. The unique identifier of the terminated thread is also collected
      +into the terminated cell, so the join construct knows which
      +threads have terminated.

      +
        rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+

      Thread joining

      + +

      Thread joining is now straightforward: all we need to do is to check whether
      +the identifier of the thread to be joined is in the terminated cell.
      +If yes, then the join statement dissolves and the joining thread
      +continues normally; if not, then the joining thread gets stuck.

      +
        rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+

      Acquire lock

      + +

      There are two cases to distinguish when a thread attempts to acquire a lock
      +(in SIMPLE any value can be used as a lock):
      +(1) The thread does not currently have the lock, in which case it has to
      +take it provided that the lock is not already taken by another thread (see
      +the side condition of the first rule).
      +(2) The thread already has the lock, in which case it just increments its
      +counter for the lock (the locks are re-entrant). These two cases are captured
      +by the two rules below:

      +
        rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    requires (notBool(V in Busy))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N => N +Int 1) ...</holds>
+

      Release lock

      + +

      Similarly, there are two corresponding cases to distinguish when a thread
      +releases a lock:
      +(1) The thread holds the lock more than once, in which case all it needs to do
      +is to decrement the lock counter.
      +(2) The thread holds the lock only once, in which case it needs to remove it
      +from its holds cell and also from the the shared busy cell,
      +so other threads can acquire it if they need to.

      +
        rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N -Int 1) ...</holds>
+    requires N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+

      Rendezvous synchronization

      + +

      In addition to synchronization through acquire and release of locks, SIMPLE
      +also provides a construct for rendezvous synchronization. A thread whose next
      +statement to execute is rendezvous(V) gets stuck until another
      +thread reaches an identical statement; when that happens, the two threads
      +drop their rendezvous statements and continue their executions. If three
      +threads happen to have an identical rendezvous statement as their next
      +statement, then precisely two of them will synchronize and the other will
      +remain blocked until another thread reaches a similar rendezvous statement.
      +The rule below is as simple as it can be. Note, however, that, again, it is
      +K's mechanism for configuration abstraction that makes it work as desired:
      +since the only cell which can multiply containing a k cell inside is
      +the thread cell, the only way to concretize the rule below to the
      +actual configuration of SIMPLE is to include each k cell in a
      +thread cell.

      +
        rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Auxiliary declarations and operations

      + +

      In this section we define all the auxiliary constructs used in the
      +above semantics.

      +

      Making declarations

      + +

      The mkDecls auxiliary construct turns a list of identifiers
      +and a list of values in a sequence of corresponding variable
      +declarations.

      +
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+

      Location lookup

      + +

      The operation below is straightforward. Note that we place it in the same
      +lookup group as the variable lookup rule defined above. This way,
      +both rules will be considered transitions when we include the lookup
      +tag in the transition option of kompile.

      +
        syntax Exp ::= lookup(Int)
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Environment recovery

      + +

      We have already discussed the environment recovery auxiliary operation in the
      +IMP++ tutorial:

      +
      // TODO: eliminate the env wrapper, like we did in IMP++
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k> <env> _ => Env </env>
+

      While theoretically sufficient, the basic definition for environment
      +recovery alone is suboptimal. Consider a loop while (E)S,
      +whose semantics (see above) was given by unrolling. S
      +is a block. Then the semantics of blocks above, together with the
      +unrolling semantics of the while loop, will yield a computation
      +structure in the k cell that increasingly grows, adding a new
      +environment recovery task right in front of the already existing sequence of
      +similar environment recovery tasks (this phenomenon is similar to the ``tail
      +recursion'' problem). Of course, when we have a sequence of environment
      +recovery tasks, we only need to keep the last one. The elegant rule below
      +does precisely that, thus avoiding the unnecessary computation explosion
      +problem:

      +
        rule (setEnv(_) => .) ~> setEnv(_)
+

      In fact, the above follows a common convention in K for recovery
      +operations of cell contents: the meaning of a computation task of the form
      +cell(C) that reaches the top of the computation is that the current
      +contents of cell cell is discarded and gets replaced with C. We
      +did not add support for these special computation tasks in our current
      +implementation of K, so we need to define them as above.

      +

      lvalue and loc

      + +

      For convenience in giving the semantics of constructs like the increment and
      +the assignment, that we want to operate the same way on variables and on
      +array elements, we used an auxiliary lvalue(E) construct which was
      +expected to evaluate to the lvalue of the expression E. This is only
      +defined when E has an lvalue, that is, when E is either a variable or
      +evaluates to an array element. lvalue(E) evaluates to a value of
      +the form loc(L), where L is the location where the value of E
      +can be found; for clarity, we use loc to structurally distinguish
      +natural numbers from location values. In giving semantics to lvalue
      +there are two cases to consider. (1) If E is a variable, then all we need
      +to do is to grab its location from the environment. (2) If E is an array
      +element, then we first evaluate the array and its index in order to identify
      +the exact location of the element of concern, and then return that location;
      +the last rule below works because its preceding context declarations ensure
      +that the array and its index are evaluated, and then the rule for array lookup
      +(defined above) rewrites the evaluated array access construct to its
      +corresponding store lookup operation.

      +
      // For parsing reasons, we prefer to allow lvalue to take a K
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+// Local variable
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+// Array element: evaluate the array and its index;
+// then the array lookup rule above applies.
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+// Finally, return the address of the desired object member
+
+  rule lvalue(lookup(L:Int) => loc(L))
+

      Initializing multiple locations

      + +

      The following operation initializes a sequence of locations with the same
      +value:

      +
        syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  requires N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  requires N <=Int M
+

      The semantics of SIMPLE is now complete. Make sure you kompile the
      +definition with the right options in order to generate the desired model.
      +No kompile options are needed if you only only want to execute the definition
      +(and thus get an interpreter), but if you want to search for a different
      +program behaviors then you need to kompile with the transition option
      +including rule groups such as lookup, increment, acquire, etc. See the
      +IMP++ tutorial for what the transition option means how to use it.

      +
      endmodule
+

      Go to Lesson 2, SIMPLE typed static

      +

      SIMPLE — Untyped

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped SIMPLE language.
      +SIMPLE is intended to be a pedagogical and research language that captures
      +the essence of the imperative programming paradigm, extended with several
      +features often encountered in imperative programming languages.
      +A program consists of a set of global variable declarations and
      +function definitions. Like in C, function definitions cannot be
      +nested and each program must have one function called main,
      +which is invoked when the program is executed. To make it more
      +interesting and to highlight some of K's strengths, SIMPLE includes
      +the following features in addition to the conventional imperative
      +expression and statement constructs:

      +
        +
      • +

        Multidimensional arrays and array references. An array evaluates
        +to an array reference, which is a special value holding a location (where
        +the elements of the array start) together with the size of the array;
        +the elements of the array can be array references themselves (particularly
        +when the array is multi-dimensional). Array references are ordinary values,
        +so they can be assigned to variables and passed/received by functions.

        +
        • +
      • +

        Functions and function values. Functions can have zero or
        +more parameters and can return abruptly using a return statement.
        +SIMPLE follows a call-by-value parameter passing style, with static scoping.
        +Function names evaluate to function abstractions, which hereby become ordinary
        +values in the language, same like the array references.

        +
        • +
      • +

        Blocks with locals. SIMPLE variables can be declared
        +anywhere, their scope being from the place where they are declared
        +until the end of the most nested enclosing block.

        +
        • +
      • +

        Input/Output. The expression read() evaluates to the
        +next value in the input buffer, and the statement write(e)
        +evaluates e and outputs its value to the output buffer. The
        +input and output buffers are lists of values.

        +
        • +
      • +

        Exceptions. SIMPLE has parametric exceptions (the value thrown as
        +an exception can be caught and bound).

        +
        • +
      • +

        Concurrency via dynamic thread creation/termination and
        +synchronization. One can spawn a thread to execute any statement.
        +The spawned thread shares with its parent its environment at creation time.
        +Threads can be synchronized via a join command which blocks the current thread
        +until the joined thread completes, via re-entrant locks which can be acquired
        +and released, as well as through rendezvous commands.

        +
        • +
      +

      Like in many other languages, some of SIMPLE's constructs can be
      +desugared into a smaller set of basic constructs. We do that at the end
      +of the syntax module, and then we only give semantics to the core constructs.

      +

      Note: This definition is commented slightly more than others, because it is
      +intended to be one of the first non-trivial definitions that the new
      +user of K sees. We recommend the beginner user to first check the
      +language definitions discussed in the K tutorial.

      +
      module SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      We start by defining the SIMPLE syntax. The language constructs discussed
      +above have the expected syntax and evaluation strategies. Recall that in K
      +we annotate the syntax with appropriate strictness attributes, thus giving
      +each language construct the desired evaluation strategy.

      +

      Identifiers

      + +

      Recall from the K tutorial that identifiers are builtin and come under the
      +syntactic category Id. The special identifier for the function
      +main belongs to all programs, and plays a special role in the semantics,
      +so we declare it explicitly. This would not be necessary if the identifiers
      +were all included automatically in semantic definitions, but that is not
      +possible because of parsing reasons (e.g., K variables used to match
      +concrete identifiers would then be ambiguously parsed as identifiers). They
      +are only included in the parser generated to parse programs (and used by the
      +kast tool). Consequently, we have to explicitly declare all the
      +concrete identifiers that play a special role in the semantics, like
      +main below.

      +
        syntax Id ::= "main" [token]
+

      Declarations

      + +

      There are two types of declarations: for variables (including arrays) and
      +for functions. We are going to allow declarations of the form
      +var x=10, a[10,10], y=23;, which is why we allow the var
      +keyword to take a list of expressions. The non-terminals used in the two
      +productions below are defined shortly.

      +
        syntax Stmt ::= "var" Exps ";"
+                | "function" Id "(" Ids ")" Block
+

      Expressions

      + +

      The expression constructs below are standard. Increment (++) takes
      +an expression rather than a variable because it can also increment an array
      +element. Recall that the syntax we define in K is what we call the syntax
      +of the semantics      : while powerful enough to define non-trivial syntaxes
      +(thanks to the underlying SDF technology that we use), we typically refrain
      +from defining precise syntaxes, that is, ones which accept precisely the
      +well-formed programs (that would not be possible anyway in general). That job
      +is deferred to type systems, which can also be defined in K. In other words,
      +we are not making any effort to guarantee syntactically that only variables
      +or array elements are passed to the increment construct, we allow any
      +expression. Nevertheless, we will only give semantics to those, so expressions
      +of the form ++5, which parse (but which will be rejected by our type
      +system in the typed version of SIMPLE later), will get stuck when executed.
      +Arrays can be multidimensional and can hold other arrays, so their
      +lookup operation takes a list of expressions as argument and applies to an
      +expression (which can in particular be another array lookup), respectively.
      +The construct sizeOf gives the size of an array in number of elements
      +of its first dimension. Note that almost all constructs are strict. The only
      +constructs which are not strict are the increment (since its first argument
      +gets updated, so it cannot be evaluated), the input read which takes no
      +arguments so strictness is irrelevant for it, the logical and and or constructs
      +which are short-circuited, the thread spawning construct which creates a new
      +thread executing the argument expression and return its unique identifier to
      +the creating thread (so it cannot just evaluate its argument in place), and the
      +assignment which is only strict in its second argument (for the same reason as
      +the increment).

      +
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+

      We also need comma-separated lists of identifiers and of expressions.
      +Moreover, we want them to be strict, that is, to evaluate to lists of results
      +whenever requested (e.g., when they appear as strict arguments of
      +the constructs above).

      +
        syntax Ids  ::= List{Id,","}           [klabel(Exps)]
+  syntax Exps ::= List{Exp,","}          [klabel(Exps), strict]  // automatically hybrid now
+  syntax Exps ::= Ids
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(Exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}    [klabel(Exps)]
+  syntax Ids ::= Bottoms
+

      Statements

      + +

      Most of the statement constructs are standard for imperative languages.
      +We syntactically distinguish between empty and non-empty blocks, because we
      +chose Stmts not to be a (;-separated) list of
      +Stmt. Variables can be declared anywhere inside a block, their scope
      +ending with the block. Expressions are allowed to be used for their side
      +effects only (followed by a semicolon ;). Functions are allowed
      +to abruptly return. The exceptions are parametric, i.e., one can throw a value
      +which is bound to the variable declared by catch. Threads can be
      +dynamically created and terminated, and can synchronize with join,
      +acquire, release and rendezvous. Note that the
      +strictness attributes obey the intended evaluation strategy of the various
      +constructs. In particular, the if-then-else construct is strict only in its
      +first argument (the if-then construct will be desugared into if-then-else),
      +while the loop constructs are not strict in any arguments. The print
      +statement construct is variadic, that is, it takes an arbitrary number of
      +arguments.

      +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+// NOTE: print strict allows non-deterministic evaluation of its arguments
+// Either keep like this but document, or otherwise make Exps seqstrict.
+// Of define and use a different expression list here, which is seqstrict.
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+

      The reason we allow Stmts as the first argument of for
      +instead of Stmt is because we want to allow more than one statement
      +to be executed when the loop is initialized. Also, as seens shorly, macros
      +may expand one statement into more statements; for example, an initialized
      +variable declaration statement var x=0; desugars into two statements,
      +namely var x; x=0;, so if we use Stmt instead of Stmts
      +in the production of for above then we risk that the macro expansion
      +of statement var x=0; happens before the macro expansion of for,
      +also shown below, in which case the latter would not apply anymore because
      +of syntactic mismatch.

      +
        syntax Stmt ::= Stmt Stmt                          [right]
+
+// I wish I were able to write the following instead, but confuses the parser.
+//
+// syntax Stmts ::= List{Stmt,""}
+// syntax Top ::= Stmt | "function" Id "(" Ids ")" Block
+// syntax Pgm ::= List{Top,""}
+//
+// With that, I could have also eliminated the empty block
+

      Desugared Syntax

      + +

      This part desugars some of SIMPLE's language constructs into core ones.
      +We only want to give semantics to core constructs, so we get rid of the
      +derived ones before we start the semantics. All desugaring macros below are
      +straightforward.

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule for(Start Cond; Step) {} => {Start while (Cond) {Step;}}
+  rule var E1:Exp, E2:Exp, Es:Exps; => var E1; var E2, Es;
+  rule var X:Id = E; => var X; X = E;
+

      For the semantics, we can therefore assume from now on that each
      +conditional has both branches, that there are only while loops, and
      +that each variable is declared alone and without any initialization as part of
      +the declaration.

      +
      endmodule
+
+
+module SIMPLE-UNTYPED
+  imports SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS
+

      Basic Semantic Infrastructure

      + +

      Before one starts adding semantic rules to a K definition, one needs to
      +define the basic semantic infrastructure consisting of definitions for
      +values and configuration. As discussed in the definitions
      +in the K tutorial, the values are needed to know when to stop applying
      +the heating rules and when to start applying the cooling rules corresponding
      +to strictness or context declarations. The configuration serves as a backbone
      +for the process of configuration abstraction which allows users to only
      +mention the relevant cells in each semantic rule, the rest of the configuration
      +context being inferred automatically. Although in some cases the configuration
      +could be automatically inferred from the rules, we believe that it is very
      +useful for language designers/semanticists to actually think of and design
      +their configuration explicitly, so the current implementation of K requires
      +one to define it.

      +

      Values

      + +

      We here define the values of the language that the various fragments of
      +programs evaluate to. First, integers and Booleans are values. As discussed,
      +arrays evaluate to special array reference values holding (1) a location from
      +where the array's elements are contiguously allocated in the store, and
      +(2) the size of the array. Functions evaluate to function values as
      +λ-abstractions (we do not need to evaluate functions to closures
      +because each function is executed in the fixed global environment and
      +function definitions cannot be nested). Like in IMP and other
      +languages, we finally tell the tool that values are K results.

      +
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+               | lambda(Ids,Stmt)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+                   | Vals  // TODO: should not need this
+

      The inclusion of values in expressions follows the methodology of
      +syntactic definitions (like, e.g., in SOS): extend the syntax of the language
      +to encompass all values and additional constructs needed to give semantics.
      +In addition to that, it allows us to write the semantic rules using the
      +original syntax of the language, and to parse them with the same (now extended
      +with additional values) parser. If writing the semantics directly on the K
      +AST, using the associated labels instead of the syntactic constructs, then one
      +would not need to include values in expressions.

      +

      Configuration

      + +

      The K configuration of SIMPLE consists of a top level cell, T,
      +holding a threads cell, a global environment map cell genv
      +mapping the global variables and function names to their locations, a shared
      +store map cell store mapping each location to some value, a set cell
      +busy holding the locks which have been acquired but not yet released
      +by threads, a set cell terminated holding the unique identifiers of
      +the threads which already terminated (needed for join), input
      +and output list cells, and a nextLoc cell holding a natural
      +number indicating the next available location. Unlike in the small languages
      +in the K tutorial, where we used the fresh predicate to generate fresh
      +locations, in larger languages, like SIMPLE, we prefer to explicitly manage
      +memory. The location counter in nextLoc models an actual physical
      +location in the store; for simplicity, we assume arbitrarily large memory and
      +no garbage collection. The threads cell contains one thread
      +cell for each existing thread in the program. Note that the thread cell has
      +multiplicity *, which means that at any given moment there could be zero,
      +one or more thread cells. Each thread cell contains a
      +computation cell k, a control cell holding the various
      +control structures needed to jump to certain points of interest in the program
      +execution, a local environment map cell env mapping the thread local
      +variables to locations in the store, and finally a holds map cell
      +indicating what locks have been acquired by the thread and not released so far
      +and how many times (SIMPLE's locks are re-entrant). The control cell
      +currently contains only two subcells, a function stack fstack which
      +is a list and an exception stack xstack which is also a list.
      +One can add more control structures in the control cell, such as a
      +stack for break/continue of loops, etc., if the language is extended with more
      +control-changing constructs. Note that all cells except for k are
      +also initialized, in that they contain a ground term of their corresponding
      +sort. The k cell is initialized with the program that will be passed
      +to the K tool, as indicated by the $PGM variable, followed by the
      +execute task (defined shortly).

      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+
+  syntax ControlCell
+  syntax ControlCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Map" color="yellow">
+                      <id color="pink"> -1 </id>
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <genv color="pink"> .Map </genv>
+                  <store color="white"> .Map </store>
+                  <busy color="cyan"> .Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                </T>
+

      Declarations and Initialization

      + +

      We start by defining the semantics of declarations (for variables,
      +arrays and functions).

      +

      Variable Declaration

      + +

      The SIMPLE syntax was desugared above so that each variable is
      +declared alone and its initialization is done as a separate statement.
      +The semantic rule below matches resulting variable declarations of the
      +form var X; on top of the k cell
      +(indeed, note that the k cell is complete, or round, to the
      +left, and is torn, or ruptured, to the right), allocates a fresh
      +location L in the store which is initialized with a special value
      +⊥ (indeed, the unit ., or nothing, is matched anywhere
      +in the map ‒note the tears at both sides‒ and replaced with the
      +mapping L ↦ ⊥), and binds X to L in the local
      +environment shadowing previous declarations of X, if any.
      +This possible shadowing of X requires us to therefore update the
      +entire environment map, which is expensive and can significantly slow
      +down the execution of larger programs. On the other hand, since we know
      +that L is not already bound in the store, we simply add the binding
      +L ↦ ⊥ to the store, thus avoiding a potentially complete
      +traversal of the the store map in order to update it. We prefer the approach
      +used for updating the store whenever possible, because, in addition to being
      +faster, it offers more true concurrency than the latter; indeed, according
      +to the concurrent semantics of K, the store is not frozen while
      +L ↦ ⊥ is added to it, while the environment is frozen during the
      +update operation Env[L/X]. The variable declaration command is
      +also removed from the top of the computation cell and the fresh location
      +counter is incremented. The undefined symbol ⊥ added in the store
      +is of sort KItem, instead of Val, on purpose; this way, the
      +store lookup rules will get stuck when one attempts to lookup an
      +uninitialized location. All the above happen in one transactional step,
      +with the rule below. Note also how configuration abstraction allows us to
      +only mention the needed cells; indeed, as the configuration above states,
      +the k and env cells are actually located within a
      +thread cell within the threads cell, but one needs
      +not mention these: the configuration context of the rule is
      +automatically transformed to match the declared configuration
      +structure.

      +
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      Array Declaration

      + +

      The K semantics of the uni-dimensional array declaration is somehow similar
      +to the above declaration of ordinary variables. First, note the
      +context declaration below, which requests the evaluation of the array
      +dimension. Once evaluated, say to a natural number N, then
      +N +Int 1 locations are allocated in the store for
      +an array of size N, the additional location (chosen to be the first
      +one allocated) holding the array reference value. The array reference
      +value array(L,N) states that the array has size N and its
      +elements are located contiguously in the store starting with location
      +L. The operation L … L' ↦ V, defined at the end of this
      +file in the auxiliary operation section, initializes each location in
      +the list L … L' to V. Note that, since the dimensions of
      +array declarations can be arbitrary expressions, this virtually means
      +that we can dynamically allocate memory in SIMPLE by means of array
      +declarations.

      +
        context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 +Int N </nextLoc>
+    requires N >=Int 0
+

      SIMPLE allows multi-dimensional arrays. For semantic simplicity, we
      +desugar them all into uni-dimensional arrays by code transformation.
      +This way, we only need to give semantics to uni-dimensional arrays.
      +First, note that the context rule above actually evaluates all the array
      +dimensions (that's why we defined the expression lists strict!):
      +Upon evaluating the array dimensions, the code generation rule below
      +desugars multi-dimensional array declaration to uni-dimensional declarations.
      +To this aim, we introduce two special unique variable identifiers,
      +$1 and $2. The first variable, $1, iterates
      +through and initializes each element of the first dimension with an array
      +of the remaining dimensions, declared as variable $2:

      +
        syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         for(var $1 = 0; $1 <= N1 - 1; ++$1) {
+           var $2[N2, Vs];
+           X[$1] = $2;
+         }
+       }
+

      Ideally, one would like to perform syntactic desugarings like the one
      +above before the actual semantics. Unfortunately, that was not possible in
      +this case because the dimension expressions of the multi-dimensional array need
      +to be evaluated first. Indeed, the desugaring rule above does not work if the
      +dimensions of the declared array are arbitrary expressions, because they can
      +have side effects (e.g., a[++x,++x]) and those side effects would be
      +propagated each time the expression is evaluated in the desugaring code (note
      +that both the loop condition and the nested multi-dimensional declaration
      +would need to evaluate the expressions given as array dimensions).

      +

      Function declaration

      + +

      Functions are evaluated to λ-abstractions and stored like any other
      +values in the store. A binding is added into the environment for the function
      +name to the location holding its body. Similarly to the C language, SIMPLE
      +only allows function declarations at the top level of the program. More
      +precisely, the subsequent semantics of SIMPLE only works well when one
      +respects this requirement. Indeed, the simplistic context-free parser
      +generated by the grammar above is more generous than we may want, in that it
      +allows function declarations anywhere any declaration is allowed, including
      +inside arbitrary blocks. However, as the rule below shows, we are not
      +storing the declaration environment with the λ-abstraction value as
      +closures do. Instead, as seen shortly, we switch to the global environment
      +whenever functions are invoked, which is consistent with our requirement that
      +functions should only be declared at the top. Thus, if one declares local
      +functions, then one may see unexpected behaviors (e.g., when one shadows a
      +global variable before declaring a local function). The type checker of
      +SIMPLE, also defined in K (see examples/simple/typed/static),
      +discards programs which do not respect this requirement.

      +
        rule <k> function F(Xs) S => . ...</k>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> lambda(Xs, S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      When we are done with the first pass (pre-processing), the computation
      +cell k contains only the token execute (see the configuration
      +declaration above, where the computation item execute was placed
      +right after the program in the k cell of the initial configuration)
      +and the cell genv is empty. In this case, we have to call
      +main() and to initialize the global environment by transferring the
      +contents of the local environment into it. We prefer to do it this way, as
      +opposed to processing all the top level declarations directly within the global
      +environment, because we want to avoid duplication of semantics: the syntax of
      +the global declarations is identical to that of their corresponding local
      +declarations, so the semantics of the latter suffices provided that we copy
      +the local environment into the global one once we are done with the
      +pre-processing. We want this separate pre-processing step precisely because
      +we want to create the global environment. All (top-level) functions end up
      +having their names bound in the global environment and, as seen below, they
      +are executed in that same global environment; all these mean, in particular,
      +that the functions "see" each other, allowing for mutual recursion, etc.

      +
        syntax KItem ::= "execute"
+  rule <k> execute => main(.Exps); </k>
+       <env> Env </env>
+       <genv> .Map => Env </genv>
+

      Expressions

      + +

      We next define the K semantics of all the expression constructs.

      +

      Variable lookup

      + +

      When a variable X is the first computational task, and X is bound to some
      +location L in the environment, and L is mapped to some value V in the
      +store, then we rewrite X into V:

      +
        rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Note that the rule above excludes reading ⊥, because ⊥ is not
      +a value and V is checked at runtime to be a value.

      +

      Variable/Array increment

      + +

      This is tricky, because we want to allow both ++x and ++a[5].
      +Therefore, we need to extract the lvalue of the expression to increment.
      +To do that, we state that the expression to increment should be wrapped
      +by the auxiliary lvalue operation and then evaluated. The semantics
      +of this auxiliary operation is defined at the end of this file. For now, all
      +we need to know is that it takes an expression and evaluates to a location
      +value. Location values, also defined at the end of the file, are integers
      +wrapped with the operation loc, to distinguish them from ordinary
      +integers.

      +
        context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I => I +Int 1) ...</store>  [group(increment)]
+

      Arithmetic operators

      + +

      There is nothing special about the following rules. They rewrite the
      +language constructs to their library counterparts when their arguments
      +become values of expected sorts:

      +
        rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 requires I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 requires I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+

      The equality and inequality constructs reduce to syntactic comparison
      +of the two argument values (which is what the equality on K terms does).

      +
        rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+

      The logical negation is clear, but the logical conjunction and disjunction
      +are short-circuited:

      +
        rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+

      Array lookup

      + +

      Untyped SIMPLE does not check array bounds (the dynamically typed version of
      +it, in examples/simple/typed/dynamic, does check for array out of
      +bounds). The first rule below desugars the multi-dimensional array access to
      +uni-dimensional array access; recall that the array access operation was
      +declared strict, so all sub-expressions involved are already values at this
      +stage. The second rule rewrites the array access to a lookup operation at a
      +precise location; we prefer to do it this way to avoid locking the store.
      +The semantics of the auxiliary lookup operation is straightforward,
      +and is defined at the end of the file.

      +
      // The [anywhere] feature is underused, because it would only be used
+// at the top of the computation or inside the lvalue wrapper. So it
+// may not be worth, or we may need to come up with a special notation
+// allowing us to enumerate contexts for [anywhere] rules.
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+

      Size of an array

      + +

      The size of the array is stored in the array reference value, and the
      +sizeOf construct was declared strict, so:

      +
        rule sizeOf(array(_,N)) => N
+

      Function call

      + +

      Function application was strict in both its arguments, so we can
      +assume that both the function and its arguments are evaluated to
      +values (the former expected to be a λ-abstraction). The first
      +rule below matches a well-formed function application on top of the
      +computation and performs the following steps atomically: it switches
      +to the function body followed by return; (for the case in
      +which the function does not use an explicit return statement); it
      +pushes the remaining computation, the current environment, and the
      +current control data onto the function stack (the remaining
      +computation can thus also be discarded from the computation cell,
      +because an unavoidable subsequent return statement ‒see
      +above‒ will always recover it from the stack); it switches the
      +current environment (which is being pushed on the function stack) to
      +the global environment, which is where the free variables in the
      +function body should be looked up; it binds the formal parameters to
      +fresh locations in the new environment, and stores the actual
      +arguments to those locations in the store (this latter step is easily
      +done by reducing the problem to variable declarations, whose semantics
      +we have already defined; the auxiliary operation mkDecls is
      +defined at the end of the file). The second rule pops the
      +computation, the environment and the control data from the function
      +stack when a return statement is encountered as the next
      +computational task, passing the returned value to the popped
      +computation (the popped computation was the context in which the
      +returning function was called). Note that the pushing/popping of the
      +control data is crucial. Without it, one may have a function that
      +contains an exception block with a return statement inside, which
      +would put the xstack cell in an inconsistent state (since the
      +exception block modifies it, but that modification should be
      +irrelevant once the function returns). We add an artificial
      +nothing value to the language, which is returned by the
      +nulary return; statements.

      +
        syntax KItem ::=  (Map,K,ControlCellFragment)
+
+  rule <k> lambda(Xs,S)(Vs:Vals) ~> K => mkDecls(Xs,Vs) S return; </k>
+       <control>
+         <fstack> .List => ListItem((Env,K,C)) ...</fstack>
+         C
+       </control>
+       <env> Env => GEnv </env>
+       <genv> GEnv </genv>
+
+  rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem((Env,K,C)) => .List ...</fstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+

      Like for division-by-zero, it is left unspecified what happens
      +when the nothing value is used in domain calculations. For
      +example, from the the perspective of the language semantics,
      +7 +Int nothing can evaluate to anything, or
      +may not evaluate at all (be undefined). If one wants to make sure that
      +such artificial values are never misused, then one needs to define a static
      +checker (also using K, like our the type checker in
      +examples/simple/typed/static) and reject programs that do.
      +Note that, unlike the undefined symbol ⊥ which had the sort K
      +instead of Val, we defined nothing to be a value. That
      +is because, as explained above, we do not want the program to get
      +stuck when nothing is returned by a function. Instead, we want the
      +behavior to be unspecified; in particular, if one is careful to never
      +use the returned value in domain computation, like it happens when we
      +call a function for its side effects (e.g., with a statement of the
      +form f(x);), then the program does not get stuck.

      +

      Read

      + +

      The read() expression construct simply evaluates to the next
      +input value, at the same time discarding the input value from the
      +in cell.

      +
        rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+

      Assignment

      + +

      In SIMPLE, like in C, assignments are expression constructs and not statement
      +constructs. To make it a statement all one needs to do is to follow it by a
      +semi-colon ; (see the semantics for expression statements below).
      +Like for the increment, we want to allow assignments not only to variables but
      +also to array elements, e.g., e1[e2] = e3 where e1 evaluates
      +to an array reference, e2 to a natural number, and e3 to any
      +value. Thus, we first compute the lvalue of the left-hand-side expression
      +that appears in an assignment, and then we do the actual assignment to the
      +resulting location:

      +
        context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+

      Statements

      + +

      We next define the K semantics of statements.

      +

      Blocks

      + +

      Empty blocks are simply discarded, as shown in the first rule below.
      +For non-empty blocks, we schedule the enclosed statement but we have to
      +make sure the environment is recovered after the enclosed statement executes.
      +Recall that we allow local variable declarations, whose scope is the block
      +enclosing them. That is the reason for which we have to recover the
      +environment after the block. This allows us to have a very simple semantics
      +for variable declarations, as we did above. One can make the two rules below
      +computational if one wants them to count as computational steps.

      +
        rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+

      The basic definition of environment recovery is straightforward and
      +given in the section on auxiliary constructs at the end of the file.

      +

      There are two common alternatives to the above semantics of blocks.
      +One is to keep track of the variables which are declared in the block and only
      +recover those at the end of the block. This way one does more work for
      +variable declarations but conceptually less work for environment recovery; we
      +say conceptually because it is not clear that it is indeed the case that
      +one does less work when AC matching is involved. The other alternative is to
      +work with a stack of environments instead of a flat environment, and push the
      +current environment when entering a block and pop it when exiting it. This
      +way, one does more work when accessing variables (since one has to search the
      +variable in the environment stack in a top-down manner), but on the other hand
      +uses smaller environments and the definition gets closer to an implementation.
      +Based on experience with dozens of language semantics and other K definitions,
      +we have found that our approach above is the best trade-off between elegance
      +and efficiency (especially since rewrite engines have built-in techniques to
      +lazily copy terms, by need, thus not creating unnecessary copies),
      +so it is the one that we follow in general.

      +

      Sequential composition

      + +

      Sequential composition is desugared into K's builtin sequentialization
      +operation (recall that, like in C, the semi-colon ; is not a
      +statement separator in SIMPLE — it is either a statement terminator or a
      +construct for a statement from an expression). Note that K allows
      +to define the semantics of SIMPLE in such a way that statements eventually
      +dissolve from the top of the computation when they are completed; this is in
      +sharp contrast to (artificially) evaluating them to a special
      +skip statement value and then getting rid of that special value, as
      +it is the case in other semantic approaches (where everything must evaluate
      +to something). This means that once S₁ completes in the rule below, S₂
      +becomes automatically the next computation item without any additional
      +(explicit or implicit) rules.

      +
        rule S1:Stmt S2:Stmt => S1 ~> S2
+

      A subtle aspect of the rule above is that S₁ is declared to have sort
      +Stmts and not Stmt. That is because desugaring macros can indeed
      +produce left associative sequential composition of statements. For example,
      +the code var x=0; x=1; is desugared to
      +(var x; x=0;) x=1;, so although originally the first term of
      +the sequential composition had sort Stmt, after desugaring it became
      +of sort Stmts. Note that the attribute [right] associated
      +to the sequential compositon production is an attribute of the syntax, and not
      +of the semantics: e.g., it tells the parser to parse
      +var x; x=0; x=1; as var x; (x=0; x=1;), but it
      +does not tell the rewrite engine to rewrite (var x; x=0;) x=1; to
      +var x; (x=0; x=1;).

      +

      Expression statements

      + +

      Expression statements are only used for their side effects, so their result
      +value is simply discarded. Common examples of expression statements are ones
      +of the form ++x;, x=e;, e1[e2]=e3;, etc.

      +
        rule _:Val; => .
+

      Conditional

      + +

      Since the conditional was declared with the strict(1) attribute, we
      +can assume that its first argument will eventually be evaluated. The rules
      +below cover the only two possibilities in which the conditional is allowed to
      +proceed (otherwise the rewriting process gets stuck).

      +
        rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+

      While loop

      + +

      The simplest way to give the semantics of the while loop is by unrolling.
      +Note, however, that its unrolling is only allowed when the while loop reaches
      +the top of the computation (to avoid non-termination of unrolling). The
      +simple while loop semantics below works because our while loops in SIMPLE are
      +indeed very basic. If we allowed break/continue of loops then we would need
      +a completely different semantics, which would also involve the control cell.

      +
        rule while (E) S => if (E) {S while(E)S}
+

      Print

      + +

      The print statement was strict, so all its arguments are now
      +evaluated (recall that print is variadic). We append each of
      +its evaluated arguments to the output buffer, and discard the residual
      +print statement with an empty list of arguments.

      +
        rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+

      Exceptions

      + +

      SIMPLE allows parametric exceptions, in that one can throw and catch a
      +particular value. The statement try S₁ catch(X) S₂
      +proceeds with the evaluation of S₁. If S₁ evaluates normally, i.e.,
      +without any exception thrown, then S₂ is discarded and the execution
      +continues normally. If S₁ throws an exception with a statement of the
      +form throw E, then E is first evaluated to some value V
      +(throw was declared to be strict), then V is bound to X, then
      +S₂ is evaluated in the new environment while the reminder of S₁ is
      +discarded, then the environment is recovered and the execution continues
      +normally with the statement following the try S₁ catch(X) S₂ statement.
      +Exceptions can be nested and the statements in the
      +catch part (S₂ in our case) can throw exceptions to the
      +upper level. One should be careful with how one handles the control data
      +structures here, so that the abrupt changes of control due to exception
      +throwing and to function returns interact correctly with each other.
      +For example, we want to allow function calls inside the statement S₁ in
      +a try S₁ catch(X) S₂ block which can throw an exception
      +that is not caught by the function but instead is propagated to the
      +try S₁ catch(X) S₂ block that called the function.
      +Therefore, we have to make sure that the function stack as well as other
      +potential control structures are also properly modified when the exception
      +is thrown to correctly recover the execution context. This can be easily
      +achieved by pushing/popping the entire current control context onto the
      +exception stack. The three rules below modularly do precisely the above.

      +
        syntax KItem ::= (Id,Stmt,K,Map,ControlCellFragment)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem((X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem((X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      The catch statement S₂ needs to be executed in the original environment,
      +but where the thrown value V is bound to the catch variable X. We here
      +chose to rely on two previously defined constructs when giving semantics to
      +the catch part of the statement: (1) the variable declaration with
      +initialization, for binding X to V; and (2) the block construct for
      +preventing X from shadowing variables in the original environment upon the
      +completion of S₂.

      +

      Threads

      + +

      SIMPLE's threads can be created and terminated dynamically, and can
      +synchronize by acquiring and releasing re-entrant locks and by rendezvous.
      +We discuss the seven rules giving the semantics of these operations below.

      +

      Thread creation

      + +

      Threads can be created by any other threads using the spawn S
      +construct. The spawn expression construct evaluates to the unique identifier
      +of the newly created thread and, at the same time, a new thread cell is added
      +into the configuration, initialized with the S statement and sharing the
      +same environment with the parent thread. Note that the newly created
      +thread cell is torn. That means that the remaining cells are added
      +and initialized automatically as described in the definition of SIMPLE's
      +configuration. This is part of K's configuration abstraction mechanism.

      +
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+             ...</thread>)
+

      Thread termination

      + +

      Dually to the above, when a thread terminates its assigned computation (the
      +contents of its k cell) is empty, so the thread can be dissolved.
      +However, since no discipline is imposed on how locks are acquired and released,
      +it can be the case that a terminating thread still holds locks. Those locks
      +must be released, so other threads attempting to acquire them do not deadlock.
      +We achieve that by removing all the locks held by the terminating thread in its
      +holds cell from the set of busy locks in the busy cell
      +(keys(H) returns the domain of the map H as a set, that is, only
      +the locks themselves ignoring their multiplicity). As seen below, a lock is
      +added to the busy cell as soon as it is acquired for the first time
      +by a thread. The unique identifier of the terminated thread is also collected
      +into the terminated cell, so the join construct knows which
      +threads have terminated.

      +
        rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+

      Thread joining

      + +

      Thread joining is now straightforward: all we need to do is to check whether
      +the identifier of the thread to be joined is in the terminated cell.
      +If yes, then the join statement dissolves and the joining thread
      +continues normally; if not, then the joining thread gets stuck.

      +
        rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+

      Acquire lock

      + +

      There are two cases to distinguish when a thread attempts to acquire a lock
      +(in SIMPLE any value can be used as a lock):
      +(1) The thread does not currently have the lock, in which case it has to
      +take it provided that the lock is not already taken by another thread (see
      +the side condition of the first rule).
      +(2) The thread already has the lock, in which case it just increments its
      +counter for the lock (the locks are re-entrant). These two cases are captured
      +by the two rules below:

      +
        rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    requires (notBool(V in Busy))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N => N +Int 1) ...</holds>
+

      Release lock

      + +

      Similarly, there are two corresponding cases to distinguish when a thread
      +releases a lock:
      +(1) The thread holds the lock more than once, in which case all it needs to do
      +is to decrement the lock counter.
      +(2) The thread holds the lock only once, in which case it needs to remove it
      +from its holds cell and also from the the shared busy cell,
      +so other threads can acquire it if they need to.

      +
        rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N -Int 1) ...</holds>
+    requires N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+

      Rendezvous synchronization

      + +

      In addition to synchronization through acquire and release of locks, SIMPLE
      +also provides a construct for rendezvous synchronization. A thread whose next
      +statement to execute is rendezvous(V) gets stuck until another
      +thread reaches an identical statement; when that happens, the two threads
      +drop their rendezvous statements and continue their executions. If three
      +threads happen to have an identical rendezvous statement as their next
      +statement, then precisely two of them will synchronize and the other will
      +remain blocked until another thread reaches a similar rendezvous statement.
      +The rule below is as simple as it can be. Note, however, that, again, it is
      +K's mechanism for configuration abstraction that makes it work as desired:
      +since the only cell which can multiply containing a k cell inside is
      +the thread cell, the only way to concretize the rule below to the
      +actual configuration of SIMPLE is to include each k cell in a
      +thread cell.

      +
        rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Auxiliary declarations and operations

      + +

      In this section we define all the auxiliary constructs used in the
      +above semantics.

      +

      Making declarations

      + +

      The mkDecls auxiliary construct turns a list of identifiers
      +and a list of values in a sequence of corresponding variable
      +declarations.

      +
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+

      Location lookup

      + +

      The operation below is straightforward. Note that we place it in the same
      +lookup group as the variable lookup rule defined above. This way,
      +both rules will be considered transitions when we include the lookup
      +tag in the transition option of kompile.

      +
        syntax Exp ::= lookup(Int)
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Environment recovery

      + +

      We have already discussed the environment recovery auxiliary operation in the
      +IMP++ tutorial:

      +
      // TODO: eliminate the env wrapper, like we did in IMP++
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k> <env> _ => Env </env>
+

      While theoretically sufficient, the basic definition for environment
      +recovery alone is suboptimal. Consider a loop while (E)S,
      +whose semantics (see above) was given by unrolling. S
      +is a block. Then the semantics of blocks above, together with the
      +unrolling semantics of the while loop, will yield a computation
      +structure in the k cell that increasingly grows, adding a new
      +environment recovery task right in front of the already existing sequence of
      +similar environment recovery tasks (this phenomenon is similar to the ``tail
      +recursion'' problem). Of course, when we have a sequence of environment
      +recovery tasks, we only need to keep the last one. The elegant rule below
      +does precisely that, thus avoiding the unnecessary computation explosion
      +problem:

      +
        rule (setEnv(_) => .) ~> setEnv(_)
+

      In fact, the above follows a common convention in K for recovery
      +operations of cell contents: the meaning of a computation task of the form
      +cell(C) that reaches the top of the computation is that the current
      +contents of cell cell is discarded and gets replaced with C. We
      +did not add support for these special computation tasks in our current
      +implementation of K, so we need to define them as above.

      +

      lvalue and loc

      + +

      For convenience in giving the semantics of constructs like the increment and
      +the assignment, that we want to operate the same way on variables and on
      +array elements, we used an auxiliary lvalue(E) construct which was
      +expected to evaluate to the lvalue of the expression E. This is only
      +defined when E has an lvalue, that is, when E is either a variable or
      +evaluates to an array element. lvalue(E) evaluates to a value of
      +the form loc(L), where L is the location where the value of E
      +can be found; for clarity, we use loc to structurally distinguish
      +natural numbers from location values. In giving semantics to lvalue
      +there are two cases to consider. (1) If E is a variable, then all we need
      +to do is to grab its location from the environment. (2) If E is an array
      +element, then we first evaluate the array and its index in order to identify
      +the exact location of the element of concern, and then return that location;
      +the last rule below works because its preceding context declarations ensure
      +that the array and its index are evaluated, and then the rule for array lookup
      +(defined above) rewrites the evaluated array access construct to its
      +corresponding store lookup operation.

      +
      // For parsing reasons, we prefer to allow lvalue to take a K
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+// Local variable
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+// Array element: evaluate the array and its index;
+// then the array lookup rule above applies.
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+// Finally, return the address of the desired object member
+
+  rule lvalue(lookup(L:Int) => loc(L))
+

      Initializing multiple locations

      + +

      The following operation initializes a sequence of locations with the same
      +value:

      +
        syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  requires N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  requires N <=Int M
+

      The semantics of SIMPLE is now complete. Make sure you kompile the
      +definition with the right options in order to generate the desired model.
      +No kompile options are needed if you only only want to execute the definition
      +(and thus get an interpreter), but if you want to search for a different
      +program behaviors then you need to kompile with the transition option
      +including rule groups such as lookup, increment, acquire, etc. See the
      +IMP++ tutorial for what the transition option means how to use it.

      +
      endmodule
+

      Go to Lesson 2, SIMPLE typed static

      +

      SIMPLE — Typed — Static

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K definition of the static semantics of the typed SIMPLE
      +language, or in other words, a type system for the typed SIMPLE
      +language in K. We do not re-discuss the various features of the
      +SIMPLE language here. The reader is referred to the untyped version of
      +the language for such discussions. We here only focus on the new and
      +interesting problems raised by the addition of type declarations, and
      +what it takes to devise a type system/checker for the language.

      +

      When designing a type system for a language, no matter in what
      +paradigm, we have to decide upon the intended typing policy. Note
      +that we can have multiple type systems for the same language, one for
      +each typing policy. For example, should we accept programs which
      +don't have a main function? Or should we allow functions that do not
      +return explicitly? Or should we allow functions whose type expects
      +them to return a value (say an int) to use a plain
      +return; statement, which returns no value, like in C?
      +And so on and so forth. Typically, there are two opposite tensions
      +when designing a type system. On the one hand, you want your type
      +system to be as permissive as possible, that is, to accept as many
      +programs that do not get stuck when executed with the untyped
      +semantics as possible; this will keep the programmers using your
      +language happy. On the other hand, you want your type system to have
      +a reasonable performance when implemented; this will keep both the
      +programmers and the implementers of your language happy. For example,
      +a type system for rejecting programs that could perform
      +division-by-zero is not expected to be feasible in general. A simple
      +guideline when designing typing policies is to imagine how the
      +semantics of the untyped language may get stuck and try to prevent
      +those situations from happening.

      +

      Before we give the K type system of SIMPLE formally, we discuss,
      +informally, the intended typing policy:

      +
        +
      • +

        Each program should contain a main() function. Indeed,
        +the untyped SIMPLE semantics will get stuck on any program which does
        +not have a main function.

        +
        • +
      • +

        Each primitive value has its own type, which can be int
        +bool, or string. There is also a type void
        +for nonexistent values, for example for the result of a function meant
        +to return no value (but only be used for its side effects, like a
        +procedure).

        +
        • +
      • +

        The syntax of untyped SIMPLE is extended to allow type
        +declarations for all the variables, including array variables. This is
        +done in a C/Java-style. For example, int x; or
        +int x=7, y=x+3;, or int[][][] a[10,20];
        +(the latter defines a 10 × 20 matrix of arrays of integers).
        +Recall from untyped SIMPLE that, unlike in C/Java, our multi-dimensional
        +arrays use comma-separated arguments, although they have the array-of-array
        +semantics.

        +
        • +
      • +

        Functions are also typed in a C/Java style. However, since in SIMPLE
        +we allow functions to be passed to and returned by other functions, we also
        +need function types. We will use the conventional higher-order arrow-notation
        +for function types, but will separate the argument types with commas. For
        +example, a function returning an array of bool elements and
        +taking as argument an array x of two-integer-argument functions
        +returning an integer, is declared using a syntax of the form
        +bool[] f(((int,int)->int)[] x) { ... }
        +and has the type ((int,int)->int)[] -> bool[].

        +
        • +
      • +

        We allow any variable declarations at the top level. Functions
        +can only be declared at the top level. Each function can only access the
        +other functions and variables declared at the top level, or its own locally
        +declared variables. SIMPLE has static scoping.

        +
        • +
      • +

        The various expression and statement constructs take only elements of
        +the expected types.

        +
        • +
      • +

        Increment and assignment can operate both on variables and on array
        +elements. For example, if f has type int->int[][] and
        +function g has the type int->int, then the
        +increment expression ++f(7)[g(2),g(3)] is valid.

        +
        • +
      • +

        Functions should only return values of their declared result
        +type. To give the programmers more flexibility, we allow functions to
        +use return; statements to terminate without returning an
        +actual value, or to not explicitly use any return statement,
        +regardless of their declared return type. This flexibility can be
        +handy when writing programs using certain functions only for their
        +side effects. Nevertheless, as the dynamic semantics shows, a return
        +value is automatically generated when an explicit return
        +statement is not encountered.

        +
        • +
      • +

        For simplicity, we here limit exceptions to only throw and catch
        +integer values. We let it as an exercise to the reader to extend the
        +semantics to allow throwing and catching arbitrary-type exceptions.
        +Like in programming languages like Java, one can go even further and
        +define a semantics where thrown exceptions are propagated through
        +try-catch statements until one of the corresponding type is found.
        +We will do this when we define the KOOL language, not here.
        +To keep the definition if SIMPLE simple, here we do not attempt to
        +reject programs which throw uncaught exceptions.

        +
        • +
      +

      Like in untyped SIMPLE, some constructs can be desugared into a
      +smaller set of basic constructs. In general, it should be clear why a
      +program does not type by looking at the top of the k cells in
      +its stuck configuration.

      +
      module SIMPLE-TYPED-STATIC-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of typed SIMPLE extends that of untyped SIMPLE with support
      +for declaring types to variables and functions.

      +
        syntax Id ::= "main" [token]
+

      Types

      + +

      Primitive, array and function types, as well as lists (or tuples) of types.
      +The lists of types are useful for function arguments.

      +
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Type "[" "]"
+                | "(" Type ")"             [bracket]
+                > Types "->" Type
+
+  syntax Types ::= List{Type,","}          [klabel(exps)]
+

      Declarations

      + +

      Variable and function declarations have the expected syntax. For variables,
      +we basically just replaced the var keyword of untyped SIMPLE with a
      +type. For functions, besides replacing the function keyword with a
      +type, we also introduce a new syntactic category for typed variables,
      +Param, and lists over it.

      +
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";"
+                | Type Id "(" Params ")" Block
+

      Expressions

      + +

      The syntax of expressions is identical to that in untyped SIMPLE,
      +except for the logical conjunction and disjunction which have
      +different strictness attributes, because they now have different
      +evaluation strategies.

      +
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict, left]
+               | Exp "||" Exp            [strict, left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+

      Note that spawn has not been declared strict. This may
      +seem unexpected, because the child thread shares the same environment
      +with the parent thread, so from a typing perspective the spawned
      +statement makes the same sense in a child thread as it makes in the
      +parent thread. The reason for not declaring it strict is because we
      +want to disallow programs where the spawned thread calls the
      +return statement, because those programs would get stuck in
      +the dynamic semantics. The type semantics of spawn below will reject
      +such programs.

      +

      We still need lists of expressions, defined below, but note that we do
      +not need lists of identifiers anymore. They have been replaced by the lists
      +of parameters.

      +
        syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+

      Statements

      + +

      The statements have the same syntax as in untyped SIMPLE, except for
      +the exceptions, which now type their parameter. Note that, unlike in untyped
      +SIMPLE, all statement constructs which have arguments and are not desugared
      +are strict, including the conditional and the while. Indeed, from a
      +typing perspective, they are all strict: first type their arguments and then
      +type the actual construct.

      +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                                  [strict]
+                | "if" "(" Exp ")" Block "else" Block      [avoid, strict]
+                | "if" "(" Exp ")" Block                   [macro]
+                | "while" "(" Exp ")" Block                [strict]
+                | "for" "(" Stmt Exp ";" Exp ")" Block     [macro]
+                | "return" Exp ";"                         [strict]
+                | "return" ";"
+                | "print" "(" Exps ")" ";"                 [strict]
+                | "try" Block "catch" "(" Param ")" Block  [strict(1)]
+                | "throw" Exp ";"                          [strict]
+                | "join" Exp ";"                           [strict]
+                | "acquire" Exp ";"                        [strict]
+                | "release" Exp ";"                        [strict]
+                | "rendezvous" Exp ";"                     [strict]
+

      Note that the sequential composition is now sequentially strict,
      +because, unlike in the dynamic semantics where statements dissolved,
      +they now reduce to the stmt type, which is a result.

      +
        syntax Stmt ::= Stmt Stmt                             [seqstrict, right]
+

      Desugaring macros

      + +

      We use the same desugaring macros like in untyped SIMPLE, but, of
      +course, including the types of the involved variables.

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule for(Start Cond; Step) {} => {Start while(Cond){Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+endmodule
+
+
+module SIMPLE-TYPED-STATIC
+  imports SIMPLE-TYPED-STATIC-SYNTAX
+  imports DOMAINS
+

      Static semantics

      + +

      Here we define the type system of SIMPLE. Like concrete semantics,
      +type systems defined in K are also executable. However, K type
      +systems turn into type checkers instead of interpreters when executed.

      +

      The typing process is done in two (overlapping) phases. In the first
      +phase the global environment is built, which contains type bindings
      +for all the globally declared variables and functions. For functions,
      +the declared types will be ``trusted'' during the first phase and
      +simply bound to their corresponding function names and placed in the
      +global type environment. At the same time, type-checking tasks that
      +the function bodies indeed respect their claimed types are generated.
      +All these tasks are (concurrently) verified during the second phase.
      +This way, all the global variable and function declarations are
      +available in the global type environment and can be used in order to
      +type-check each function code. This is consistent with the semantics
      +of untyped SIMPLE, where functions can access all the global variables
      +and can call any other function declared in the same program. The
      +two phases may overlap because of the K concurrent semantics. For
      +example, a function task can be started while the first phase is still
      +running; moreover, it may even complete before the first phase does,
      +namely when all the global variables and functions that it needs have
      +already been processed and made available in the global environment by
      +the first phase task.

      +

      Extended syntax and results

      + +

      The idea is to start with a configuration holding the program to type
      +in one of its cells, then apply rewrite rules on it mixing types and
      +language syntax, and eventually obtain a type instead of the original
      +program. In other words, the program reduces to its type using
      +the K rules giving the type system of the language. In doing so,
      +additional typing tasks for function bodies are generated and solved
      +the same way. If this rewriting process gets stuck, then we say that
      +the program is not well-typed. Otherwise the program is well-typed
      +(by definition). We did not need types for statements and for blocks
      +as part of the typed SIMPLE syntax, because programmers are not allowed
      +to use such types explicitly. However, we are going to need them in the
      +type system, because blocks and statements reduce to them.

      +

      We start by allowing types to be used inside expressions and statements in
      +our language. This way, types can be used together with language syntax in
      +subsequent K rules without any parsing errors. Like in the type system of
      +IMP++ in the K tutorial, we prefer to group the block and statement types
      +under one syntactic sub-category of types, because this allows us to more
      +compactly state that certain terms can be either blocks or statements. Also,
      +since programs and fragments of program will reduce to their types, in order
      +for the strictness and context declarations to be executable we state that
      +types are results (same like we did in the IMP++ tutorial).

      +
        syntax Exp ::= Type
+  syntax Exps ::= Types
+  syntax BlockOrStmtType ::= "block" | "stmt"
+  syntax Type ::= BlockOrStmtType
+  syntax Block ::= BlockOrStmtType
+  syntax KResult ::= Type
+                   | Types    //TODO: remove this, eventually
+

      Configuration

      + +

      The configuration of our type system consists of a tasks cell
      +holding various typing task cells, and a global type environment.
      +Each task includes a k cell holding the code to type, a tenv
      +cell holding the local type environment, and a return cell holding
      +the return type of the currently checked function. The latter is needed in
      +order to check whether return statements return values of the expected type.
      +Initially, the program is placed in a k cell inside a
      +task cell. Since the cells with multiplicity ? are not
      +included in the initial configuration, the task cell holding
      +the original program in its k cell will contain no other
      +subcells.

      +
        configuration <T color="yellow">
+                  <tasks color="orange">
+                    <task multiplicity="*" color="yellow" type="Set">
+                      <k color="green"> $PGM:Stmt </k>
+                      <tenv multiplicity="?" color="cyan"> .Map </tenv>
+                      <returnType multiplicity="?" color="black"> void </returnType>
+                    </task>
+                  </tasks>
+//                  <br/>
+                  <gtenv color="blue"> .Map </gtenv>
+                </T>
+

      Variable declarations

      + +

      Variable declarations type as statements, that is, they reduce to the
      +type stmt. There are only two cases that need to be
      +considered: when a simple variable is declared and when an array
      +variable is declared. The macros at the end of the syntax module
      +above take care of reducing other variable declarations, including
      +ones where the declared variables are initialized, to only these two
      +cases. The first case has two subcases: when the variable declaration
      +is global (i.e., the task cell contains only the k
      +cell), in which case it is added to the global type environment
      +checking at the same time that the variable has not been already
      +declared; and when the variable declaration is local (i.e., a
      +tenv cell is available), in which case it is simply added to
      +the local type environment, possibly shadowing previous homonymous
      +variables. The third case reduces to the second, incrementally moving
      +the array dimension into the type until the array becomes a simple
      +variable.

      +
        rule <task> <k> T:Type X:Id; => stmt ...</k> </task>
+       <gtenv> Rho (.Map => X |-> T) </gtenv>
+    requires notBool(X in keys(Rho))
+  rule <k> T:Type X:Id; => stmt ...</k> <tenv> Rho => Rho[X <- T] </tenv>
+
+  context _:Type _::Exp[HOLE::Exps];
+// The rule below may need to sort E to Exp in the future, if the
+// parser gets stricter; without that information, it may not be able
+// to complete the LHS into T E[int,Ts],.Exps; (and similarly for the RHS)
+  rule T:Type E:Exp[int,Ts:Types]; => T[] E[Ts];
+// I want to write the rule below as _:Type (E:Exp[.Types] => E),
+// but the list completion seems to not work well with that.
+  rule T:Type E:Exp[.Types]; => T E;
+

      Function declarations

      + +

      Functions are allowed to be declared only at the top level (the
      +task cell holds only its k subcell). Each function
      +declaration reduces to a variable declaration (a binding of its name
      +to its declared function type), but also adds a task into the
      +tasks cell. The task consists of a typing of the statement
      +declaring all the function parameters followed by the function body,
      +together with the expected return type of the function. The
      +getTypes and mkDecls functions, defined at the end of
      +the file in the section on auxiliary operations, extracts the list of
      +types and makes a sequence of variable declarations from a list of
      +function parameters, respectively. Note that, although in the dynamic
      +semantics we include a terminating return statement at the
      +end of the function body to eliminate from the analysis the case when
      +the function does not provide an explicit return, we do not need to
      +include such a similar return statement here. That's because
      +the return statements type to stmt anyway, and the
      +entire code of the function body needs to type anyway.

      +
        rule <task> <k> T:Type F:Id(Ps:Params) S => getTypes(Ps)->T F; ...</k> </task>
+       (.Bag => <task>
+               <k> mkDecls(Ps) S </k> <tenv> .Map </tenv> <returnType> T </returnType>
+             </task>)
+

      Checking if main() exists}

      + +

      Once the entire program is processed (generating appropriate tasks
      +to type check its function bodies), we can dissolve the main
      +task cell (the one holding only a k subcell). Since
      +we want to enforce that programs include a main function, we also
      +generate a function task executing main() to ensure that it
      +types (remove this task creation if you do not want your type system
      +to reject programs without a main function).

      +
        rule <task> <k> stmt => main(.Exps); </k> (.Bag => <tenv> .Map </tenv>) </task>
+

      Collecting the terminated tasks

      + +

      Similarly, once a non-main task (i.e., one which contains a
      +tenv subcells) is completed using the subsequent rules (i.e.,
      +its k cell holds only the block or stmt
      +type), we can dissolve its corresponding cell. Note that it is
      +important to ensure that we only dissolve tasks containing a
      +tenv cell with the rule below, because the main task should
      +not dissolve this way! It should do what the above rule says.
      +In the end, there should be no task cell left in the configuration
      +when the program correctly type checks.

      +
        rule <task>... <k> _:BlockOrStmtType </k> <tenv> _ </tenv> ...</task> => .Bag
+

      Basic values

      + +

      The first three rewrite rules below reduce the primitive values to
      +their types, as we typically do when we define type systems in K.

      +
        rule _:Int => int
+  rule _:Bool => bool
+  rule _:String => string
+

      Variable lookup

      + +

      There are three cases to distinguish for variable lookup: (1) if the
      +variable is bound in the local type environment, then look its type up
      +there; (2) if a local environment exists and the variable is not bound
      +in it, then look its type up in the global environment; (3) finally,
      +if there is no local environment, meaning that we are executing the
      +top-level pass, then look the variable's type up in the global
      +environment, too.

      +
        rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</tenv>
+
+  rule <k> X:Id => T ...</k> <tenv> Rho </tenv> <gtenv>... X |-> T ...</gtenv>
+    requires notBool(X in keys(Rho))
+
+  rule <task> <k> X:Id => T ...</k> </task> <gtenv>... X |-> T ...</gtenv>
+

      Increment

      + +

      We want the increment operation to apply to any lvalue, including
      +array elements, not only to variables. For that reason, we define a
      +special context extracting the type of the argument of the increment
      +operation only if that argument is an lvalue. Otherwise the rewriting
      +process gets stuck. The operation ltype is defined at the
      +end of this file, in the auxiliary operation section. It essentially
      +acts as a filter, getting stuck if its argument is not an lvalue and
      +letting it reduce otherwise. The type of the lvalue is expected to be
      +an integer in order to be allowed to be incremented, as seen in the
      +rule ++ int => int below.

      +
        context ++(HOLE => ltype(HOLE))
+  rule ++ int => int
+

      Common expression constructs

      + +

      The rules below are straightforward and self-explanatory:

      +
        rule int + int => int
+  rule string + string => string
+  rule int - int => int
+  rule int * int => int
+  rule int / int => int
+  rule int % int => int
+  rule - int => int
+  rule int < int => bool
+  rule int <= int => bool
+  rule int > int => bool
+  rule int >= int => bool
+  rule T:Type == T => bool
+  rule T:Type != T => bool
+  rule bool && bool => bool
+  rule bool || bool => bool
+  rule ! bool => bool
+

      Array access and size

      + +

      Array access requires each index to type to an integer, and the
      +array type to be at least as deep as the number of indexes:

      +
      // NOTE:
+// We used to need parentheses in the RHS, to avoid capturing Ts as an attribute
+// Let's hope that is not a problem anymore.
+
+  rule (T[])[int, Ts:Types] => T[Ts]
+  rule T:Type[.Types] => T
+

      sizeOf only needs to check that its argument is an array:

      +
        rule sizeOf(_T[]) => int
+

      Input/Output

      + +

      The read expression construct types to an integer, while print types
      +to a statement provided that all its arguments type to integers or
      +strings.

      +
        rule read() => int
+
+  rule print(T:Type, Ts => Ts); requires T ==K int orBool T ==K string
+  rule print(.Types); => stmt
+

      Assignment

      + +

      The special context and the rule for assignment below are similar
      +to those for increment: the LHS of the assignment must be an lvalue
      +and, in that case, it must have the same type as the RHS, which then
      +becomes the type of the assignment.

      +
        context (HOLE => ltype(HOLE)) = _
+  rule T:Type = T => T
+

      Function application and return

      + +

      Function application requires the type of the function and the
      +types of the passed values to be compatible. Note that a special case
      +is needed to handle the no-argument case:

      +
        rule (Ts:Types -> T)(Ts) => T requires Ts =/=K .Types
+  rule (void -> T)(.Types) => T
+

      The returned value must have the same type as the declared
      +function return type. If an empty return is encountered, than
      +we should check that we are in a function (and not a thread)
      +context, that is, a return cell must be available:

      +
        rule <k> return T:Type; => stmt ...</k> <returnType> T </returnType>
+  rule <k> return; => stmt ...</k> <returnType> _ </returnType>
+

      Blocks

      + +

      To avoid having to recover type environments after blocks, we prefer
      +to start a new task for block body, making sure that the new task
      +is passed the same type environment and return cells. The value
      +returned by return statements must have the same type as
      +stated in the return cell. The print variadic
      +function is allowed to only print integers and strings. The thrown
      +exceptions can only have integer type.

      +
        rule {} => block
+
+  rule <task> <k> {S} => block ...</k> <tenv> Rho </tenv> R </task>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> R </task>)
+

      Expression statement

      + +
        rule _:Type; => stmt
+

      Conditional and while loop

      + +
        rule if (bool) block else block => stmt
+  rule while (bool) block => stmt
+

      Exceptions

      + +

      We currently force the parameters of exceptions to only be integers.
      +Moreover, for simplicity, we assume that integer exceptions can be
      +thrown from anywhere, including from functions which do not define
      +any try-catch block (with the currently unchecked ‒also for
      +simplicity‒ expectation that the caller functions would catch those
      +exceptions).

      +
        rule try block catch(int X:Id) {S} => {int X; S}
+  rule try block catch(int X:Id) {} => {int X;}
+  rule throw int; => stmt
+

      Concurrency

      + +

      Nothing special about typing the concurrency constructs, except that
      +we do not want the spawned thread to return, so we do not include any
      +return cell in the new task cell for the thread statement.
      +Same like with the functions above, we do not check for thrown
      +exceptions which are not caught.

      +
        rule <k> spawn S => int ...</k> <tenv> Rho </tenv>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> </task>)
+  rule join int; => stmt
+  rule acquire _:Type; => stmt
+  rule release _:Type; => stmt
+  rule rendezvous _:Type; => stmt
+
+  rule _:BlockOrStmtType _:BlockOrStmtType => stmt
+

      Auxiliary constructs

      + +

      The function mkDecls turns a list of parameters into a
      +list of variable declarations.

      +
        syntax Stmt ::= mkDecls(Params)  [function]
+  rule mkDecls(T:Type X:Id, Ps:Params) => T X; mkDecls(Ps)
+  rule mkDecls(.Params) => {}
+

      The ltype context allows only expressions which have an
      +lvalue to evaluate.

      +
        syntax LValue ::= Id
+  rule isLValue(_:Exp[_:Exps]) => true
+  syntax Exp ::= LValue  // K should be able to infer this
+                         // if not added, then it gets stuck with an Id on k cell
+
+// Instead of the second LValue production above you can use a rule:
+//  rule isLValue(_:Exp[_:Exps]) => true
+
+  syntax Exp ::= ltype(Exp)
+//  context ltype(HOLE:LValue)
+// The above context does not work due to some error, so we write instead
+  context ltype(HOLE) requires isLValue(HOLE)
+

      The function getTypes is the same as in SIMPLE typed dynamic.

      +
        syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types   // I would like to not use .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+
+endmodule
+

      Go to Lesson 3, SIMPLE typed dynamic

      +

      SIMPLE — Typed — Dynamic

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K dynamic semantics of the typed SIMPLE language.
      +It is very similar to the semantics of the untyped SIMPLE, the
      +difference being that we now dynamically check the typing policy
      +described in the static semantics of typed SIMPLE. Because of the
      +dynamic nature of the semantics, we can also perform some additional
      +checks which were not possible in the static semantics, such as
      +memory leaks due to accessing an array out of its bounds. We will
      +highlight the differences between the dynamically typed and the
      +untyped SIMPLE as we proceed with the semantics. We recommend the
      +reader to consult the typing policy and the syntax of types discussed
      +in the static semantics of the typed SIMPLE language.

      +
      module SIMPLE-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of typed SIMPLE extends that of untyped SIMPLE with support
      +for declaring types to variables and functions.

      +

      The syntax below is identical to that of the static semantics of typed
      +SIMPLE. However, the K strictness attributes are like those of the untyped
      +SIMPLE, to capture the desired evaluation strategies of the various language
      +constructs.

      +
        syntax Id ::= "main" [token]
+

      Types

      + +
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+  syntax Types ::= List{Type,","}        [klabel(exps)]
+

      Declarations

      + +
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";"
+                | Type Id "(" Params ")" Block
+

      Expressions

      + +
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+

      Like in the static semantics, there is no need for lists of identifiers
      +(because we now have lists of parameters).

      +
        syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+

      Statements

      + +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"
+                | "try" Block "catch" "(" Param ")" Block
+            | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+

      The same desugaring macros like in the statically typed SIMPLE.

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule for(Start Cond; Step) {} => {Start while(Cond){Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+endmodule
+
+
+module SIMPLE-TYPED-DYNAMIC
+  imports SIMPLE-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS
+

      Semantics

      + +

      Values and results

      + +

      These are similar to those of untyped SIMPLE, except that the array
      +references and the function abstrations now also hold their types.
      +These types are needed in order to easily compute the type of any
      +value in the language (see the auxiliary typeOf operation at
      +the end of this module).

      +
        syntax Val ::= Int | Bool | String
+               | array(Type,Int,Int)
+               | lambda(Type,Params,Stmt)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+                   | Vals  // TODO: should not need this
+

      Configuration

      + +

      The configuration is almost identical to that of untyped SIMPLE,
      +except for a return cell inside the control cell.
      +This return cell will hold, like in the static semantics of
      +typed SIMPLE, the expected type of the value returned by the function
      +being executed. The contents of this cell will be set whenever a
      +function is invoked and will be checked whenever the evaluation of the
      +function body encounters an explicit return statement.

      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+
+  syntax ControlCell
+  syntax ControlCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" color="yellow" type="Map">
+                      <id color="pink"> 0 </id>
+                      <k color="green"> ($PGM:Stmt ~> execute) </k>
+//                      <br/>
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                        <returnType color="LimeGreen"> void </returnType>
+                       </control>
+//                      <br/>
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                    </thread>
+                  </threads>
+//                  <br/>
+                  <genv color="pink"> .Map </genv>
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set</busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                </T>
+

      Declarations and Initialization

      + +

      Variable Declaration

      + +

      The undefined construct is now parameterized by a type.
      +A main difference between untyped SIMPLE and dynamically typed SIMPLE
      +is that the latter assigns a type to each of its locations and that
      +type cannot be changed during the execution of the program. We do not
      +do any memory management in our semantic definitions here, so
      +locations cannot be reclaimed, garbage collected and/or reused. Each
      +location corresponds precisely to an allocated variable or array
      +element, whose type was explicitly or implicitly declared in the
      +program and does not change. It is therefore safe to type each
      +location and then never allow that type to change. The typed
      +undefined values effectively assign both a type and an undefined value
      +to a location.

      +
        syntax KItem ::= undefined(Type)  [latex(\bot_{#1})]
+
+  rule <k> T:Type X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+

      Array Declaration

      + +

      The dynamic semantics of typed array declarations is similar to that
      +in untyped SIMPLE, but we have to make sure that we associate the
      +right type to the allocated locations.

      +
        rule <k> T:Type X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(T, L +Int 1, N)
+                          (L +Int 1)...(L +Int N) |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    when N >=Int 0
+
+  context _:Type _::Exp[HOLE::Exps];
+

      The desugaring of multi-dimensional arrays into unidimensional
      +ones is also similar to that in untyped SIMPLE, although we have to
      +make sure that all the declared variables have the right types. The
      +auxiliary operation T<Vs>, defined at the end of the file,
      +adds the length of Vs dimensions to the type T.

      +
      // TODO: Check the desugaring below to be consistent with the one for untyped simple
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule T:Type X:Id[N1:Int, N2:Int, Vs:Vals];
+    => T[]<Vs> X[N1];
+       {
+         T[][]<Vs> $1=X;
+         for(int $2=0; $2 <= N1 - 1; ++$2) {
+           T X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+

      Function declaration

      + +

      Store all function parameters, as well as the return type, as part
      +of the lambda abstraction. In the spirit of dynamic typing, we will
      +make sure that parameters are well typed when the function is invoked.

      +
        rule <k> T:Type F:Id(Ps:Params) S => . ...</k>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> lambda(T, Ps, S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      Calling main()

      + +

      When done with the first pass, call main().

      +
        syntax KItem ::= "execute"
+  rule <k> execute => main(.Exps); </k>
+       <env> Env </env>
+       <genv> .Map => Env </genv>
+

      Expressions

      + +

      Variable lookup

      + +
        rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Variable/Array increment

      + +
        context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+

      Arithmetic operators

      + +
        rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+

      Array lookup

      + +

      Check array bounds, as part of the dynamic typing policy.

      +
      // Same comment as for simple untyped regarding [anywhere]
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+// Same comment as for simple untyped regarding [anywhere]
+  rule array(_:Type, L:Int, M:Int)[N:Int] => lookup(L +Int N)
+    when N >=Int 0 andBool N <Int M  [anywhere]
+

      Size of an array

      + +
        rule sizeOf(array(_,_,N)) => N
+

      Function call

      + +

      Define function call and return together, to see their relationship.
      +Note that the operation mkDecls now declares properly typed
      +instantiated variables, and that the semantics of return also
      +checks that that type of the returned value is expected one.

      +
        syntax KItem ::= (Type,Map,K,ControlCellFragment)
+
+  rule <k> lambda(T,Ps,S)(Vs:Vals) ~> K => mkDecls(Ps,Vs) S return; </k>
+       <control>
+         <fstack> .List => ListItem((T',Env,K,C)) ...</fstack>
+         <returnType> T' => T </returnType>
+         C
+       </control>
+       <env> Env => GEnv </env>
+       <genv> GEnv </genv>
+
+  rule <k> return V:Val; ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem((T',Env,K,C)) => .List ...</fstack>
+         <returnType> T => T' </returnType>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+    when typeOf(V) ==K T   // check the type of the returned value
+

      Like the undefined above, nothing also gets
      +tagged with a type now. The empty return statement is
      +completed to return the nothing value tagged as expected.

      +
        syntax Val ::= nothing(Type)
+  rule <k> return; => return nothing(T); ...</k> <returnType> T </returnType>
+

      Read

      + +
        rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+

      Assignment

      + +

      The assignment now checks that the type of the assigned location is
      +preserved:

      +
        context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (V' => V) ...</store>
+    when typeOf(V) ==K typeOf(V')  [group(assignment)]
+

      Statements

      + +

      Blocks

      + +
        rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+

      Sequential composition

      + +
        rule S1:Stmt S2:Stmt => S1 ~> S2
+

      Expression statements

      + +
        rule _:Val; => .
+

      Conditional

      + +
        rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+

      While loop

      + +
        rule while (E) S => if (E) {S while(E)S}
+

      Print

      + +

      We only allow printing integers and strings:

      +
        rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    when typeOf(V) ==K int orBool typeOf(V) ==K string  [group(print)]
+  rule print(.Vals); => .
+

      Exceptions

      + +

      Exception parameters are now typed, but note that the semantics below
      +works correctly only when the thrown exception has the same type as
      +the innermost try-catch paramete. To keep things simple, for the time
      +being we can assume that SIMPLE only throws and catches integer
      +values, in which case our semantics below works fine:

      +
        syntax KItem ::= (Param,Stmt,K,Map,ControlCellFragment)  // Param instead of Id
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(P) S2 => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem((P, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { T X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem((T:Type X:Id, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      Threads

      + +

      Thread creation

      + +
         rule <thread>...
+          <k> spawn S => !T:Int +Int 1 ...</k>
+          <env> Env </env>
+        ...</thread>
+        (.Bag => <thread>...
+                <k> S </k>
+                <env> Env </env>
+                <id> !T +Int 1 </id>
+              ...</thread>)
+

      Thread termination

      + +
         rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+        <busy> Busy => Busy -Set keys(H) </busy>
+        <terminated>... .Set => SetItem(T) ...</terminated>
+

      Thread joining

      + +
         rule <k> join T:Int; => . ...</k>
+        <terminated>... SetItem(T) ...</terminated>
+

      Acquire lock

      + +
         rule <k> acquire V:Val; => . ...</k>
+        <holds>... .Map => V |-> 0 ...</holds>
+        <busy> Busy (.Set => SetItem(V)) </busy>
+     when (notBool(V in Busy:Set))  [group(acquire)]
+
+   rule <k> acquire V; => . ...</k>
+        <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+

      Release lock

      + +
         rule <k> release V:Val; => . ...</k>
+        <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+      when N >Int 0
+
+   rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+        <busy>... SetItem(V) => .Set ...</busy>
+

      Rendezvous synchronization

      + +
         rule <k> rendezvous V:Val; => . ...</k>
+        <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Auxiliary declarations and operations

      + +

      Turns a list of parameters and a list of instance values for them
      +into a list of variable declarations.

      +
        syntax Stmt ::= mkDecls(Params,Vals)  [function]
+  rule mkDecls((T:Type X:Id, Ps:Params), (V:Val, Vs:Vals))
+    => T X=V; mkDecls(Ps,Vs)
+  rule mkDecls(.Params,.Vals) => {}
+

      Location lookup.

      +
        syntax Exp ::= lookup(Int)  // see NOTES.md for why Exp instead of KItem
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+

      Environment recovery.

      +
      // TODO: same comment regarding setEnv(...) as for simple untyped
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+

      lvalue and loc

      +
        syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k>  <env>... X |-> L:Int ...</env>
+
+  //context lvalue(_[HOLE])
+  //context lvalue(HOLE[_])
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+

      Adds the corresponding depth to an array type

      +
        syntax Type ::= Type "<" Vals ">"  [function]
+  rule T:Type<_,Vs:Vals> => T[]<Vs>
+  rule T:Type<.Vals> => T
+

      Sequences of locations.

      +
        syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  when N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  when N <=Int M
+
+// Type of a value.
+  syntax Type ::= typeOf(K)  [function]
+  rule typeOf(_:Int) => int
+  rule typeOf(_:Bool) => bool
+  rule typeOf(_:String) => string
+  rule typeOf(array(T,_,_)) => (T[])   // () needed! K parses [] as "no tags"
+  rule typeOf(lambda(T,Ps,_)) => getTypes(Ps) -> T
+  rule typeOf(undefined(T)) => T
+  rule typeOf(nothing(T)) => T
+

      List of types of a parameter.

      +
        syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types   // I would like to not use .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+endmodule
+

      KOOL — Untyped

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped KOOL language. KOOL
      +is aimed at being a pedagogical and research language that captures
      +the essence of the object-oriented programming paradigm. Its untyped
      +variant discussed here is simpler than the typed one, ignoring several
      +intricate aspects of types in the presence of objects. A program
      +consists of a set of class declarations. Each class can extend at
      +most one other class (KOOL is single-inheritance). A class can
      +declare a set of fields and a set of methods, all public and called
      +the class' members. Specifically, KOOL includes the
      +following features:

      +
        +
      • +

        Class declarations, where a class may or may not explicitly
        +extend another class. In case a class does not explicitly extend
        +another class, then it is assumed that it extends the default top-most
        +and empty (i.e., no members) class called Object. Each class
        +is required to declare precisely one homonymous method, called its
        +constructor. Each valid program should contain one class
        +named Main, whose constructor, Main(), takes no
        +arguments. The execution of a program consists of creating an object
        +instance of class Main and invoking the constructor
        +Main() on it, that is, of executing new Main();.

        +
        • +
      • +

        All features of SIMPLE (see examples/simple/untyped),
        +i.e., multidimensional arrays, function (here called "method")
        +abstractions with call-by-value parameter passing style and static
        +scoping, blocks with locals, input/output, parametric exceptions, and
        +concurrency via dynamic thread creation/termination and synchronization.
        +The only change in the syntax of SIMPLE when imported in KOOL is the
        +function declaration keyword, function, which is changed into
        +method. The exact same desugaring macros from SIMPLE are
        +also included in KOOL. We can think of KOOL's classes as embedding
        +SIMPLE programs (extended with OO constructs, as discussed next).

        +
        • +
      • +

        Object creation using the new C(e1,...,en)
        +expression construct. An object instance of class C is first
        +created and then the constructor C(e1,...,en) is implicitly
        +called on that object. KOOL only allows (and requires) one
        +constructor per class. The class constructor can be called either
        +implicitly during a new object creation for the class, or explicitly.
        +The superclass constructor is not implicitly invoked when a
        +class constructor is invoked; if you want to invoke the superclass
        +constructor from a subclass constructor then you have to do it
        +explicitly.

        +
        • +
      • +

        An expression construct this, which evaluates to the
        +current object.

        +
        • +
      • +

        An expression construct super, which is used (only) in
        +combination with member lookup (see next) to refer to a superclass
        +field or method.

        +
        • +
      • +

        A member lookup expression construct e.x, where e
        +is an expression (either an expression expected to evaluate to an object
        +or the super construct) and x is a class member name,
        +that is, a field or a method name.

        +
        • +
      • +

        Expression constructs e instanceOf C and
        +(C) e, where e is an expression expected
        +to evaluate to an object and C a class name. The former
        +tells whether the class of e is a subclass of C,
        +that is, whether e can be used as an instance of C,
        +and the latter changes the class of e to C. These
        +operations always succeed: the former returns a Boolean value, while
        +the latter changes the current class of e to C
        +regardless of whether it is safe to do so or not. The typed version
        +of KOOL will check the safety of casting by ensuring that the instance
        +class of the object is a subclass of C. In untyped KOOL we
        +do not want to perform this check because we want to allow the
        +programmer maximum of flexibility: if one always accesses only
        +available members, then the program can execute successfully despite
        +the potentially unsafe cast.

        +
        • +
      +

      There are some specific aspects of KOOL that need to be discussed.

      +

      First, KOOL is higher-order, allowing function abstractions to be
      +treated like any other values in the language. For example, if
      +m is a method of object e then e.m
      +evaluates to the corresponding function abstraction. The function
      +abstraction is in fact a closure, because in addition to the method
      +parameters and body it also encapsulates the object value (i.e., the
      +environment of the object together with its current class—see below)
      +that e evaluates to. This way, function abstractions can be
      +invoked anywhere and have the capability to change the state of their
      +object. For example, if m is a method of object e
      +which increments a field c of e when invoked, and if
      +getm is another method of e which simply returns
      +m when invoked, then the double application
      +(e.getm())() has the same effect as e.m(), that is,
      +increments the counter c of e. Note that the
      +higher-order nature of KOOL was not originally planned; it came as a
      +natural consequence of evaluating methods to closures and we decided
      +to keep it. If you do not like it then do not use it.

      +

      Second, since all the fields and methods are public in KOOL and since
      +they can be redeclared in subclasses, it is not immediately clear how
      +to lookup the member x when we write e.x and
      +e is different from super. We distinguish two cases,
      +depending on whether e.x occurs in a method invocation
      +context (i.e., e.x(...)) or in a field context. KOOL has
      +dynamic method dispatch, so if e.x is invoked as a method
      +then x will be searched for starting with the instance class of
      +the object value to which e evaluates. If e.x
      +occurs in a non-method-invocation context then x will be
      +treated as a field (although it may hold a method closure due to the
      +higher-order nature of KOOL) and thus will be searched starting with
      +the current class of the object value of e (which, because of
      +this and casting, may be different from its instance class).
      +In order to achieve the above, each object value will consist of a
      +pair holding the current class of the object and an environment stack
      +with one layer for each class in the object's instance class hierarchy.

      +

      Third, although KOOL is dynamic method dispatch, its capabilities
      +described above are powerful enough to allow us to mimic static
      +method dispatch. For example, suppose that you want to invoke method
      +m() statically. Then all you need to do is to declare a
      +local variable and bind it to m, for example var staticm = m;, and
      +then call staticm(). This works because
      +staticm is first bound to the method closure that m
      +evaluates to, and then looked up as any local variable when invoked.
      +We only enable the dynamic method dispatch when we have an object
      +member on an application position, e.g., m().

      +

      In what follows, we limit our comments to the new, KOOL-specific
      +aspects of the language. We refer the reader to the untyped SIMPLE
      +language for documentation on the the remaining features, because
      +those were all borrowed from SIMPLE.

      +
      module KOOL-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of KOOL extends that of SIMPLE with object-oriented
      +constructs. We removed from the K annotated syntax of SIMPLE two
      +constructs, namely the one for function declarations (because we want
      +to call them methods now) and the one for function application
      +(because application is not strict in the first argument
      +anymore—needs to initiate dynamic method dispatch). The additional
      +syntax includes:

      +
        +
      • First, we need a new dedicated identifier, Object, for
        +the default top-most class.
        • +
      • Second, we rename the function keyword of SIMPLE into method.
        • +
      • Third, we add syntax for class declarations together with a
        +macro making classes which extend nothing to extend Object.
        • +
      • Fourth, we change the strictness attribute of application
        +into strict(2).
        • +
      • Finally, we add syntax and corresponding strictness
        +for the KOOL object-oriented constructs.
        • +
      +
        syntax Id ::= "Object" [token] | "Main" [token]
+
+  syntax Stmt ::= "var" Exps ";"
+                | "method" Id "(" Ids ")" Block  // called "function" in SIMPLE
+                | "class" Id Block               // KOOL
+                | "class" Id "extends" Id Block  // KOOL
+
+  syntax Exp ::= Int | Bool | String | Id
+               | "this"                                 // KOOL
+               | "super"                                // KOOL
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]    // KOOL
+               | "(" Id ")" Exp          [strict(2)]    // KOOL  cast
+               | "new" Id "(" Exps ")"   [strict(2)]    // KOOL
+               | Exp "." Id                             // KOOL
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict(2)]    // was strict in SIMPLE
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Ids  ::= List{Id,","}
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+
+  syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+

      Old desugaring rules, from SIMPLE

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule var E1::Exp, E2::Exp, Es::Exps; => var E1; var E2, Es;       [anywhere]
+  rule var X::Id = E; => var X; X = E;                              [anywhere]
+

      New desugaring rule

      +
        rule class C:Id S => class C extends Object S                     // KOOL
+
+endmodule
+

      Semantics

      + +

      We first discuss the new configuration of KOOL, which extends that of
      +SIMPLE. Then we include the semantics of the constructs borrowed from
      +SIMPLE unchanged; we refrain from discussing those, because they were
      +already discussed in the K definition of SIMPLE. Then we discuss
      +changes to SIMPLE's semantics needed for the more general meaning of
      +the previous SIMPLE constructs (for example for thread spawning,
      +assignment, etc.). Finally, we discuss in detail the
      +semantics of the additional KOOL constructs.

      +
      module KOOL-UNTYPED
+  imports KOOL-UNTYPED-SYNTAX
+  imports DOMAINS
+

      Configuration

      + +

      KOOL removes one cell and adds two nested cells to the configuration
      +of SIMPLE. The cell which is removed is the one holding the global
      +environment, because a KOOL program consists of a set of classes only,
      +with no global declarations. In fact, since informally speaking each
      +KOOL class now includes a SIMPLE program, it is safe to say that the
      +global variables in SIMPLE became class fields in KOOL. Let us now
      +discuss the new cells that are added to the configuration of SIMPLE.

      +
        +
      • +

        The cell crntObj holds data pertaining to the current
        +object, that is, the object environment in which the code in cell
        +k executes: crntClass holds the current class (which
        +can change as methods of the current object are invoked);
        +envStack holds the stack of environments as a list,
        +each layer corresponding to one class in the objects' instance class
        +hierarchy; location, which is optional, holds the location in
        +the store where the current object is or has to be located (this is
        +useful both for method closures and for the semantics of object
        +creation).

        +
        • +
      • +

        The cell classes holds all the declared classes, each
        +class being held in its own class cell which contains a name
        +(className), a parent (extends), and the actual
        +member declarations (declarations).

        +
        • +
      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+  syntax EnvCell
+  syntax ControlCell
+  syntax EnvStackCell
+  syntax CrntObjCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="yellow">
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      //<br/> // TODO(KORE): support latex annotations #1799
+                        <crntObj color="Fuchsia">  // KOOL
+                           <crntClass> Object </crntClass>
+                           <envStack> .List </envStack>
+                           <location multiplicity="?"> .K </location>
+                        </crntObj>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                      <id color="pink"> 0 </id>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <classes color="Fuchsia">        // KOOL
+                     <classData multiplicity="*" type="Map" color="Fuchsia">
+                        // the Map has as its key the first child of the cell,
+                        // in this case the className cell.
+                        <className color="Fuchsia"> Main </className>
+                        <baseClass color="Fuchsia"> Object </baseClass>
+                        <declarations color="Fuchsia"> .K </declarations>
+                     </classData>
+                  </classes>
+                </T>
+

      Unchanged Semantics from untyped SIMPLE

      + +

      The semantics below is taken over from SIMPLE unchanged.
      +The semantics of function declaration and invocation, including the
      +use of the special lambda abstraction value, needs to change
      +in order to account for the fact that methods are now invoked into
      +their object's environment. The semantics of function return actually
      +stays unchanged. Also, the semantics of program initialization is
      +different: now we have to create an instance of the Main
      +class which also calls the constructor Main(), while in
      +SIMPLE we only had to invoke the function Main().
      +Finally, the semantics of thread spawning needs to change, too: the
      +parent thread needs to also share its object environment with the
      +spawned thread (in addition to its local environment, like in SIMPLE).
      +This is needed in order to be able to spawn method invokations under
      +dynamic method dispatch; for example, spawn { run(); }
      +will need to look up the method run() in the newly created
      +thread, operation which will most likely fail unless the child thread
      +sees the object environment of the parent thread. Note that the
      +spawn statement of KOOL is more permissive than the threads
      +of Java. In fact, the latter can be implemented in terms of our
      +spawn—see the program threads.kool for a sketch.

      +

      Below is a subset of the values of SIMPLE, which are also values
      +of KOOL. We will add other values later in the semantics, such as
      +object and method closures.

      +
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax KResult ::= Vals
+

      The semantics below are taken verbatim from the untyped SIMPLE
      +definition.

      +
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
+
+  context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    when N >=Int 0
+
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         var $1=X;
+         for(var $2=0; $2 <= N1 - 1; ++$2) {
+           var X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
+
+  rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+
+  context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
+
+  rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+
+
+  rule sizeOf(array(_,N)) => N
+

      The semantics of function application needs to change into dynamic
      +method dispatch invocation, which is defined shortly. However,
      +interestingly, the semantics of return stays unchanged.

      +
        rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem(fstackFrame(Env,K,XS,<crntObj> CO </crntObj>)) => .List ...</fstack>
+         <xstack> _ => XS </xstack>
+         <crntObj> _ => CO </crntObj>
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+
+
+  rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
+
+  context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+
+
+  rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
+
+  rule S1::Stmt S2::Stmt => S1 ~> S2
+
+  rule _:Val; => .
+
+  rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
+  rule while (E) S => if (E) {S while(E)S}
+
+  rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+
+
+  syntax KItem ::= xstackFrame(Id,Stmt,K,Map,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Id,Stmt,K,Map,K)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem(xstackFrame(X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem(xstackFrame(X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      Thread spawning needs a new semantics, because we want the child
      +thread to also share the object environment with its parent. The new
      +semantics of thread spawning will be defined shortly. However,
      +interestingly, the other concurrency constructs keep their semantics
      +from SIMPLE unchanged.

      +
        // TODO(KORE): ..Bag should be . throughout this definition #1772
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+  /*
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .)
+  */
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
+  rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
+  rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    when (notBool(V in Busy:Set))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
+  rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+    when N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
+  rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Unchanged auxiliary operations from untyped SIMPLE

      + +
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+
+  // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookup(Int)
+  /*
+  syntax KItem ::= lookup(Int)
+  */
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+  // TODO: How can we make sure that the second rule above applies before the first one?
+  //       Probably we'll deal with this using strategies, eventually.
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
+
+  syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  when N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  when N <=Int M
+

      Changes to the existing untyped SIMPLE semantics

      + +

      When we extend a language, sometimes we need to do more than just add
      +new language constructs and semantics for them. Sometimes we want to
      +also extend the semantics of existing language constructs, in order to
      +get more from them.

      +

      Program initialization

      + +

      In SIMPLE, once all the global declarations were processed, the
      +function main() was invoked. In KOOL, the global
      +declarations are classes, and their specific semantics is given
      +shortly; essentially, they are pre-processed one by one and added
      +into the class cell structure in the configuration.
      +Once all the classes are processed, the computation item
      +execute, which was placed right after the program in the
      +initial configuration, is reached. In SIMPLE, the program was
      +initialized by calling the method main(). In KOOL, the
      +program is initialized by creating an object instance of class
      +Main. This will also implicitly call the method
      +Main() (the Main class constructor). The emptiness
      +of the env cell below is just a sanity check, to make sure
      +that the user has not declared anything but classes at the top level
      +of the program.

      +
        syntax KItem ::= "execute"
+  rule <k> execute => new Main(.Exps); </k> <env> .Map </env>
+

      The semantics of new (defined below) requires the
      +execution of all the class' declarations (and also of its
      +superclasses').

      +

      Object and method closures

      + +

      Before we can define the semantics of method application (previously
      +called function application in SIMPLE), we need to add two more values
      +to the language, namely object and method closures:

      +
        syntax Val ::= objectClosure(Id, List)
+               | methodClosure(Id,Int,Ids,Stmt)
+

      An object value consists of an objectClosure-wrapped bag
      +containing the current class of the object and the environment stack
      +of the object. The current class of an object will always be one of
      +the classes mapped to an environment in the environment stack of the
      +object. A method closure encapsulates the method's parameters and
      +code (last two arguments), as well as the object context in which the
      +method code should execute. This object context includes the current
      +class of the object (the first argument of methodClosure) and
      +the object environment stack (located in the object stored at the
      +location specified as the second argument of methodClosure).

      +

      Method application

      + +

      KOOL has a complex mechanism to invoke methods, because it allows both
      +dynamic method dispatch and methods as first-class-citizen values (the
      +latter making it a higher-order language). The invocation mechanism
      +will be defined later. What is sufficient to know for now is that
      +the two arguments of the application construct eventually reduce to
      +values, the first being a method closure and the latter a list of
      +values. The semantics of the method closure application is then as
      +expected: the local environment and control are stacked, then we
      +switch to method closure's class and object environment and execute
      +the method body. The mkDecls construct is the one that came
      +with the unchanged semantics of SIMPLE above.

      +
        syntax KItem ::= fstackFrame(Map,K,List,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Map,K,K)
+
+  rule <k> methodClosure(Class,OL,Xs,S)(Vs:Vals) ~> K
+           => mkDecls(Xs,Vs) S return; </k>
+       <env> Env => .Map </env>
+       <store>... OL |-> objectClosure(_, EnvStack)...</store>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+          <xstack> XS </xstack>
+          <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj' </crntObj>))
+          ...</fstack>
+          <crntObj> Obj' => <crntClass> Class </crntClass> <envStack> EnvStack </envStack> </crntObj>
+       </control>
+

      Spawn

      + +

      We want to extend the semantics of spawn to also share the
      +current object environment with the child thread, in addition to the
      +current environment. This extension will allow us to also use method
      +invocations in the spawned statements, which will be thus looked up as
      +expected, using dynamic method dispatch. This lookup operation would
      +fail if the child thread did not have access to its parent's object
      +environment.

      +
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+         <crntObj> Obj </crntObj>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+               <crntObj> Obj </crntObj>
+             ...</thread>)
+

      Semantics of the new KOOL constructs

      + +

      Class declaration

      + +

      Initially, the classes forming the program are moved into their
      +corresponding cells:

      +
        rule <k> class Class1 extends Class2 { S } => . ...</k>
+       <classes>... (.Bag => <classData>
+                            <className> Class1 </className>
+                            <baseClass> Class2 </baseClass>
+                            <declarations> S </declarations>
+                        </classData>)
+       ...</classes>
+

      Method declaration

      + +

      Like in SIMPLE, method names are added to the environment and bound
      +to their code. However, unlike in SIMPLE where each function was
      +executed in the same environment, namely the program global
      +environment, a method in KOOL needs to be executed into its object's
      +environment. Thus, methods evaluate to closures, which encapsulate
      +their object's context (i.e., the current class and environment stack
      +of the object) in addition to method's parameters and body. This
      +approach to bind method names to method closures in the environment
      +will also allow objects to pass their methods to other objects, to
      +dynamically change their methods by assigning them other method
      +closures, and even to allow all these to be done from other objects.
      +This gives the KOOL programmer a lot of power; one should use this
      +power wisely, though, because programs can become easily hard to
      +understand and reason about if one overuses these features.

      +
        rule <k> method F:Id(Xs:Ids) S => . ...</k>
+       <crntClass> Class:Id </crntClass>
+       <location> OL:Int </location>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> methodClosure(Class,OL,Xs,S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      New

      + +

      The semantics of new consists of two actions: memory
      +allocation for the new object and execution of the corresponding
      +constructor. Then the created object is returned as the result of the
      +new operation; the value returned by the constructor, if any,
      +is discarded. The current environment and object are stored onto the
      +stack and recovered after new (according to the semantics of
      +return borrowed from SIMPLE, when the statement
      +return this; in the rule below is reached and evaluated),
      +because the object creation part of new will destroy them.
      +The rule below also initializes the object creation process by
      +emptying the local environment and the current object, and allocating
      +a location in the store where the created object will be eventually
      +stored (this is what the storeObj task after the object
      +creation task in the rule below will do—its rule is defined
      +shortly). The location where the object will be stored is also made
      +available in the crntObj cell, so that method closures can
      +refer to it (see rule above).

      +
        syntax KItem ::= "envStackFrame" "(" Id "," Map ")"
+
+  rule <k> new Class:Id(Vs:Vals) ~> K
+           => create(Class) ~> storeObj ~> Class(Vs); return this; </k>
+       <env> Env => .Map </env>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control> <xstack> XS </xstack>
+         <crntObj> Obj
+                   => <crntClass> Object </crntClass>
+                      <envStack> ListItem(envStackFrame(Object, .Map)) </envStack>
+                      <location> L </location>
+         </crntObj>
+         <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj </crntObj>)) ...</fstack>
+       </control>
+

      The creation of a new object (the memory allocation part only) is
      +a recursive process, requiring to first create an object for the
      +superclass. A memory object representation is a layered structure:
      +for each class on the path from the instance class to the root of the
      +hierarchy there is a layer including the memory allocated for the
      +members (both fields and methods) of that class.

      +
        syntax KItem ::= create(Id)
+
+  rule <k> create(Class:Id)
+           => create(Class1) ~> setCrntClass(Class) ~> S ~> addEnvLayer ...</k>
+       <className> Class </className>
+       <baseClass> Class1:Id </baseClass>
+       <declarations> S </declarations>
+
+  rule <k> create(Object) => . ...</k>
+

      The next operation sets the current class of the current object.
      +This is necessary to be done at each layer, because the current class
      +of the object is enclosed as part of the method closures (see the
      +semantics of method declarations above).

      +
        syntax KItem ::= setCrntClass(Id)
+
+  rule <k> setCrntClass(C) => . ...</k>
+       <crntClass> _ => C </crntClass>
+

      The next operation adds a new tagged environment layer to the
      +current object and gets ready for the next layer by clearing the
      +environment (note that create expects the environment to be
      +empty).

      +
        syntax KItem ::= "addEnvLayer"
+
+  rule <k> addEnvLayer => . ...</k>
+       <env> Env => .Map </env>
+       <crntClass> Class:Id </crntClass>
+       <envStack> .List => ListItem(envStackFrame(Class, Env)) ...</envStack>
+

      The following operation stores the created object at the location
      +reserved by new. Note that the location reserved by
      +new was temporarily stored in the crntObj cell
      +precisely for this purpose. Now that the newly created object is
      +stored at its location and that all method closures are aware of it,
      +the location is unnecessary and thus we delete it from the
      +crntObj cell.

      +
        syntax KItem ::= "storeObj"
+
+  rule <k> storeObj => . ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> (<location> L:Int </location> => .Bag) </crntObj>
+       <store>... .Map => L |-> objectClosure(CC, ES) ...</store>
+

      Self reference

      + +

      The semantics of this is straightforward: evaluate to the
      +current object.

      +
        rule <k> this => objectClosure(CC, ES) ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> </crntObj>
+

      Object member access

      + +

      We can access an object member (field or method) either explicitly,
      +using the construct e.x, or implicitly, using only the member
      +name x directly. The borrowed semantics of SIMPLE will
      +already lookup a sole name in the local environment. The first rule
      +below reduces implicit member access to explicit access when the name
      +cannot be found in the local environment. There are two cases to
      +analyze for explicit object member access, depending upon whether the
      +object is a proper object or it is just a redirection to the parent
      +class via the construct super. In the first case, we
      +evaluate the object expression and lookup the member starting with the
      +current class (static scoping). Note the use of the conditional
      +evaluation context. In the second case, we just lookup the member
      +starting with the superclass of the current class. In both cases,
      +the lookupMember task eventually yields a lookup(L)
      +task for some appropriate location L, which will be further
      +solved with the corresponding rule borrowed from SIMPLE. Note that the
      +current object is not altered by super, so future method
      +invocations see the entire object, as needed for dynamic method dispatch.

      +
        rule <k> X:Id => this . X ...</k> <env> Env:Map </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id when (HOLE =/=K super)
+
+// TODO: explain how Assoc matching has been replaced with two rules here.
+// Maybe also improve it a bit.
+
+/*  rule objectClosure(<crntClass> Class:Id </crntClass>
+                     <envStack>... envStackFrame(Class,EnvC) EStack </envStack>)
+       . X:Id
+    => lookupMember(envStackFrame(Class,EnvC) EStack, X) */
+
+  rule objectClosure(Class:Id, ListItem(envStackFrame(Class,Env)) EStack)
+       . X:Id
+    => lookupMember(ListItem(envStackFrame(Class,Env)) EStack, X)
+  rule objectClosure(Class:Id, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+       . _X:Id
+    when Class =/=K Class'
+
+/*  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,EnvC) EStack </envStack> */
+  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> super . _X ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+

      Method invocation

      + +

      Unlike in SIMPLE, in KOOL application was declared strict only in its
      +second argument. That is because we want to ensure dynamic method
      +dispatch when the first argument is a method access. As a
      +consequence, we need to consider all the cases of interest for the
      +first argument and to explicitly say what to do in each case. In all
      +cases except for method access in a proper object (i.e., not
      +super), we want the same behavior for the first argument as
      +if it was not in a method invocation position. When it is a member
      +access (the third rule below), we look it up starting with the
      +instance class of the corresponding object. This ensures dynamic
      +dispatch for methods; it actually dynamically dispatches field
      +accesses, too, which is correct in KOOL, because one can assign method
      +closures to fields and the field appeared in a method invocation
      +context. The last context declaration below says that method
      +applications or array accesses are also allowed as first argument to
      +applications; that is because methods are allowed to return methods
      +and arrays are allowed to hold methods in KOOL, since it is
      +higher-order. If that is the case, then we want to evaluate the
      +method call or the array access.

      +
        rule <k> (X:Id => V)(_:Exps) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  rule <k> (X:Id => this . X)(_:Exps) ...</k>
+       <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id(_) when HOLE =/=K super
+
+  rule (objectClosure(_, EStack) . X
+    => lookupMember(EStack, X:Id))(_:Exps)
+
+/*  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,_) EStack </envStack> */
+  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> (super . _X)(_:Exps) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+
+  // TODO(KORE): fix getKLabel #1801
+  rule (A:Exp(B:Exps))(C:Exps) => A(B) ~> #freezerFunCall(C)
+  rule (A:Exp[B:Exps])(C:Exps) => A[B] ~> #freezerFunCall(C)
+  rule V:Val ~> #freezerFunCall(C:Exps) => V(C)
+  syntax KItem ::= "#freezerFunCall" "(" K ")"
+  /*
+  context HOLE(_:Exps)
+    when getKLabel(HOLE) ==K #klabel(`_(_)`) orBool getKLabel(HOLE) ==K #klabel(`_[_]`)
+  */
+

      Eventually, each of the rules above produces a lookup(L)
      +task as a replacement for the method. When that happens, we just
      +lookup the value at location L:

      +
        rule <k> (lookup(L) => V)(_:Exps) ...</k>  <store>... L |-> V:Val ...</store>
+    [group(lookup)]
+

      The value V looked up above is expected to be a method closure,
      +in which case the semantics of method application given above will
      +apply. Otherwise, the execution will get stuck.

      +

      Instance Of

      + +

      It searches the object environment for a layer corresponding to the
      +desired class. It returns true iff it can find the class,
      +otherwise it returns false; it only gets stuck when its first
      +argument does not evaluate to an object.

      +
        rule objectClosure(_, ListItem(envStackFrame(C,_)) _)
+       instanceOf C => true
+
+  rule objectClosure(_, (ListItem(envStackFrame(C,_)) => .List) _)
+       instanceOf C'  when C =/=K C'
+//TODO: remove the sort cast ::Id of C above, when sort inference bug fixed
+
+  rule objectClosure(_, .List) instanceOf _ => false
+

      Cast

      + +

      In untyped KOOL, we prefer to not check the validity of casting. In
      +other words, any cast is allowed on any object, simply changing the
      +current class of the object to the desired class. The execution will
      +get stuck later if one attempts to access a field which is not
      +available. Moreover, the execution may complete successfully even
      +in the presence of invalid casts, provided that each accessed member
      +during the current execution is, or happens to be, available.

      +
        rule (C) objectClosure(_ , EnvStack) => objectClosure(C ,EnvStack)
+

      KOOL-specific auxiliary declarations and operations

      + +

      Here we define all the auxiliary constructs used in the above
      +KOOL-specific semantics (those used in the SIMPLE fragment
      +have already been defined in a corresponding section above).

      +

      Objects as lvalues

      + +

      The current machinery borrowed with the semantics of SIMPLE allows us
      +to enrich the set of lvalues, this way allowing new means to assign
      +values to locations. In KOOL, we want object member names to be
      +lvalues, so that we can assign values to them using the already
      +existing machinery. The first rule below ensures that the object is
      +always explicit, the evaluation context enforces the object to be
      +evaluated, and finally the second rule initiates the lookup for the
      +member's location based on the current class of the object.

      +
        rule <k> lvalue(X:Id => this . X) ...</k>  <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context lvalue((HOLE . _)::Exp)
+
+/*  rule lvalue(objectClosure(<crntClass> C </crntClass>
+                            <envStack>... envStackFrame(C,EnvC) EStack </envStack>)
+              . X
+              => lookupMember(<envStack> envStackFrame(C,EnvC) EStack </envStack>,
+                              X))  */
+  rule lvalue(objectClosure(Class, ListItem(envStackFrame(Class,Env)) EStack)
+              . X
+              => lookupMember(ListItem(envStackFrame(Class,Env)) EStack,
+                              X))
+  rule lvalue(objectClosure(Class, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+              . _X)
+    when Class =/=K Class'
+

      Lookup member

      + +

      It searches for the given member in the given environment stack,
      +starting with the most concrete class and going up in the hierarchy.

      +
        // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookupMember(List, Id)  [function]
+  /*
+  syntax KItem ::= lookupMember(EnvStackCell,Id)  [function]
+  */
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env>... X|->L ...</env>) ...</envStack>, X)
+//    => lookup(L)
+  rule lookupMember(ListItem(envStackFrame(_, X|->L _)) _, X)
+    => lookup(L)
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env> Env </env>) => .List ...</envStack>, X)
+//    when notBool(X in keys(Env))
+  rule lookupMember(ListItem(envStackFrame(_, Env)) Rest, X) =>
+       lookupMember(Rest, X)
+    when notBool(X in keys(Env))
+//TODO: beautify the above
+
+endmodule
+

      Go to Lesson 2, KOOL typed dynamic.

      +

      KOOL — Untyped

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped KOOL language. KOOL
      +is aimed at being a pedagogical and research language that captures
      +the essence of the object-oriented programming paradigm. Its untyped
      +variant discussed here is simpler than the typed one, ignoring several
      +intricate aspects of types in the presence of objects. A program
      +consists of a set of class declarations. Each class can extend at
      +most one other class (KOOL is single-inheritance). A class can
      +declare a set of fields and a set of methods, all public and called
      +the class' members. Specifically, KOOL includes the
      +following features:

      +
        +
      • +

        Class declarations, where a class may or may not explicitly
        +extend another class. In case a class does not explicitly extend
        +another class, then it is assumed that it extends the default top-most
        +and empty (i.e., no members) class called Object. Each class
        +is required to declare precisely one homonymous method, called its
        +constructor. Each valid program should contain one class
        +named Main, whose constructor, Main(), takes no
        +arguments. The execution of a program consists of creating an object
        +instance of class Main and invoking the constructor
        +Main() on it, that is, of executing new Main();.

        +
        • +
      • +

        All features of SIMPLE (see examples/simple/untyped),
        +i.e., multidimensional arrays, function (here called "method")
        +abstractions with call-by-value parameter passing style and static
        +scoping, blocks with locals, input/output, parametric exceptions, and
        +concurrency via dynamic thread creation/termination and synchronization.
        +The only change in the syntax of SIMPLE when imported in KOOL is the
        +function declaration keyword, function, which is changed into
        +method. The exact same desugaring macros from SIMPLE are
        +also included in KOOL. We can think of KOOL's classes as embedding
        +SIMPLE programs (extended with OO constructs, as discussed next).

        +
        • +
      • +

        Object creation using the new C(e1,...,en)
        +expression construct. An object instance of class C is first
        +created and then the constructor C(e1,...,en) is implicitly
        +called on that object. KOOL only allows (and requires) one
        +constructor per class. The class constructor can be called either
        +implicitly during a new object creation for the class, or explicitly.
        +The superclass constructor is not implicitly invoked when a
        +class constructor is invoked; if you want to invoke the superclass
        +constructor from a subclass constructor then you have to do it
        +explicitly.

        +
        • +
      • +

        An expression construct this, which evaluates to the
        +current object.

        +
        • +
      • +

        An expression construct super, which is used (only) in
        +combination with member lookup (see next) to refer to a superclass
        +field or method.

        +
        • +
      • +

        A member lookup expression construct e.x, where e
        +is an expression (either an expression expected to evaluate to an object
        +or the super construct) and x is a class member name,
        +that is, a field or a method name.

        +
        • +
      • +

        Expression constructs e instanceOf C and
        +(C) e, where e is an expression expected
        +to evaluate to an object and C a class name. The former
        +tells whether the class of e is a subclass of C,
        +that is, whether e can be used as an instance of C,
        +and the latter changes the class of e to C. These
        +operations always succeed: the former returns a Boolean value, while
        +the latter changes the current class of e to C
        +regardless of whether it is safe to do so or not. The typed version
        +of KOOL will check the safety of casting by ensuring that the instance
        +class of the object is a subclass of C. In untyped KOOL we
        +do not want to perform this check because we want to allow the
        +programmer maximum of flexibility: if one always accesses only
        +available members, then the program can execute successfully despite
        +the potentially unsafe cast.

        +
        • +
      +

      There are some specific aspects of KOOL that need to be discussed.

      +

      First, KOOL is higher-order, allowing function abstractions to be
      +treated like any other values in the language. For example, if
      +m is a method of object e then e.m
      +evaluates to the corresponding function abstraction. The function
      +abstraction is in fact a closure, because in addition to the method
      +parameters and body it also encapsulates the object value (i.e., the
      +environment of the object together with its current class—see below)
      +that e evaluates to. This way, function abstractions can be
      +invoked anywhere and have the capability to change the state of their
      +object. For example, if m is a method of object e
      +which increments a field c of e when invoked, and if
      +getm is another method of e which simply returns
      +m when invoked, then the double application
      +(e.getm())() has the same effect as e.m(), that is,
      +increments the counter c of e. Note that the
      +higher-order nature of KOOL was not originally planned; it came as a
      +natural consequence of evaluating methods to closures and we decided
      +to keep it. If you do not like it then do not use it.

      +

      Second, since all the fields and methods are public in KOOL and since
      +they can be redeclared in subclasses, it is not immediately clear how
      +to lookup the member x when we write e.x and
      +e is different from super. We distinguish two cases,
      +depending on whether e.x occurs in a method invocation
      +context (i.e., e.x(...)) or in a field context. KOOL has
      +dynamic method dispatch, so if e.x is invoked as a method
      +then x will be searched for starting with the instance class of
      +the object value to which e evaluates. If e.x
      +occurs in a non-method-invocation context then x will be
      +treated as a field (although it may hold a method closure due to the
      +higher-order nature of KOOL) and thus will be searched starting with
      +the current class of the object value of e (which, because of
      +this and casting, may be different from its instance class).
      +In order to achieve the above, each object value will consist of a
      +pair holding the current class of the object and an environment stack
      +with one layer for each class in the object's instance class hierarchy.

      +

      Third, although KOOL is dynamic method dispatch, its capabilities
      +described above are powerful enough to allow us to mimic static
      +method dispatch. For example, suppose that you want to invoke method
      +m() statically. Then all you need to do is to declare a
      +local variable and bind it to m, for example var staticm = m;, and
      +then call staticm(). This works because
      +staticm is first bound to the method closure that m
      +evaluates to, and then looked up as any local variable when invoked.
      +We only enable the dynamic method dispatch when we have an object
      +member on an application position, e.g., m().

      +

      In what follows, we limit our comments to the new, KOOL-specific
      +aspects of the language. We refer the reader to the untyped SIMPLE
      +language for documentation on the the remaining features, because
      +those were all borrowed from SIMPLE.

      +
      module KOOL-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of KOOL extends that of SIMPLE with object-oriented
      +constructs. We removed from the K annotated syntax of SIMPLE two
      +constructs, namely the one for function declarations (because we want
      +to call them methods now) and the one for function application
      +(because application is not strict in the first argument
      +anymore—needs to initiate dynamic method dispatch). The additional
      +syntax includes:

      +
        +
      • First, we need a new dedicated identifier, Object, for
        +the default top-most class.
        • +
      • Second, we rename the function keyword of SIMPLE into method.
        • +
      • Third, we add syntax for class declarations together with a
        +macro making classes which extend nothing to extend Object.
        • +
      • Fourth, we change the strictness attribute of application
        +into strict(2).
        • +
      • Finally, we add syntax and corresponding strictness
        +for the KOOL object-oriented constructs.
        • +
      +
        syntax Id ::= "Object" [token] | "Main" [token]
+
+  syntax Stmt ::= "var" Exps ";"
+                | "method" Id "(" Ids ")" Block  // called "function" in SIMPLE
+                | "class" Id Block               // KOOL
+                | "class" Id "extends" Id Block  // KOOL
+
+  syntax Exp ::= Int | Bool | String | Id
+               | "this"                                 // KOOL
+               | "super"                                // KOOL
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]    // KOOL
+               | "(" Id ")" Exp          [strict(2)]    // KOOL  cast
+               | "new" Id "(" Exps ")"   [strict(2)]    // KOOL
+               | Exp "." Id                             // KOOL
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict(2)]    // was strict in SIMPLE
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Ids  ::= List{Id,","}
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+
+  syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+

      Old desugaring rules, from SIMPLE

      +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule var E1::Exp, E2::Exp, Es::Exps; => var E1; var E2, Es;       [anywhere]
+  rule var X::Id = E; => var X; X = E;                              [anywhere]
+

      New desugaring rule

      +
        rule class C:Id S => class C extends Object S                     // KOOL
+
+endmodule
+

      Semantics

      + +

      We first discuss the new configuration of KOOL, which extends that of
      +SIMPLE. Then we include the semantics of the constructs borrowed from
      +SIMPLE unchanged; we refrain from discussing those, because they were
      +already discussed in the K definition of SIMPLE. Then we discuss
      +changes to SIMPLE's semantics needed for the more general meaning of
      +the previous SIMPLE constructs (for example for thread spawning,
      +assignment, etc.). Finally, we discuss in detail the
      +semantics of the additional KOOL constructs.

      +
      module KOOL-UNTYPED
+  imports KOOL-UNTYPED-SYNTAX
+  imports DOMAINS
+

      Configuration

      + +

      KOOL removes one cell and adds two nested cells to the configuration
      +of SIMPLE. The cell which is removed is the one holding the global
      +environment, because a KOOL program consists of a set of classes only,
      +with no global declarations. In fact, since informally speaking each
      +KOOL class now includes a SIMPLE program, it is safe to say that the
      +global variables in SIMPLE became class fields in KOOL. Let us now
      +discuss the new cells that are added to the configuration of SIMPLE.

      +
        +
      • +

        The cell crntObj holds data pertaining to the current
        +object, that is, the object environment in which the code in cell
        +k executes: crntClass holds the current class (which
        +can change as methods of the current object are invoked);
        +envStack holds the stack of environments as a list,
        +each layer corresponding to one class in the objects' instance class
        +hierarchy; location, which is optional, holds the location in
        +the store where the current object is or has to be located (this is
        +useful both for method closures and for the semantics of object
        +creation).

        +
        • +
      • +

        The cell classes holds all the declared classes, each
        +class being held in its own class cell which contains a name
        +(className), a parent (extends), and the actual
        +member declarations (declarations).

        +
        • +
      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+  syntax EnvCell
+  syntax ControlCell
+  syntax EnvStackCell
+  syntax CrntObjCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="yellow">
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      //<br/> // TODO(KORE): support latex annotations #1799
+                        <crntObj color="Fuchsia">  // KOOL
+                           <crntClass> Object </crntClass>
+                           <envStack> .List </envStack>
+                           <location multiplicity="?"> .K </location>
+                        </crntObj>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                      <id color="pink"> 0 </id>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <classes color="Fuchsia">        // KOOL
+                     <classData multiplicity="*" type="Map" color="Fuchsia">
+                        // the Map has as its key the first child of the cell,
+                        // in this case the className cell.
+                        <className color="Fuchsia"> Main </className>
+                        <baseClass color="Fuchsia"> Object </baseClass>
+                        <declarations color="Fuchsia"> .K </declarations>
+                     </classData>
+                  </classes>
+                </T>
+

      Unchanged Semantics from untyped SIMPLE

      + +

      The semantics below is taken over from SIMPLE unchanged.
      +The semantics of function declaration and invocation, including the
      +use of the special lambda abstraction value, needs to change
      +in order to account for the fact that methods are now invoked into
      +their object's environment. The semantics of function return actually
      +stays unchanged. Also, the semantics of program initialization is
      +different: now we have to create an instance of the Main
      +class which also calls the constructor Main(), while in
      +SIMPLE we only had to invoke the function Main().
      +Finally, the semantics of thread spawning needs to change, too: the
      +parent thread needs to also share its object environment with the
      +spawned thread (in addition to its local environment, like in SIMPLE).
      +This is needed in order to be able to spawn method invokations under
      +dynamic method dispatch; for example, spawn { run(); }
      +will need to look up the method run() in the newly created
      +thread, operation which will most likely fail unless the child thread
      +sees the object environment of the parent thread. Note that the
      +spawn statement of KOOL is more permissive than the threads
      +of Java. In fact, the latter can be implemented in terms of our
      +spawn—see the program threads.kool for a sketch.

      +

      Below is a subset of the values of SIMPLE, which are also values
      +of KOOL. We will add other values later in the semantics, such as
      +object and method closures.

      +
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax KResult ::= Vals
+

      The semantics below are taken verbatim from the untyped SIMPLE
      +definition.

      +
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
+
+  context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    when N >=Int 0
+
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         var $1=X;
+         for(var $2=0; $2 <= N1 - 1; ++$2) {
+           var X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
+
+  rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+
+  context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
+
+  rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+
+
+  rule sizeOf(array(_,N)) => N
+

      The semantics of function application needs to change into dynamic
      +method dispatch invocation, which is defined shortly. However,
      +interestingly, the semantics of return stays unchanged.

      +
        rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem(fstackFrame(Env,K,XS,<crntObj> CO </crntObj>)) => .List ...</fstack>
+         <xstack> _ => XS </xstack>
+         <crntObj> _ => CO </crntObj>
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+
+
+  rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
+
+  context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+
+
+  rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
+
+  rule S1::Stmt S2::Stmt => S1 ~> S2
+
+  rule _:Val; => .
+
+  rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
+  rule while (E) S => if (E) {S while(E)S}
+
+  rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+
+
+  syntax KItem ::= xstackFrame(Id,Stmt,K,Map,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Id,Stmt,K,Map,K)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem(xstackFrame(X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem(xstackFrame(X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      Thread spawning needs a new semantics, because we want the child
      +thread to also share the object environment with its parent. The new
      +semantics of thread spawning will be defined shortly. However,
      +interestingly, the other concurrency constructs keep their semantics
      +from SIMPLE unchanged.

      +
        // TODO(KORE): ..Bag should be . throughout this definition #1772
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+  /*
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .)
+  */
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
+  rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
+  rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    when (notBool(V in Busy:Set))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
+  rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+    when N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
+  rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Unchanged auxiliary operations from untyped SIMPLE

      + +
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+
+  // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookup(Int)
+  /*
+  syntax KItem ::= lookup(Int)
+  */
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+  // TODO: How can we make sure that the second rule above applies before the first one?
+  //       Probably we'll deal with this using strategies, eventually.
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
+
+  syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  when N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  when N <=Int M
+

      Changes to the existing untyped SIMPLE semantics

      + +

      When we extend a language, sometimes we need to do more than just add
      +new language constructs and semantics for them. Sometimes we want to
      +also extend the semantics of existing language constructs, in order to
      +get more from them.

      +

      Program initialization

      + +

      In SIMPLE, once all the global declarations were processed, the
      +function main() was invoked. In KOOL, the global
      +declarations are classes, and their specific semantics is given
      +shortly; essentially, they are pre-processed one by one and added
      +into the class cell structure in the configuration.
      +Once all the classes are processed, the computation item
      +execute, which was placed right after the program in the
      +initial configuration, is reached. In SIMPLE, the program was
      +initialized by calling the method main(). In KOOL, the
      +program is initialized by creating an object instance of class
      +Main. This will also implicitly call the method
      +Main() (the Main class constructor). The emptiness
      +of the env cell below is just a sanity check, to make sure
      +that the user has not declared anything but classes at the top level
      +of the program.

      +
        syntax KItem ::= "execute"
+  rule <k> execute => new Main(.Exps); </k> <env> .Map </env>
+

      The semantics of new (defined below) requires the
      +execution of all the class' declarations (and also of its
      +superclasses').

      +

      Object and method closures

      + +

      Before we can define the semantics of method application (previously
      +called function application in SIMPLE), we need to add two more values
      +to the language, namely object and method closures:

      +
        syntax Val ::= objectClosure(Id, List)
+               | methodClosure(Id,Int,Ids,Stmt)
+

      An object value consists of an objectClosure-wrapped bag
      +containing the current class of the object and the environment stack
      +of the object. The current class of an object will always be one of
      +the classes mapped to an environment in the environment stack of the
      +object. A method closure encapsulates the method's parameters and
      +code (last two arguments), as well as the object context in which the
      +method code should execute. This object context includes the current
      +class of the object (the first argument of methodClosure) and
      +the object environment stack (located in the object stored at the
      +location specified as the second argument of methodClosure).

      +

      Method application

      + +

      KOOL has a complex mechanism to invoke methods, because it allows both
      +dynamic method dispatch and methods as first-class-citizen values (the
      +latter making it a higher-order language). The invocation mechanism
      +will be defined later. What is sufficient to know for now is that
      +the two arguments of the application construct eventually reduce to
      +values, the first being a method closure and the latter a list of
      +values. The semantics of the method closure application is then as
      +expected: the local environment and control are stacked, then we
      +switch to method closure's class and object environment and execute
      +the method body. The mkDecls construct is the one that came
      +with the unchanged semantics of SIMPLE above.

      +
        syntax KItem ::= fstackFrame(Map,K,List,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Map,K,K)
+
+  rule <k> methodClosure(Class,OL,Xs,S)(Vs:Vals) ~> K
+           => mkDecls(Xs,Vs) S return; </k>
+       <env> Env => .Map </env>
+       <store>... OL |-> objectClosure(_, EnvStack)...</store>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+          <xstack> XS </xstack>
+          <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj' </crntObj>))
+          ...</fstack>
+          <crntObj> Obj' => <crntClass> Class </crntClass> <envStack> EnvStack </envStack> </crntObj>
+       </control>
+

      Spawn

      + +

      We want to extend the semantics of spawn to also share the
      +current object environment with the child thread, in addition to the
      +current environment. This extension will allow us to also use method
      +invocations in the spawned statements, which will be thus looked up as
      +expected, using dynamic method dispatch. This lookup operation would
      +fail if the child thread did not have access to its parent's object
      +environment.

      +
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+         <crntObj> Obj </crntObj>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+               <crntObj> Obj </crntObj>
+             ...</thread>)
+

      Semantics of the new KOOL constructs

      + +

      Class declaration

      + +

      Initially, the classes forming the program are moved into their
      +corresponding cells:

      +
        rule <k> class Class1 extends Class2 { S } => . ...</k>
+       <classes>... (.Bag => <classData>
+                            <className> Class1 </className>
+                            <baseClass> Class2 </baseClass>
+                            <declarations> S </declarations>
+                        </classData>)
+       ...</classes>
+

      Method declaration

      + +

      Like in SIMPLE, method names are added to the environment and bound
      +to their code. However, unlike in SIMPLE where each function was
      +executed in the same environment, namely the program global
      +environment, a method in KOOL needs to be executed into its object's
      +environment. Thus, methods evaluate to closures, which encapsulate
      +their object's context (i.e., the current class and environment stack
      +of the object) in addition to method's parameters and body. This
      +approach to bind method names to method closures in the environment
      +will also allow objects to pass their methods to other objects, to
      +dynamically change their methods by assigning them other method
      +closures, and even to allow all these to be done from other objects.
      +This gives the KOOL programmer a lot of power; one should use this
      +power wisely, though, because programs can become easily hard to
      +understand and reason about if one overuses these features.

      +
        rule <k> method F:Id(Xs:Ids) S => . ...</k>
+       <crntClass> Class:Id </crntClass>
+       <location> OL:Int </location>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> methodClosure(Class,OL,Xs,S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      New

      + +

      The semantics of new consists of two actions: memory
      +allocation for the new object and execution of the corresponding
      +constructor. Then the created object is returned as the result of the
      +new operation; the value returned by the constructor, if any,
      +is discarded. The current environment and object are stored onto the
      +stack and recovered after new (according to the semantics of
      +return borrowed from SIMPLE, when the statement
      +return this; in the rule below is reached and evaluated),
      +because the object creation part of new will destroy them.
      +The rule below also initializes the object creation process by
      +emptying the local environment and the current object, and allocating
      +a location in the store where the created object will be eventually
      +stored (this is what the storeObj task after the object
      +creation task in the rule below will do—its rule is defined
      +shortly). The location where the object will be stored is also made
      +available in the crntObj cell, so that method closures can
      +refer to it (see rule above).

      +
        syntax KItem ::= "envStackFrame" "(" Id "," Map ")"
+
+  rule <k> new Class:Id(Vs:Vals) ~> K
+           => create(Class) ~> storeObj ~> Class(Vs); return this; </k>
+       <env> Env => .Map </env>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control> <xstack> XS </xstack>
+         <crntObj> Obj
+                   => <crntClass> Object </crntClass>
+                      <envStack> ListItem(envStackFrame(Object, .Map)) </envStack>
+                      <location> L </location>
+         </crntObj>
+         <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj </crntObj>)) ...</fstack>
+       </control>
+

      The creation of a new object (the memory allocation part only) is
      +a recursive process, requiring to first create an object for the
      +superclass. A memory object representation is a layered structure:
      +for each class on the path from the instance class to the root of the
      +hierarchy there is a layer including the memory allocated for the
      +members (both fields and methods) of that class.

      +
        syntax KItem ::= create(Id)
+
+  rule <k> create(Class:Id)
+           => create(Class1) ~> setCrntClass(Class) ~> S ~> addEnvLayer ...</k>
+       <className> Class </className>
+       <baseClass> Class1:Id </baseClass>
+       <declarations> S </declarations>
+
+  rule <k> create(Object) => . ...</k>
+

      The next operation sets the current class of the current object.
      +This is necessary to be done at each layer, because the current class
      +of the object is enclosed as part of the method closures (see the
      +semantics of method declarations above).

      +
        syntax KItem ::= setCrntClass(Id)
+
+  rule <k> setCrntClass(C) => . ...</k>
+       <crntClass> _ => C </crntClass>
+

      The next operation adds a new tagged environment layer to the
      +current object and gets ready for the next layer by clearing the
      +environment (note that create expects the environment to be
      +empty).

      +
        syntax KItem ::= "addEnvLayer"
+
+  rule <k> addEnvLayer => . ...</k>
+       <env> Env => .Map </env>
+       <crntClass> Class:Id </crntClass>
+       <envStack> .List => ListItem(envStackFrame(Class, Env)) ...</envStack>
+

      The following operation stores the created object at the location
      +reserved by new. Note that the location reserved by
      +new was temporarily stored in the crntObj cell
      +precisely for this purpose. Now that the newly created object is
      +stored at its location and that all method closures are aware of it,
      +the location is unnecessary and thus we delete it from the
      +crntObj cell.

      +
        syntax KItem ::= "storeObj"
+
+  rule <k> storeObj => . ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> (<location> L:Int </location> => .Bag) </crntObj>
+       <store>... .Map => L |-> objectClosure(CC, ES) ...</store>
+

      Self reference

      + +

      The semantics of this is straightforward: evaluate to the
      +current object.

      +
        rule <k> this => objectClosure(CC, ES) ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> </crntObj>
+

      Object member access

      + +

      We can access an object member (field or method) either explicitly,
      +using the construct e.x, or implicitly, using only the member
      +name x directly. The borrowed semantics of SIMPLE will
      +already lookup a sole name in the local environment. The first rule
      +below reduces implicit member access to explicit access when the name
      +cannot be found in the local environment. There are two cases to
      +analyze for explicit object member access, depending upon whether the
      +object is a proper object or it is just a redirection to the parent
      +class via the construct super. In the first case, we
      +evaluate the object expression and lookup the member starting with the
      +current class (static scoping). Note the use of the conditional
      +evaluation context. In the second case, we just lookup the member
      +starting with the superclass of the current class. In both cases,
      +the lookupMember task eventually yields a lookup(L)
      +task for some appropriate location L, which will be further
      +solved with the corresponding rule borrowed from SIMPLE. Note that the
      +current object is not altered by super, so future method
      +invocations see the entire object, as needed for dynamic method dispatch.

      +
        rule <k> X:Id => this . X ...</k> <env> Env:Map </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id when (HOLE =/=K super)
+
+// TODO: explain how Assoc matching has been replaced with two rules here.
+// Maybe also improve it a bit.
+
+/*  rule objectClosure(<crntClass> Class:Id </crntClass>
+                     <envStack>... envStackFrame(Class,EnvC) EStack </envStack>)
+       . X:Id
+    => lookupMember(envStackFrame(Class,EnvC) EStack, X) */
+
+  rule objectClosure(Class:Id, ListItem(envStackFrame(Class,Env)) EStack)
+       . X:Id
+    => lookupMember(ListItem(envStackFrame(Class,Env)) EStack, X)
+  rule objectClosure(Class:Id, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+       . _X:Id
+    when Class =/=K Class'
+
+/*  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,EnvC) EStack </envStack> */
+  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> super . _X ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+

      Method invocation

      + +

      Unlike in SIMPLE, in KOOL application was declared strict only in its
      +second argument. That is because we want to ensure dynamic method
      +dispatch when the first argument is a method access. As a
      +consequence, we need to consider all the cases of interest for the
      +first argument and to explicitly say what to do in each case. In all
      +cases except for method access in a proper object (i.e., not
      +super), we want the same behavior for the first argument as
      +if it was not in a method invocation position. When it is a member
      +access (the third rule below), we look it up starting with the
      +instance class of the corresponding object. This ensures dynamic
      +dispatch for methods; it actually dynamically dispatches field
      +accesses, too, which is correct in KOOL, because one can assign method
      +closures to fields and the field appeared in a method invocation
      +context. The last context declaration below says that method
      +applications or array accesses are also allowed as first argument to
      +applications; that is because methods are allowed to return methods
      +and arrays are allowed to hold methods in KOOL, since it is
      +higher-order. If that is the case, then we want to evaluate the
      +method call or the array access.

      +
        rule <k> (X:Id => V)(_:Exps) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  rule <k> (X:Id => this . X)(_:Exps) ...</k>
+       <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id(_) when HOLE =/=K super
+
+  rule (objectClosure(_, EStack) . X
+    => lookupMember(EStack, X:Id))(_:Exps)
+
+/*  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,_) EStack </envStack> */
+  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> (super . _X)(_:Exps) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+
+  // TODO(KORE): fix getKLabel #1801
+  rule (A:Exp(B:Exps))(C:Exps) => A(B) ~> #freezerFunCall(C)
+  rule (A:Exp[B:Exps])(C:Exps) => A[B] ~> #freezerFunCall(C)
+  rule V:Val ~> #freezerFunCall(C:Exps) => V(C)
+  syntax KItem ::= "#freezerFunCall" "(" K ")"
+  /*
+  context HOLE(_:Exps)
+    when getKLabel(HOLE) ==K #klabel(`_(_)`) orBool getKLabel(HOLE) ==K #klabel(`_[_]`)
+  */
+

      Eventually, each of the rules above produces a lookup(L)
      +task as a replacement for the method. When that happens, we just
      +lookup the value at location L:

      +
        rule <k> (lookup(L) => V)(_:Exps) ...</k>  <store>... L |-> V:Val ...</store>
+    [group(lookup)]
+

      The value V looked up above is expected to be a method closure,
      +in which case the semantics of method application given above will
      +apply. Otherwise, the execution will get stuck.

      +

      Instance Of

      + +

      It searches the object environment for a layer corresponding to the
      +desired class. It returns true iff it can find the class,
      +otherwise it returns false; it only gets stuck when its first
      +argument does not evaluate to an object.

      +
        rule objectClosure(_, ListItem(envStackFrame(C,_)) _)
+       instanceOf C => true
+
+  rule objectClosure(_, (ListItem(envStackFrame(C,_)) => .List) _)
+       instanceOf C'  when C =/=K C'
+//TODO: remove the sort cast ::Id of C above, when sort inference bug fixed
+
+  rule objectClosure(_, .List) instanceOf _ => false
+

      Cast

      + +

      In untyped KOOL, we prefer to not check the validity of casting. In
      +other words, any cast is allowed on any object, simply changing the
      +current class of the object to the desired class. The execution will
      +get stuck later if one attempts to access a field which is not
      +available. Moreover, the execution may complete successfully even
      +in the presence of invalid casts, provided that each accessed member
      +during the current execution is, or happens to be, available.

      +
        rule (C) objectClosure(_ , EnvStack) => objectClosure(C ,EnvStack)
+

      KOOL-specific auxiliary declarations and operations

      + +

      Here we define all the auxiliary constructs used in the above
      +KOOL-specific semantics (those used in the SIMPLE fragment
      +have already been defined in a corresponding section above).

      +

      Objects as lvalues

      + +

      The current machinery borrowed with the semantics of SIMPLE allows us
      +to enrich the set of lvalues, this way allowing new means to assign
      +values to locations. In KOOL, we want object member names to be
      +lvalues, so that we can assign values to them using the already
      +existing machinery. The first rule below ensures that the object is
      +always explicit, the evaluation context enforces the object to be
      +evaluated, and finally the second rule initiates the lookup for the
      +member's location based on the current class of the object.

      +
        rule <k> lvalue(X:Id => this . X) ...</k>  <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context lvalue((HOLE . _)::Exp)
+
+/*  rule lvalue(objectClosure(<crntClass> C </crntClass>
+                            <envStack>... envStackFrame(C,EnvC) EStack </envStack>)
+              . X
+              => lookupMember(<envStack> envStackFrame(C,EnvC) EStack </envStack>,
+                              X))  */
+  rule lvalue(objectClosure(Class, ListItem(envStackFrame(Class,Env)) EStack)
+              . X
+              => lookupMember(ListItem(envStackFrame(Class,Env)) EStack,
+                              X))
+  rule lvalue(objectClosure(Class, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+              . _X)
+    when Class =/=K Class'
+

      Lookup member

      + +

      It searches for the given member in the given environment stack,
      +starting with the most concrete class and going up in the hierarchy.

      +
        // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookupMember(List, Id)  [function]
+  /*
+  syntax KItem ::= lookupMember(EnvStackCell,Id)  [function]
+  */
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env>... X|->L ...</env>) ...</envStack>, X)
+//    => lookup(L)
+  rule lookupMember(ListItem(envStackFrame(_, X|->L _)) _, X)
+    => lookup(L)
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env> Env </env>) => .List ...</envStack>, X)
+//    when notBool(X in keys(Env))
+  rule lookupMember(ListItem(envStackFrame(_, Env)) Rest, X) =>
+       lookupMember(Rest, X)
+    when notBool(X in keys(Env))
+//TODO: beautify the above
+
+endmodule
+

      Go to Lesson 2, KOOL typed dynamic.

      +

      KOOL — Typed — Dynamic

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K dynamic semantics of the typed KOOL language. It is
      +very similar to the semantics of the untyped KOOL, the difference
      +being that we now check the typing policy dynamically. Since we have
      +to now declare the types of variables and methods, we adopt a syntax
      +for those which is close to Java. Like in the semantics of
      +untyped KOOL, where we borrowed almost all the semantics of untyped
      +SIMPLE, we are going to also borrow much of the semantics of
      +dynamically typed SIMPLE here. We will highlight the differences
      +between the dynamically typed and the untyped KOOL as we proceed with
      +the semantics. In general, the type policy of the typed KOOL language
      +is similar to that of Java. You may find it useful to also read
      +the discussion in the preamble of the static semantics of typed KOOL
      +before proceeding.

      +
      module KOOL-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      Like for the untyped KOOL language, the syntax of typed KOOL extends
      +that of typed SIMPLE with object-oriented constructs.
      +The syntax below was produced by copying and modifying/extending the
      +syntax of dynamically typed SIMPLE. In fact, the only change we made
      +to the existing syntax of dynamically typed SIMPLE was to change the
      +strictness of the application construct like in untyped KOOL, from
      +strict to strict(2) (because application is not
      +strict in the first argument anymore due to dynamic method dispatch).
      +The KOOL-specific syntactic extensions are identical to those in
      +untyped KOOL.

      +
        syntax Id ::= "Object" [token] | "Main" [token]
+

      Types

      + +
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Id                              // KOOL class
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+  // TODO(KORE): drop klabel once issues #1913 are fixed
+  syntax Types ::= List{Type,","}   [klabel(_,_::Types)]
+  /*
+  syntax Types ::= List{Type,","}
+  */
+

      Declarations

      + +
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";" [avoid]
+                | Type Id "(" Params ")" Block    // stays like in typed SIMPLE
+                | "class" Id Block                // KOOL
+                | "class" Id "extends" Id Block   // KOOL
+

      Expressions

      + +
        syntax Exp ::= Int | Bool | String | Id
+               | "this"                                 // KOOL
+               | "super"                                // KOOL
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]    // KOOL
+               | "(" Id ")" Exp          [strict(2)]    // KOOL  cast
+               | "new" Id "(" Exps ")"   [strict(2)]    // KOOL
+               | Exp "." Id                             // KOOL
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict(2)]    // was strict in SIMPLE
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+

      Statements

      + +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"
+                | "try" Block "catch" "(" Param ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+

      Desugaring macros

      + +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S::Stmt} => {Start while(Cond){S Step;}}
+  rule T::Type E1::Exp, E2::Exp, Es::Exps; => T E1; T E2, Es;           [anywhere]
+  rule T::Type X::Id = E; => T X; X = E;                                [anywhere]
+
+  rule class C:Id S => class C extends Object S                     // KOOL
+
+endmodule
+

      Semantics

      + +

      We first discuss the new configuration, then we include the semantics of
      +the constructs borrowed from SIMPLE which stay unchanged, then those
      +whose semantics had to change, and finally the semantics of the
      +KOOL-specific constructs.

      +
      module KOOL-TYPED-DYNAMIC
+  imports KOOL-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS
+

      Configuration

      + +

      The configuration of dynamically typed KOOL is almost identical to
      +that of its untyped variant. The only difference is the cell
      +return, inside the control cell, whose role is to
      +hold the expected return type of the invoked method. That is because
      +we want to dynamically check that the value that a method returns has
      +the expected type.

      +
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+  syntax EnvCell
+  syntax ControlCellFragment
+  syntax EnvStackCell
+  syntax CrntObjCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="yellow">
+                      <k color="green"> ($PGM:Stmt ~> execute) </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                        <returnType color="LimeGreen"> void </returnType>  // KOOL
+                      //<br/> // TODO(KORE): support latex annotations #1799
+                        <crntObj color="Fuchsia">  // KOOL
+                           <crntClass> Object </crntClass>
+                           <envStack> .List </envStack>
+                           <location multiplicity="?"> .K </location>
+                        </crntObj>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                      <id color="pink"> 0 </id>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <classes color="Fuchsia">        // KOOL
+                     <classData multiplicity="*" type="Map" color="Fuchsia">
+                        <className color="Fuchsia"> Main </className>
+                        <baseClass color="Fuchsia"> Object </baseClass>
+                        <declarations color="Fuchsia"> .K </declarations>
+                     </classData>
+                  </classes>
+                </T>
+

      Unchanged semantics from dynamically typed SIMPLE

      + +

      The semantics below is taken over from dynamically typed SIMPLE
      +unchanged. Like for untyped KOOL, the semantics of function/method
      +declaration and invocation, and of program initialization needs to
      +change. Moreover, due to subtyping, the semantics of several imported
      +SIMPLE constructs can be made more general, such as that of the
      +return statement, that of the assignment, and that of the exceptions.
      +We removed all these from the imported semantics of SIMPLE below and
      +gave their modified semantics right after, together with the extended
      +semantics of thread spawning (which is identical to that of untyped
      +KOOL).

      +
        syntax Val ::= Int | Bool | String
+               | array(Type,Int,Int)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax KResult ::= Vals
+
+
+  syntax KItem ::= undefined(Type)  [latex(\bot_{#1})]
+
+  rule <k> T:Type X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
+
+  rule <k> T:Type X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(T, L +Int 1, N)
+                          (L +Int 1)...(L +Int N) |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    requires N >=Int 0
+
+  context _:Type _::Exp[HOLE::Exps];
+
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule T:Type X:Id[N1:Int, N2:Int, Vs:Vals];
+    => T[]<Vs> X[N1];
+       {
+         T[][]<Vs> $1=X;
+         for(int $2=0; $2 <= N1 - 1; ++$2) {
+           T X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
+
+  rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+
+  context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
+
+  rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 requires I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 requires I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(_:Type, L:Int, M:Int)[N:Int] => lookup(L +Int N)
+    requires N >=Int 0 andBool N <Int M  [anywhere]
+
+  rule sizeOf(array(_,_,N)) => N
+
+
+  syntax Val ::= nothing(Type)
+  rule <k> return; => return nothing(T); ...</k> <returnType> T </returnType>
+
+
+  rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
+
+  context (HOLE => lvalue(HOLE)) = _
+
+
+  rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
+
+  rule S1:Stmt S2:Stmt => S1 ~> S2
+
+
+  rule _:Val; => .
+
+
+  rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
+
+  rule while (E) S => if (E) {S while(E)S}
+
+
+  rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    requires typeOf(V) ==K int orBool typeOf(V) ==K string  [group(print)]
+  rule print(.Vals); => .
+
+
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
+  rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
+  rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    requires (notBool(V in Busy:Set))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
+  rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+    requires N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
+  rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+

      Unchanged auxiliary operations from dynamically typed SIMPLE

      + +
        syntax Stmt ::= mkDecls(Params,Vals)  [function]
+  rule mkDecls((T:Type X:Id, Ps:Params), (V:Val, Vs:Vals))
+    => T X=V; mkDecls(Ps,Vs)
+  rule mkDecls(.Params,.Vals) => {}
+
+  syntax Exp ::= lookup(Int)
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+  rule <k> lvalue(X:Id => loc(L)) ...</k>  <env>... X |-> L:Int ...</env>
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
+  syntax Type ::= Type "<" Vals ">"  [function]
+  rule T:Type<_,Vs:Vals> => T[]<Vs>
+  rule T:Type<.Vals> => T
+
+  syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  requires N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  requires N <=Int M
+
+  syntax Type ::= typeOf(K)  [function]
+  rule typeOf(_:Int) => int
+  rule typeOf(_:Bool) => bool
+  rule typeOf(_:String) => string
+  rule typeOf(array(T,_,_)) => (T[])
+  rule typeOf(undefined(T)) => T
+  rule typeOf(nothing(T)) => T
+
+  syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+

      Changes to the existing dynamically typed SIMPLE semantics

      + +

      We extend/change the semantics of several SIMPLE constructs in order
      +to take advantage of the richer KOOL semantic infrastructure and thus
      +get more from the existing SIMPLE constructs.

      +

      Program initialization

      + +

      Like in untyped KOOL.

      +
        syntax KItem ::= "execute"
+  rule <k> execute => new Main(.Exps); </k> <env> .Map </env>
+

      Method application

      + +

      The only change to untyped KOOL's values is that method closures are
      +now typed (their first argument holds their type):

      +
       syntax Val ::= objectClosure(Id,List)
+              | methodClosure(Type,Id,Int,Params,Stmt)
+

      The type held by a method clossure will be the entire type of the
      +method, not only its result type like the lambda-closure of typed
      +SIMPLE. The reason for this change comes from the the need to
      +dynamically upcast values when passed to contexts where values of
      +superclass types are expected; since we want method closures to be
      +first-class-citizen values in our language, we have to be able to
      +dynamically upcast them, and in order to do that elegantly it is
      +convenient to store the entire ``current type'' of the method closure
      +instead of just its result type. Note that this was unnecessary in
      +the semantics of the dynamically typed SIMPLE language.

      +

      Method closure application needs to also set a new return type in
      +the return cell, like in dynamically typed SIMPLE, in order
      +for the values returned by its body to be checked against the return
      +type of the method. To do this correctly, we also need to stack the
      +current status of the return cell and then pop it when the
      +method returns. We have to do the same with the current object
      +environment, so we group them together in the stack frame.

      +
        syntax KItem ::= fstackFrame(Map, K, List, Type, K)
+
+  rule <k> methodClosure(_->T,Class,OL,Ps,S)(Vs:Vals) ~> K
+           => mkDecls(Ps,Vs) S return; </k>
+       <env> Env => .Map </env>
+       <store>... OL |-> objectClosure(_, EStack)...</store>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+          <fstack> .List => ListItem(fstackFrame(Env, K, XS, T', <crntObj> Obj' </crntObj>)) ...</fstack>
+          <xstack> XS </xstack>
+          <returnType> T' => T </returnType>
+          <crntObj> Obj' => <crntClass> Class </crntClass> <envStack> EStack </envStack> </crntObj>
+       </control>
+

      At method return, we have to check that the type of the returned
      +value is a subtype of the expected return type. Moreover, if that is
      +the case, then we also upcast the returned value to one of the
      +expected type. The computation item unsafeCast(V,T) changes
      +the typeof V to T without any additional checks; however, it only
      +does it when V is an object or a method, otherwise it returns V
      +unchanged.

      +
        rule <k> return V:Val; ~> _
+           => subtype(typeOf(V), T) ~> true? ~> unsafeCast(V, T) ~> K
+       </k>
+       <control>
+         <fstack> ListItem(fstackFrame(Env, K, XS, RT, <crntObj> CO </crntObj>)) => .List ...</fstack>
+         <xstack> _ => XS </xstack>
+         <returnType> T:Type => RT </returnType>
+         <crntObj> _ => CO </crntObj>
+       </control>
+       <env> _ => Env </env>
+

      Assignment

      + +

      Typed KOOL allows to assign subtype instance values to supertype
      +lvalues. The semantics of assignment below is similar in spirit to
      +dynamically typed SIMPLE's, but a check is performed that the assigned
      +value's type is a subtype of the location's type. If that is the
      +case, then the assigned value is returned as a result and stored, but
      +it is upcast appropriately first, so the context will continue to see
      +a value of the expected type of the location. Note that the type of a
      +location is implicit in the type of its contents and it never changes
      +during the execution of a program; its type is assigned when the
      +location is allocated and initialized, and then only type-preserving
      +values are allowed to be stored in each location.

      +
        rule <k> loc(L) = V:Val
+           => subtype(typeOf(V),typeOf(V')) ~> true?
+              ~> unsafeCast(V, typeOf(V')) ...</k>
+       <store>... L |-> (V' => unsafeCast(V, typeOf(V'))) ...</store>
+    [group(assignment)]
+

      Typed exceptions

      + +

      Exceptions are propagated now until a catch that can handle them is
      +encountered.

      +
        syntax KItem ::= xstackFrame(Param, Stmt, K, Map, K)
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(P) S2 => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem(xstackFrame(P, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _
+        => if (subtype(typeOf(V),T)) { T X = V; S2 } else { throw V; } ~> K
+       </k>
+       <control>
+         <xstack> ListItem(xstackFrame(T:Type X:Id, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+

      Spawn

      + +

      Like in untyped KOOL.

      +
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+         <crntObj> Obj </crntObj>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+               <crntObj> Obj </crntObj>
+             ...</thread>)
+

      Semantics of the new KOOL constructs

      + +

      Class declaration

      + +

      Like in untyped KOOL.

      +
        rule <k> class Class1 extends Class2 { S } => . ...</k>
+       <classes>... (.Bag => <classData>
+                            <className> Class1 </className>
+                            <baseClass> Class2 </baseClass>
+                            <declarations> S </declarations>
+                        </classData>)
+       ...</classes>
+

      Method declaration

      + +

      Methods are now typed and we need to store their types in their
      +closures, so that their type contract can be checked at invocation
      +time. The rule below is conceptually similar to that of untyped KOOL;
      +the only difference is the addition of the types.

      +
        rule <k> T:Type F:Id(Ps:Params) S => . ...</k>
+       <crntClass> C </crntClass>
+       <location> OL </location>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L|->methodClosure(getTypes(Ps)->T,C,OL,Ps,S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+

      New

      + +

      The semantics of new in dynamically typed KOOL is also
      +similar to that in untyped KOOL, the main difference being the
      +management of the return types. Indeed, when a new object is created
      +we also have to stack the current type in the return cell in
      +order to be recovered after the creation of the new object. Only the
      +first rule below needs to be changed; the others are identical to
      +those in untyped KOOL.

      +
        syntax KItem ::= envStackFrame(Id, Map)
+
+  rule <k> new Class:Id(Vs:Vals) ~> K
+           => create(Class) ~> (storeObj ~> ((Class(Vs)); return this;)) </k>
+       <env> Env => .Map </env>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+         <xstack> XS </xstack>
+         <crntObj> Obj
+                   => <crntClass> Object </crntClass>
+                      <envStack> ListItem(envStackFrame(Object, .Map)) </envStack>
+                      <location> L </location>
+         </crntObj>
+         <returnType> T => Class </returnType>
+         <fstack> .List => ListItem(fstackFrame(Env, K, XS, T, <crntObj>Obj</crntObj>)) ...</fstack>
+       </control>
+
+  syntax KItem ::= create(Id)
+
+  rule <k> create(Class:Id)
+           => create(Class1) ~> setCrntClass(Class) ~> S ~> addEnvLayer ...</k>
+       <className> Class </className>
+       <baseClass> Class1:Id </baseClass>
+       <declarations> S </declarations>
+
+  rule <k> create(Object) => . ...</k>
+
+  syntax KItem ::= setCrntClass(Id)
+
+  rule <k> setCrntClass(C) => . ...</k>
+       <crntClass> _ => C </crntClass>
+
+  syntax KItem ::= "addEnvLayer"
+
+  rule <k> addEnvLayer => . ...</k>
+       <env> Env => .Map </env>
+       <crntClass> Class:Id </crntClass>
+       <envStack> .List => ListItem(envStackFrame(Class, Env)) ...</envStack>
+
+  syntax KItem ::= "storeObj"
+
+  rule <k> storeObj => . ...</k>
+       <crntObj>
+         <crntClass> Class </crntClass>
+         <envStack> EStack </envStack>
+         (<location> L:Int </location> => .Bag)
+       </crntObj>
+       <store>... .Map => L |-> objectClosure(Class, EStack) ...</store>
+

      Self reference

      + +

      Like in untyped KOOL.

      +
        rule <k> this => objectClosure(Class, EStack) ...</k>
+       <crntObj>
+         <crntClass> Class </crntClass>
+         <envStack> EStack </envStack>
+         ...
+       </crntObj>
+

      Object member access

      + +

      Like in untyped KOOL.

      +
        rule <k> X:Id => this . X ...</k> <env> Env:Map </env>
+    requires notBool(X in keys(Env))
+
+  context HOLE . _::Id requires (HOLE =/=K super)
+
+/*  rule objectClosure(<crntObj> <crntClass> Class:Id </crntClass>
+                     <envStack>... ListItem((Class,EnvC:EnvCell)) EStack </envStack> </crntObj>)
+       . X:Id
+    => lookupMember(<envStack> ListItem((Class,EnvC)) EStack </envStack>, X) */
+  rule objectClosure(Class:Id,
+                     ListItem(envStackFrame(Class,Env)) EStack)
+       . X:Id
+    => lookupMember(ListItem(envStackFrame(Class,Env)) EStack, X)
+  rule objectClosure(Class:Id,
+                     (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack)
+       . _X:Id
+    requires Class =/=K Class'
+
+/*  rule <k> super . X => lookupMember(<envStack>EStack</envStack>, X) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... ListItem((Class,EnvC:EnvCell)) EStack </envStack> */
+  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> super . _X ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack </envStack>
+    requires Class =/=K Class'
+

      Method invocation

      + +

      The method lookup is the same as in untyped KOOL.

      +
        rule <k> (X:Id => V)(_:Exps) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  rule <k> (X:Id => this . X)(_:Exps) ...</k>
+       <env> Env </env>
+    requires notBool(X in keys(Env))
+
+  context HOLE._::Id(_) requires HOLE =/=K super
+
+  rule (objectClosure(_, EStack) . X
+    => lookupMember(EStack, X:Id))(_:Exps)
+
+/*  rule <k> (super . X
+            => lookupMember(<envStack>EStack</envStack>,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... ListItem((Class,_)) EStack </envStack> */
+  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> (super . _X)(_:Exps)...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack </envStack>
+    requires Class =/=K Class'
+
+  // TODO(KORE): fix getKLabel #1801
+  rule (A:Exp(B:Exps))(C:Exps) => A(B) ~> #freezerFunCall(C)
+  rule (A:Exp[B:Exps])(C:Exps) => A[B] ~> #freezerFunCall(C)
+  rule V:Val ~> #freezerFunCall(C:Exps) => V(C)
+  syntax KItem ::= "#freezerFunCall" "(" K ")"
+  /*
+  context HOLE(_:Exps)
+    requires getKLabel HOLE ==KLabel '_`(_`) orBool getKLabel HOLE ==KLabel '_`[_`]
+  */
+
+  rule <k> (lookup(L) => V)(_:Exps) ...</k>  <store>... L |-> V:Val ...</store>
+    [group(lookup)]
+

      Instance of

      + +

      Like in untyped KOOL.

      +
        rule objectClosure(_, ListItem(envStackFrame(C,_)) _)
+       instanceOf C => true
+
+  rule objectClosure(_, (ListItem(envStackFrame(C::Id,_)) => .List) _)
+       instanceOf C'  requires C =/=K C'
+
+  rule objectClosure(_, .List) instanceOf _ => false
+

      Cast

      + +

      Unlike in untyped KOOL, in typed KOOL we actually check that the object
      +can indeed be cast to the claimed type.

      +
        rule (C:Id) objectClosure(Irrelevant, EStack)
+    => objectClosure(Irrelevant, EStack) instanceOf C ~> true?
+       ~> objectClosure(C, EStack)
+

      KOOL-specific auxiliary declarations and operations

      + +

      Objects as lvalues

      + +

      Like in untyped KOOL.

      +
        rule <k> lvalue(X:Id => this . X) ...</k>  <env> Env </env>
+    requires notBool(X in keys(Env))
+
+  context lvalue((HOLE . _)::Exp)
+
+/*  rule lvalue(objectClosure(<crntObj> <crntClass> C </crntClass>
+                            <envStack>... ListItem((C,EnvC:EnvCell)) EStack </envStack> </crntObj>)
+              . X
+              => lookupMember(<envStack> ListItem((C,EnvC)) EStack </envStack>,
+                              X)) */
+  rule lvalue(objectClosure(C:Id,
+                            ListItem(envStackFrame(C,Env)) EStack)
+              . X
+              => lookupMember(ListItem(envStackFrame(C,Env)) EStack,
+                              X))
+  rule lvalue(objectClosure(C,
+                            (ListItem(envStackFrame(C',_)) => .List) _EStack)
+              . _X)
+    requires C =/=K C'
+

      Lookup member

      + +

      Like in untyped KOOL.

      +
        syntax Exp ::= lookupMember(List,Id)  [function]
+
+  rule lookupMember(ListItem(envStackFrame(_, X |-> L _)) _, X) => lookup(L)
+
+  // TODO: fix rule below as shown once we support functions with deep rewrites
+  // rule lookupMember(<envStack> ListItem((_, <env> Env </env>)) => .List
+  //                     ...</envStack>, X)
+  //   requires notBool(X in keys(Env))
+  rule lookupMember(ListItem(envStackFrame(_, Env)) L, X)
+    => lookupMember(L, X)
+    requires notBool(X in keys(Env))
+

      typeOf for the additional values}

      + +
        rule typeOf(objectClosure(C,_)) => C
+  rule typeOf(methodClosure(T:Type,_,_,_Ps:Params,_)) => T
+

      Subtype checking

      + +

      The subclass relation induces a subtyping relation.

      +
        syntax Exp ::= subtype(Types,Types)
+
+  rule subtype(T:Type, T) => true
+
+  rule <k> subtype(C1:Id, C:Id) => subtype(C2, C) ...</k>
+       <className> C1 </className>
+       <baseClass> C2:Id </baseClass>
+    requires C1 =/=K C
+
+  rule subtype(Object,Class:Id) => false
+    requires Class =/=K Object
+
+  rule subtype(Ts1->T2,Ts1'->T2') => subtype(((T2)::Type,Ts1'),((T2')::Type,Ts1))
+
+// Note that the following rule would be wrong!
+//  rule subtype(T[],T'[]) => subtype(T,T')
+
+  rule subtype((T:Type,Ts),(T':Type,Ts')) => subtype(T,T') && subtype(Ts,Ts')
+    requires Ts =/=K .Types
+  rule subtype(.Types,.Types) => true
+

      Unsafe Casting

      + +

      Performs unsafe casting. One should only use it in combination with
      +the subtype relation above.

      +
        syntax Val ::= unsafeCast(Val,Type)  [function]
+
+  rule unsafeCast(objectClosure(_,EStack), C:Id)
+    => objectClosure(C,EStack)
+
+  rule unsafeCast(methodClosure(_T',C,OL,Ps,S), T) => methodClosure(T,C,OL,Ps,S)
+
+  rule unsafeCast(V:Val, T:Type) => V  requires typeOf(V) ==K T
+

      Generic guard

      + +

      A generic computational guard: it allows the computation to continue
      +only if a prefix guard evaluates to true.

      +
        syntax KItem ::= "true?"
+  rule true ~> true? => .
+
+endmodule
+

      Go to Lesson 3, KOOL typed static.

      +

      KOOL — Typed — Static

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K static semantics of the typed KOOL language.
      +It extends the static semantics of typed SIMPLE with static semantics
      +for the object-oriented constructs. Also, the static semantics of
      +some of the existing SIMPLE constructs need to change, in order to
      +become more generous with regards to the set of accepted programs,
      +mostly due to subtyping. For example, the assignment construct
      +x = e required that both the variable x and the
      +expression e had the same type in SIMPLE. In KOOL, the type
      +of e can be a subtype of the type of x.
      +Specifically, we define the following typing policy for KOOL,
      +everything else not mentioned below borrowing its semantics from
      +SIMPLE:

      +
        +
      • +

        Each class C yields a homonymous type, which can be
        +explicitly used in programs to type variables and methods, possibly in
        +combination with other types.

        +
        • +
      • +

        Since now we have user-defined types, we check that each type
        +used in a KOOL program is well-formed, that is, it is constructed only
        +from primitive and class types corresponding to declared classes.

        +
        • +
      • +

        Class members and their types form a class type
        +environment        . Each class will have such a type environment.
        +Each member in a class is allowed to be declared only once. Since in
        +KOOL we allow methods to be assigned to fields, we make no distinction
        +between field and method members; in other words, we reject programs
        +declaring both a field and a method with the same name.

        +
        • +
      • +

        If an identifier is not found in the local type environment, it
        +will be searched for in the current class type environment. If not
        +there, then it will be searched for in its superclass' type
        +environment. And so on and so forth. If not found until the
        +Object class is reached, a typing error is reported.

        +
        • +
      • +

        The assignment allows variables to be assigned values of
        +more concrete types. The result type of the assignment expression
        +construct will be the (more abstract) type of the assigned variable,
        +and not the (more concrete) type of the expression, like in Java.

        +
        • +
      • +

        Exceptions are changed (from SIMPLE) to allow throwing and
        +catching only objects, like in Java. Also, unlike in SIMPLE, we do
        +not check whether the type of the thrown exception matches the type of
        +the caught variable, because exceptions can be caught by other
        +try/catch blocks, even by ones in other methods. To avoid
        +having to annotate each method with what exceptions it can throw, we
        +prefer to not check the type safety of exceptions (although this is an
        +excellent homework!). We only check that the try block
        +type-checks and that the catch block type-checks after we bind
        +the caught variable to its claimed type.

        +
        • +
      • +

        Class declarations are not allowed to have any cycles in their
        +extends relation. Such cycles would lead to non-termination of
        +new, as it actually does in the dynamic semantics of KOOL
        +where no such circularity checks are performed.

        +
        • +
      • +

        Methods overriding other methods should be in the right subtyping
        +relationship with the overridden methods: co-variant in the codomain
        +and contra-variant in the domain.

        +
        • +
      +
      module KOOL-TYPED-STATIC-SYNTAX
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of statically typed KOOL is identical to that of
      +dynamically typed KOOL, they both taking as input the same programs.
      +What differs is the K strictness attributes. Like in statically
      +typed SIMPLE, almost all language constructs are strict now, since we
      +want each to type its arguments almost all the time. Like in the
      +other two KOOL definitions, we prefer to copy and then modify/extend
      +the syntax of statically typed SIMPLE.

      +

      Note: This paragraph is old, now we can do things better. We keep
      +it here only for historical reasons, to see how much we used to suffer 😃

      +

      Annoying K-tool technical problem:
      +Currently, the K tool treats the "non-terminal" productions (i.e.,
      +productions consisting of just one non-terminal), also called
      +"subsorting" production, differently from the other productions.
      +Specifically, it does not insert a node in the AST for them. This may
      +look desirable at first, but it has a big problem: it does not allow
      +us to treat the subsort differently in different context. For
      +example, since we want Id to be both a type (a class name) and a
      +program variable, and since we want expressions to reduce to their
      +types, we are in an impossible situations in which we do not know how
      +to treat an identifier in the semantics: as a type, i.e., a result of
      +computations, or as a program variable, i.e., a non-result. Ideally,
      +we would like to tag the identifiers at parse-time with their local
      +interpretation, but that, unfortunately, is not possible with the
      +current parsing capabilities of the K tool, because it requires to
      +insert additional information in the AST for the subsort productions.
      +This will be fixed soon. Until then, unfortunately, we have to do the
      +job of the parser manually. Instead of subsorting Id directly
      +to Type, we "wrap" it first, say with a wrapper called
      +class(...), exactly how the parser should have done.
      +The major drawback of this is that all the typed KOOL programs
      +in kool/typed/programs need to also be modified to always
      +declare class types accordingly. The modified programs can be found
      +in kool/typed/static/programs. So make sure you execute the
      +static semantics of KOOL using the modified programs. To avoid seeing
      +the wrapper in the generated documentation, we associate it an
      +"invisibility" latex attribute below.

      +
        syntax Id ::= "Object" [token] | "Main" [token]
+

      Types

      + +
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Id                     [klabel("class"), symbol, avoid]  // see next
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+
+  syntax Types ::= List{Type,","}        [klabel(exps)]
+

      Declarations

      + +
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";" [avoid]
+                | Type Id "(" Params ")" Block
+                | "class" Id Block
+                | "class" Id "extends" Id Block
+

      Expressions

      + +
        syntax FieldReference ::= Exp "." Id          [strict(1)]
+  syntax ArrayReference ::= Exp "[" Exps "]"    [strict]
+
+  syntax Exp ::= Int | Bool | String | Id
+               | "this"
+               | "super"
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]
+               | "(" Id ")" Exp          [strict(2)]
+               | "new" Id "(" Exps ")"   [strict(2)]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict, left]
+               | Exp "||" Exp            [strict, left]
+               > "spawn" Block  // not strict: to check return and exceptions
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Exp ::= FieldReference | ArrayReference
+  syntax priority _.__KOOL-TYPED-STATIC-SYNTAX > _[_]_KOOL-TYPED-STATIC-SYNTAX > _(_)_KOOL-TYPED-STATIC-SYNTAX
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+

      Statements

      + +
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                                 [strict]
+                | "if" "(" Exp ")" Block "else" Block     [avoid, strict]
+                | "if" "(" Exp ")" Block                  [macro]
+                | "while" "(" Exp ")" Block               [strict]
+                | "for" "(" Stmt Exp ";" Exp ")" Block    [macro]
+                | "return" Exp ";"                        [strict]
+                | "return" ";"
+                | "print" "(" Exps ")" ";"                [strict]
+                | "try" Block "catch" "(" Param ")" Block [strict(1)]
+                | "throw" Exp ";"                         [strict]
+                | "join" Exp ";"                          [strict]
+                | "acquire" Exp ";"                       [strict]
+                | "release" Exp ";"                       [strict]
+                | "rendezvous" Exp ";"                    [strict]
+
+  syntax Stmt ::= Stmt Stmt                            [seqstrict, right]
+

      Desugaring macros

      + +
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+  rule class C:Id S => class C extends Object S
+
+endmodule
+

      Static semantics

      + +

      We first discuss the configuration, then give the static semantics
      +taken over unchanged from SIMPLE, then discuss the static semantics of
      +SIMPLE syntactic constructs that needs to change, and in the end we
      +discuss the static semantics and additional checks specifically
      +related to the KOOL proper syntax.

      +
      module KOOL-TYPED-STATIC
+  imports KOOL-TYPED-STATIC-SYNTAX
+  imports DOMAINS
+

      Configuration

      + +

      The configuration of our type system consists of a tasks
      +cell with the same meaning like in statically typed SIMPLE, of an
      +out cell streamed to the standard output that will be used to
      +display typing error messages, and of a cell classes holding
      +data about each class in a separate class cell. The
      +task cells now have two additional optional subcells, namely
      +ctenvT and inClass. The former holds a temporary
      +class type environment; its contents will be transferred into the
      +ctenv cell of the corresponding class as soon as all the
      +fields and methods in the task are processed. In fact, there will be
      +three types of tasks in the subsequent semantics, each determined by
      +the subset of cells that it holds:

      +
        +
      1. +

        Main task, holding only a k cell holding the
        +original program as a set of classes. The role of this task is to
        +process each class, generating a class task (see next) for each.

        +
        2. +
      2. +

        Class task, holding k, ctenvT, and
        +inClass subcells. The role of this task type is to process
        +a class' contents, generating a class type environment in the
        +ctenvT cell and a method task (see next) for each method in
        +the class. To avoid interference with object member lookup rules
        +below, it is important to add the class type environment to a class
        +atomically; this is the reason for which we use ctenvT
        +temporary cells within class tasks (instead of adding each member
        +incrementally to the class' type environment).

        +
        3. +
      3. +

        Method task, holding k, tenv and
        +return cells. These tasks are similar to SIMPLE's function
        +tasks, so we do not discuss them here any further.

        +
        4. +
      +

      Each class cell hods its name (in the className
      +cell) and the name of the class it extends (in the extends
      +cell), as well as its type environment (in the ctenv cell)
      +and the set of all its superclasses (in the extendsAll cell).
      +The later is useful for example for checking whether there are cycles
      +in the class extends relation.

      +
        configuration <T multiplicity="?" color="yellow">
+                  <tasks color="orange" multiplicity="?">
+                    <task multiplicity="*" color="yellow" type="Set">
+                      <k color="green"> $PGM:Stmt </k>
+                      <tenv multiplicity="?" color="cyan"> .Map </tenv>
+                      <ctenvT multiplicity="?" color="blue"> .Map </ctenvT>
+                      <returnType multiplicity="?" color="black"> void </returnType>
+                      <inClass multiplicity="?" color="Fuchsia"> .K </inClass>
+                    </task>
+                  </tasks>
+//                  <br/>
+                  <classes color="Fuchsia">
+                    <classData multiplicity="*" type="Map">
+                      <className color="Fuchsia"> Object </className>
+                      <baseClass color="Fuchsia"> .K </baseClass>
+                      <baseClasses color="Fuchsia"> .Set </baseClasses>
+                      <ctenv multiplicity="?" color="blue"> .Map </ctenv>
+                    </classData>
+                  </classes>
+                </T>
+                <output color="brown" stream="stdout"> .List </output>
+

      Unchanged semantics from statically typed SIMPLE

      + +

      The syntax and rules below are borrowed unchanged from statically
      +typed SIMPLE, so we do not discuss them much here.

      +
        syntax Exp ::= Type
+  syntax Exps ::= Types
+  syntax BlockOrStmtType ::= "block" | "stmt"
+  syntax Type ::= BlockOrStmtType
+  syntax Block ::= BlockOrStmtType
+  syntax KResult ::= Type
+                   | Types  // TODO: should not be needed
+
+
+  context _:Type _::Exp[HOLE::Exps];
+
+  rule T:Type E:Exp[int,Ts:Types]; => T[] E[Ts];
+  rule T:Type E:Exp[.Types]; => T E;
+
+
+  rule <task>... <k> _:BlockOrStmtType </k> <tenv> _ </tenv> ...</task> => .Bag
+
+
+  rule _:Int => int
+  rule _:Bool => bool
+  rule _:String => string
+
+
+  rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</tenv>
+
+
+  context ++(HOLE => ltype(HOLE))
+  rule ++ int => int
+  rule int + int => int
+  rule string + string => string
+  rule int - int => int
+  rule int * int => int
+  rule int / int => int
+  rule int % int => int
+  rule - int => int
+  rule int < int => bool
+  rule int <= int => bool
+  rule int > int => bool
+  rule int >= int => bool
+  rule T:Type == T => bool
+  rule T:Type != T => bool
+  rule bool && bool => bool
+  rule bool || bool => bool
+  rule ! bool => bool
+
+
+  rule (T[])[int, Ts:Types] => T[Ts]
+  rule T:Type[.Types] => T
+
+  rule sizeOf(_T[]) => int
+
+
+  rule read() => int
+
+  rule print(T:Type, Ts => Ts); when T ==K int orBool T ==K string
+  rule print(.Types); => stmt
+
+
+  context (HOLE => ltype(HOLE)) = _
+
+
+  rule <k> return; => stmt ...</k> <returnType> _ </returnType>
+
+
+  rule {} => block
+
+  rule <task> <k> {S:Stmt} => block ...</k> <tenv> Rho </tenv> R </task>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> R </task>)
+
+  rule _:Type; => stmt
+  rule if (bool) block else block => stmt
+  rule while (bool) block => stmt
+
+  rule join int; => stmt
+  rule acquire _:Type; => stmt
+  rule release _:Type; => stmt
+  rule rendezvous _:Type; => stmt
+
+  syntax Stmt ::= BlockOrStmtType
+  rule _:BlockOrStmtType _:BlockOrStmtType => stmt
+

      Unchanged auxiliary operations from dynamically typed SIMPLE

      + +
        syntax Stmt ::= mkDecls(Params)  [function]
+  rule mkDecls(T:Type X:Id, Ps:Params) => T X; mkDecls(Ps)
+  rule mkDecls(.Params) => {}
+
+  syntax LValue ::= Id
+                  | FieldReference
+                  | ArrayReference
+  syntax Exp ::= LValue
+
+  syntax Exp ::= ltype(Exp)
+// We would like to say:
+//  context ltype(HOLE:LValue)
+// but we currently cannot type the HOLE
+  context ltype(HOLE) when isLValue(HOLE)
+
+// OLD approach:
+//  syntax Exp ::= ltype(Exp)  [function]
+//  rule ltype(X:Id) => X
+//  rule ltype(E:Exp [Es:Exps]) => E[Es]
+
+  syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+

      Changes to the existing statically typed SIMPLE semantics

      + +

      Below we give the new static semantics for language constructs that
      +come from SIMPLE, but whose SIMPLE static semantics was too
      +restrictive or too permissive and thus had to change.

      +

      Local variable declaration

      + +

      Since we can define new types in KOOL (corresponding to classes), the
      +variable declaration needs to now check that the claimed types exist.
      +The operation checkType, defined at the end of this module,
      +checks whether the argument type is correct (it actually works with
      +lists of types as well).

      +
        rule <k> T:Type X:Id; => checkType(T) ~> stmt ...</k>
+       <tenv> Rho => Rho[X <- T] </tenv>
+

      Class member declaration

      + +

      In class tasks, variable declarations mean class member declarations.
      +Since we reduce method declarations to variable declarations (see
      +below), a variable declaration in a class task can mean either a field
      +or a method declaration. Unlike local variable declarations, which
      +can shadow previous homonymous local or member declarations, member
      +declarations are regarded as a set, so we disallow multiple
      +declarations for the same member (one could improve upon this, like in
      +Java, by treating members with different types or number of arguments
      +as different, etc., but we do not do it here). We also issue an error
      +message if one attempts to redeclare the same class member. The
      +framed variable declaration in the second rule below should be read
      +"stuck". In fact, it is nothing but a unary operation called
      +stuck, which takes a K-term as argument and does nothing
      +with it; this stuck operation is displayed as a frame in this
      +PDF document because of its latex attribute (see the ASCII .k file,
      +at the end of this module).

      +
        rule <k> T:Type X:Id; => checkType(T) ~> stmt ...</k>
+       <ctenvT> Rho (.Map => X |-> T) </ctenvT>
+    when notBool(X in keys(Rho))
+
+  rule <k> T:Type X:Id; => stuck(T X;) ...</k>
+       <ctenvT>... X |-> _ ...</ctenvT>
+       <inClass> C:Id </inClass>
+//       <br/>
+       <output>... .List => ListItem("Member \"" +String Id2String(X)
+                              +String "\" declared twice in class \""
+                              +String Id2String(C) +String "\"!\n") </output>
+

      Method declaration

      + +

      A method declaration requires two conceptual checks to be performed:
      +first, that the method's type is consistent with the type of the
      +homonymous method that it overrides, if any; and second, that its body
      +types correctly. At the same time, it should also be added to the
      +type environment of its class. The first conceptual task is performed
      +using the checkMethod operation defined below, and the second
      +by generating a corresponding method task. To add it to the class
      +type environment, we take advantage of the fact that KOOL is higher
      +order and reduce the problem to a field declaration problem, which we
      +have already defined. The role of the ctenvT cell in the
      +rule below is to structurally ensure that the method declaration takes
      +place in a class task (we do not want to allow methods to be declared,
      +for example, inside other methods).

      +
        rule <k> T:Type F:Id(Ps:Params) S
+        => checkMethod(F, getTypes(Ps)->T, C')
+           ~> getTypes(Ps)->T F; ...</k>
+//       <br/>
+       <inClass> C </inClass>
+       <ctenvT> _ </ctenvT> // to ensure we are in a class pass
+       <className> C </className>
+       <baseClass> C' </baseClass>
+//       <br/>
+       (.Bag => <task>
+               <k> mkDecls(Ps) S </k>
+               <inClass> C </inClass>
+               <tenv> .Map </tenv>
+               <returnType> T </returnType>
+             </task>)
+

      Assignment

      + +

      A more concrete value is allowed to be assigned to a more abstract
      +variable. The operation checkSubtype is defined at the end
      +of the module and it also works with pairs of lists of types.

      +
        rule T:Type = T':Type => checkSubtype(T', T) ~> T
+

      Method invocation and return

      + +

      Methods can be applied on values of more concrete types than their
      +arguments:

      +
        rule (Ts:Types -> T:Type) (Ts':Types) => checkSubtype(Ts',Ts) ~> T
+

      Similarly, we allow values of more concrete types to be returned by
      +methods:

      +
        rule <k> return T:Type; => checkSubtype(T,T') ~> stmt ...</k>
+       <returnType> T':Type </returnType>
+

      Exceptions

      + +

      Exceptions can throw and catch values of any types. Since unlike in Java
      +KOOL's methods do not declare the exception types that they can throw,
      +we cannot test the full type safety of exceptions. Instead, we
      +only check that the try and the catch statements
      +type correctly.

      +
        rule try block catch(T:Type X:Id) S => {T X; S}
+  rule throw _T:Type ; => stmt
+

      Spawn

      + +

      The spawned cell needs to also be passed the parent's class.

      +
      // explain why
+
+  rule <k> spawn S:Block => int ...</k>
+       <tenv> Rho </tenv>
+       <inClass> C </inClass>
+       (.Bag => <task>
+               <k> S </k>
+               <tenv> Rho </tenv>
+               <inClass> C </inClass>
+             </task>)
+

      Semantics of the new KOOL constructs

      + +

      Class declaration

      + +

      We process each class in the main task, adding the corresponding data
      +into its class cell and also adding a class task for it. We
      +also perform some well-formedness checks on the class hierarchy.

      +

      Initiate class processing
      +We create a class cell and a class task for each task. Also, we start
      +the class task with a check that the class it extends is declared
      +(this delays the task until that class is processed using another
      +instance of this rule).

      +
      // There seems to be some error with the configuration concretization,
+// as the rule below does not work when rewriting . to both the task
+// and the class cells; I had to include two separate . rewrites
+
+// TODO: the following fails krun; see #2117
+  rule <task> <k> class C:Id extends C':Id { S:Stmt } => stmt ...</k> </task>
+       (.Bag => <classData>...
+               <className> C </className>
+               <baseClass> C' </baseClass>
+             ...</classData>)
+//       <br/>
+       (.Bag => <task>
+                <k> checkType(`class`(C')) ~> S </k>
+                <inClass> C </inClass>
+                <ctenvT> .Map </ctenvT>
+             </task>)
+
+// You may want to try the thing below, but that failed, too
+/*
+syntax Type ::= "stmtStop"
+
+  rule <tasks>...
+       <task> <k> class C:Id extends C':Id { S:Stmt } => stmtStop ...</k> </task>
+       (.Bag => <task>
+                <k> checkType(`class`(C')) ~> S </k>
+                <inClass> C </inClass>
+                <ctenvT> .Map </ctenvT>
+             </task>)
+       ...</tasks>
+       <classes>...
+       .Bag => <classData>...
+               <className> C </className>
+               <baseClass> C' </baseClass>
+             ...</classData>
+       ...</classes>
+//       <br/>
+*/
+

      Check for unique class names

      + +
        rule (<T>...
+          <className> C </className>
+          <className> C </className>
+        ...</T> => .Bag)
+       <output>... .List => ListItem("Class \"" +String Id2String(C)
+                                  +String "\" declared twice!\n") </output>
+

      Check for cycles in class hierarchy
      +We check for cycles in the class hierarchy by transitively closing the
      +class extends relation using the extendsAll cells, and
      +checking that a class will never appear in its own extendsAll
      +cell. The first rule below initiates the transitive closure of the
      +superclass relation, the second transitively closes it, and the third
      +checks for cycles.

      +
        rule <baseClass> C </baseClass>
+       <baseClasses> .Set => SetItem(C) </baseClasses>  [priority(25)]
+
+  rule <classData>...
+         <baseClasses> SetItem(C) Cs:Set (.Set => SetItem(C')) </baseClasses>
+       ...</classData>
+       <classData>... <className>C</className> <baseClass>C'</baseClass> ...</classData>
+    when notBool(C' in (SetItem(C) Cs))  [priority(25)]
+
+  rule (<T>...
+          <className> C </className>
+          <baseClasses>... SetItem(C) ...</baseClasses>
+        ...</T> => .Bag)
+       <output>... .List => ListItem("Class \"" +String Id2String(C)
+                                  +String "\" is in a cycle!\n") </output>
+    [group(inheritance-cycle), priority(25)]
+

      New

      + +

      To type new we only need to check that the class constructor
      +can be called with arguments of the given types, so we initiate a call
      +to the constructor method in the corresponding class. If that
      +succeeds, meaning that it types to stmt, then we discard the
      +stmt type and produce instead the corresponding class type of
      +the new object. The auxiliary discard operation is defined
      +also at the end of this module.

      +
        rule new C:Id(Ts:Types) => `class`(C) . C (Ts) ~> discard ~> `class`(C)
+

      Self reference

      + +

      The typing rule for this is straightforward: reduce to the
      +current class type.

      +
        rule <k> this => `class`(C) ...</k>
+       <inClass> C:Id </inClass>
+

      Super

      + +

      Similarly, super types to the parent class type.
      +Note that for typing concerns, super can be considered as an object
      +(recall that this was not the case in the dynamic semantics).

      +
         rule <k> super => `class`(C') ...</k>
+        <inClass> C:Id </inClass>
+        <className> C </className>
+        <baseClass> C':Id </baseClass>
+

      Object member access

      + +

      There are several cases to consider here. First, if we are in a class
      +task, we should lookup the member into the temporary class type
      +environemnt in cell ctenvT. That is because we want to allow
      +initialized field declarations in classes, such as int x=10;.
      +This is desugared to a declaration of x, which is added to
      +ctenvT during the class task processing, followed by an
      +assignment of x to 10. In order for the assignment to type
      +check, we need to know that x has been declared with type
      +int; this information can only be found in the
      +ctenvT cell. Second, we should redirect non-local variable
      +lookups in method tasks to corresponding member accesses (the
      +local variables are handled by the rule borrowed from SIMPLE).
      +This is what the second rule below does. Third, we should allow
      +object member accesses as lvalues, which is done by the third rule
      +below. These last two rules therefore ensure that each necessary
      +object member access is explicitly allowed for evaluation. Recall
      +from the annotated syntax module above that the member access
      +operation is strict in the object. That means that the object is
      +expected to evaluate to a class type. The next two rules below define
      +the actual member lookup operation, moving the search to the
      +superclass when the member is not found in the current class. Note
      +that this works because we create the class type environments
      +atomically; thus, a class either has its complete type environment
      +available, in which case these rules can safely apply, or its cell
      +ctenv is not yet available, in which case these rules have to
      +wait. Finally, the sixth rule below reports an error when the
      +Object class is reached.

      +
        rule <k> X:Id => T ...</k>
+       <ctenvT>... X |-> T ...</ctenvT>
+
+  rule <k> X:Id => this . X ...</k>
+       <tenv> Rho </tenv>
+    when notBool(X in keys(Rho))
+
+// OLD approach:
+//  rule ltype(E:Exp . X:Id) => E . X
+
+  rule <k> `class`(C:Id) . X:Id => T ...</k>
+       <className> C </className>
+       <ctenv>... X |-> T:Type ...</ctenv>
+
+  rule <k> `class`(C1:Id => C2) . X:Id ...</k>
+       <className> C1 </className>
+       <baseClass> C2:Id </baseClass>
+       <ctenv> Rho </ctenv>
+    when notBool(X in keys(Rho))
+
+  rule <k> `class`(Object) . X:Id => stuck(`class`(Object) . X) ...</k>
+       <inClass> C:Id </inClass>
+//      <br/>
+       <output>... .List => ListItem("Member \"" +String Id2String(X)
+                              +String "\" not declared! (see class \""
+                              +String Id2String(C) +String "\")\n") </output>
+

      Instance of and casting

      + +

      As it is hard to check statically whether casting is always safe,
      +the programmer is simply trusted from a typing perspective. We only
      +do some basic upcasting and downcasting checks, to reject casts which
      +will absolutely fail. However, dynamic semantics or implementations
      +of the language need to insert runtime checks for downcasting to be safe.

      +
        rule `class`(_C1:Id) instanceOf _C2:Id => bool
+  rule (C:Id) `class`(C) => `class`(C)
+  rule <k> (C2:Id) `class`(C1:Id) => `class`(C2) ...</k>
+       <className> C1 </className>
+       <baseClasses>...SetItem(C2)...</baseClasses>    // upcast
+  rule <k> (C2:Id) `class`(C1:Id) => `class`(C2) ...</k>
+       <className> C2 </className>
+       <baseClasses>...SetItem(C1)...</baseClasses>    // downcast
+  rule <k> (C2) `class`(C1:Id) => stuck((C2) `class`(C1)) ...</k>
+       <classData>...
+         <className> C1 </className>
+         <baseClasses> S1 </baseClasses>
+       ...</classData>
+       <classData>...
+         <className> C2 </className>
+         <baseClasses> S2 </baseClasses>
+       ...</classData>
+       <output>... .List => ListItem("Classes \"" +String Id2String(C1)
+                              +String "\" and \"" +String Id2String(C2)
+                              +String "\" are incompatible!\n") </output>
+    when notBool(C1 in S2) andBool notBool(C2 in S1)
+

      Cleanup tasks

      + +

      Finally, we need to clean up the terminated tasks. Each of the three
      +types of tasks is handled differently. The main task is replaced by a
      +method task holding new main();, which will ensure that a
      +main class with a main() method actually exists
      +(first rule below). A class task moves its temporary class type
      +environment into its class' cell, and then it dissolves itself (second
      +rule). A method task simply dissolves when terminated (third rule);
      +the presence of the tenv cell in that rule ensures that that
      +task is a method task.
      +Finally, when all the tasks are cleaned up, we can also remove the
      +tasks cell, issuing a corresponding message. Note that
      +checking for cycles or duplicate methods can still be performed after
      +the tasks cell has been removed.

      +
      // discard main task when done, issuing a "new main();" command to
+// make sure that the class main and the method main() are declared.
+
+  rule <task> <k> stmt => new Main(.Exps); </k>
+              (.Bag => <tenv> .Map </tenv>
+                    <returnType> void </returnType>
+                    <inClass> Main </inClass>)
+       </task>
+
+// discard class task when done, adding a ctenv in class
+
+  rule (<task>
+          <k> stmt </k>
+          <ctenvT> Rho </ctenvT>
+          <inClass> C:Id </inClass>
+        </task> => .Bag)
+        <className> C </className>
+        (.Bag => <ctenv> Rho </ctenv>)
+
+// discard method task when done
+
+  rule <task>...
+         <k> stmt </k>
+         <tenv> _ </tenv>  // only to ensure that this is a method task
+       ...</task> => .Bag
+
+// cleanup tasks and output a success message when done
+
+  rule (<T>... <tasks> .Bag </tasks> ...</T> => .Bag)
+       <output>... .List => ListItem("Type checked!\n") </output>
+

      KOOL-specific auxiliary declarations and operations

      + +

      Subtype checking

      + +

      The subclass relation introduces a subtyping relation.

      +
        syntax KItem ::= checkSubtype(Types,Types)
+
+  rule checkSubtype(T:Type, T) => .
+
+  rule <k> checkSubtype(`class`(C:Id), `class`(C':Id)) => . ...</k>
+       <className> C </className>
+       <baseClasses>... SetItem(C') ...</baseClasses>
+
+  rule checkSubtype(Ts1->T2,Ts1'->T2')
+    => checkSubtype(((T2)::Type,Ts1'),((T2')::Type,Ts1))
+
+// note that the following rule would be wrong!
+//  rule checkSubtype(T[],T'[]) => checkSubtype(T,T')
+
+  rule checkSubtype((T:Type,Ts),(T':Type,Ts'))
+    => checkSubtype(T,T') ~> checkSubtype(Ts,Ts')
+    when Ts =/=K .Types
+
+  rule checkSubtype(.Types,.Types) => .
+  rule checkSubtype(.Types,void) => .
+

      Checking well-formedness of types

      + +

      Since now any Id can be used as the type of a class, we need to
      +check that the types used in the program actually exists

      +
        syntax KItem ::= checkType(Types)
+
+  rule checkType(T:Type,Ts:Types) => checkType(T) ~> checkType(Ts)
+    when Ts =/=K .Types
+  rule checkType(.Types) => .
+  rule checkType(int) => .
+  rule checkType(bool) => .
+  rule checkType(string) => .
+  rule checkType(void) => .
+  rule <k> checkType(`class`(C:Id)) => . ...</k> <className> C </className>
+  rule checkType(`class`(Object)) => .
+  rule checkType(Ts:Types -> T:Type) => checkType(T,Ts)
+  rule checkType(T:Type[]) => checkType(T)
+

      Checking correct overiding of methods

      + +

      The checkMethod operation below searches to see whether
      +the current method overrides some other method in some superclass.
      +If yes, then it issues an additional check that the new method's type
      +is more concrete than the overridden method's. The types T and T'
      +below can only be function types. See the definition of
      +checkSubtype on function types at the end of this module (it
      +is co-variant in the codomain and contra-variant in the domain).

      +
        syntax KItem ::= checkMethod(Id,Type,Id)
+
+  rule <k> checkMethod(F:Id, T:Type, C:Id) => checkSubtype(T, T') ...</k>
+       <className> C </className>
+       <ctenv>... F |-> T':Type ...</ctenv>
+
+  rule <k> checkMethod(F:Id, _T:Type, (C:Id => C')) ...</k>
+       <className> C </className>
+       <baseClass> C':Id </baseClass>
+       <ctenv> Rho </ctenv>
+    when notBool(F in keys(Rho))
+
+  rule checkMethod(_:Id,_,Object) => .
+

      Generic operations which could be part of the K framework

      + +
        syntax KItem ::= stuck(K)  [latex(\framebox{${#1}$})]
+
+  syntax KItem ::= "discard"
+  rule _:KResult ~> discard => .
+
+endmodule
+

      FUN — Untyped — Environment

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped FUN language.
      +FUN is a pedagogical and research language that captures the essence
      +of the functional programming paradigm, extended with several features
      +often encountered in functional programming languages.
      +Like many functional languages, FUN is an expression language, that
      +is, everything, including the main program, is an expression.
      +Functions can be declared anywhere and are first class values in the
      +language.
      +FUN is call-by-value here, but it has been extended (as student
      +homework assignments) with other parameter-passing styles.
      +To make it more interesting and to highlight some of K's strengths,
      +FUN includes the following features:

      +
        +
      • +

        The basic builtin data-types of integers, booleans and strings.

        +
        • +
      • +

        Builtin lists, which can hold any elements, including other lists.
        +Lists are enclosed in square brackets and their elements are
        +comma-separated; e.g., [1,2,3].

        +
        • +
      • +

        User-defined data-types, by means of constructor terms.
        +Constructor names start with a capital letter (while any other
        +identifier in the language starts with a lowercase letter), and they
        +can be followed by an arbitrary number of comma-separated arguments
        +enclosed in parentheses; parentheses are not needed when the
        +constructor takes no arguments.
        +For example, Pair(5,7) is a constructor term holding two
        +numbers, Cons(1,Cons(2,Cons(3,Nil))) is a list-like
        +constructor term holding 3 elements, and
        +Tree(Tree(Leaf(1), Leaf(2)), Leaf(3)) is a tree-like
        +constructor term holding 3 elements.
        +In the untyped version of the FUN language, no type checking or
        +inference is performed to ensure that the data constructors are used
        +correctly.
        +The execution will simply get stuck when they are misused.
        +Moreover, since no type checking is performed, the data-types are not
        +even declared in the untyped version of FUN.

        +
        • +
      • +

        Functions and let/letrec binders can take
        +multiple space-separated arguments, but these are desugared to
        +ones that only take one argument, by currying. For example, the
        +expressions

        +
        fun x y -> x y
+let x y = y in x
+

        are desugared, respectively, into the following expressions:

        +
        fun x -> fun y -> x y
+let x = fun y -> y in x
+
        • +
      • +

        Functions can be defined using pattern matching over the
        +available data-types. For example, the program

        +
        letrec max = fun [h] -> h
+             |   [h|t] -> let x = max t
+                          in  if h > x then h else x
+in max [1, 3, 5, 2, 4, 0, -1, -5]
+

        defines a function max that calculates the maximum element of
        +a non-empty list, and the function

        +
        letrec ack = fun Pair(0,n) -> n + 1
+             |   Pair(m,0) -> ack Pair(m - 1, 1)
+             |   Pair(m,n) -> ack Pair(m - 1, ack Pair(m, n - 1))
+in ack Pair(2,3)
+

        calculates the Ackermann function applied to a particular pair of numbers.
        +Patterns can be nested. Patterns can currently only be used in function
        +definitions, and not directly in let/letrec binders.
        +For example, this is not allowed:

        +
        letrec Pai(x,y) = Pair(1,2) in x+y
+

        But this is allowed:

        +
        let f Pair(x,y) = x+y in f Pair(1,2)
+

        because it is first reduced to

        +
        let f = fun Pair(x,y) -> x+y in f Pair(1,2)
+

        by uncurrying of the let binder, and pattern matching is
        +allowed in function arguments.

        +
        • +
      • +

        We include a callcc construct, for two reasons: first,
        +several functional languages support this construct; second, some
        +semantic frameworks have difficulties defining it. Not K.

        +
        • +
      • +

        Finally, we include mutables by means of referencing an
        +expression, getting the reference of a variable, dereferencing and
        +assignment. We include these for the same reasons as above: there are
        +languages which have them, and they are not easy to define in some
        +semantic frameworks.

        +
        • +
      +

      Like in many other languages, some of FUN's constructs can be
      +desugared into a smaller set of basic constructs. We do that as usual,
      +using macros, and then we only give semantics to the core constructs.

      +

      Note:
      +We recommend the reader to first consult the dynamic semantics of the
      +LAMBDA++ language in the first part of the K Tutorial.
      +To keep the comments below small and focused, we will not re-explain
      +functional or K features that have already been explained in there.

      +

      Syntax

      + +
      //require "modules/pattern-matching.k"
+
+module FUN-UNTYPED-COMMON
+  imports DOMAINS-SYNTAX
+

      FUN is an expression language. The constructs below fall into
      +several categories: names, arithmetic constructs, conventional
      +functional constructs, patterns and pattern matching, data constructs,
      +lists, references, and call-with-current-continuation (callcc).
      +The arithmetic constructs are standard; they are present in almost all
      +our K language definitions. The meaning of FUN's constructs are
      +discussed in more depth when we define their semantics in the next
      +module.

      +

      The Syntactic Constructs

      + +

      We start with the syntactic definition of FUN names.
      +We have several categories of names: ones to be used for functions and
      +variables, others to be used for data constructors, others for types and
      +others for type variables. We will introduce them as needed, starting
      +with the former category. We prefer the names of variables and functions
      +to start with lower case letters. We take the freedom to tacitly introduce
      +syntactic lists/sequences for each nonterminal for which we need them:

      +
        syntax Name                                      [token]
+  syntax Names ::= List{Name,","}                  [klabel(exps)]
+

      Expression constructs will be defined throughtout the syntax module.
      +Below are the very basic ones, namely the builtins, the names, and the
      +parentheses used as brackets for grouping. Lists of expressions are
      +declared strict, so all expressions in the list get evaluated whenever
      +the list is on a position which can be evaluated:

      +
        syntax Exp ::= Int | Bool | String | Name
+               | "(" Exp ")"                       [bracket]
+  syntax Exps  ::= List{Exp,","}                   [strict, klabel(exps)]
+  syntax Val
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= List{Val,","}                    [klabel(exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}              [klabel(exps)]
+

      We next define the syntax of arithmetic constructs, together with
      +their relative priorities and left-/non-associativities. We also
      +tag all these rules as members of a new group, "arith", so we can more easily
      +define global syntax priorities later (at the end of the syntax module).

      +
        syntax Exp ::= left:
+                 Exp "*" Exp                       [strict, group(arith)]
+               | Exp "/" Exp                       [strict, group(arith)]
+               | Exp "%" Exp                       [strict, group(arith)]
+               > left:
+                 Exp "+" Exp                       [strict, left, group(arith)]
+               | Exp "^" Exp                       [strict, left, group(arith)]
+// left attribute should not be necessary; currently a parsing bug
+               | Exp "-" Exp                       [strict, prefer, group(arith)]
+// the "prefer" attribute above is to not parse x-1 as x(-1)
+// Due to some parsing problems, we currently cannot add unary minus:
+               | "-" Exp                           [strict, group(arith)]
+               > non-assoc:
+                 Exp "<" Exp                       [strict, group(arith)]
+               | Exp "<=" Exp                      [strict, group(arith)]
+               | Exp ">" Exp                       [strict, group(arith)]
+               | Exp ">=" Exp                      [strict, group(arith)]
+               | Exp "==" Exp                      [strict, group(arith)]
+               | Exp "!=" Exp                      [strict, group(arith)]
+               > "!" Exp                           [strict, group(arith)]
+               > Exp "&&" Exp                      [strict(1), left, group(arith)]
+               > Exp "||" Exp                      [strict(1), left, group(arith)]
+

      The conditional construct has the expected evaluation strategy,
      +stating that only the first argument is evaluate:

      +
        syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+

      FUN's builtin lists are formed by enclosing comma-separated
      +sequences of expressions (i.e., terms of sort Exps) in square
      +brackets. The list constructor cons adds a new element to the
      +top of the list, head and tail get the first element
      +and the tail sublist of a list if they exist, respectively, and get
      +stuck otherwise, and null?? tests whether a list is empty or
      +not; syntactically, these are just expression constants.
      +In function patterns, we are also going to allow patterns following the
      +usual head/tail notation; for example, the pattern [x_1,...,x_n|t]
      +binds x_1, ..., x_n to the first elements of the matched list,
      +and t to the list formed with the remaining elements. We define list
      +patterns as ordinary expression constructs, although we will make sure that
      +we do not give them semantics if they appear in any other place then in a
      +function case pattern.

      +
        syntax Exp ::= "[" Exps "]"                             [strict, klabel(list)]
+               | "head" [macro] | "tail" [macro] | "null?" [macro]
+               | "[" Exps "|" Exp "]"
+  syntax Val ::= "[" Vals "]"                             [klabel(list)]
+  syntax Cons ::= "cons"
+  syntax Val ::= Cons
+  syntax Val ::= Cons Val                                 [klabel(apply)]
+

      Data constructors start with capital letters and they may or may
      +not have arguments. We need to use the attribute "prefer" to make
      +sure that, e.g., Cons(a) parses as constructor Cons with
      +argument a, and not as the expression Cons (because
      +constructor names are also expressions) regarded as a function applied
      +to the expression a. Also, note that the constructor is strict
      +in its second argument, because we want to evaluate its arguments but
      +not the constuctor name itsef.

      +
        syntax ConstructorName                         [token]
+  syntax Exp ::= ConstructorName
+               | ConstructorName "(" Exps ")"    [prefer, strict(2), klabel(constructor)]
+  syntax Val ::= ConstructorName "(" Vals ")"    [klabel(constructor)]
+

      A function is essentially a |-separated ordered
      +sequence of cases, each case of the form pattern -> expression,
      +preceded by the language construct fun. Patterns will be defined
      +shortly, both for the builtin lists and for user-defined constructors.
      +Recall that the syntax we define in K is not meant to serve as a
      +ultimate parser for the defined language, but rather as a convenient
      +notation for K abstract syntax trees, which we prefer when we write
      +the semantic rules. It is therefore often the case that we define a
      +more ``generous'' syntax than we want to allow programs to use.
      +We do it here, too. Specifically, the syntax of Cases
      +below allows any expressions to appear as pattern. This syntactic
      +relaxation permits many wrong programs to be parsed, but that is not a
      +problem because we are not going to give semantics to wrong combinations,
      +so those programs will get stuck; moreover, our type inferencer will reject
      +those programs anyway. Function application is just concatenation of
      +expressions, without worrying about type correctness. Again, the type
      +system will reject type-incorrect programs.

      +
        syntax Exp ::= "fun" Cases
+               | Exp Exp                              [strict, left, klabel(apply)]
+// NOTE: We would like eventually to also have Exp "(" Exps ")
+  syntax Case  ::= Exp "->" Exp
+  syntax Cases ::= List{Case, "|"}
+

      The let and letrec binders have the usual syntax
      +and functional meaning. We allow multiple and-separated bindings.
      +Like for the function cases above, we allow a more generous syntax for
      +the left-hand sides of bindings, noting that the semantics will get stuck
      +on incorrect bindings and that the type system will reject those programs.

      +
        syntax Exp ::= "let" Bindings "in" Exp
+               | "letrec" Bindings "in" Exp                 [prefer]
+// The "prefer" attribute for letrec currently needed due to tool bug,
+// to make sure that "letrec" is not parsed as "let rec".
+  syntax Binding  ::= Exp "=" Exp
+  syntax Bindings ::= List{Binding,"and"}
+

      References are first class values in FUN. The construct ref
      +takes an expression, evaluates it, and then it stores the resulting value
      +at a fresh location in the store and returns that reference. Syntactically,
      +ref is just an expression constant. The construct &
      +takes a name as argument and evaluates to a reference, namely the store
      +reference where the variable passed as argument stores its value; this
      +construct is a bit controversial and is further discussed in the
      +environment-based semantics of the FUN language, where we desugar
      +ref to it. The construct @ takes a reference
      +and evaluates to the value stored there. The construct := takes
      +two expressions, the first expected to evaluate to a reference; the value
      +of its second argument will be stored at the location to which the first
      +points (the old value is thus lost). Finally, since expression evaluation
      +now has side effects, it makes sense to also add a sequential composition
      +construct, which is sequentially strict. This evaluates to the value of
      +its second argument; the value of the first argument is lost (which has
      +therefore been evaluated only for its side effects.

      +
        syntax Exp ::= "ref"                             [macro]
+               | "&" Name
+               | "@" Exp                                     [strict]
+               | Exp ":=" Exp                                [strict]
+               | Exp ";" Exp                       [strict(1), right]
+

      Call-with-current-continuation, named callcc in FUN, is a
      +powerful control operator that originated in the Scheme programming
      +language, but it now exists in many other functional languages. It works
      +by evaluating its argument, expected to evaluate to a function, and by
      +passing the current continuation, or evaluation context (or computation,
      +in K terminology), as a special value to it. When/If this special value
      +is invoked, the current context is discarded and replaced with the one
      +held by the special value and the computation continues from there.
      +It is like taking a snapshot of the execution context at some moment
      +in time and then, when desired, being able to get back in time to that
      +point. If you like games, it is like saving the game now (so you can
      +work on your homework!) and then continuing the game tomorrow or whenever
      +you wish. To issustrate the strength of callcc, we also
      +allow exceptions in FUN by means of a conventional try-catch
      +construct, which will desugar to callcc. We also need to
      +introduce the special expression contant throw, but we need to
      +use it as a function argument name in the desugaring macro, so we define
      +it as a name instead of as an expression constant:

      +
        syntax Exp ::= "try" Exp "catch" "(" Name ")" Exp [macro]
+  syntax Val ::= "callcc"
+  syntax Name ::= "throw" [token]
+

      Finally, FUN also allows polymorphic datatype declarations. These
      +will be useful when we define the type system later on.

      +
        syntax Exp ::= "datatype" Type "=" TypeCases Exp [macro]
+// NOTE: In a future version of K, we want the datatype declaration
+// to be a construct by itself, but that is not possible currently
+// because K's parser wronly identifies the __ operation allowing
+// a declaration to appear in front of an expression with the function
+// application construct, giving ambiguous parsing errors.
+

      We next need to define the syntax of types and type cases that appear
      +in datatype declarations.

      +

      Like in many functional languages, type parameters/variables in
      +user-defined types are quoted identifiers.

      +
        syntax TypeVar                        [token]
+  syntax TypeVars ::= List{TypeVar,","} [klabel(types)]
+

      Types can be basic types, function types, or user-defined
      +parametric types. In the dynamic semantics we are going to simply ignore
      +all the type declations, so here the syntax of types below is only useful
      +for generating the desired parser. To avoid syntactic ambiguities with
      +the arrow construct for function cases, we use the symbol --> as
      +a constructor for function types:

      +
        syntax TypeName [token]
+  syntax Type ::= "int" | "bool" | "string"
+                | Type "-->" Type                            [right]
+                | "(" Type ")"                             [bracket]
+                | TypeVar
+                | TypeName             [klabel(TypeName), avoid]
+                | Type TypeName   [klabel(Type-TypeName), symbol, macro]
+                | "(" Types ")" TypeName                    [prefer]
+  syntax Types ::= List{Type,","} [klabel(types)]
+  syntax Types ::= TypeVars
+
+  syntax TypeCase ::= ConstructorName
+                    | ConstructorName "(" Types ")"
+  syntax TypeCases ::= List{TypeCase,"|"}     [klabel(_|TypeCase_)]
+

      Additional Priorities

      + +
        syntax priorities @__FUN-UNTYPED-COMMON
+                  > apply
+                  > arith
+                  > _:=__FUN-UNTYPED-COMMON
+                  > let_in__FUN-UNTYPED-COMMON
+                    letrec_in__FUN-UNTYPED-COMMON
+                    if_then_else__FUN-UNTYPED-COMMON
+                  > _;__FUN-UNTYPED-COMMON
+                  > fun__FUN-UNTYPED-COMMON
+                  > datatype_=___FUN-UNTYPED-COMMON
+endmodule
+
+module FUN-UNTYPED-MACROS
+  imports FUN-UNTYPED-COMMON
+

      Desugaring macros

      + +

      We desugar the list non-constructor operations to functions matching
      +over list patterns. In order to do that we need some new variables; for
      +those, we follow the same convention like in the K tutorial, where we
      +added them as new identifier constructs starting with the character $,
      +so we can easily recognize them when we debug or trace the semantics.

      +
        syntax Name ::= "$h" [token] | "$t" [token]
+  rule head => fun [$h|$t] -> $h
+  rule tail => fun [$h|$t] -> $t
+  rule null? => fun [.Exps] -> true | [$h|$t] -> false
+

      Multiple-head list patterns desugar into successive one-head patterns:

      +
        rule [E1,E2,Es:Exps|T] => [E1|[E2,Es|T]]                   [anywhere]
+

      Uncurrying of multiple arguments in functions and binders:

      +
        rule P1 P2 -> E => P1 -> fun P2 -> E                       [anywhere]
+  rule F P = E => F = fun P -> E                             [anywhere]
+

      We desugar the try-catch construct into callcc:

      +
        syntax Name ::= "$k" [token] | "$v" [token]
+  rule try E catch(X) E'
+    => callcc (fun $k -> (fun throw -> E)(fun X -> $k E'))
+

      For uniformity, we reduce all types to their general form:

      +
        rule `Type-TypeName`(T:Type, Tn:TypeName) => (T) Tn
+

      The dynamic semantics ignores all the type declarations:

      +
        rule datatype _T = _TCs E => E
+
+endmodule
+
+
+module FUN-UNTYPED-SYNTAX
+  imports FUN-UNTYPED-COMMON
+  imports BUILTIN-ID-TOKENS
+
+  syntax Name ::= r"[a-z][_a-zA-Z0-9]*"           [token, prec(2)]
+                | #LowerId                        [token]
+  syntax ConstructorName ::= #UpperId             [token]
+  syntax TypeVar  ::= r"['][a-z][_a-zA-Z0-9]*"    [token]
+  syntax TypeName ::= Name                        [token]
+endmodule
+

      Semantics

      + +

      The semantics below is environment-based. A substitution-based
      +definition of FUN is also available, but that drops the &
      +construct as explained above.

      +
      module FUN-UNTYPED
+  imports FUN-UNTYPED-COMMON
+  imports FUN-UNTYPED-MACROS
+  imports DOMAINS
+  //imports PATTERN-MATCHING
+

      Configuration

      + +

      The k, env, and store cells are standard
      +(see, for example, the definition of LAMBDA++ or IMP++ in the first
      +part of the K tutorial).

      +
        configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <env color="violet"> .Map </env>
+                  <store color="white"> .Map </store>
+                </T>
+

      Values and results

      + +

      We only define integers, Booleans and strings as values here, but will
      +add more values later.

      +
        syntax Val ::= Int | Bool | String
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+

      Lookup

      + +
        rule <k> X:Name => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V ...</store>
+

      Arithmetic expressions

      + +
        rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule I1 + I2 => I1 +Int I2
+  rule S1 ^ S2 => S1 +String S2
+  rule I1 - I2 => I1 -Int I2
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+

      Conditional

      + +
        rule if  true then E else _ => E
+  rule if false then _ else E => E
+

      Lists

      + +

      We have already declared the syntactic list of expressions strict, so
      +we can assume that all the elements that appear in a FUN list are
      +evaluated. The only thing left to do is to state that a list of
      +values is a value itself, that is, that the list square-bracket
      +construct is indeed a constructor, and to give the semantics of
      +cons. Since cons is a builtin function and is
      +expected to take two arguments, we have to also state that
      +cons itself is a value (specifically, a function/closure
      +value, but we do not need that level of detail here), and also that
      +cons applied to a value is a value (specifically, it would be
      +a function/closure value that expects the second, list argument):

      +
        rule cons V:Val [Vs:Vals] => [V,Vs]
+

      Data Constructors

      + +

      Constructors take values as arguments and produce other values:

      +
        syntax Val ::= ConstructorName
+

      Functions and Closures

      + +

      Like in the environment-based semantics of LAMBDA++ in the first part
      +of the K tutorial, functions evaluate to closures. A closure includes
      +the current environment besides the function contents; the environment
      +will be used at execution time to lookup all the variables that appear
      +free in the function body (we want static scoping in FUN).

      +
        syntax Val ::= closure(Map,Cases)
+  rule <k> fun Cases => closure(Rho,Cases) ...</k>  <env> Rho </env>
+

      Note: The reader may want to get familiar with
      +how the pre-defined pattern matching works before proceeding.
      +The best way to do that is to consult
      +k/include/modules/pattern-matching.k.

      +

      We distinguish two cases when the closure is applied.
      +If the first pattern matches, then we pick the first case: switch to
      +the closed environment, get the matching map and bind all its
      +variables, and finally evaluate the function body of the first case,
      +making sure that the environment is properly recovered afterwards.
      +If the first pattern does not match, then we drop it and thus move on
      +to the next one.

      +
        rule (. => getMatching(P, V)) ~> closure(_, P->_ | _) V:Val
+  rule <k> matchResult(M:Map) ~> closure(Rho, _->E | _) _
+           => bindMap(M) ~> E ~> setEnv(Rho') ...</k>
+       <env> Rho' => Rho </env>
+  rule (matchFailure => .) ~> closure(_, (_->_ | Cs:Cases => Cs)) _
+//  rule <k> closure(Rho, P->E | _) V:Val
+//           => bindMap(getMatching(P,V)) ~> E ~> setEnv(Rho') ...</k>
+//       <env> Rho' => Rho </env>  when isMatching(P,V)
+//  rule closure(_, (P->_ | Cs:Cases => Cs)) V:Val  when notBool isMatching(P,V)
+

      Let and Letrec

      + +

      To highlight the similarities and differences between let and
      +letrec, we prefer to give them direct semantics instead of
      +to desugar them like in LAMBDA. See the formal definitions of
      +bindTo, bind, and assignTo at the end of
      +this module. Informally, bindTo(Xs, Es) first
      +evaluates the expressions Es in Exps in the current
      +environment (i.e., it is strict in its second argument), then it binds
      +the variables in Xs in Names to new locations and adds
      +those bindings to the environment, and finally writes the values
      +previously obtained after evaluating the expressions Es to those
      +new locations; bind(Xs) does only the bindings of
      +Xs to new locations and adds those bindings to the environment;
      +and assignTo(Xs,Es) evaluates the expressions
      +Es in the current environment and then it writes the resulting
      +values to the locations to which the variables Xs are already
      +bound to in the environment.

      +

      Therefore, let Xs = Es in E first
      +evaluates Es in the current environment, then adds new
      +bindings for Xs to fresh locations in the environment, then
      +writes the values of Es to those locations, and finally
      +evaluates E in the new environment, making sure that the
      +environment is properly recovered after the evaluation of E.
      +On the other hand, letrec does the same things but in a
      +different order: it first adds new bindings for Xs to fresh
      +locations in the environment, then it evaluates Es in the new
      +environment, then it writes the resulting values to their
      +corresponding locations, and finally it evaluates E and
      +recovers the environment. The crucial difference is that the
      +expressions Es now see the locations of the variables Xs
      +in the environment, so if they are functions, which is typically the
      +case with letrec, their closures will encapsulate in their
      +environments the bindings of all the bound variables, including
      +themselves (thus, we may have a closure value stored at location
      +L, whose environment contains a binding of the form
      +F ↦ L; this way, the closure can invoke
      +itself).

      +
        rule <k> let Bs in E
+        => bindTo(names(Bs),exps(Bs)) ~> E ~> setEnv(Rho) ...</k>
+       <env> Rho </env>
+
+  rule <k> letrec Bs in E
+        => bind(names(Bs))~>assignTo(names(Bs),exps(Bs))~>E~>setEnv(Rho)...</k>
+       <env> Rho </env>
+

      Recall that our syntax allows let and letrec to
      +take any expression in place of its binding. This allows us to use
      +the already existing function application construct to bind names to
      +functions, such as, e.g., let x y = y in ....
      +The desugaring macro in the syntax module uncurries such declarations,
      +and then the semantic rules above only work when the remaining
      +bindings are identifiers, so the semantics will get stuck on programs
      +that misuse the let and letrec binders.

      +

      References

      + +

      The semantics of references is self-explanatory, except maybe for the
      +desugaring rule of ref, which is further discussed. Note
      +that &X grabs the location of X from the environment.
      +Sequential composition, which is needed only to accumulate the
      +side effects due to assignments, was strict in the first argument.
      +Once evaluated, its first argument is simply discarded:

      +
        syntax Name ::= "$x" [token]
+  rule ref => fun $x -> & $x
+  rule <k> & X => L ...</k>  <env>... X |-> L ...</env>
+  rule <k> @ L:Int => V:Val ...</k>  <store>... L |-> V ...</store>
+  rule <k> L:Int := V:Val => V ...</k>  <store>... L |-> (_=>V) ...</store>
+  rule _V:Val; E => E
+

      The desugaring rule of ref (first rule above) works
      +because & takes a variable and returns its location (like in C).
      +Note that some ``pure'' functional programming researchers strongly dislike
      +the & construct, but favor ref. We refrain from having
      +a personal opinion on this issue here, but support & in the
      +environment-based definition of FUN because it is, technically speaking,
      +more powerful than ref. From a language design perspective, it
      +would be equally easy to drop & and instead give a direct
      +semantics to ref. In fact, this is precisely what we do in the
      +substitution-based definition of FUN, because there appears to be no way
      +to give a substitution-based definition to the & construct.

      +

      Callcc

      + +

      As we know it from the LAMBDA++ tutorial, call-with-current-continuation
      +is quite easy to define in K. We first need to define a special
      +value wrapping an execution context, that is, an environment saying
      +where the variables should be looked up, and a computation structure
      +saying what is left to execute (in a substitution-based definition,
      +this special value would be even simpler, as it would only need to
      +wrap the computation structure---see, for example, the
      +substitution-based semantics of LAMBDA++ in the the first part of the
      +K tutorial, or the substitution-based definition of FUN). Then
      +callcc creates such a value containing the current
      +environment and the current remaining computation, and passes it to
      +its argument function. When/If invoked, the special value replaces
      +the current execution context with its own and continues the execution
      +normally.

      +
        syntax Val ::= cc(Map,K)
+  rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k>  <env> Rho </env>
+  rule <k> cc(Rho,K) V:Val ~> _ => V ~> K </k>  <env> _ => Rho </env>
+

      Auxiliary operations

      + +

      Environment recovery

      + +

      The environment recovery operation is the same as for the LAMBDA++
      +language in the K tutorial and many other languages provided with the
      +K distribution. The first ``anywhere'' rule below shows an elegant
      +way to achieve the benefits of tail recursion in K.

      +
        syntax KItem ::= setEnv(Map)  // TODO: get rid of env
+  //rule (setEnv(_) => .) ~> setEnv(_)  [anywhere]
+  rule <k> _:Val ~> (setEnv(Rho) => .) ...</k> <env> _ => Rho </env>
+

      bindTo, bind and assignTo

      + +

      The meaning of these operations has already been explained when we
      +discussed the let and letrec language constructs
      +above.

      +
        syntax KItem ::= bindTo(Names,Exps)         [strict(2)]
+                 | bindMap(Map)
+                 | bind(Names)
+
+  rule (. => getMatchingAux(Xs,Vs)) ~> bindTo(Xs:Names,Vs:Vals)
+  rule matchResult(M:Map) ~> bindTo(_:Names, _:Vals) => bindMap(M)
+
+  rule bindMap(.Map) => .
+  rule <k> bindMap((X:Name |-> V:Val => .Map) _:Map) ...</k>
+       <env> Rho => Rho[X <- !L:Int] </env>
+       <store>... .Map => !L |-> V ...</store>
+
+  rule bind(.Names) => .
+  rule <k> bind(X:Name,Xs => Xs) ...</k>
+       <env> Rho => Rho[X <- !_L:Int] </env>
+
+  syntax KItem ::= assignTo(Names,Exps)  [strict(2)]
+
+  rule <k> assignTo(.Names,.Vals) => . ...</k>
+  rule <k> assignTo((X:Name,Xs => Xs),(V:Val,Vs:Vals => Vs)) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... .Map => L |-> V ...</store>
+

      Getters

      + +

      The following auxiliary operations extract the list of identifiers
      +and of expressions in a binding, respectively.

      +
        syntax Names ::= names(Bindings)  [function]
+  rule names(.Bindings) => .Names
+  rule names(X:Name=_ and Bs) => (X,names(Bs))::Names
+
+  syntax Exps ::= exps(Bindings)  [function]
+  rule exps(.Bindings) => .Exps
+  rule exps(_:Name=E and Bs) => E,exps(Bs)
+
+  /* Extra kore stuff */
+  syntax KResult ::= Vals
+  syntax Exps ::= Names
+  syntax Names ::= Bottoms
+
+  /* Matching */
+  syntax MatchResult ::= getMatching(Exp, Val)                      [function]
+                       | getMatchingAux(Exps, Vals)                 [function]
+                       | mergeMatching(MatchResult, MatchResult)    [function]
+                       | matchResult(Map)
+                       | "matchFailure"
+
+  rule getMatching(C:ConstructorName(Es:Exps), C(Vs:Vals)) => getMatchingAux(Es, Vs)
+  rule getMatching([Es:Exps], [Vs:Vals])                   => getMatchingAux(Es, Vs)
+  rule getMatching(C:ConstructorName, C) => matchResult(.Map)
+  rule getMatching(B:Bool, B)            => matchResult(.Map)
+  rule getMatching(I:Int, I)             => matchResult(.Map)
+  rule getMatching(S:String, S)          => matchResult(.Map)
+  rule getMatching(N:Name, V:Val) => matchResult(N |-> V)
+  rule getMatching(_, _) => matchFailure        [owise]
+
+  rule getMatchingAux((E:Exp, Es:Exps), (V:Val, Vs:Vals)) => mergeMatching(getMatching(E, V), getMatchingAux(Es, Vs))
+  rule getMatchingAux(.Exps, .Vals)                       => matchResult(.Map)
+  rule getMatchingAux(_, _) => matchFailure     [owise]
+
+  rule mergeMatching(matchResult(M1:Map), matchResult(M2:Map)) => matchResult(M1 M2)
+    requires intersectSet(keys(M1), keys(M2)) ==K .Set
+  //rule mergeMatching(_, _) => matchFailure      [owsie]
+  rule mergeMatching(matchResult(_:Map), matchFailure) => matchFailure
+  rule mergeMatching(matchFailure, matchResult(_:Map)) => matchFailure
+  rule mergeMatching(matchFailure, matchFailure)       => matchFailure
+

      Besides the generic decomposition rules for patterns and values,
      +we also want to allow [head|tail] matching for lists, so we add
      +the following custom pattern decomposition rule:

      +
        rule getMatching([H:Exp | T:Exp], [V:Val, Vs:Vals])
+    => getMatchingAux((H, T), (V, [Vs]))
+endmodule
+

      Go to Lesson 2, FUN untyped, Substitution-Based.

      +

      FUN — Untyped — Environment

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of the untyped FUN language.
      +FUN is a pedagogical and research language that captures the essence
      +of the functional programming paradigm, extended with several features
      +often encountered in functional programming languages.
      +Like many functional languages, FUN is an expression language, that
      +is, everything, including the main program, is an expression.
      +Functions can be declared anywhere and are first class values in the
      +language.
      +FUN is call-by-value here, but it has been extended (as student
      +homework assignments) with other parameter-passing styles.
      +To make it more interesting and to highlight some of K's strengths,
      +FUN includes the following features:

      +
        +
      • +

        The basic builtin data-types of integers, booleans and strings.

        +
        • +
      • +

        Builtin lists, which can hold any elements, including other lists.
        +Lists are enclosed in square brackets and their elements are
        +comma-separated; e.g., [1,2,3].

        +
        • +
      • +

        User-defined data-types, by means of constructor terms.
        +Constructor names start with a capital letter (while any other
        +identifier in the language starts with a lowercase letter), and they
        +can be followed by an arbitrary number of comma-separated arguments
        +enclosed in parentheses; parentheses are not needed when the
        +constructor takes no arguments.
        +For example, Pair(5,7) is a constructor term holding two
        +numbers, Cons(1,Cons(2,Cons(3,Nil))) is a list-like
        +constructor term holding 3 elements, and
        +Tree(Tree(Leaf(1), Leaf(2)), Leaf(3)) is a tree-like
        +constructor term holding 3 elements.
        +In the untyped version of the FUN language, no type checking or
        +inference is performed to ensure that the data constructors are used
        +correctly.
        +The execution will simply get stuck when they are misused.
        +Moreover, since no type checking is performed, the data-types are not
        +even declared in the untyped version of FUN.

        +
        • +
      • +

        Functions and let/letrec binders can take
        +multiple space-separated arguments, but these are desugared to
        +ones that only take one argument, by currying. For example, the
        +expressions

        +
        fun x y -> x y
+let x y = y in x
+

        are desugared, respectively, into the following expressions:

        +
        fun x -> fun y -> x y
+let x = fun y -> y in x
+
        • +
      • +

        Functions can be defined using pattern matching over the
        +available data-types. For example, the program

        +
        letrec max = fun [h] -> h
+             |   [h|t] -> let x = max t
+                          in  if h > x then h else x
+in max [1, 3, 5, 2, 4, 0, -1, -5]
+

        defines a function max that calculates the maximum element of
        +a non-empty list, and the function

        +
        letrec ack = fun Pair(0,n) -> n + 1
+             |   Pair(m,0) -> ack Pair(m - 1, 1)
+             |   Pair(m,n) -> ack Pair(m - 1, ack Pair(m, n - 1))
+in ack Pair(2,3)
+

        calculates the Ackermann function applied to a particular pair of numbers.
        +Patterns can be nested. Patterns can currently only be used in function
        +definitions, and not directly in let/letrec binders.
        +For example, this is not allowed:

        +
        letrec Pai(x,y) = Pair(1,2) in x+y
+

        But this is allowed:

        +
        let f Pair(x,y) = x+y in f Pair(1,2)
+

        because it is first reduced to

        +
        let f = fun Pair(x,y) -> x+y in f Pair(1,2)
+

        by uncurrying of the let binder, and pattern matching is
        +allowed in function arguments.

        +
        • +
      • +

        We include a callcc construct, for two reasons: first,
        +several functional languages support this construct; second, some
        +semantic frameworks have difficulties defining it. Not K.

        +
        • +
      • +

        Finally, we include mutables by means of referencing an
        +expression, getting the reference of a variable, dereferencing and
        +assignment. We include these for the same reasons as above: there are
        +languages which have them, and they are not easy to define in some
        +semantic frameworks.

        +
        • +
      +

      Like in many other languages, some of FUN's constructs can be
      +desugared into a smaller set of basic constructs. We do that as usual,
      +using macros, and then we only give semantics to the core constructs.

      +

      Note:
      +We recommend the reader to first consult the dynamic semantics of the
      +LAMBDA++ language in the first part of the K Tutorial.
      +To keep the comments below small and focused, we will not re-explain
      +functional or K features that have already been explained in there.

      +

      Syntax

      + +
      //require "modules/pattern-matching.k"
+
+module FUN-UNTYPED-COMMON
+  imports DOMAINS-SYNTAX
+

      FUN is an expression language. The constructs below fall into
      +several categories: names, arithmetic constructs, conventional
      +functional constructs, patterns and pattern matching, data constructs,
      +lists, references, and call-with-current-continuation (callcc).
      +The arithmetic constructs are standard; they are present in almost all
      +our K language definitions. The meaning of FUN's constructs are
      +discussed in more depth when we define their semantics in the next
      +module.

      +

      The Syntactic Constructs

      + +

      We start with the syntactic definition of FUN names.
      +We have several categories of names: ones to be used for functions and
      +variables, others to be used for data constructors, others for types and
      +others for type variables. We will introduce them as needed, starting
      +with the former category. We prefer the names of variables and functions
      +to start with lower case letters. We take the freedom to tacitly introduce
      +syntactic lists/sequences for each nonterminal for which we need them:

      +
        syntax Name                                      [token]
+  syntax Names ::= List{Name,","}                  [klabel(exps)]
+

      Expression constructs will be defined throughtout the syntax module.
      +Below are the very basic ones, namely the builtins, the names, and the
      +parentheses used as brackets for grouping. Lists of expressions are
      +declared strict, so all expressions in the list get evaluated whenever
      +the list is on a position which can be evaluated:

      +
        syntax Exp ::= Int | Bool | String | Name
+               | "(" Exp ")"                       [bracket]
+  syntax Exps  ::= List{Exp,","}                   [strict, klabel(exps)]
+  syntax Val
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= List{Val,","}                    [klabel(exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}              [klabel(exps)]
+

      We next define the syntax of arithmetic constructs, together with
      +their relative priorities and left-/non-associativities. We also
      +tag all these rules as members of a new group, "arith", so we can more easily
      +define global syntax priorities later (at the end of the syntax module).

      +
        syntax Exp ::= left:
+                 Exp "*" Exp                       [strict, group(arith)]
+               | Exp "/" Exp                       [strict, group(arith)]
+               | Exp "%" Exp                       [strict, group(arith)]
+               > left:
+                 Exp "+" Exp                       [strict, left, group(arith)]
+               | Exp "^" Exp                       [strict, left, group(arith)]
+// left attribute should not be necessary; currently a parsing bug
+               | Exp "-" Exp                       [strict, prefer, group(arith)]
+// the "prefer" attribute above is to not parse x-1 as x(-1)
+// Due to some parsing problems, we currently cannot add unary minus:
+               | "-" Exp                           [strict, group(arith)]
+               > non-assoc:
+                 Exp "<" Exp                       [strict, group(arith)]
+               | Exp "<=" Exp                      [strict, group(arith)]
+               | Exp ">" Exp                       [strict, group(arith)]
+               | Exp ">=" Exp                      [strict, group(arith)]
+               | Exp "==" Exp                      [strict, group(arith)]
+               | Exp "!=" Exp                      [strict, group(arith)]
+               > "!" Exp                           [strict, group(arith)]
+               > Exp "&&" Exp                      [strict(1), left, group(arith)]
+               > Exp "||" Exp                      [strict(1), left, group(arith)]
+

      The conditional construct has the expected evaluation strategy,
      +stating that only the first argument is evaluate:

      +
        syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+

      FUN's builtin lists are formed by enclosing comma-separated
      +sequences of expressions (i.e., terms of sort Exps) in square
      +brackets. The list constructor cons adds a new element to the
      +top of the list, head and tail get the first element
      +and the tail sublist of a list if they exist, respectively, and get
      +stuck otherwise, and null?? tests whether a list is empty or
      +not; syntactically, these are just expression constants.
      +In function patterns, we are also going to allow patterns following the
      +usual head/tail notation; for example, the pattern [x_1,...,x_n|t]
      +binds x_1, ..., x_n to the first elements of the matched list,
      +and t to the list formed with the remaining elements. We define list
      +patterns as ordinary expression constructs, although we will make sure that
      +we do not give them semantics if they appear in any other place then in a
      +function case pattern.

      +
        syntax Exp ::= "[" Exps "]"                             [strict, klabel(list)]
+               | "head" [macro] | "tail" [macro] | "null?" [macro]
+               | "[" Exps "|" Exp "]"
+  syntax Val ::= "[" Vals "]"                             [klabel(list)]
+  syntax Cons ::= "cons"
+  syntax Val ::= Cons
+  syntax Val ::= Cons Val                                 [klabel(apply)]
+

      Data constructors start with capital letters and they may or may
      +not have arguments. We need to use the attribute "prefer" to make
      +sure that, e.g., Cons(a) parses as constructor Cons with
      +argument a, and not as the expression Cons (because
      +constructor names are also expressions) regarded as a function applied
      +to the expression a. Also, note that the constructor is strict
      +in its second argument, because we want to evaluate its arguments but
      +not the constuctor name itsef.

      +
        syntax ConstructorName                         [token]
+  syntax Exp ::= ConstructorName
+               | ConstructorName "(" Exps ")"    [prefer, strict(2), klabel(constructor)]
+  syntax Val ::= ConstructorName "(" Vals ")"    [klabel(constructor)]
+

      A function is essentially a |-separated ordered
      +sequence of cases, each case of the form pattern -> expression,
      +preceded by the language construct fun. Patterns will be defined
      +shortly, both for the builtin lists and for user-defined constructors.
      +Recall that the syntax we define in K is not meant to serve as a
      +ultimate parser for the defined language, but rather as a convenient
      +notation for K abstract syntax trees, which we prefer when we write
      +the semantic rules. It is therefore often the case that we define a
      +more ``generous'' syntax than we want to allow programs to use.
      +We do it here, too. Specifically, the syntax of Cases
      +below allows any expressions to appear as pattern. This syntactic
      +relaxation permits many wrong programs to be parsed, but that is not a
      +problem because we are not going to give semantics to wrong combinations,
      +so those programs will get stuck; moreover, our type inferencer will reject
      +those programs anyway. Function application is just concatenation of
      +expressions, without worrying about type correctness. Again, the type
      +system will reject type-incorrect programs.

      +
        syntax Exp ::= "fun" Cases
+               | Exp Exp                              [strict, left, klabel(apply)]
+// NOTE: We would like eventually to also have Exp "(" Exps ")
+  syntax Case  ::= Exp "->" Exp
+  syntax Cases ::= List{Case, "|"}
+

      The let and letrec binders have the usual syntax
      +and functional meaning. We allow multiple and-separated bindings.
      +Like for the function cases above, we allow a more generous syntax for
      +the left-hand sides of bindings, noting that the semantics will get stuck
      +on incorrect bindings and that the type system will reject those programs.

      +
        syntax Exp ::= "let" Bindings "in" Exp
+               | "letrec" Bindings "in" Exp                 [prefer]
+// The "prefer" attribute for letrec currently needed due to tool bug,
+// to make sure that "letrec" is not parsed as "let rec".
+  syntax Binding  ::= Exp "=" Exp
+  syntax Bindings ::= List{Binding,"and"}
+

      References are first class values in FUN. The construct ref
      +takes an expression, evaluates it, and then it stores the resulting value
      +at a fresh location in the store and returns that reference. Syntactically,
      +ref is just an expression constant. The construct &
      +takes a name as argument and evaluates to a reference, namely the store
      +reference where the variable passed as argument stores its value; this
      +construct is a bit controversial and is further discussed in the
      +environment-based semantics of the FUN language, where we desugar
      +ref to it. The construct @ takes a reference
      +and evaluates to the value stored there. The construct := takes
      +two expressions, the first expected to evaluate to a reference; the value
      +of its second argument will be stored at the location to which the first
      +points (the old value is thus lost). Finally, since expression evaluation
      +now has side effects, it makes sense to also add a sequential composition
      +construct, which is sequentially strict. This evaluates to the value of
      +its second argument; the value of the first argument is lost (which has
      +therefore been evaluated only for its side effects.

      +
        syntax Exp ::= "ref"                             [macro]
+               | "&" Name
+               | "@" Exp                                     [strict]
+               | Exp ":=" Exp                                [strict]
+               | Exp ";" Exp                       [strict(1), right]
+

      Call-with-current-continuation, named callcc in FUN, is a
      +powerful control operator that originated in the Scheme programming
      +language, but it now exists in many other functional languages. It works
      +by evaluating its argument, expected to evaluate to a function, and by
      +passing the current continuation, or evaluation context (or computation,
      +in K terminology), as a special value to it. When/If this special value
      +is invoked, the current context is discarded and replaced with the one
      +held by the special value and the computation continues from there.
      +It is like taking a snapshot of the execution context at some moment
      +in time and then, when desired, being able to get back in time to that
      +point. If you like games, it is like saving the game now (so you can
      +work on your homework!) and then continuing the game tomorrow or whenever
      +you wish. To issustrate the strength of callcc, we also
      +allow exceptions in FUN by means of a conventional try-catch
      +construct, which will desugar to callcc. We also need to
      +introduce the special expression contant throw, but we need to
      +use it as a function argument name in the desugaring macro, so we define
      +it as a name instead of as an expression constant:

      +
        syntax Exp ::= "try" Exp "catch" "(" Name ")" Exp [macro]
+  syntax Val ::= "callcc"
+  syntax Name ::= "throw" [token]
+

      Finally, FUN also allows polymorphic datatype declarations. These
      +will be useful when we define the type system later on.

      +
        syntax Exp ::= "datatype" Type "=" TypeCases Exp [macro]
+// NOTE: In a future version of K, we want the datatype declaration
+// to be a construct by itself, but that is not possible currently
+// because K's parser wronly identifies the __ operation allowing
+// a declaration to appear in front of an expression with the function
+// application construct, giving ambiguous parsing errors.
+

      We next need to define the syntax of types and type cases that appear
      +in datatype declarations.

      +

      Like in many functional languages, type parameters/variables in
      +user-defined types are quoted identifiers.

      +
        syntax TypeVar                        [token]
+  syntax TypeVars ::= List{TypeVar,","} [klabel(types)]
+

      Types can be basic types, function types, or user-defined
      +parametric types. In the dynamic semantics we are going to simply ignore
      +all the type declations, so here the syntax of types below is only useful
      +for generating the desired parser. To avoid syntactic ambiguities with
      +the arrow construct for function cases, we use the symbol --> as
      +a constructor for function types:

      +
        syntax TypeName [token]
+  syntax Type ::= "int" | "bool" | "string"
+                | Type "-->" Type                            [right]
+                | "(" Type ")"                             [bracket]
+                | TypeVar
+                | TypeName             [klabel(TypeName), avoid]
+                | Type TypeName   [klabel(Type-TypeName), symbol, macro]
+                | "(" Types ")" TypeName                    [prefer]
+  syntax Types ::= List{Type,","} [klabel(types)]
+  syntax Types ::= TypeVars
+
+  syntax TypeCase ::= ConstructorName
+                    | ConstructorName "(" Types ")"
+  syntax TypeCases ::= List{TypeCase,"|"}     [klabel(_|TypeCase_)]
+

      Additional Priorities

      + +
        syntax priorities @__FUN-UNTYPED-COMMON
+                  > apply
+                  > arith
+                  > _:=__FUN-UNTYPED-COMMON
+                  > let_in__FUN-UNTYPED-COMMON
+                    letrec_in__FUN-UNTYPED-COMMON
+                    if_then_else__FUN-UNTYPED-COMMON
+                  > _;__FUN-UNTYPED-COMMON
+                  > fun__FUN-UNTYPED-COMMON
+                  > datatype_=___FUN-UNTYPED-COMMON
+endmodule
+
+module FUN-UNTYPED-MACROS
+  imports FUN-UNTYPED-COMMON
+

      Desugaring macros

      + +

      We desugar the list non-constructor operations to functions matching
      +over list patterns. In order to do that we need some new variables; for
      +those, we follow the same convention like in the K tutorial, where we
      +added them as new identifier constructs starting with the character $,
      +so we can easily recognize them when we debug or trace the semantics.

      +
        syntax Name ::= "$h" [token] | "$t" [token]
+  rule head => fun [$h|$t] -> $h
+  rule tail => fun [$h|$t] -> $t
+  rule null? => fun [.Exps] -> true | [$h|$t] -> false
+

      Multiple-head list patterns desugar into successive one-head patterns:

      +
        rule [E1,E2,Es:Exps|T] => [E1|[E2,Es|T]]                   [anywhere]
+

      Uncurrying of multiple arguments in functions and binders:

      +
        rule P1 P2 -> E => P1 -> fun P2 -> E                       [anywhere]
+  rule F P = E => F = fun P -> E                             [anywhere]
+

      We desugar the try-catch construct into callcc:

      +
        syntax Name ::= "$k" [token] | "$v" [token]
+  rule try E catch(X) E'
+    => callcc (fun $k -> (fun throw -> E)(fun X -> $k E'))
+

      For uniformity, we reduce all types to their general form:

      +
        rule `Type-TypeName`(T:Type, Tn:TypeName) => (T) Tn
+

      The dynamic semantics ignores all the type declarations:

      +
        rule datatype _T = _TCs E => E
+
+endmodule
+
+
+module FUN-UNTYPED-SYNTAX
+  imports FUN-UNTYPED-COMMON
+  imports BUILTIN-ID-TOKENS
+
+  syntax Name ::= r"[a-z][_a-zA-Z0-9]*"           [token, prec(2)]
+                | #LowerId                        [token]
+  syntax ConstructorName ::= #UpperId             [token]
+  syntax TypeVar  ::= r"['][a-z][_a-zA-Z0-9]*"    [token]
+  syntax TypeName ::= Name                        [token]
+endmodule
+

      Semantics

      + +

      The semantics below is environment-based. A substitution-based
      +definition of FUN is also available, but that drops the &
      +construct as explained above.

      +
      module FUN-UNTYPED
+  imports FUN-UNTYPED-COMMON
+  imports FUN-UNTYPED-MACROS
+  imports DOMAINS
+  //imports PATTERN-MATCHING
+

      Configuration

      + +

      The k, env, and store cells are standard
      +(see, for example, the definition of LAMBDA++ or IMP++ in the first
      +part of the K tutorial).

      +
        configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <env color="violet"> .Map </env>
+                  <store color="white"> .Map </store>
+                </T>
+

      Values and results

      + +

      We only define integers, Booleans and strings as values here, but will
      +add more values later.

      +
        syntax Val ::= Int | Bool | String
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+

      Lookup

      + +
        rule <k> X:Name => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V ...</store>
+

      Arithmetic expressions

      + +
        rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule I1 + I2 => I1 +Int I2
+  rule S1 ^ S2 => S1 +String S2
+  rule I1 - I2 => I1 -Int I2
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+

      Conditional

      + +
        rule if  true then E else _ => E
+  rule if false then _ else E => E
+

      Lists

      + +

      We have already declared the syntactic list of expressions strict, so
      +we can assume that all the elements that appear in a FUN list are
      +evaluated. The only thing left to do is to state that a list of
      +values is a value itself, that is, that the list square-bracket
      +construct is indeed a constructor, and to give the semantics of
      +cons. Since cons is a builtin function and is
      +expected to take two arguments, we have to also state that
      +cons itself is a value (specifically, a function/closure
      +value, but we do not need that level of detail here), and also that
      +cons applied to a value is a value (specifically, it would be
      +a function/closure value that expects the second, list argument):

      +
        rule cons V:Val [Vs:Vals] => [V,Vs]
+

      Data Constructors

      + +

      Constructors take values as arguments and produce other values:

      +
        syntax Val ::= ConstructorName
+

      Functions and Closures

      + +

      Like in the environment-based semantics of LAMBDA++ in the first part
      +of the K tutorial, functions evaluate to closures. A closure includes
      +the current environment besides the function contents; the environment
      +will be used at execution time to lookup all the variables that appear
      +free in the function body (we want static scoping in FUN).

      +
        syntax Val ::= closure(Map,Cases)
+  rule <k> fun Cases => closure(Rho,Cases) ...</k>  <env> Rho </env>
+

      Note: The reader may want to get familiar with
      +how the pre-defined pattern matching works before proceeding.
      +The best way to do that is to consult
      +k/include/modules/pattern-matching.k.

      +

      We distinguish two cases when the closure is applied.
      +If the first pattern matches, then we pick the first case: switch to
      +the closed environment, get the matching map and bind all its
      +variables, and finally evaluate the function body of the first case,
      +making sure that the environment is properly recovered afterwards.
      +If the first pattern does not match, then we drop it and thus move on
      +to the next one.

      +
        rule (. => getMatching(P, V)) ~> closure(_, P->_ | _) V:Val
+  rule <k> matchResult(M:Map) ~> closure(Rho, _->E | _) _
+           => bindMap(M) ~> E ~> setEnv(Rho') ...</k>
+       <env> Rho' => Rho </env>
+  rule (matchFailure => .) ~> closure(_, (_->_ | Cs:Cases => Cs)) _
+//  rule <k> closure(Rho, P->E | _) V:Val
+//           => bindMap(getMatching(P,V)) ~> E ~> setEnv(Rho') ...</k>
+//       <env> Rho' => Rho </env>  when isMatching(P,V)
+//  rule closure(_, (P->_ | Cs:Cases => Cs)) V:Val  when notBool isMatching(P,V)
+

      Let and Letrec

      + +

      To highlight the similarities and differences between let and
      +letrec, we prefer to give them direct semantics instead of
      +to desugar them like in LAMBDA. See the formal definitions of
      +bindTo, bind, and assignTo at the end of
      +this module. Informally, bindTo(Xs, Es) first
      +evaluates the expressions Es in Exps in the current
      +environment (i.e., it is strict in its second argument), then it binds
      +the variables in Xs in Names to new locations and adds
      +those bindings to the environment, and finally writes the values
      +previously obtained after evaluating the expressions Es to those
      +new locations; bind(Xs) does only the bindings of
      +Xs to new locations and adds those bindings to the environment;
      +and assignTo(Xs,Es) evaluates the expressions
      +Es in the current environment and then it writes the resulting
      +values to the locations to which the variables Xs are already
      +bound to in the environment.

      +

      Therefore, let Xs = Es in E first
      +evaluates Es in the current environment, then adds new
      +bindings for Xs to fresh locations in the environment, then
      +writes the values of Es to those locations, and finally
      +evaluates E in the new environment, making sure that the
      +environment is properly recovered after the evaluation of E.
      +On the other hand, letrec does the same things but in a
      +different order: it first adds new bindings for Xs to fresh
      +locations in the environment, then it evaluates Es in the new
      +environment, then it writes the resulting values to their
      +corresponding locations, and finally it evaluates E and
      +recovers the environment. The crucial difference is that the
      +expressions Es now see the locations of the variables Xs
      +in the environment, so if they are functions, which is typically the
      +case with letrec, their closures will encapsulate in their
      +environments the bindings of all the bound variables, including
      +themselves (thus, we may have a closure value stored at location
      +L, whose environment contains a binding of the form
      +F ↦ L; this way, the closure can invoke
      +itself).

      +
        rule <k> let Bs in E
+        => bindTo(names(Bs),exps(Bs)) ~> E ~> setEnv(Rho) ...</k>
+       <env> Rho </env>
+
+  rule <k> letrec Bs in E
+        => bind(names(Bs))~>assignTo(names(Bs),exps(Bs))~>E~>setEnv(Rho)...</k>
+       <env> Rho </env>
+

      Recall that our syntax allows let and letrec to
      +take any expression in place of its binding. This allows us to use
      +the already existing function application construct to bind names to
      +functions, such as, e.g., let x y = y in ....
      +The desugaring macro in the syntax module uncurries such declarations,
      +and then the semantic rules above only work when the remaining
      +bindings are identifiers, so the semantics will get stuck on programs
      +that misuse the let and letrec binders.

      +

      References

      + +

      The semantics of references is self-explanatory, except maybe for the
      +desugaring rule of ref, which is further discussed. Note
      +that &X grabs the location of X from the environment.
      +Sequential composition, which is needed only to accumulate the
      +side effects due to assignments, was strict in the first argument.
      +Once evaluated, its first argument is simply discarded:

      +
        syntax Name ::= "$x" [token]
+  rule ref => fun $x -> & $x
+  rule <k> & X => L ...</k>  <env>... X |-> L ...</env>
+  rule <k> @ L:Int => V:Val ...</k>  <store>... L |-> V ...</store>
+  rule <k> L:Int := V:Val => V ...</k>  <store>... L |-> (_=>V) ...</store>
+  rule _V:Val; E => E
+

      The desugaring rule of ref (first rule above) works
      +because & takes a variable and returns its location (like in C).
      +Note that some ``pure'' functional programming researchers strongly dislike
      +the & construct, but favor ref. We refrain from having
      +a personal opinion on this issue here, but support & in the
      +environment-based definition of FUN because it is, technically speaking,
      +more powerful than ref. From a language design perspective, it
      +would be equally easy to drop & and instead give a direct
      +semantics to ref. In fact, this is precisely what we do in the
      +substitution-based definition of FUN, because there appears to be no way
      +to give a substitution-based definition to the & construct.

      +

      Callcc

      + +

      As we know it from the LAMBDA++ tutorial, call-with-current-continuation
      +is quite easy to define in K. We first need to define a special
      +value wrapping an execution context, that is, an environment saying
      +where the variables should be looked up, and a computation structure
      +saying what is left to execute (in a substitution-based definition,
      +this special value would be even simpler, as it would only need to
      +wrap the computation structure---see, for example, the
      +substitution-based semantics of LAMBDA++ in the the first part of the
      +K tutorial, or the substitution-based definition of FUN). Then
      +callcc creates such a value containing the current
      +environment and the current remaining computation, and passes it to
      +its argument function. When/If invoked, the special value replaces
      +the current execution context with its own and continues the execution
      +normally.

      +
        syntax Val ::= cc(Map,K)
+  rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k>  <env> Rho </env>
+  rule <k> cc(Rho,K) V:Val ~> _ => V ~> K </k>  <env> _ => Rho </env>
+

      Auxiliary operations

      + +

      Environment recovery

      + +

      The environment recovery operation is the same as for the LAMBDA++
      +language in the K tutorial and many other languages provided with the
      +K distribution. The first ``anywhere'' rule below shows an elegant
      +way to achieve the benefits of tail recursion in K.

      +
        syntax KItem ::= setEnv(Map)  // TODO: get rid of env
+  //rule (setEnv(_) => .) ~> setEnv(_)  [anywhere]
+  rule <k> _:Val ~> (setEnv(Rho) => .) ...</k> <env> _ => Rho </env>
+

      bindTo, bind and assignTo

      + +

      The meaning of these operations has already been explained when we
      +discussed the let and letrec language constructs
      +above.

      +
        syntax KItem ::= bindTo(Names,Exps)         [strict(2)]
+                 | bindMap(Map)
+                 | bind(Names)
+
+  rule (. => getMatchingAux(Xs,Vs)) ~> bindTo(Xs:Names,Vs:Vals)
+  rule matchResult(M:Map) ~> bindTo(_:Names, _:Vals) => bindMap(M)
+
+  rule bindMap(.Map) => .
+  rule <k> bindMap((X:Name |-> V:Val => .Map) _:Map) ...</k>
+       <env> Rho => Rho[X <- !L:Int] </env>
+       <store>... .Map => !L |-> V ...</store>
+
+  rule bind(.Names) => .
+  rule <k> bind(X:Name,Xs => Xs) ...</k>
+       <env> Rho => Rho[X <- !_L:Int] </env>
+
+  syntax KItem ::= assignTo(Names,Exps)  [strict(2)]
+
+  rule <k> assignTo(.Names,.Vals) => . ...</k>
+  rule <k> assignTo((X:Name,Xs => Xs),(V:Val,Vs:Vals => Vs)) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... .Map => L |-> V ...</store>
+

      Getters

      + +

      The following auxiliary operations extract the list of identifiers
      +and of expressions in a binding, respectively.

      +
        syntax Names ::= names(Bindings)  [function]
+  rule names(.Bindings) => .Names
+  rule names(X:Name=_ and Bs) => (X,names(Bs))::Names
+
+  syntax Exps ::= exps(Bindings)  [function]
+  rule exps(.Bindings) => .Exps
+  rule exps(_:Name=E and Bs) => E,exps(Bs)
+
+  /* Extra kore stuff */
+  syntax KResult ::= Vals
+  syntax Exps ::= Names
+  syntax Names ::= Bottoms
+
+  /* Matching */
+  syntax MatchResult ::= getMatching(Exp, Val)                      [function]
+                       | getMatchingAux(Exps, Vals)                 [function]
+                       | mergeMatching(MatchResult, MatchResult)    [function]
+                       | matchResult(Map)
+                       | "matchFailure"
+
+  rule getMatching(C:ConstructorName(Es:Exps), C(Vs:Vals)) => getMatchingAux(Es, Vs)
+  rule getMatching([Es:Exps], [Vs:Vals])                   => getMatchingAux(Es, Vs)
+  rule getMatching(C:ConstructorName, C) => matchResult(.Map)
+  rule getMatching(B:Bool, B)            => matchResult(.Map)
+  rule getMatching(I:Int, I)             => matchResult(.Map)
+  rule getMatching(S:String, S)          => matchResult(.Map)
+  rule getMatching(N:Name, V:Val) => matchResult(N |-> V)
+  rule getMatching(_, _) => matchFailure        [owise]
+
+  rule getMatchingAux((E:Exp, Es:Exps), (V:Val, Vs:Vals)) => mergeMatching(getMatching(E, V), getMatchingAux(Es, Vs))
+  rule getMatchingAux(.Exps, .Vals)                       => matchResult(.Map)
+  rule getMatchingAux(_, _) => matchFailure     [owise]
+
+  rule mergeMatching(matchResult(M1:Map), matchResult(M2:Map)) => matchResult(M1 M2)
+    requires intersectSet(keys(M1), keys(M2)) ==K .Set
+  //rule mergeMatching(_, _) => matchFailure      [owsie]
+  rule mergeMatching(matchResult(_:Map), matchFailure) => matchFailure
+  rule mergeMatching(matchFailure, matchResult(_:Map)) => matchFailure
+  rule mergeMatching(matchFailure, matchFailure)       => matchFailure
+

      Besides the generic decomposition rules for patterns and values,
      +we also want to allow [head|tail] matching for lists, so we add
      +the following custom pattern decomposition rule:

      +
        rule getMatching([H:Exp | T:Exp], [V:Val, Vs:Vals])
+    => getMatchingAux((H, T), (V, [Vs]))
+endmodule
+

      Go to Lesson 2, FUN untyped, Substitution-Based.

      +

      FUN — Untyped — Substitution

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the substitution-based definition of FUN. For additional
      +explanations regarding the semantics of the various FUN constructs,
      +the reader should consult the emvironment-based definition of FUN.

      +

      Syntax

      + +
      require "substitution.md"
+//require "modules/pattern-matching.k"
+
+module FUN-UNTYPED-COMMON
+  imports DOMAINS-SYNTAX
+

      The Syntactic Constructs

      + +
        syntax Name
+  syntax Names ::= List{Name,","}
+
+  syntax Exp ::= Int | Bool | String | Name
+               | "(" Exp ")"                       [bracket]
+  syntax Exps  ::= List{Exp,","}                   [strict]
+  syntax Val
+  syntax Vals ::= List{Val,","}
+
+  syntax Exp ::= left:
+                 Exp "*" Exp                       [strict, arith]
+               | Exp "/" Exp                       [strict, arith]
+               | Exp "%" Exp                       [strict, arith]
+               > left:
+                 Exp "+" Exp                       [strict, left, arith]
+               | Exp "^" Exp                       [strict, left, arith]
+               | Exp "-" Exp                       [strict, prefer, arith]
+               | "-" Exp                           [strict, arith]
+               > non-assoc:
+                 Exp "<" Exp                       [strict, arith]
+               | Exp "<=" Exp                      [strict, arith]
+               | Exp ">" Exp                       [strict, arith]
+               | Exp ">=" Exp                      [strict, arith]
+               | Exp "==" Exp                      [strict, arith]
+               | Exp "!=" Exp                      [strict, arith]
+               > "!" Exp                           [strict, arith]
+               > Exp "&&" Exp                      [strict(1), left, arith]
+               > Exp "||" Exp                      [strict(1), left, arith]
+
+  syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+
+  syntax Exp ::= "[" Exps "]"                      [strict]
+               | "cons" |  "head" | "tail" | "null?"
+               | "[" Exps "|" Exp "]"
+  syntax Val ::= "[" Vals "]"
+
+  syntax ConstructorName
+  syntax Exp ::= ConstructorName
+               | ConstructorName "(" Exps ")"      [prefer, strict(2)]
+  syntax Val ::= ConstructorName "(" Vals ")"
+
+  syntax Exp ::= "fun" Cases
+               | Exp Exp                           [strict, left]
+  syntax Case  ::= Exp "->" Exp                    [binder]
+// NOTE: The binder attribute above is the only difference between this
+// module and the syntax module of environment-based FUN.  We need
+// to fix a bug in order to import modules and override the attributes
+// of operations.
+  syntax Cases ::= List{Case, "|"}
+
+  syntax Exp ::= "let" Bindings "in" Exp
+               | "letrec" Bindings "in" Exp                 [prefer]
+  syntax Binding  ::= Exp "=" Exp
+  syntax Bindings ::= List{Binding,"and"}
+
+  syntax Exp ::= "ref"
+               | "&" Name
+               | "@" Exp                           [strict]
+               | Exp ":=" Exp                      [strict]
+               | Exp ";" Exp                       [strict(1), right]
+
+  syntax Exp ::= "callcc"
+               | "try" Exp "catch" "(" Name ")" Exp
+  syntax Name ::= "throw" [token]
+
+  syntax Exp ::= "datatype" Type "=" TypeCases Exp
+
+  syntax TypeVar
+  syntax TypeVars ::= List{TypeVar,","}
+
+  syntax TypeName
+  syntax Type ::= "int" | "bool" | "string"
+                | Type "-->" Type                            [right]
+                | "(" Type ")"                             [bracket]
+                | TypeVar
+                | TypeName             [klabel(TypeName), avoid]
+                | Type TypeName   [klabel(Type-TypeName), onlyLabel]
+                | "(" Types ")" TypeName                    [prefer]
+  syntax Types ::= List{Type,","}
+  syntax Types ::= TypeVars
+
+  syntax TypeCase ::= ConstructorName
+                    | ConstructorName "(" Types ")"
+  syntax TypeCases ::= List{TypeCase,"|"}     [klabel(_|TypeCase_)]
+

      Additional Priorities

      + +
        syntax priorities @__FUN-UNTYPED-COMMON
+                  > ___FUN-UNTYPED-COMMON
+                  > arith
+                  > _:=__FUN-UNTYPED-COMMON
+                  > let_in__FUN-UNTYPED-COMMON
+                    letrec_in__FUN-UNTYPED-COMMON
+                    if_then_else__FUN-UNTYPED-COMMON
+                  > _;__FUN-UNTYPED-COMMON
+                  > fun__FUN-UNTYPED-COMMON
+                  > datatype_=___FUN-UNTYPED-COMMON
+endmodule
+
+module FUN-UNTYPED-MACROS
+  imports FUN-UNTYPED-COMMON
+

      Desugaring macros

      + +
        rule P1 P2 -> E => P1 -> fun P2 -> E                       [macro-rec]
+  rule F P = E => F = fun P -> E                             [macro-rec]
+
+  rule [E1,E2,Es:Exps|T] => [E1|[E2,Es|T]]                   [macro-rec]
+
+//  rule 'TypeName(Tn:TypeName) => (.TypeVars) Tn              [macro]
+  rule `Type-TypeName`(T:Type, Tn:TypeName) => (T) Tn          [macro]
+
+  syntax Name ::= "$h" | "$t"
+  rule head => fun [$h|$t] -> $h                             [macro]
+  rule tail => fun [$h|$t] -> $t                             [macro]
+  rule null? => fun [.Exps] -> true | [$h|$t] -> false       [macro]
+
+  syntax Name ::= "$k" | "$v"
+  rule try E catch(X) E'
+    => callcc (fun $k -> (fun throw -> E)(fun X -> $k E'))   [macro]
+
+  rule datatype _T = _TCs E => E                               [macro]
+

      mu needed for letrec, but we put it here so we can also write
      +programs with mu in them, which is particularly useful for testing.

      +
        syntax Exp ::= "mu" Case
+
+endmodule
+
+
+module FUN-UNTYPED-SYNTAX
+  imports FUN-UNTYPED-COMMON
+  imports BUILTIN-ID-TOKENS
+
+  syntax Name ::= r"[a-z][_a-zA-Z0-9]*"            [token, prec(2)]
+                | #LowerId                         [token]
+  syntax ConstructorName ::= #UpperId              [token]
+  syntax TypeVar  ::= r"['][a-z][_a-zA-Z0-9]*"     [token]
+  syntax TypeName ::= Name                         [token]
+endmodule
+

      Semantics

      + +
      module FUN-UNTYPED
+  imports FUN-UNTYPED-COMMON
+  imports FUN-UNTYPED-MACROS
+  imports DOMAINS
+  imports SUBSTITUTION
+  //imports PATTERN-MATCHING
+
+  configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <store color="white"> .Map </store>
+                </T>
+

      Both Name and functions are values now:

      +
        syntax Val ::= Int | Bool | String | Name
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax Exps ::= Names
+  syntax Vals ::= Names
+
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule I1 + I2 => I1 +Int I2
+  rule S1 ^ S2 => S1 +String S2
+  rule I1 - I2 => I1 -Int I2
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+  rule if  true then E else _ => E
+  rule if false then _ else E => E
+
+  rule isVal(cons) => true
+  rule isVal(cons _V:Val) => true
+  rule cons V:Val [Vs:Vals] => [V,Vs]
+
+  syntax Val ::= ConstructorName
+
+  rule isVal(fun _) => true
+  syntax KVar ::= Name
+  syntax Name ::= freshName(Int)    [freshGenerator, function]
+  rule freshName(I:Int) => {#parseToken("Name", "#" +String Int2String(I))}:>Name
+
+  rule (. => getMatching(P, V)) ~> (fun P->_ | _) V:Val
+  rule matchResult(M:Map) ~> (fun _->E | _) _ => E[M]
+  rule (matchFailure => .) ~> (fun (_->_ | Cs:Cases => Cs)) _
+//  rule (fun P->E | _) V:Val => E[getMatching(P,V)]  when isMatching(P,V)
+//  rule (fun (P->_ | Cs:Cases => Cs)) V:Val  when notBool isMatching(P,V)
+

      We can reduce multiple bindings to one list binding, and then
      +apply the usual desugaring of let into function application.
      +It is important that the rule below is a macro, so let is eliminated
      +immediately, otherwise it may interfere in ugly ways with substitution.

      +
        rule let Bs in E => ((fun [names(Bs)] -> E) [exps(Bs)])    [macro]
+

      We only give the semantics of one-binding letrec.
      +Multipe bindings are left as an exercise.

      +
        // changed because of parsing error
+  //rule mu X:Name -> E => E[(mu X -> E) / X]
+  rule mu X:Name -> E => E[X |-> (mu X -> E)]
+  rule letrec F:Name = E in E' => let F = (mu F -> E) in E'  [macro]
+

      We cannot have & anymore, but we can give direct
      +semantics to ref. We also have to declare ref to
      +be a value, so that we will never heat on it.

      +
      //  rule <k> & X => L ...</k>  <env>... X |-> L </env>
+  rule isVal(ref) => true
+  rule <k> ref V:Val => !L:Int ...</k> <store>... .Map => !L |-> V ...</store>
+  rule <k> @ L:Int => V:Val ...</k>  <store>... L |-> V ...</store>
+  rule <k> L:Int := V:Val => V ...</k>  <store>... L |-> (_=>V) ...</store>
+  rule _V:Val; E => E
+
+  syntax Val ::= cc(K)
+  rule isVal(callcc) => true
+  rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+  rule <k> cc(K) V:Val ~> _ => V ~> K </k>
+

      Auxiliary getters

      +
        syntax Names ::= names(Bindings)  [function]
+  rule names(.Bindings) => .Names
+  rule names(X:Name=_ and Bs) => X,names(Bs)
+
+  syntax Exps ::= exps(Bindings)  [function]
+  rule exps(.Bindings) => .Exps
+  rule exps(_:Name=E and Bs) => E,exps(Bs)
+
+  /* Extra kore stuff */
+  syntax KResult ::= Vals
+  syntax Exps ::= Names
+
+  /* Matching */
+  syntax MatchResult ::= getMatching(Exp, Val)                      [function]
+                       | getMatchingAux(Exps, Vals)                 [function]
+                       | mergeMatching(MatchResult, MatchResult)    [function]
+                       | matchResult(Map)
+                       | "matchFailure"
+
+  rule getMatching(C:ConstructorName(Es:Exps), C(Vs:Vals)) => getMatchingAux(Es, Vs)
+  rule getMatching([Es:Exps], [Vs:Vals])                   => getMatchingAux(Es, Vs)
+  rule getMatching(C:ConstructorName, C) => matchResult(.Map)
+  rule getMatching(B:Bool, B)            => matchResult(.Map)
+  rule getMatching(I:Int, I)             => matchResult(.Map)
+  rule getMatching(S:String, S)          => matchResult(.Map)
+  rule getMatching(N:Name, V:Val) => matchResult(N |-> V)
+  rule getMatching(_, _) => matchFailure        [owise]
+
+  rule getMatchingAux((E:Exp, Es:Exps), (V:Val, Vs:Vals)) => mergeMatching(getMatching(E, V), getMatchingAux(Es, Vs))
+  rule getMatchingAux(.Exps, .Vals)                       => matchResult(.Map)
+  rule getMatchingAux(_, _) => matchFailure     [owise]
+
+  rule mergeMatching(matchResult(M1:Map), matchResult(M2:Map)) => matchResult(M1 M2)
+    requires intersectSet(keys(M1), keys(M2)) ==K .Set
+  //rule mergeMatching(_, _) => matchFailure      [owsie]
+  rule mergeMatching(matchResult(_:Map), matchFailure) => matchFailure
+  rule mergeMatching(matchFailure, matchResult(_:Map)) => matchFailure
+  rule mergeMatching(matchFailure, matchFailure)       => matchFailure
+

      Besides the generic decomposition rules for patterns and values,
      +we also want to allow [head|tail] matching for lists, so we add
      +the following custom pattern decomposition rule:

      +
        rule getMatching([H:Exp | T:Exp], [V:Val, Vs:Vals])
+    => getMatchingAux((H, T), (V, [Vs]))
+endmodule
+

      LOGIK

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of LOGIK, a trivial language
      +capturing the essence of the logic programming paradigm. In this
      +definition, we explicitly focus on simplicity and mathematical
      +clarity, not on advanced logic programming features or performance.
      +Those are covered in the LOGIK++ extension under examples/logik++.

      +

      Specifically, a LOGIK program consists of a sequence of Horn clauses
      +of the form

      +
      P :- P1, P2, ..., Pn .
+

      followed by a query of the form

      +
      ?- Q1, Q2, ..., Qm .
+

      where P, P1, P2, ..., Pn, Q1, Q2,
      +..., Qm are literals. The
      +symbol :- is read "if". A literal has the form
      +p(T1,T2,...,Tk), where p is a predicate symbol
      +and where T1,T2,...,Tk are terms. Terms are built as
      +usual, with operation symbols and variables. A common
      +convention in logic programming languages, also adopted here, is that
      +variables are capitalized and operation symbols are not. Operations
      +with zero arguments are called constants and are written without
      +parentheses, that is, c instead of c(). Horn
      +clauses without conditions, called facts, are written
      +without :-, that is, P. instead of P :- ..

      +

      For example, the LOGIK program below gives a few facts about a
      +parent predicate, then several clauses defining some useful
      +predicates including an ancestor predicate, and finally a
      +query asking for those who both have ancestors and are ancestors
      +themselves in the parent relation:

      +
      parent(david,john).
+parent(jim,david).
+parent(steve,jim).
+parent(nathan,steve).
+
+grandparent(A,B):-
+  parent(A,X),
+  parent(X,B).
+
+ancestor(A,B):-
+  parent(A,X),
+  parents(X,B).
+
+parents(X,X).
+parents(A,B):-
+  ancestor(A,B).
+
+both(X) :- ancestor(A,X), ancestor(X,B).
+
+?- both(X).
+

      Above, we only have constant operation symbols, so these and variables
      +are the only terms that can be used in predicates. As expected, the
      +LOGIK program above will give us three solutions for X:
      +david, steve, and jim. If we inline the
      +both(X) predicate in the query, that is, if we replace the
      +query with ?- ancestor(A,X), ancestor(X,B). then we get
      +10 solutions, one for for each triple A, X, and
      +B satisfying both predicates ancestor(A,X) and
      +ancestor(X,B).

      +

      As another example, the program below defines an append
      +predicate followed by a simple goal:

      +
      append(nil,L,L).
+append(cons(H,T),L,cons(H,Z)) :- append(T,L,Z).
+
+?- append(cons(a,nil), cons(b,nil), V).
+

      Besides the predicate symbol append, the program above also
      +includes a constant symbol nil and a binary operation symbol
      +cons. Additionally, the query also includes two more
      +constants, a and b. The capitalized identifiers are
      +all variables. As expected, the LOGIK program above yields only one
      +solution, namely V = cons(a,cons(b,nil)). On the other hand,
      +if we change the query to:

      +
      ?- append(L1, cons(a,L2), cons(a,cons(b,cons(a,nil)))).
+

      then LOGIK yields two solutions: one where L1 is
      +cons(a,cons(b,nil)) and L2 is nil,
      +and another where L1 is nil and L2 is
      +cons(a,cons(b,nil)).

      +

      The programs above all generated ground solutions, that is,
      +solutions where the query variables are mapped to ground terms (i.e.,
      +terms without variables). Let us now consider the following query:

      +
      ?- append(cons(a,nil), Y, Z).
+

      There are obviously infinitely many ground solutions for the query
      +above, e.g.,
      +Y = nil and Z = cons(a,nil),
      +Y = cons(a,nil) and Z = cons(a,cons(a,nil)),
      +Y = cons(b,nil) and Z = cons(a,cons(b,nil)),
      +Y = cons(c,cons(b,nil)) and Z = cons(a,cons(c,cons(b,nil))),
      +etc. However, all the ground solutions for the query above can be
      +elegantly characterized by the property that Z is bound to a list
      +starting with a and followed by the list that Y is
      +bound to. This property can in fact be described as a symbolic solution
      +to the query: Z = cons(a,Y) or, equivalently,
      +Y = Symb and Z = cons(a,Symb). It is possible to
      +define a ``more general than'' relation on such symbolic solutions,
      +in the sense that the more particular solution can be obtained as a
      +specialization/substitution of the more general one, and then it can
      +be shown that the above is the most general solution to the
      +stated query. Logic programming languages, including our LOGIK,
      +attempt to always compute such most general solutions.

      +

      Logic programming languages are highly non-deterministic, in that
      +several Horn clauses may be used at the same time, each possibly
      +resulting in a different solution. Implementations of logic
      +programming languages consist of complex, optimized search and
      +indexing algorithms, which we are not concerned with here. Instead,
      +we here take advantage of K's builtin support for search.
      +Specifically, to find all the solutions of a LOGIK program, we have to
      +use krun with the option --search. However, note
      +that some programs have infinitely many solutions which cannot relate
      +to each other by the "more general" relation. For example, the query

      +
      ?- append(L1, cons(a,L2), L3) .
+

      To address such cases and terminate, logic programming languages allow
      +the user to choose how many solutions to be computed and displayed.
      +In LOGIK, we can use the --bound option of krun for
      +this purpose.

      +

      Finally, note that some queries have no solution. In some cases that
      +is easy to detect by exhaustive analysis, such as for the following
      +query:

      +
      ?- append(cons(a,L1), L2, cons(b,L3)).
+

      Logic programming languages, including LOGIK, terminate in such cases
      +and report a no solution answer. However, there are cases where
      +exhaustive analysis is not sufficient, such as for the query:

      +
      ?- append(cons(a,L), nil, L).
+

      In such cases, logic programming languages do not terminate. While
      +one may devise techniques to detect non-termination in some cases,
      +one cannot do it in general (same like for all Turing-complete
      +languages).

      +
      require "unification.k"
+
+module LOGIK-COMMON
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of LOGIK is straightforward: a program is a sequence of
      +Horn clauses followed by a query:

      +
        syntax Literal
+  syntax Term ::= Literal | Literal "(" Terms ")"
+  syntax Terms ::= List{Term,","}
+  syntax Clause ::= Term ":-" Terms "." | Term "."
+  syntax Query ::= "?-" Terms "."
+  syntax Pgm ::= Query | Clause Pgm
+endmodule
+
+module LOGIK-SYNTAX
+  imports LOGIK-COMMON
+  imports BUILTIN-ID-TOKENS
+

      Variables and literals are defined as tokens following the conventions
      +used in Prolog (variables start with _ or capital letter, while literals
      +start with lower case letters):

      +
        syntax #KVariable ::= r"[A-Z_][A-Za-z0-9_]*"   [token, prec(2)]
+                      | #UpperId                 [token]
+  syntax Term ::= #KVariable [klabel(#SemanticCastToTerm)]
+  syntax Literal ::= r"[a-z][a-zA-Z0-9_]*"                 [token]
+                   | #LowerId                              [token]
+endmodule
+
+module LOGIK
+  imports LOGIK-COMMON
+  imports DOMAINS
+  imports UNIFICATION
+

      Unification is at the core of logic programming. Here we are
      +going to use the predefined unification procedure (the same one we
      +used in the type inferencers in Tutorial 5).

      +

      Configuration

      + +

      The configuration stores each clause in its own cell for easy access,
      +and the most general unifier in a cell named mgu, same like
      +the type inferencers. The k cell holds the query and the
      +fresh cell holds a fresh clause instance to be attempted on
      +the next query item. To more easily read the solutions, we add a
      +second top-level cell, solution. Both top cells are
      +optional. Indeed, we start with the main top cell and, when a
      +solution is found, we move it into the solution cell and
      +discard the main cell.

      +
        configuration <T color="yellow" multiplicity="?">
+                  <k color="green"> $PGM:Pgm </k>
+                  <fresh color="orange"> .K </fresh>
+                  <clauses color="red">
+                    <clause color="pink" multiplicity="*"> .K </clause>
+                  </clauses>
+                  <mgu> .K </mgu>
+                </T>
+                <solution multiplicity="?"> .K </solution>
+

      Pre- and post-processing

      + +

      Before we launch the semantics, we first scan the given program and
      +place each clause in its own cell, and then place the query in the
      +k cell and initialize the mgu with the variables from the query.

      +

      Note that we put a fresh instance of the clause to avoid interference with
      +the query variables. By a "fresh instance" of a clause we mean one whose
      +variables are renamed with fresh names; we need that in order to avoid
      +undesired unification conflicts due to particular names chosen for
      +variables in the original program, as well as conflicts due to
      +subsequent uses of the same clause. It is safe to rename the
      +variables in a clause, because clauses are universally quantified in
      +their variables. This process of creating a fresh instance of a
      +clause is similar to how we created fresh instances of type schemas in
      +the higher-order type inferencer discussed in Tutorial 5. Indeed, we
      +can safely regard clauses as "clause schemas" comprising infinitely
      +many instances, one for each context.

      +
        rule <k> C:Clause Pgm => Pgm </k>
+       (.Bag => <clause> #renameVariables(C) </clause>)
+
+  rule <k> ?- Ls:Terms. => Ls ...</k>
+       <mgu> _ => #variablesMap(#variables(Ls)) </mgu>
+

      We also sequentialize the goals for easier processing:

      +
        rule L:Term, Ls:Terms => L ~> Ls
+  rule .Terms => .
+

      When all the goals are solved, indicated by the empty k
      +cell, the calculated most general unifier (mgu) is in the mgu
      +cell. In that case, to ease reading of the final solution we move the
      +mgu in the solution cell and delete the rest of the
      +configuration:

      +
      rule <T>... <k> . </k> <mgu> Theta </mgu> ...</T>
+  => <solution> Theta </solution>
+

      Since we are not interested in seeing the failed attempts to solve
      +the query, we collapse all the error configurations into an empty
      +configuration (recall that both top-level cells in the configuration
      +were declared optional). This way, if we see an empty configuration
      +when we search for all solutions, we know that some attempts failed
      +(but we do not know which ones).

      +
      // this would be nice, but we need feedback from the external unifier
+// for this.
+//  rule <T>... <mgu> _:MguError </mgu> ...</T> => .
+

      Semantics

      + +

      Once all the infrastructure is in place, the actual semantics of LOGIK
      +is quite simple. All we have to do is to pick some (fresh instance of
      +a) clause, then unify its conclusion with the first query literal, and
      +then replace that literal with condition of the clause. The intuition
      +here is the following: to satisfy the first literal in the query, we
      +need to find some instance of some clause that matches it, and then to
      +similarly show that we can satisfy the conditions of that clause.
      +Mathematically, this is an instance of the proof principle called
      +resolution: if p ∨ q and ¬ p ∨ r hold, then so does
      +q ∨ r. We let it as an exercise to the reader to see how the two
      +relate (hint: assume the negation of the goal together with all the
      +clauses, and then derive false).

      +

      The following two rules are tightly connected and they together
      +perform the following core task: pick a fresh instance of a clause
      +which unifies with the first goal item, then add its conditions as new
      +goals.

      +

      Pick a clause and generate a fresh instance of it when the
      +fresh cell is empty:

      +
        rule <fresh> . => #renameVariables(C) </fresh> <clause> C </clause>
+       <k> T:Term ...</k>
+  requires #unifiable(T,head(C))
+
+  syntax Term ::= head(Clause) [function]
+  rule head(L.) => L
+  rule head(L:-_.) => L
+

      If the goal is unifiable with the fresh clause's head, replace the goal
      +with the clause body, and empty the fresh cell (so that
      +another clause can be chosen using the rule above):

      +
        rule <k> L:Term => . ...</k>
+       <fresh>  L:Term . => . </fresh>
+
+  rule <k> L:Term :KItem => Ls ...</k>
+       <fresh>  L:Term :- Ls:Terms. => . </fresh>
+

      Note that there is no problem if a clause is chosen whose
      +conclusion literal does not unify with the first goal literal.
      +The search
      +option of krun will systematically try all clauses, so no
      +solution is missed. Of course, the above is not the most efficient
      +way to implement a logic programming language, but recall that our
      +objective here was to present a simple and mathematically clean
      +solution. We encourage the interested reader to consult the LOGIK++
      +language definition for a more efficient definition of a richer logic
      +programming language.

      +
      endmodule
+

      LOGIK

      + +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      + +

      This is the K semantic definition of LOGIK, a trivial language
      +capturing the essence of the logic programming paradigm. In this
      +definition, we explicitly focus on simplicity and mathematical
      +clarity, not on advanced logic programming features or performance.
      +Those are covered in the LOGIK++ extension under examples/logik++.

      +

      Specifically, a LOGIK program consists of a sequence of Horn clauses
      +of the form

      +
      P :- P1, P2, ..., Pn .
+

      followed by a query of the form

      +
      ?- Q1, Q2, ..., Qm .
+

      where P, P1, P2, ..., Pn, Q1, Q2,
      +..., Qm are literals. The
      +symbol :- is read "if". A literal has the form
      +p(T1,T2,...,Tk), where p is a predicate symbol
      +and where T1,T2,...,Tk are terms. Terms are built as
      +usual, with operation symbols and variables. A common
      +convention in logic programming languages, also adopted here, is that
      +variables are capitalized and operation symbols are not. Operations
      +with zero arguments are called constants and are written without
      +parentheses, that is, c instead of c(). Horn
      +clauses without conditions, called facts, are written
      +without :-, that is, P. instead of P :- ..

      +

      For example, the LOGIK program below gives a few facts about a
      +parent predicate, then several clauses defining some useful
      +predicates including an ancestor predicate, and finally a
      +query asking for those who both have ancestors and are ancestors
      +themselves in the parent relation:

      +
      parent(david,john).
+parent(jim,david).
+parent(steve,jim).
+parent(nathan,steve).
+
+grandparent(A,B):-
+  parent(A,X),
+  parent(X,B).
+
+ancestor(A,B):-
+  parent(A,X),
+  parents(X,B).
+
+parents(X,X).
+parents(A,B):-
+  ancestor(A,B).
+
+both(X) :- ancestor(A,X), ancestor(X,B).
+
+?- both(X).
+

      Above, we only have constant operation symbols, so these and variables
      +are the only terms that can be used in predicates. As expected, the
      +LOGIK program above will give us three solutions for X:
      +david, steve, and jim. If we inline the
      +both(X) predicate in the query, that is, if we replace the
      +query with ?- ancestor(A,X), ancestor(X,B). then we get
      +10 solutions, one for for each triple A, X, and
      +B satisfying both predicates ancestor(A,X) and
      +ancestor(X,B).

      +

      As another example, the program below defines an append
      +predicate followed by a simple goal:

      +
      append(nil,L,L).
+append(cons(H,T),L,cons(H,Z)) :- append(T,L,Z).
+
+?- append(cons(a,nil), cons(b,nil), V).
+

      Besides the predicate symbol append, the program above also
      +includes a constant symbol nil and a binary operation symbol
      +cons. Additionally, the query also includes two more
      +constants, a and b. The capitalized identifiers are
      +all variables. As expected, the LOGIK program above yields only one
      +solution, namely V = cons(a,cons(b,nil)). On the other hand,
      +if we change the query to:

      +
      ?- append(L1, cons(a,L2), cons(a,cons(b,cons(a,nil)))).
+

      then LOGIK yields two solutions: one where L1 is
      +cons(a,cons(b,nil)) and L2 is nil,
      +and another where L1 is nil and L2 is
      +cons(a,cons(b,nil)).

      +

      The programs above all generated ground solutions, that is,
      +solutions where the query variables are mapped to ground terms (i.e.,
      +terms without variables). Let us now consider the following query:

      +
      ?- append(cons(a,nil), Y, Z).
+

      There are obviously infinitely many ground solutions for the query
      +above, e.g.,
      +Y = nil and Z = cons(a,nil),
      +Y = cons(a,nil) and Z = cons(a,cons(a,nil)),
      +Y = cons(b,nil) and Z = cons(a,cons(b,nil)),
      +Y = cons(c,cons(b,nil)) and Z = cons(a,cons(c,cons(b,nil))),
      +etc. However, all the ground solutions for the query above can be
      +elegantly characterized by the property that Z is bound to a list
      +starting with a and followed by the list that Y is
      +bound to. This property can in fact be described as a symbolic solution
      +to the query: Z = cons(a,Y) or, equivalently,
      +Y = Symb and Z = cons(a,Symb). It is possible to
      +define a ``more general than'' relation on such symbolic solutions,
      +in the sense that the more particular solution can be obtained as a
      +specialization/substitution of the more general one, and then it can
      +be shown that the above is the most general solution to the
      +stated query. Logic programming languages, including our LOGIK,
      +attempt to always compute such most general solutions.

      +

      Logic programming languages are highly non-deterministic, in that
      +several Horn clauses may be used at the same time, each possibly
      +resulting in a different solution. Implementations of logic
      +programming languages consist of complex, optimized search and
      +indexing algorithms, which we are not concerned with here. Instead,
      +we here take advantage of K's builtin support for search.
      +Specifically, to find all the solutions of a LOGIK program, we have to
      +use krun with the option --search. However, note
      +that some programs have infinitely many solutions which cannot relate
      +to each other by the "more general" relation. For example, the query

      +
      ?- append(L1, cons(a,L2), L3) .
+

      To address such cases and terminate, logic programming languages allow
      +the user to choose how many solutions to be computed and displayed.
      +In LOGIK, we can use the --bound option of krun for
      +this purpose.

      +

      Finally, note that some queries have no solution. In some cases that
      +is easy to detect by exhaustive analysis, such as for the following
      +query:

      +
      ?- append(cons(a,L1), L2, cons(b,L3)).
+

      Logic programming languages, including LOGIK, terminate in such cases
      +and report a no solution answer. However, there are cases where
      +exhaustive analysis is not sufficient, such as for the query:

      +
      ?- append(cons(a,L), nil, L).
+

      In such cases, logic programming languages do not terminate. While
      +one may devise techniques to detect non-termination in some cases,
      +one cannot do it in general (same like for all Turing-complete
      +languages).

      +
      require "unification.k"
+
+module LOGIK-COMMON
+  imports DOMAINS-SYNTAX
+

      Syntax

      + +

      The syntax of LOGIK is straightforward: a program is a sequence of
      +Horn clauses followed by a query:

      +
        syntax Literal
+  syntax Term ::= Literal | Literal "(" Terms ")"
+  syntax Terms ::= List{Term,","}
+  syntax Clause ::= Term ":-" Terms "." | Term "."
+  syntax Query ::= "?-" Terms "."
+  syntax Pgm ::= Query | Clause Pgm
+endmodule
+
+module LOGIK-SYNTAX
+  imports LOGIK-COMMON
+  imports BUILTIN-ID-TOKENS
+

      Variables and literals are defined as tokens following the conventions
      +used in Prolog (variables start with _ or capital letter, while literals
      +start with lower case letters):

      +
        syntax #KVariable ::= r"[A-Z_][A-Za-z0-9_]*"   [token, prec(2)]
+                      | #UpperId                 [token]
+  syntax Term ::= #KVariable [klabel(#SemanticCastToTerm)]
+  syntax Literal ::= r"[a-z][a-zA-Z0-9_]*"                 [token]
+                   | #LowerId                              [token]
+endmodule
+
+module LOGIK
+  imports LOGIK-COMMON
+  imports DOMAINS
+  imports UNIFICATION
+

      Unification is at the core of logic programming. Here we are
      +going to use the predefined unification procedure (the same one we
      +used in the type inferencers in Tutorial 5).

      +

      Configuration

      + +

      The configuration stores each clause in its own cell for easy access,
      +and the most general unifier in a cell named mgu, same like
      +the type inferencers. The k cell holds the query and the
      +fresh cell holds a fresh clause instance to be attempted on
      +the next query item. To more easily read the solutions, we add a
      +second top-level cell, solution. Both top cells are
      +optional. Indeed, we start with the main top cell and, when a
      +solution is found, we move it into the solution cell and
      +discard the main cell.

      +
        configuration <T color="yellow" multiplicity="?">
+                  <k color="green"> $PGM:Pgm </k>
+                  <fresh color="orange"> .K </fresh>
+                  <clauses color="red">
+                    <clause color="pink" multiplicity="*"> .K </clause>
+                  </clauses>
+                  <mgu> .K </mgu>
+                </T>
+                <solution multiplicity="?"> .K </solution>
+

      Pre- and post-processing

      + +

      Before we launch the semantics, we first scan the given program and
      +place each clause in its own cell, and then place the query in the
      +k cell and initialize the mgu with the variables from the query.

      +

      Note that we put a fresh instance of the clause to avoid interference with
      +the query variables. By a "fresh instance" of a clause we mean one whose
      +variables are renamed with fresh names; we need that in order to avoid
      +undesired unification conflicts due to particular names chosen for
      +variables in the original program, as well as conflicts due to
      +subsequent uses of the same clause. It is safe to rename the
      +variables in a clause, because clauses are universally quantified in
      +their variables. This process of creating a fresh instance of a
      +clause is similar to how we created fresh instances of type schemas in
      +the higher-order type inferencer discussed in Tutorial 5. Indeed, we
      +can safely regard clauses as "clause schemas" comprising infinitely
      +many instances, one for each context.

      +
        rule <k> C:Clause Pgm => Pgm </k>
+       (.Bag => <clause> #renameVariables(C) </clause>)
+
+  rule <k> ?- Ls:Terms. => Ls ...</k>
+       <mgu> _ => #variablesMap(#variables(Ls)) </mgu>
+

      We also sequentialize the goals for easier processing:

      +
        rule L:Term, Ls:Terms => L ~> Ls
+  rule .Terms => .
+

      When all the goals are solved, indicated by the empty k
      +cell, the calculated most general unifier (mgu) is in the mgu
      +cell. In that case, to ease reading of the final solution we move the
      +mgu in the solution cell and delete the rest of the
      +configuration:

      +
      rule <T>... <k> . </k> <mgu> Theta </mgu> ...</T>
+  => <solution> Theta </solution>
+

      Since we are not interested in seeing the failed attempts to solve
      +the query, we collapse all the error configurations into an empty
      +configuration (recall that both top-level cells in the configuration
      +were declared optional). This way, if we see an empty configuration
      +when we search for all solutions, we know that some attempts failed
      +(but we do not know which ones).

      +
      // this would be nice, but we need feedback from the external unifier
+// for this.
+//  rule <T>... <mgu> _:MguError </mgu> ...</T> => .
+

      Semantics

      + +

      Once all the infrastructure is in place, the actual semantics of LOGIK
      +is quite simple. All we have to do is to pick some (fresh instance of
      +a) clause, then unify its conclusion with the first query literal, and
      +then replace that literal with condition of the clause. The intuition
      +here is the following: to satisfy the first literal in the query, we
      +need to find some instance of some clause that matches it, and then to
      +similarly show that we can satisfy the conditions of that clause.
      +Mathematically, this is an instance of the proof principle called
      +resolution: if p ∨ q and ¬ p ∨ r hold, then so does
      +q ∨ r. We let it as an exercise to the reader to see how the two
      +relate (hint: assume the negation of the goal together with all the
      +clauses, and then derive false).

      +

      The following two rules are tightly connected and they together
      +perform the following core task: pick a fresh instance of a clause
      +which unifies with the first goal item, then add its conditions as new
      +goals.

      +

      Pick a clause and generate a fresh instance of it when the
      +fresh cell is empty:

      +
        rule <fresh> . => #renameVariables(C) </fresh> <clause> C </clause>
+       <k> T:Term ...</k>
+  requires #unifiable(T,head(C))
+
+  syntax Term ::= head(Clause) [function]
+  rule head(L.) => L
+  rule head(L:-_.) => L
+

      If the goal is unifiable with the fresh clause's head, replace the goal
      +with the clause body, and empty the fresh cell (so that
      +another clause can be chosen using the rule above):

      +
        rule <k> L:Term => . ...</k>
+       <fresh>  L:Term . => . </fresh>
+
+  rule <k> L:Term :KItem => Ls ...</k>
+       <fresh>  L:Term :- Ls:Terms. => . </fresh>
+

      Note that there is no problem if a clause is chosen whose
      +conclusion literal does not unify with the first goal literal.
      +The search
      +option of krun will systematically try all clauses, so no
      +solution is missed. Of course, the above is not the most efficient
      +way to implement a logic programming language, but recall that our
      +objective here was to present a simple and mathematically clean
      +solution. We encourage the interested reader to consult the LOGIK++
      +language definition for a more efficient definition of a richer logic
      +programming language.

      +
      endmodule
+

      Projects using K

      + +

      A list of projects using the K framework. If you are working on something interesting, and you want to share it with the community,
      +let us know on our socials, and we will feature you on this list.

      +
      + + +
      +
        +
      • +

        KAVM (Feb 2022 - Present)

        +

        The Algorand Virtual Machine and TEAL Semantics in K
        +KAVM leverages the K Framework to empower Algorand smart contracts' developers
        +with property-based testing and formal verification.

        +
        • +
      • +

        KPlutus (2016 - Present)

        +

        The K Semantics of Plutus-Core

        +
        • +
      • +

        Dedukti (Mar 2021 - Present)

        +

        This project aims to translate real K semantics into Dedukti.

        +
        • +
      • +

        KWasm (Aug 2015 - Present)

        +

        KWasm is the K semantics of WebAssembly.
        +WebAssembly is a low-level (but simple and streamlined) assembly language that was originally developed to provide a fast execution engine for browser-based tools.
        +More recently, it has been used in several blockchain smart-contract platforms as the underlying language for executing financial agreements.
        +KWasm has been used for measuring coverage of test-suites over Wasm code and verifying programs which are compiled to Wasm.

        +
        • +
      • +

        KEVM (Sep 2017 - Present)

        +

        KEVM is the K semantics of the Ethereum Virtual Machine.
        +It passes all the Ethereum Test Suite, and is used for verifying EVM programs.

        +
        • +
      • +

        IELE (Oct 2016 - Present)

        +

        IELE is the underlying VM integrated into the Cardano blockchain.
        +IELE is a register-based VM (inspired by LLVM), which attempts to avoid many of the missteps in design present in EVM.

        +
        • +
      • +

        K-Michelson (Oct 2019 - Present)

        +

        K-Michelson is the K semantics of Michelson blockchain programming language, which powers the Tezos blockchain.
        +KMichelson provides additional testing tools for developers, including a unit-testing framework which is extendable to symbolic property testing.

        +
        • +
      • +

        C (Jul 2010 - Present)

        +

        The K semantics of the C programming language specifies the translation, linking, and execution semantics of the C language according to the official C standard.
        +It has been used to build tools like RV-Match, which detects undefined behaviors in users programs by running their test-suites through the C semantics.

        +
        • +
      +

      Archived

      + + +
      +
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      FAQ

      +

      General questions

      +

      What is K?

      +

      [13 Dec, 2013] K is a framework for defining programming languages. Once you define a language, K gives you a series of tools for that language, such as: a parser, an interpreter, a state-space explorer (like a model-checker for reachability), and even a deductive program verifier. We continuously work on making these tools better and on adding new tools.

      +

      What is a language definition?

      +

      [13 Dec, 2013] A language definition consists of two parts: syntax and semantics. The syntax is defined using a BNF-style, enriched with several features to ease the semantics. The semantics tells what each language construct is meant to do. This way, a language definition says both how the programs in your language should look like and also what they mean, or how they execute.

      +

      What is the difference between a definition and an implementation?

      +

      [13 Dec, 2013] No difference in K. We think of K definitions as formal, rigorous implementations of the language. In fact, many users of K have no background on programming language semantics, they think of K as a domain-specific language for implementing programming languages. The benefit of implementing your language in K is that you can make use of the tools that K offers, which is not possible when you implement your language in a conventional programming language.

      +

      Why K?

      +

      [13 Dec, 2013] There was and still is a considerable amount of effort spent by many scientists on developing parsing, model-checking, program verification and other formal program analysis techniques. Most of these techniques are language independent, yet a considerable amount of effort is then spent on developing language-specific tools based on these techniques. For example, developing a model-checker or a program verifier for Java, or C, or Python, is a serious endeavor, that only very few highly-skilled people can attempt. We believe that all these language-specific tools can be automatically derived from the K language definition, so that language designers spend the time only once to define their language and then get not only an implementation of their language, but also all the other tools, essentially for free.

      +

      What is the difference between K and ...

      +

      SDF

      +

      [13 Dec, 2013]sdf is a parser generator. Simply speaking, it takes as input a grammar written in the SDF format and a text, and creates the abstract-syntax tree of that text corresponding to the grammar specification. K currently uses SDF for its parsing needs, but we integrated it into a more complex environment suitable for semantic definitions. Using the same language specification, we generate multiple parsers for different purposes: parse programs, parse rewrite rules, etc. Another difference is that we changed a bit the syntax of the grammar specification. We adopted a BNF-style notation whereas SDF uses an algebraic specification, but we keept the same disambiguation system with priorities and associativity filters.

      +

      Maude

      +

      PLT Redex

      +

      [16 Dec, 2013]plt redex is a language definitional framework based on reduction semantics with evaluation contexts, a type of Structural Operational Semantics. A PLT definition consists of the syntax for the language (including the syntax of the execution configuration, if needed), followed by a syntax for evaluation contexts which allows identifying the next reducible expression (redex). The rules can specify the parts of the context (and abstract parts of it using variables), and can alter both the redex and the context. PLT Redex offers a suite of tools built on top of the Racket Scheme-based IDE to help visualize and explore executions. K borrows from PLT Redex the idea of evaluation contexts, and extends it further allowing more complex conditions be put on them. A distinctive difference between Redex and K is the fact that in K evaluation contexts are used only for the computational fragment of the executing configuration, the rules applying modulo the configuration abstraction. This, for example, allows K to more easily specify synchronous communication of agents or threads.

      +

      Spoofax

      +

      Rascal

      +

      OTT

      +

      ATL and Model-Driven Engineering

      +

      [14 Dec, 2013]atl (Atlas Transformation Language) falls in the Model-Driven Engineering (MDE) field and includes a model transformation language and toolkit. ATL is also based on rules, which provide a means to produce a target model Mb conforming to a meta-model MMb, from a source model Ma conforming to a meta-model MMa. It should not be difficult to define such model transformations using K, this way effectively using the target meta-model MMb to give semantics to the source meta-model MMa. Moreover, if MMa and MMb have K semantics themselves, then the K tool can be used for proving the conformance of the transformation. Note, however, that K does not currently supply any explicit support for meta-model technologies, such as EMF (Eclipse Modelling Framework), etc.

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      + + + + K + + Semantic Framework + + + Download +
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      K is a rewrite-based +executable semantic framework in which programming languages, type +systems and formal analysis tools can be defined using configurations +and rules. Configurations organize the state in units called cells, +which are labeled and can be nested. K rewrite rules make it explicit +which parts of the term are read-only, write-only, read-write, or +unused. This makes K suitable for defining truly concurrent languages +even in the presence of sharing. Computations are represented as +syntactic extensions of the original language abstract syntax, using a +nested list structure which sequentializes computational tasks, such +as program fragments. Computations are like any other terms in a +rewriting environment: they can be matched, moved from one place to +another, modified, or deleted. This makes K suitable for defining +control-intensive features such as abrupt termination, exceptions, or +call/cc.

      +

      K Tool Download

      +
        +
      • Install from the latest K GitHub Release.
        • +
      • Try our Editor Support page for links to K syntax highlighting definitions for various popular editors/IDEs. Please feel free to contribute.
        • +
      • Build or browse the code on GitHub, where you can also report bugs.
        • +
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      Learn K

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      + +
      +
      + + + + + + + + + + + + diff --git a/install b/install new file mode 100644 index 00000000000..10cdb520878 --- /dev/null +++ b/install @@ -0,0 +1,69 @@ +#!/usr/bin/env bash + +{ # Prevent execution if this script was only partially downloaded + set -e + + GREEN='\033[0;32m' + RED='\033[0;31m' + NC='\033[0m' + + NIX_POST_INSTALL_MESSAGE="To ensure that the necessary environment variables are set, either log in again, or type + + . /nix/var/nix/profiles/default/etc/profile.d/nix-daemon.sh + +in your shell." + + oops() { + >&2 echo -e "${RED}error:${NC} $1" + exit 1 + } + + [[ "$(id -u)" -eq 0 ]] && oops "Please run this script as a regular user" + + # Check if nix is already installed and is at least version 2.4 + if command -v nix > /dev/null; then + NIX_VERSION=$(nix --version | sed -ne 's/[^0-9]*\(\([0-9]\.\)\)/\1/p') + NIX_VERSION_MINOR=$(echo "$NIX_VERSION" | cut -d. -f2) + if [ "$NIX_VERSION_MINOR" -gt "3" ]; then + echo -e "Detected nix version $NIX_VERSION. Proceeding with K framework install." + else + oops "It appears that you have a version of nix on your system that's too old. The K framework installer requires nix >=2.4. Please update nix and try again." + fi + else + read -p "It appears that you don't have nix installed. Since the K framework needs nix for distribution, this script will attempt to install nix first. Would you like to proceed? [y/N]" -n 1 -r + echo + if [[ $REPLY =~ ^[Yy]$ ]] + then + echo "Downloading nix and running the installer..." + curl --proto '=https' --tlsv1.2 -sSf -L https://install.determinate.systems/nix | sh -s -- install --no-confirm + if [ -e '/nix/var/nix/profiles/default/etc/profile.d/nix-daemon.sh' ]; then + . '/nix/var/nix/profiles/default/etc/profile.d/nix-daemon.sh' + else + oops "Could not source nix." + fi + NIX_FRESH_INSTALL=true + else + oops "Cannot proceed with the installation without nix." + fi + fi + + PREV_KUP_INSTALL=$(nix profile list --experimental-features 'nix-command flakes' | awk '/packages\..*\.kup/ {print $1}') + if ! [[ -z "$PREV_KUP_INSTALL" ]]; then + echo "Removing previous K framework installer versions ..." + GC_DONT_GC=1 nix profile remove $PREV_KUP_INSTALL \ + --experimental-features 'nix-command flakes' + fi + + echo "Installing the K framework installer utility (kup) ..." + + GC_DONT_GC=1 nix profile install github:runtimeverification/kup#kup \ + --option extra-substituters 'https://k-framework.cachix.org' \ + --option extra-trusted-public-keys 'k-framework.cachix.org-1:jeyMXB2h28gpNRjuVkehg+zLj62ma1RnyyopA/20yFE=' \ + --experimental-features 'nix-command flakes' + + echo -e "${GREEN}All set!${NC}" + if [ -n "$NIX_FRESH_INSTALL" ]; then + echo -e "$NIX_POST_INSTALL_MESSAGE" + fi + +} diff --git a/k-distribution/INSTALL/index.html b/k-distribution/INSTALL/index.html new file mode 100644 index 00000000000..fce8c208591 --- /dev/null +++ b/k-distribution/INSTALL/index.html @@ -0,0 +1,638 @@ + + + + + + + + + + + + + + +Installing the K Framework | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Installing the K Framework

      +

      Fast Installation (preferred)

      +

      If you're on a system that supports Nix, +use this command to install via Nix:

      +
      shell
      bash <(curl https://kframework.org/install)
+kup install k
+
      +

      You can update K with:

      +
      shell
      kup update k
+
      +

      And list available versions with:

      +
      shell
      kup list
+
      +

      This will take care of all the dependencies and specific versions used by K. +Note that the first run will take longer to fetch all the libraries and compile +sources. (30m to 1h)

      +

      If you are on Apple Silicon, kup is currently the only way to install K +because of upstream issues in the general Haskell ecosystem.

      +

      Install through packages

      +

      We currently strive to provide packages for the following platforms:

      +
        +
      • Ubuntu Focal Fossa (20.04) and Jammy Jellyfish (22.04)
        • +
      • Debian Bookworm
        • +
      • Arch Linux
        • +
      • macOS Catalina (10.15), Big Sur (11) and Monterey (12) via Homebrew
        • +
      • Docker Images
        • +
      +

      Pre-installation Notes

      +
        +
      • +

        We do not currently support running K natively on Windows. To use K on +Windows 10, you are encouraged to install the +Windows Subsystem for Linux (version 2) +and follow the instructions for installing Ubuntu Focal.

        +

        If you have already installed WSL, before proceeding, you will need to +enter the WSL environment. You can do this by:

        +
          +
        1. opening up the command prompt (accessible by searching cmd or +command prompt from the start menu);
          2. +
        2. using the wsl.exe command to access the WSL environment.
          3. +
        +
        • +
      • +

        To use K in other non-linux environments (e.g. Windows 8 or earlier), +you will need to use a virtual machine (VM) software. We assume you have:

        +
          +
        1. Created a virtual machine
          2. +
        2. Installed a Linux distribution (e.g. Ubuntu Focal Fossa) on your +virtual machine
          3. +
        +

        Consult your virtual machine software if you need help with the above +steps. We recommend the free VirtualBox virtual machine software.

        +

        Before proceeding, follow the virtual machine softare UI to start your +Linux virtual machine and enter the command line environment.

        +
        • +
      • +

        WSL and virtual machine users should be aware that, if you use your web +browser to download the package, you will need to make it accessible to +the command line environment. For this reason, we recommend downloading the +package from the command line directly using a tool like wget. For +example, you could copy the package download URL and then type:

        +
        wget <package-download-url>
+
        +

        where <package-download-url> is replaced by the URL you just copied.

        +
        • +
      • +

        K depends on version 4.8.15 of Z3, which may not be supplied by package +managers. If this is the case, it should be built and installed from source +following the +instructions in +the Z3 repository. Other versions (older and newer) are not supported by K, +and may lead to incorrect behaviour or performance issues.

        +
        • +
      +

      Downloading Packages

      +

      Download the appropriate package from the GitHub, via the +Releases page. +Releases are generated as often as possible from master build.

      +

      Installing Packages

      +

      For version X.Y.Z, distribution DISTRO, and package ID ID, the following +instructions tell you how to install on each system. Note that this typically +requires about ~1.4GB of dependencies and will take some time.

      +
        +
      • On Linux systems, K will typically be installed under /usr.
        • +
      • On macOS/brew, K will typically be installed under /usr/local.
        • +
      +

      Ubuntu Focal (20.04)

      +
      sh
      sudo apt install ./kframework_amd64_ubuntu_focal.deb
+
      +

      Ubuntu Jammy (22.04)

      +
      sh
      sudo apt install ./kframework_amd64_ubuntu_jammy.deb
+
      +

      Debian Bookworm

      +
      sh
      sudo apt install ./kframework_amd64_debian_bookworm.deb
+
      +

      Arch Linux

      +
      sh
      pacman -U ./kframework_arch_x86_64.pkg.tar.zst
+
      +

      MacOS X Mojave/Homebrew

      +

      Homebrew (or just brew) is a third-party package manager +for MacOS. +If you have not installed brew, you must do so before installing the K +Framework brew package.

      +

      With brew installed, do the following to install the K Framework brew package +(with build number BN):

      +
      sh
      brew install kframework--X.Y.Z.ID.bottle.BN.tar.gz -v
+
      +

      Note: we do not yet have a MacOS Monterey bottle. The existing brew package +also does not work on M1 Silicon Macs. If you have either of these systems, +you should build from source for the time being.

      +

      Homebrew Alternate Installation

      +

      To directly install the latest K Framework brew package without needing to +download it separately, do the following:

      +
      sh
      brew install kframework/k/kframework
+
      +

      Or, to streamline future K Framework upgrades, you can tap the K Framework +package repository. This lets future installations/upgrades/etc... use the +unprefixed package name.

      +
      sh
      brew tap kframework/k
+brew install kframework
+
      +

      Docker Images

      +

      Docker images with K pre-installed are available at the +runtimeverification/kframework-k Docker Hub repository.

      +

      Each release at COMMIT_ID has an image associated with it at +runtimeverificationinc/kframework-k:ubuntu-focal-COMMIT_ID.

      +

      To run the image directly:

      +
      sh
      docker run -it runtimeverificationinc/kframework-k:ubuntu-focal-COMMIT_ID
+
      +

      and to make a Docker Image based on it, use the following line in your +Dockerfile:

      +
      Dockerfile
      FROM runtimeverificationinc/kframework-k:ubuntu-focal-COMMIT_ID
+
      +

      We also create Ubuntu 22.04 images with the ubuntu-jammy-COMMIT_ID tags.

      +

      Testing Packages

      +

      The easiest way to test the K package is to copy a K tutorial language and +check if you can compile and run an included example.

      +
        +
      1. +

        Start by copying the K tutorial to some work directory +(e.g. $HOME/pl-tutorial) from the K distribution root. Using a Linux +package, this command typically will be like:

        +
        sh
        $ cp -R /usr/share/kframework/pl-tutorial $HOME/pl-tutorial
+
        +

        On macOS/brew, this command typically will be like:

        +
        sh
        $ cp -R /usr/local/share/kframework/pl-tutorial $HOME/pl-tutorial
+
        +

        This step is needed because sometimes only the root user can run the +examples in the default installation directory.

        +
        2. +
      2. +

        Now you can try to run some programs:

        +
        sh
        $ cd $HOME/pl-tutorial/2_languages/1_simple/1_untyped
+$ make kompile
+$ krun tests/diverse/factorial.simple
+
        +
        3. +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/domains/index.html b/k-distribution/include/kframework/builtin/domains/index.html new file mode 100644 index 00000000000..325577fb7d3 --- /dev/null +++ b/k-distribution/include/kframework/builtin/domains/index.html @@ -0,0 +1,3920 @@ + + + + + + + + + + + + + + +Basic Builtin Types in K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Basic Builtin Types in K

      +

      A major piece of the K prelude consists of a series of modules that contain +implementations of basic data types and language features in K. You do not need +to require this file yourself; it is required automatically in every K +definition unless --no-prelude is passed to kompile. K may not work correctly +if some of these modules do not exist or do not declare certain functions.

      +

      Note that some functions in the K prelude functions are not total, that is, +they are not defined on all possible input values. When you invoke such a +function on an undefined input, the behavior is undefined. In particular, when +this happens, interpreters generated by the K LLVM backend may crash.

      +
      k
      require "kast.md"
+
      +

      Default Modules

      +

      K declares certain modules that contain most of the builtins you usually want +when defining a language in K. In particular, this includes integers, booleans, +strings, identifiers, I/O, lists, maps, and sets. The DOMAINS-SYNTAX module +is designed to be imported by the syntax module of the language and contains +only the program-level syntax of identifiers, integers, booleans, and strings. +The DOMAINS module contains the rest of the syntax, including builtin +functions over those and the remaining types.

      +

      Note that not all modules are included in DOMAINS. A few less-common modules +are not, including ARRAY, COLLECTIONS, FLOAT, STRING-BUFFER, BYTES, +K-REFLECTION, MINT, and STRATEGY.

      +
      k
      module DOMAINS-SYNTAX
+  imports SORT-K
+  imports ID-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+  imports STRING-SYNTAX
+endmodule
+
+module DOMAINS
+  imports DOMAINS-SYNTAX
+  imports INT
+  imports BOOL
+  imports STRING
+  imports BASIC-K
+  imports LIST
+  imports K-IO
+  imports MAP
+  imports SET
+  imports ID
+  imports K-EQUAL
+endmodule
+
      +

      Arrays

      +

      Provided here is an implementation for fixed-sized, contiguous maps from Int +to KItem. In some previous versions of K, the Array type was a builtin type +backed by mutable arrays of objects. However, in modern K, the Array type is +implemented by means of the List type; users should not access this interface +directly and should instead make only of the functions listed below. Users of +this module should import only the ARRAY module.

      +
      k
      module ARRAY-SYNTAX
+  imports private LIST
+
+  syntax Array [hook(ARRAY.Array)]
+
      +

      Array lookup

      +

      You can look up an element in an Array by its index in O(log(N)) time. Note +that the base of the logarithm is a relatively high number and thus the time is +effectively constant.

      +
      k
        syntax KItem ::= Array "[" Int "]" [function, hook(ARRAY.lookup)]
+
      +

      Array update

      +

      You can create a new Array with a new value for a key in O(log(N)) time, or +effectively constant.

      +
      k
        syntax Array ::= Array "[" key: Int "<-" value: KItem "]" [function, hook(ARRAY.update), klabel(_[_<-_]), symbol]
+
      +

      Array reset

      +

      You can create a new Array where a particular key is reset to its default +value in O(log(N)) time, or effectively constant.

      +
      k
        syntax Array ::= Array "[" Int "<-" "undef" "]" [function, hook(ARRAY.remove)]
+
      +

      Multiple array update

      +

      You can create a new Array from a List L of size N where the N +elements starting at index are replaced with the contents of L, in +O(N*log(K)) time (where K is the size of the array), or effectively linear. +Having index + N > K yields an exception.

      +
      k
        syntax Array ::= updateArray(Array, index: Int, List) [function, hook(ARRAY.updateAll)]
+
      +

      Array fill

      +

      You can create a new Array where the length elements starting at index +are replaced with value, in O(length*log(N)) time, or effectively linear.

      +
      k
        syntax Array ::= fillArray(Array, index: Int, length: Int, value: KItem) [function, hook(ARRAY.fill)]
+
      +

      Array range check

      +

      You can test whether an integer is within the bounds of an array in O(1) time.

      +
      k
        syntax Bool ::= Int "in_keys" "(" Array ")" [function, total, hook(ARRAY.in_keys)]
+
      +
      k
      endmodule
+
+module ARRAY-IN-K [private]
+  imports public ARRAY-SYNTAX
+  imports private LIST
+  imports private K-EQUAL
+  imports private INT
+  imports private BOOL
+
      +

      Array creation

      +

      You can create an array with length elements where each element is +initialized to value in O(1) time. Note that the array is stored in a manner +where only the highest element that is actually modified is given a value +in its internal representation, which means that subsequent array operations +may incur a one-time O(N) resizing cost, possibly amortized across multiple +operations.

      +
      k
        syntax Array ::= makeArray(length: Int, value: KItem) [function, hook(ARRAY.make), public]
+
      +

      Implementation of Arrays

      +

      The remainder of this section consists of an implementation in K of the +operations listed above. Users of the ARRAY module should not make use +of any of the syntax defined in any of these modules.

      +
      k
        syntax Array ::= arr(List, Int, KItem)
+
+  rule makeArray(I::Int, D::KItem) => arr(.List, I, D)
+
+  rule arr(L::List, _, _       ) [ IDX::Int ] => L[IDX] requires 0 <=Int IDX andBool IDX  <Int size(L)
+  rule arr(_      , _, D::KItem) [ _        ] => D      [owise]
+
+  syntax List ::= ensureOffsetList(List, Int, KItem) [function]
+  rule ensureOffsetList(L::List, IDX::Int, D::KItem) => L makeList(IDX +Int 1 -Int size(L), D) requires         IDX >=Int size(L)
+  rule ensureOffsetList(L::List, IDX::Int, _::KItem) => L                                      requires notBool IDX >=Int size(L)
+
+  rule arr(L::List, I::Int, D::KItem) [ IDX::Int <- VAL::KItem ] => arr(ensureOffsetList(L, IDX, D) [ IDX <- VAL ], I, D)
+
+  rule arr(L::List, I::Int, D::KItem) [ IDX::Int <- undef ] => arr(L, I, D) [ IDX <- D ]
+
+  rule updateArray(arr(L::List, I::Int, D::KItem), IDX::Int, L2::List) => arr(updateList(ensureOffsetList(L, IDX +Int size(L2) -Int 1, D), IDX, L2), I, D)
+
+  rule fillArray(arr(L::List, I::Int, D::KItem), IDX::Int, LEN::Int, VAL::KItem) => arr(fillList(ensureOffsetList(L, IDX +Int LEN -Int 1, D), IDX, LEN, VAL), I, D)
+
+  rule IDX::Int in_keys(arr(_, I::Int, _)) => IDX >=Int 0 andBool IDX <Int I
+endmodule
+
+module ARRAY-SYMBOLIC [symbolic]
+  imports ARRAY-IN-K
+endmodule
+
+module ARRAY-KORE [kore]
+  imports ARRAY-IN-K
+endmodule
+
+module ARRAY
+  imports ARRAY-SYMBOLIC
+  imports ARRAY-KORE
+endmodule
+
      +

      Maps

      +

      Provided here is the syntax of an implementation of immutable, associative, +commutative maps from KItem to KItem. This type is hooked to an +implementation of maps provided by the backend. For more information on +matching on maps and allowable patterns for doing so, refer to K's +user documentation.

      +
      k
      module MAP
+  imports private BOOL-SYNTAX
+  imports private INT-SYNTAX
+  imports private LIST
+  imports private SET
+
+  syntax Map [hook(MAP.Map)]
+
      +

      Map concatenation

      +

      The Map sort represents a generalized associative array. Each key can be +paired with an arbitrary value, and can be used to reference its associated +value. Multiple bindings for the same key are not allowed.

      +

      You can construct a new Map consisting of key/value pairs of two Maps. The +result is #False if the maps have keys in common (in particular, this will +yield an exception during concrete execution). This operation is O(Nlog(M)) +where N is the size of the smaller map, when it appears on the right hand side. +When it appears on the left hand side and all variables are bound, it is +O(Nlog(M)) where M is the size of the map it is matching and N is the number +of elements being matched. When it appears on the left hand side containing +variables not bound elsewhere in the term, it is O(N^K) where N is the size of +the map it is matching and K is the number of unbound keys being matched. In +other words, one unbound variable is linear, two is quadratic, three is cubic, +etc.

      +
      k
        syntax Map ::= Map Map                        [left, function, hook(MAP.concat), klabel(_Map_), symbol, assoc, comm, unit(.Map), element(_|->_), index(0), format(%1%n%2)]
+
      +

      Map unit

      +

      The map with zero elements is represented by .Map.

      +
      k
        syntax Map ::= ".Map"                         [function, total, hook(MAP.unit), klabel(.Map), symbol, latex(\dotCt{Map})]
+
      +

      Map elements

      +

      An element of a Map is constructed via the |-> operator. The key is on the +left and the value is on the right.

      +
      k
        syntax Map ::= KItem "|->" KItem                      [function, total, hook(MAP.element), klabel(_|->_), symbol, latex({#1}\mapsto{#2}), injective]
+
+  syntax priorities _|->_ > _Map_ .Map
+  syntax non-assoc _|->_
+
      +

      Map lookup

      +

      You can look up the value associated with the key of a map in O(log(N)) time. +Note that the base of the logarithm is a relatively high number and thus the +time is effectively constant. The value is #False if the key is not in the +map (in particular, this will yield an exception during concrete execution).

      +
      k
        syntax KItem ::= Map "[" KItem "]"                    [function, hook(MAP.lookup), klabel(Map:lookup), symbol]
+
      +

      Map lookup with default

      +

      You can also look up the value associated with the key of a map using a +total function that assigns a specific default value if the key is not present +in the map. This operation is also O(log(N)), or effectively constant.

      +
      k
        syntax KItem ::= Map "[" KItem "]" "orDefault" KItem      [function, total, hook(MAP.lookupOrDefault), klabel(Map:lookupOrDefault)]
+
      +

      Map update

      +

      You can insert a key/value pair into a map in O(log(N)) time, or effectively +constant.

      +
      k
        syntax Map ::= Map "[" key: KItem "<-" value: KItem "]"           [function, total, klabel(Map:update), symbol, hook(MAP.update), prefer]
+
      +

      Map delete

      +

      You can remove a key/value pair from a map via its key in O(log(N)) time, or +effectively constant.

      +
      k
        syntax Map ::= Map "[" KItem "<-" "undef" "]"     [function, total, hook(MAP.remove), klabel(_[_<-undef]), symbol]
+
      +

      Map difference

      +

      You can remove the key/value pairs in a map that are present in another map in +O(N*log(M)) time (where M is the size of the first map and N is the size of the +second), or effectively linear. Note that only keys whose value is the same +in both maps are removed. To remove all the keys in one map from another map, +you can say removeAll(M1, keys(M2)).

      +
      k
        syntax Map ::= Map "-Map" Map                 [function, total, hook(MAP.difference), latex({#1}-_{\it Map}{#2})]
+
      +

      Multiple map update

      +

      You can update a map by adding all the key/value pairs in the second map in +O(N*log(M)) time (where M is the size of the first map and N is the size of the +second map), or effectively linear. If any keys are present in both maps, the +value from the second map overwrites the value in the first. This function is +total, which is distinct from map concatenation, a partial function only +defined on maps with disjoint keys.

      +
      k
        syntax Map ::= updateMap(Map, Map)            [function, total, hook(MAP.updateAll)]
+
      +

      Multiple map removal

      +

      You can remove a Set of keys from a map in O(N*log(M)) time (where M is the +size of the Map and N is the size of the Set), or effectively linear.

      +
      k
        syntax Map ::= removeAll(Map, Set)            [function, total, hook(MAP.removeAll)]
+
      +

      Map keys (as Set)

      +

      You can get a Set of all the keys in a Map in O(N) time.

      +
      k
        syntax Set ::= keys(Map)                      [function, total, hook(MAP.keys)]
+
      +

      Map keys (as List)

      +

      You can get a List of all the keys in a Map in O(N) time.

      +
      k
        syntax List ::= "keys_list" "(" Map ")"       [function, hook(MAP.keys_list)]
+
      +

      Map key membership

      +

      You can check whether a key is present in a map in O(1) time.

      +
      k
        syntax Bool ::= KItem "in_keys" "(" Map ")"       [function, total, hook(MAP.in_keys)]
+
      +

      Map values (as List)

      +

      You can get a List of all the values in a map in O(N) time.

      +
      k
        syntax List ::= values(Map)                   [function, hook(MAP.values)]
+
      +

      Map size

      +

      You can get the number of key/value pairs in a map in O(1) time.

      +
      k
        syntax Int ::= size(Map)                      [function, total, hook(MAP.size), klabel(sizeMap)]
+
      +

      Map inclusion

      +

      You can determine whether a Map is a strict subset of another Map in O(N) +time (where N is the size of the first map). Only keys that are bound to the +same value are considered equal.

      +
      k
        syntax Bool ::= Map "<=Map" Map               [function, total, hook(MAP.inclusion)]
+
      +

      Map choice

      +

      You can get an arbitrarily chosen key of a Map in O(1) time. The same key +will always be returned for the same map, but no guarantee is given that two +different maps will return the same element, even if they are similar.

      +
      k
        syntax KItem ::= choice(Map)                      [function, hook(MAP.choice), klabel(Map:choice)]
+
      +

      Implementation of Maps

      +

      The remainder of this section contains lemmas used by the Java and Haskell +backend to simplify expressions of sort Map. They do not affect the semantics +of maps, merely describing additional rules that the backend can use to +simplify terms.

      +
      k
      endmodule
+
+module MAP-KORE-SYMBOLIC [kore,symbolic,haskell]
+  imports MAP
+  imports private K-EQUAL
+  imports private BOOL
+
+  rule #Ceil(@M:Map [@K:KItem]) => {(@K in_keys(@M)) #Equals true} #And #Ceil(@M) #And #Ceil(@K) [simplification]
+
+  // Symbolic update
+
+  // Adding the definedness condition `notBool (K in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K |-> _ M:Map) [ K <- V ] => (K |-> V M) [simplification]
+  rule M:Map [ K <- V ] => (K |-> V M) requires notBool (K in_keys(M)) [simplification]
+  rule M:Map [ K <- _ ] [ K <- V ] => M [ K <- V ] [simplification]
+  // Adding the definedness condition `notBool (K1 in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K1 also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K1 |-> V1 M:Map) [ K2 <- V2 ] => (K1 |-> V1 (M [ K2 <- V2 ])) requires K1 =/=K K2 [simplification]
+
+  // Symbolic remove
+  rule (K |-> _ M:Map) [ K <- undef ] => M ensures notBool (K in_keys(M)) [simplification]
+  rule M:Map [ K <- undef ] => M requires notBool (K in_keys(M)) [simplification]
+  // Adding the definedness condition `notBool (K1 in_keys(M))` in the ensures clause of the following rule would be redundant
+  // because K1 also appears in the rhs, preserving the case when it's #Bottom.
+  rule (K1 |-> V1 M:Map) [ K2 <- undef ] => (K1 |-> V1 (M [ K2 <- undef ])) requires K1 =/=K K2 [simplification]
+
+  // Symbolic lookup
+  rule (K  |->  V M:Map) [ K ]  => V ensures notBool (K in_keys(M)) [simplification]
+  rule (K1 |-> _V M:Map) [ K2 ] => M [K2] requires K1 =/=K K2 ensures notBool (K1 in_keys(M)) [simplification]
+  rule (_MAP:Map [ K  <-  V1 ]) [ K ]  => V1 [simplification]
+  rule ( MAP:Map [ K1 <- _V1 ]) [ K2 ] => MAP [ K2 ] requires K1 =/=K K2 [simplification]
+
+  rule (K  |->  V M:Map) [  K ] orDefault _ => V ensures notBool (K in_keys(M)) [simplification]
+  rule (K1 |-> _V M:Map) [ K2 ] orDefault D => M [K2] orDefault D requires K1 =/=K K2 ensures notBool (K1 in_keys(M)) [simplification]
+  rule (_MAP:Map [ K  <-  V1 ]) [ K ] orDefault _ => V1 [simplification]
+  rule ( MAP:Map [ K1 <- _V1 ]) [ K2 ] orDefault D => MAP [ K2 ] orDefault D requires K1 =/=K K2 [simplification]
+  rule .Map [ _ ] orDefault D => D [simplification]
+
+  // Symbolic in_keys
+  rule K in_keys(_M [ K <- undef ]) => false [simplification]
+  rule K in_keys(_M [ K <- _ ]) => true [simplification]
+  rule K1 in_keys(M [ K2 <- _ ]) => true requires K1 ==K K2 orBool K1 in_keys(M) [simplification]
+  rule K1 in_keys(M [ K2 <- _ ]) => K1 in_keys(M) requires K1 =/=K K2 [simplification]
+
+  rule {false #Equals @Key in_keys(.Map)} => #Ceil(@Key) [simplification]
+  rule {@Key in_keys(.Map) #Equals false} => #Ceil(@Key) [simplification]
+  rule {false #Equals @Key in_keys(Key' |-> Val @M)} => #Ceil(@Key) #And #Ceil(Key' |-> Val @M) #And #Not({@Key #Equals Key'}) #And {false #Equals @Key in_keys(@M)} [simplification]
+  rule {@Key in_keys(Key' |-> Val @M) #Equals false} => #Ceil(@Key) #And #Ceil(Key' |-> Val @M) #And #Not({@Key #Equals Key'}) #And {@Key in_keys(@M) #Equals false} [simplification]
+
+/*
+// The rule below is automatically generated by the frontend for every sort
+// hooked to MAP.Map. It is left here to serve as documentation.
+
+  rule #Ceil(@M:Map (@K:KItem |-> @V:KItem)) => {(@K in_keys(@M)) #Equals false} #And #Ceil(@M) #And #Ceil(@K) #And #Ceil(@V)
+    [simplification]
+*/
+endmodule
+
+module MAP-SYMBOLIC
+  imports MAP-KORE-SYMBOLIC
+endmodule
+
      +

      Range Maps

      +

      Provided here is the syntax of an implementation of immutable, associative, +commutative range maps from Int to KItem. This type is hooked to an +implementation of range maps provided by the LLVM backend. +Currently, this type is not supported by other backends. +Although the underlying range map data structure supports any key sort, the +current implementation by the backend only supports Int keys due to +limitations of the underlying ordering function.

      +
      k
      module RANGEMAP
+  imports private BOOL-SYNTAX
+  imports private INT-SYNTAX
+  imports private LIST
+  imports private SET
+
+
      +

      Range, bounded inclusively below and exclusively above.

      +
      k
        syntax Range ::= "[" KItem "," KItem ")"    [klabel(Rangemap:Range), symbol]
+  
+  syntax RangeMap [hook(RANGEMAP.RangeMap)]
+
      +

      Range map concatenation

      +

      The RangeMap sort represents a map whose keys are stored as ranges, bounded +inclusively below and exclusively above. Contiguous or overlapping ranges that +map to the same value are merged into a single range.

      +

      You can construct a new RangeMap consisting of range/value pairs of two +RangeMaps. If the RangeMaps have overlapping ranges an exception will be +thrown during concrete execution. This operation is O(N*log(M)) (where N is +the size of the smaller map and M is the size of the larger map).

      +
      k
        syntax RangeMap ::= RangeMap RangeMap                        [left, function, hook(RANGEMAP.concat), klabel(_RangeMap_), symbol, assoc, comm, unit(.RangeMap), element(_r|->_), index(0), format(%1%n%2)]
+
      +

      Range map unit

      +

      The RangeMap with zero elements is represented by .RangeMap.

      +
      k
        syntax RangeMap ::= ".RangeMap"                         [function, total, hook(RANGEMAP.unit), klabel(.RangeMap), symbol, latex(\dotCt{RangeMap})]
+
      +

      Range map elements

      +

      An element of a RangeMap is constructed via the r|-> operator. The range +of keys is on the left, and the value is on the right.

      +
      k
        syntax RangeMap ::= Range "r|->" KItem                      [function, hook(RANGEMAP.elementRng), klabel(_r|->_), symbol, latex({#1}\mapsto{#2}), injective]
+
+  syntax priorities _r|->_ > _RangeMap_ .RangeMap
+  syntax non-assoc _r|->_
+
      +

      Range map lookup

      +

      You can look up the value associated with a key of a RangeMap in O(log(N)) +time (where N is the size of the RangeMap). This will yield an exception +during concrete execution if the key is not in the range map.

      +
      k
        syntax KItem ::= RangeMap "[" KItem "]"                    [function, hook(RANGEMAP.lookup), klabel(RangeMap:lookup), symbol]
+
      +

      Range map lookup with default

      +

      You can also look up the value associated with a key of a RangeMap using a +total function that assigns a specific default value if the key is not present +in the RangeMap. This operation is also O(log(N)) (where N is the size of +the range map).

      +
      k
        syntax KItem ::= RangeMap "[" KItem "]" "orDefault" KItem      [function, total, hook(RANGEMAP.lookupOrDefault), klabel(RangeMap:lookupOrDefault)]
+
      +

      Range map lookup for range of key

      +

      You can look up for the range that a key of a RangeMap is stored in in +O(log(N)) time (where N is the size of the RangeMap). This will yield an +exception during concrete execution if the key is not in the range map.

      +
      k
        syntax Range ::= "find_range" "(" RangeMap "," KItem ")"                    [function, hook(RANGEMAP.find_range), klabel(RangeMap:find_range)]
+
      +

      Range map update

      +

      You can insert a range/value pair into a RangeMap in O(log(N)) time (where N +is the size of the RangeMap). Any ranges adjacent to or overlapping with the +range to be inserted will be updated accordingly.

      +
      k
        syntax RangeMap ::= RangeMap "[" keyRange: Range "<-" value: KItem "]"           [function, klabel(RangeMap:update), symbol, hook(RANGEMAP.updateRng), prefer]
+
      +

      Range map delete

      +

      You can remove a range/value pair from a RangeMap in O(log(N)) time (where N +is the size of the RangeMap). If all or any part of the range is present in +the range map, it will be removed.

      +
      k
        syntax RangeMap ::= RangeMap "[" Range "<-" "undef" "]"     [function, hook(RANGEMAP.removeRng), klabel(_r[_<-undef]), symbol]
+
      +

      Range map difference

      +

      You can remove the range/value pairs in a RangeMap that are also present in +another RangeMap in O(max{M,N}*log(M)) time (where M is the size of the +first RangeMap and N is the size of the second RangeMap). Note that only +the parts of overlapping ranges whose value is the same in both range maps +will be removed.

      +
      k
        syntax RangeMap ::= RangeMap "-RangeMap" RangeMap                 [function, total, hook(RANGEMAP.difference), latex({#1}-_{\it RangeMap}{#2})]
+
      +

      Multiple range map update

      +

      You can update a RangeMap by adding all the range/value pairs in the second +RangeMap in O(N*log(M+N)) time (where M is the size of the first RangeMap +and N is the size of the second RangeMap). If any ranges are overlapping, +the value from the second range map overwrites the value in the first for the +parts where ranges are overlapping. This function is total, which is distinct +from range map concatenation, a partial function only defined on range maps +with non overlapping ranges.

      +
      k
        syntax RangeMap ::= updateRangeMap(RangeMap, RangeMap)            [function, total, hook(RANGEMAP.updateAll)]
+
      +

      Multiple range map removal

      +

      You can remove a Set of ranges from a RangeMap in O(N*log(M)) time (where +M is the size of the RangeMap and N is the size of the Set). For every +range in the set, all or any part of it that is present in the range map will +be removed.

      +
      k
        syntax RangeMap ::= removeAll(RangeMap, Set)            [function, hook(RANGEMAP.removeAll)]
+
      +

      Range map keys (as Set)

      +

      You can get a Set of all the ranges in a RangeMap in O(N) time (where N +is the size of the RangeMap).

      +
      k
        syntax Set ::= keys(RangeMap)                      [function, total, hook(RANGEMAP.keys)]
+
      +

      Range map keys (as List)

      +

      You can get a List of all the ranges in a RangeMap in O(N) time (where N +is the size of the RangeMap).

      +
      k
        syntax List ::= "keys_list" "(" RangeMap ")"       [function, hook(RANGEMAP.keys_list)]
+
      +

      Range map key membership

      +

      You can check whether a key is present in a RangeMap in O(log(N)) time (where +N is the size of the RangeMap).

      +
      k
        syntax Bool ::= KItem "in_keys" "(" RangeMap ")"       [function, total, hook(RANGEMAP.in_keys)]
+
      +

      Range map values (as List)

      +

      You can get a List of all values in a RangeMap in O(N) time (where N is the +size of the RangeMap).

      +
      k
        syntax List ::= values(RangeMap)                   [function, hook(RANGEMAP.values)]
+
      +

      Range map size

      +

      You can get the number of range/value pairs in a RangeMap in O(1) time.

      +
      k
        syntax Int ::= size(RangeMap)                      [function, total, hook(RANGEMAP.size), klabel(sizeRangeMap)]
+
      +

      Range map inclusion

      +

      You can determine whether a RangeMap is a strict subset of another RangeMap +in O(M+N) time (where M is the size of the first RangeMap and N is the size +of the second RangeMap). Only keys within equal or overlapping ranges that +are bound to the same value are considered equal.

      +
      k
        syntax Bool ::= RangeMap "<=RangeMap" RangeMap               [function, total, hook(RANGEMAP.inclusion)]
+
      +

      Range map choice

      +

      You can get an arbitrarily chosen key of a RangeMap in O(1) time. The same +key will always be returned for the same range map, but no guarantee is given +that two different range maps will return the same element, even if they are +similar.

      +
      k
        syntax KItem ::= choice(RangeMap)                      [function, hook(RANGEMAP.choice), klabel(RangeMap:choice)]
+endmodule
+
      +

      Sets

      +

      Provided here is the syntax of an implementation of immutable, associative, +commutative sets of KItem. This type is hooked to an implementation of sets +provided by the backend. For more information on matching on sets and allowable +patterns for doing so, refer to K's +user documentation.

      +
      k
      module SET
+  imports private INT-SYNTAX
+  imports private BASIC-K
+
+  syntax Set [hook(SET.Set)]
+
      +

      Set concatenation

      +

      The Set sort represents a mathematical set (A collection of unique items). +The sets are nilpotent, i.e., the concatenation of two sets containing elements +in common is #False (note however, this may be silently allowed during +concrete execution). If you intend to add an element to a set that might +already be present in the set, use the |Set operator instead.

      +

      The concatenation operator is O(Nlog(M)) where N is the size of the smaller +set, when it appears on the right hand side. When it appears on the left hand +side and all variables are bound, it is O(Nlog(M)) where M is the size of the +set it is matching and N is the number of elements being matched. When it +appears on the left hand side containing variables not bound elsewhere in the +term, it is O(N^K) where N is the size of the set it is matching and K is the +number of unbound keys being mached. In other words, one unbound variable is +linear, two is quadratic, three is cubic, etc.

      +
      k
        syntax Set ::= Set Set                  [left, function, hook(SET.concat), klabel(_Set_), symbol, assoc, comm, unit(.Set), idem, element(SetItem), format(%1%n%2)]
+
      +

      Set unit

      +

      The set with zero elements is represented by .Set.

      +
      k
        syntax Set ::= ".Set"                   [function, total, hook(SET.unit), klabel(.Set), symbol, latex(\dotCt{Set})]
+
      +

      Set elements

      +

      An element of a Set is constructed via the SetItem operator.

      +
      k
        syntax Set ::= SetItem(KItem)               [function, total, hook(SET.element), klabel(SetItem), symbol, injective]
+
      +

      Set union

      +

      You can compute the union of two sets in O(N*log(M)) time (Where N is the size +of the smaller set). Note that the base of the logarithm is a relatively high +number and thus the time is effectively linear. The union consists of all the +elements present in either set.

      +
      k
        syntax Set ::= Set "|Set" Set              [left, function, total, hook(SET.union), comm]
+  rule S1:Set |Set S2:Set => S1 (S2 -Set S1) [concrete]
+
      +

      Set intersection

      +

      You can compute the intersection of two sets in O(N*log(M)) time (where N +is the size of the smaller set), or effectively linear. The intersection +consists of all the elements present in both sets.

      +
      k
        syntax Set ::= intersectSet(Set, Set)   [function, total, hook(SET.intersection), comm]
+
      +

      Set complement

      +

      You can compute the relative complement of two sets in O(N*log(M)) time (where +N is the size of the second set), or effectively linear. This is the set of +elements in the first set that are not present in the second set.

      +
      k
        syntax Set ::= Set "-Set" Set           [function, total, hook(SET.difference), latex({#1}-_{\it Set}{#2}), klabel(Set:difference), symbol]
+
      +

      Set membership

      +

      You can compute whether an element is a member of a set in O(1) time.

      +
      k
        syntax Bool ::= KItem "in" Set              [function, total, hook(SET.in), klabel(Set:in), symbol]
+
      +

      Set inclusion

      +

      You can determine whether a Set is a strict subset of another Set in O(N) +time (where N is the size of the first set).

      +
      k
        syntax Bool ::= Set "<=Set" Set         [function, total, hook(SET.inclusion)]
+
      +

      Set size

      +

      You can get the number of elements (the cardinality) of a set in O(1) time.

      +
      k
        syntax Int ::= size(Set)                [function, total, hook(SET.size)]
+
      +

      Set choice

      +

      You can get an arbitrarily chosen element of a Set in O(1) time. The same +element will always be returned for the same set, but no guarantee is given +that two different sets will return the same element, even if they are similar.

      +
      k
        syntax KItem ::= choice(Set)                [function, hook(SET.choice), klabel(Set:choice)]
+
      +
      k
      endmodule
+
      +

      Implementation of Sets

      +

      The following lemmas are simplifications that the Haskell backend can +apply to simplify expressions of sort Set.

      +
      k
      module SET-KORE-SYMBOLIC [kore,symbolic,haskell]
+  imports SET
+  imports private K-EQUAL
+  imports private BOOL
+
+  //Temporarly rule for #Ceil simplification, should be generated in front-end
+
+// Matching for this version not implemented.
+  // rule #Ceil(@S1:Set @S2:Set) =>
+  //        {intersectSet(@S1, @S2) #Equals .Set} #And #Ceil(@S1) #And #Ceil(@S2)
+  //   [simplification]
+
+//simpler version
+  rule #Ceil(@S:Set SetItem(@E:KItem)) =>
+         {(@E in @S) #Equals false} #And #Ceil(@S) #And #Ceil(@E)
+    [simplification]
+
+  // -Set simplifications
+  rule S              -Set .Set           => S          [simplification]
+  rule .Set           -Set  _             => .Set       [simplification]
+  rule SetItem(X)     -Set (S SetItem(X)) => .Set
+                               ensures notBool (X in S) [simplification]
+  rule S              -Set (S SetItem(X)) => .Set
+                               ensures notBool (X in S) [simplification]
+  rule (S SetItem(X)) -Set S              => SetItem(X)
+                               ensures notBool (X in S) [simplification]
+  rule (S SetItem(X)) -Set SetItem(X)     => S
+                               ensures notBool (X in S) [simplification]
+  // rule SetItem(X)     -Set S              => SetItem(X)
+  //                            requires notBool (X in S)  [simplification]
+  // rule (S1 SetItem(X)) -Set (S2 SetItem(X))  => S1 -Set S2
+  //                             ensures notBool (X in S1)
+  //                             andBool notBool (X in S2) [simplification]
+
+
+
+  // |Set simplifications
+  rule S    |Set .Set => S    [simplification, comm]
+  rule S    |Set S    => S    [simplification]
+
+  rule (S SetItem(X)) |Set SetItem(X) => S SetItem(X)
+                             ensures notBool (X in S) [simplification, comm]
+  // Currently disabled, see runtimeverification/haskell-backend#3301
+  // rule (S SetItem(X)) |Set S          => S SetItem(X)
+  //                            ensures notBool (X in S) [simplification, comm]
+
+  // intersectSet simplifications
+  rule intersectSet(.Set, _   ) => .Set    [simplification, comm]
+  rule intersectSet( S  , S   ) =>  S      [simplification]
+
+  rule intersectSet( S SetItem(X), SetItem(X))     => SetItem(X)
+                                                        ensures notBool (X in S)      [simplification, comm]
+  // Currently disabled, see runtimeverification/haskell-backend#3294
+  // rule intersectSet( S SetItem(X) , S)             => S ensures notBool (X in S)      [simplification, comm]
+  rule intersectSet( S1 SetItem(X), S2 SetItem(X)) => intersectSet(S1, S2) SetItem(X)
+                                                        ensures notBool (X in S1)
+                                                        andBool notBool (X in S2)     [simplification]
+
+  // membership simplifications
+  rule _E in .Set           => false   [simplification]
+  rule E  in (S SetItem(E)) => true
+              ensures notBool (E in S) [simplification]
+
+// These two rules would be sound but impose a giant overhead on `in` evaluation:
+  // rule E1 in (S SetItem(E2)) => true requires E1 in S
+  //                                 ensures notBool (E2 in S) [simplification]
+  // rule E1 in (S SetItem(E2)) => E1 in S requires E1 =/=K E2
+  //                                 ensures notBool (E2 in S) [simplification]
+
+  rule X in ((SetItem(X) S) |Set  _            ) => true
+                                    ensures notBool (X in S) [simplification]
+  rule X in ( _             |Set (SetItem(X) S)) => true
+                                    ensures notBool (X in S) [simplification]
+
+endmodule
+
+module SET-SYMBOLIC
+  imports SET-KORE-SYMBOLIC
+endmodule
+
      +

      Lists

      +

      Provided here is the syntax of an implementation of immutable, associative +lists of KItem. This type is hooked to an implementation of lists provided +by the backend. For more information on matching on lists and allowable +patterns for doing so, refer to K's +user documentation.

      +
      k
      module LIST
+  imports private INT-SYNTAX
+  imports private BASIC-K
+
+  syntax List [hook(LIST.List)]
+
      +

      List concatenation

      +

      The List sort is an ordered collection that may contain duplicate elements. +They are backed by relaxed radix balanced trees, which means that they support +efficiently adding elements to both sides of the list, concatenating two lists, +indexing, and updating elements.

      +

      The concatenation operator is O(log(N)) (where N is the size of the longer +list) when it appears on the right hand side. When it appears on the left hand +side, it is O(N), where N is the number of elements matched on the front and +back of the list.

      +
      k
        syntax List ::= List List               [left, function, total, hook(LIST.concat), klabel(_List_), symbol, smtlib(smt_seq_concat), assoc, unit(.List), element(ListItem), format(%1%n%2)]
+
      +

      List unit

      +

      The list with zero elements is represented by .List.

      +
      k
        syntax List ::= ".List"                 [function, total, hook(LIST.unit), klabel(.List), symbol, smtlib(smt_seq_nil), latex(\dotCt{List})]
+
      +

      List elements

      +

      An element of a List is constucted via the ListItem operator.

      +
      k
        syntax List ::= ListItem(KItem)             [function, total, hook(LIST.element), klabel(ListItem), symbol, smtlib(smt_seq_elem)]
+
      +

      List indexing

      +

      You can get an element of a list by its integer offset in O(log(N)) time, or +effectively constant. Positive indices are 0-indexed from the beginning of the +list, and negative indices are -1-indexed from the end of the list. In other +words, 0 is the first element and -1 is the last element.

      +
      k
        syntax KItem ::= List "[" Int "]"           [function, hook(LIST.get), klabel(List:get), symbol]
+
      +

      List update

      +

      You can create a new List with a new value at a particular index in +O(log(N)) time, or effectively constant.

      +
      k
        syntax List ::= List "[" index: Int "<-" value: KItem "]" [function, hook(LIST.update), klabel(List:set)]
+
      +

      List of identical elements

      +

      You can create a list with length elements, each containing value, in O(N) +time.

      +
      k
        syntax List ::= makeList(length: Int, value: KItem) [function, hook(LIST.make)]
+
      +

      Multiple list update

      +

      You can create a new List which is equal to dest except the N elements +starting at index are replaced with the contents of src in O(N*log(K)) time +(where K is the size of destand N is the size of src), or effectively linear. Having index + N > K yields an exception.

      +
      k
        syntax List ::= updateList(dest: List, index: Int, src: List) [function, hook(LIST.updateAll)]
+
      +

      List fill

      +

      You can create a new List where the length elements starting at index +are replaced with value, in O(length*log(N)) time, or effectively linear.

      +
      k
        syntax List ::= fillList(List, index: Int, length: Int, value: KItem) [function, hook(LIST.fill)]
+
      +

      List slicing

      +

      You can compute a new List by removing fromFront elements from the front +of the list and fromBack elements from the back of the list in +O((fromFront+fromBack)*log(N)) time, or effectively linear.

      +
      k
        syntax List ::= range(List, fromFront: Int, fromBack: Int)   [function, hook(LIST.range), klabel(List:range), symbol]
+
      +

      List membership

      +

      You can compute whether an element is in a list in O(N) time. For repeated +comparisons, it is much better to first convert to a set using List2Set.

      +
      k
        syntax Bool ::= KItem "in" List             [function, total, hook(LIST.in), klabel(_inList_)]
+
      +

      List size

      +

      You can get the number of elements of a list in O(1) time.

      +
      k
        syntax Int ::= size(List)               [function, total, hook(LIST.size), klabel (sizeList), smtlib(smt_seq_len)]
+
      +
      k
      endmodule
+
      +

      Collection Conversions

      +

      It is possible to convert from a List to a Set or from a Set to a list. +Converting from a List to a Set and back will not provide the same list; +duplicates will have been removed and the list may be reordered. Converting +from a Set to a List and back will generate the same set.

      +

      Note that because sets are unordered and lists are ordered, converting from a +Set to a List will generate some arbitrary ordering of elements, which may +be different from the natural ordering you might assume, or may not. Two +equal sets are guaranteed to generate the same ordering, but no guarantee is +otherwise provided about what the ordering will be. In particular, adding an +element to a set may completely reorder the elements already in the set, when +it is converted to a list.

      +
      k
      module COLLECTIONS
+  imports LIST
+  imports SET
+  imports MAP
+
+  syntax List ::= Set2List(Set) [function, total, hook(SET.set2list)]
+  syntax Set ::= List2Set(List) [function, total, hook(SET.list2set)]
+
+endmodule
+
      +

      Booleans

      +

      Provided here is the syntax of an implementation of boolean algebra in K. +This type is hooked to an implementation of booleans provided by the backend. +Note that this algebra is different from the builtin truth in matching logic. +You can, however, convert from the truth of the Bool sort to the truth in +matching logic via the expression {B #Equals true}.

      +

      The boolean values are true and false.

      +
      k
      module SORT-BOOL
+  syntax Bool [hook(BOOL.Bool)]
+endmodule
+
+module BOOL-SYNTAX
+  imports SORT-BOOL
+  syntax Bool ::= "true"  [token]
+  syntax Bool ::= "false" [token]
+endmodule
+
+module BOOL-COMMON
+  imports private BASIC-K
+  imports BOOL-SYNTAX
+
      +

      Basic boolean arithmetic

      +

      You can:

      +
        +
      • Negate a boolean value.
        • +
      • AND two boolean values.
        • +
      • XOR two boolean values.
        • +
      • OR two boolean values.
        • +
      • IMPLIES two boolean values (i.e., P impliesBool Q is the same as +notBool P orBool Q)
        • +
      • Check equality of two boolean values.
        • +
      • Check inequality of two boolean values.
        • +
      +

      Note that only andThenBool and orElseBool are short-circuiting. andBool +and orBool may be short-circuited in concrete backends, but in symbolic +ackends, both arguments will be evaluated.

      +
      k
        syntax Bool ::= "notBool" Bool          [function, total, klabel(notBool_), symbol, smt-hook(not), group(boolOperation), latex(\neg_{\scriptstyle\it Bool}{#1}), hook(BOOL.not)]
+                > Bool "andBool" Bool     [function, total, klabel(_andBool_), symbol, left, smt-hook(and), group(boolOperation), latex({#1}\wedge_{\scriptstyle\it Bool}{#2}), hook(BOOL.and)]
+                | Bool "andThenBool" Bool [function, total, klabel(_andThenBool_), symbol, left, smt-hook(and), group(boolOperation), hook(BOOL.andThen)]
+                | Bool "xorBool" Bool     [function, total, klabel(_xorBool_), symbol, left, smt-hook(xor), group(boolOperation), hook(BOOL.xor)]
+                | Bool "orBool" Bool      [function, total, klabel(_orBool_), symbol, left, smt-hook(or), group(boolOperation), latex({#1}\vee_{\scriptstyle\it Bool}{#2}), hook(BOOL.or)]
+                | Bool "orElseBool" Bool  [function, total, klabel(_orElseBool_), symbol, left, smt-hook(or), group(boolOperation), hook(BOOL.orElse)]
+                | Bool "impliesBool" Bool [function, total, klabel(_impliesBool_), symbol, left, smt-hook(=>), group(boolOperation), hook(BOOL.implies)]
+                > left:
+                  Bool "==Bool" Bool      [function, total, klabel(_==Bool_), symbol, left, comm, smt-hook(=), hook(BOOL.eq)]
+                | Bool "=/=Bool" Bool     [function, total, klabel(_=/=Bool_), symbol, left, comm, smt-hook(distinct), hook(BOOL.ne)]
+
      +

      Implementation of Booleans

      +

      The remainder of this section consists of an implementation in K of the +operations listed above.

      +
      k
        rule notBool true => false
+  rule notBool false => true
+
+  rule true andBool B:Bool => B:Bool
+  rule B:Bool andBool true => B:Bool
+  rule false andBool _:Bool => false
+  rule _:Bool andBool false => false
+
+  rule true andThenBool K::Bool => K
+  rule K::Bool andThenBool true => K
+  rule false andThenBool _ => false
+  rule _ andThenBool false => false
+
+  rule false xorBool B:Bool => B:Bool
+  rule B:Bool xorBool false => B:Bool
+  rule B:Bool xorBool B:Bool => false
+
+  rule true orBool _:Bool => true
+  rule _:Bool orBool true => true
+  rule false orBool B:Bool => B
+  rule B:Bool orBool false => B
+
+  rule true orElseBool _ => true
+  rule _ orElseBool true => true
+  rule false orElseBool K::Bool => K
+  rule K::Bool orElseBool false => K
+
+  rule true impliesBool B:Bool => B
+  rule false impliesBool _:Bool => true
+  rule _:Bool impliesBool true => true
+  rule B:Bool impliesBool false => notBool B
+
+  rule B1:Bool =/=Bool B2:Bool => notBool (B1 ==Bool B2)
+endmodule
+
+module BOOL-KORE [kore, symbolic]
+  imports BOOL-COMMON
+
+  rule {true #Equals notBool @B} => {false #Equals @B} [simplification]
+  rule {notBool @B #Equals true} => {@B #Equals false} [simplification]
+  rule {false #Equals notBool @B} => {true #Equals @B} [simplification]
+  rule {notBool @B #Equals false} => {@B #Equals true} [simplification]
+
+  rule {true #Equals @B1 andBool @B2} => {true #Equals @B1} #And {true #Equals @B2} [simplification]
+  rule {@B1 andBool @B2 #Equals true} => {@B1 #Equals true} #And {@B2 #Equals true} [simplification]
+  rule {false #Equals @B1 orBool @B2} => {false #Equals @B1} #And {false #Equals @B2} [simplification]
+  rule {@B1 orBool @B2 #Equals false} => {@B1 #Equals false} #And {@B2 #Equals false} [simplification]
+endmodule
+
+module BOOL
+  imports BOOL-COMMON
+  imports BOOL-KORE
+endmodule
+
      +

      Integers

      +

      Provided here is the syntax of an implementation of arbitrary-precision +integer arithmetic in K. This type is hooked to an implementation of integers +provided by the backend. For a fixed-width integer type, see the MINT module +below.

      +

      The UNSIGNED-INT-SYNTAX module provides a syntax of whole numbers in K. +This is useful because often programming languages implement the sign of an +integer as a unary operator rather than part of the lexical syntax of integers. +However, you can also directly reference integers with a sign using the +INT-SYNTAX module.

      +
      k
      module UNSIGNED-INT-SYNTAX
+  syntax Int [hook(INT.Int)]
+  syntax Int ::= r"[0-9]+" [prefer, token, prec(2)]
+endmodule
+
+module INT-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  syntax Int ::= r"[\\+-]?[0-9]+" [prefer, token, prec(2)]
+endmodule
+
+module INT-COMMON
+  imports INT-SYNTAX
+  imports private BOOL
+
      +

      Integer arithmetic

      +

      You can:

      +
        +
      • Compute the bitwise complement ~Int of an integer value in twos-complement.
        • +
      • Compute the exponentiation ^Int of two integers.
        • +
      • Compute the exponentiation of two integers modulo another integer (^%Int). +A ^%Int B C is equal in value to (A ^Int B) %Int C, but has a better +asymptotic complexity.
        • +
      • Compute the product *Int of two integers.
        • +
      • Compute the quotient /Int or modulus %Int of two integers using +t-division, which rounds towards zero. Division by zero is #False.
        • +
      • Compute the quotient divInt or modulus modInt of two integers using +Euclidean division, in which the remainder is always non-negative. Division +by zero is #False.
        • +
      • Compute the sum +Int or difference -Int of two integers.
        • +
      • Compute the arithmetic right shift >>Int of two integers. Shifting by a +negative quantity is #False.
        • +
      • Compute the left shift of two integers. Shifting by a negative quantity is +#False.
        • +
      • Compute the bitwise and of two integers in twos-complement.
        • +
      • Compute the bitwise xor of two integers in twos-complement.
        • +
      • Compute the bitwise inclusive-or of two integers in twos-complement.
        • +
      +
      k
        syntax Int ::= "~Int" Int                     [function, klabel(~Int_), symbol, total, latex(\mathop{\sim_{\scriptstyle\it Int}}{#1}), hook(INT.not), smtlib(notInt)]
+               > left:
+                 Int "^Int" Int                 [function, klabel(_^Int_), symbol, left, smt-hook(^), latex({#1}\mathrel{{\char`\^}_{\!\scriptstyle\it Int}}{#2}), hook(INT.pow)]
+               | Int "^%Int" Int Int            [function, klabel(_^%Int__), symbol, left, smt-hook((mod (^ #1 #2) #3)), hook(INT.powmod)]
+               > left:
+                 Int "*Int" Int                 [function, total, klabel(_*Int_), symbol, left, comm, smt-hook(*), latex({#1}\mathrel{\ast_{\scriptstyle\it Int}}{#2}), hook(INT.mul)]
+               /* FIXME: translate /Int and %Int into smtlib */
+               /* /Int and %Int implement t-division, which rounds towards 0 */
+               | Int "/Int" Int                 [function, klabel(_/Int_), symbol, left, smt-hook(div), latex({#1}\mathrel{\div_{\scriptstyle\it Int}}{#2}), hook(INT.tdiv)]
+               | Int "%Int" Int                 [function, klabel(_%Int_), symbol, left, smt-hook(mod), latex({#1}\mathrel{\%_{\scriptstyle\it Int}}{#2}), hook(INT.tmod)]
+               /* divInt and modInt implement e-division according to the Euclidean division theorem, therefore the remainder is always positive */
+               | Int "divInt" Int               [function, klabel(_divInt_), symbol, left, smt-hook(div), hook(INT.ediv)]
+               | Int "modInt" Int               [function, klabel(_modInt_), symbol, left, smt-hook(mod), hook(INT.emod)]
+               > left:
+                 Int "+Int" Int                 [function, total, klabel(_+Int_), symbol, left, comm, smt-hook(+), latex({#1}\mathrel{+_{\scriptstyle\it Int}}{#2}), hook(INT.add)]
+               | Int "-Int" Int                 [function, total, klabel(_-Int_), symbol, left, smt-hook(-), latex({#1}\mathrel{-_{\scriptstyle\it Int}}{#2}), hook(INT.sub)]
+               > left:
+                 Int ">>Int" Int                [function, klabel(_>>Int_), symbol, left, latex({#1}\mathrel{\gg_{\scriptstyle\it Int}}{#2}), hook(INT.shr), smtlib(shrInt)]
+               | Int "<<Int" Int                [function, klabel(_<<Int_), symbol, left, latex({#1}\mathrel{\ll_{\scriptstyle\it Int}}{#2}), hook(INT.shl), smtlib(shlInt)]
+               > left:
+                 Int "&Int" Int                 [function, total, klabel(_&Int_), symbol, left, comm, latex({#1}\mathrel{\&_{\scriptstyle\it Int}}{#2}), hook(INT.and), smtlib(andInt)]
+               > left:
+                 Int "xorInt" Int               [function, total, klabel(_xorInt_), symbol, left, comm, latex({#1}\mathrel{\oplus_{\scriptstyle\it Int}}{#2}), hook(INT.xor), smtlib(xorInt)]
+               > left:
+                 Int "|Int" Int                 [function, total, klabel(_|Int_), symbol, left, comm, latex({#1}\mathrel{|_{\scriptstyle\it Int}}{#2}), hook(INT.or), smtlib(orInt)]
+
      +

      Integer minimum and maximum

      +

      You can compute the minimum and maximum minInt and maxInt of two integers.

      +
      k
        syntax Int ::= "minInt" "(" Int "," Int ")"   [function, total, smt-hook((ite (< #1 #2) #1 #2)), hook(INT.min)]
+               | "maxInt" "(" Int "," Int ")"   [function, total, smt-hook((ite (< #1 #2) #2 #1)), hook(INT.max)]
+
      +

      Absolute value

      +

      You can compute the absolute value absInt of an integer.

      +
      k
        syntax Int ::= absInt ( Int )                 [function, total, smt-hook((ite (< #1 0) (- 0 #1) #1)), hook(INT.abs)]
+
      +

      Log base 2

      +

      You can compute the log base 2, rounded towards zero, of an integer. The log +base 2 of an integer is equal to the index of the highest bit set in the +representation of a positive integer. Log base 2 of zero or a negative number +is #False.

      +
      k
        syntax Int ::= log2Int ( Int )                [function, hook(INT.log2)]
+
      +

      Bit slicing

      +

      You can compute the value of a range of bits in the twos-complement +representation of an integer, as interpeted either unsigned or signed, of an +integer. index is offset from 0 and length is the number of bits, starting +with index, that should be read. The number is assumed to be represented +in little endian notation with each byte going from least significant to +most significant. In other words, 0 is the least-significant bit, and each +successive bit is more significant than the last.

      +
      k
        syntax Int ::= bitRangeInt           ( Int, index: Int, length: Int ) [function, hook(INT.bitRange)]
+               | signExtendBitRangeInt ( Int, index: Int, length: Int ) [function, hook(INT.signExtendBitRange)]
+
      +

      Integer comparisons

      +

      You can compute whether two integers are less than or equal to, less than, +greater than or equal to, greater than, equal, or unequal to another integer.

      +
      k
        syntax Bool ::= Int "<=Int" Int         [function, total, klabel(_<=Int_), symbol, smt-hook(<=), latex({#1}\mathrel{\leq_{\scriptstyle\it Int}}{#2}), hook(INT.le)]
+                | Int "<Int" Int          [function, total, klabel(_<Int_), symbol, smt-hook(<), latex({#1}\mathrel{<_{\scriptstyle\it Int}}{#2}), hook(INT.lt)]
+                | Int ">=Int" Int         [function, total, klabel(_>=Int_), symbol, smt-hook(>=), latex({#1}\mathrel{\geq_{\scriptstyle\it Int}}{#2}), hook(INT.ge)]
+                | Int ">Int" Int          [function, total, klabel(_>Int_), symbol, smt-hook(>), latex({#1}\mathrel{>_{\scriptstyle\it Int}}{#2}), hook(INT.gt)]
+                | Int "==Int" Int         [function, total, klabel(_==Int_), symbol, comm, smt-hook(=), latex({#1}\mathrel{{=}{=}_{\scriptstyle\it Int}}{#2}), hook(INT.eq)]
+                | Int "=/=Int" Int        [function, total, klabel(_=/=Int_), symbol, comm, smt-hook(distinct), latex({#1}\mathrel{{=}{/}{=}_{\scriptstyle\it Int}}{#2}), hook(INT.ne)]
+
      +

      Divides

      +

      You can compute whether one integer evenly divides another. This is the +case when the second integer modulo the first integer is equal to zero.

      +
      k
        syntax Bool ::= Int "dividesInt" Int    [function]
+
      +

      Random integers

      +

      You can, on concrete backends, compute a pseudorandom integer, or seed the +pseudorandom number generator. These operations are represented as +uninterpreted functions on symbolic backends.

      +
      k
        syntax Int ::= randInt(Int) [function, hook(INT.rand), impure]
+  syntax K ::= srandInt(Int) [function, hook(INT.srand), impure]
+
      +

      Implementation of Integers

      +

      The remainder of this section consists of an implementation in K of some +of the operators above, as well as lemmas used by the Java and Haskell backend +to simplify expressions of sort Int. They do not affect the semantics of +integers, merely describing additional rules that the backend can use to +simplify terms.

      +
      k
      endmodule
+
+module INT-SYMBOLIC [symbolic]
+  imports INT-COMMON
+  imports INT-SYMBOLIC-KORE
+  imports private BOOL
+
+  // Arithmetic Normalization
+  rule I +Int 0 => I [simplification]
+  rule I -Int 0 => I [simplification]
+
+  rule X modInt N => X requires 0 <=Int X andBool X <Int N [simplification]
+  rule X   %Int N => X requires 0 <=Int X andBool X <Int N [simplification]
+
+  // Bit-shifts
+  rule X <<Int 0 => X [simplification]
+  rule 0 <<Int _ => 0 [simplification]
+  rule X >>Int 0 => X [simplification]
+  rule 0 >>Int _ => 0 [simplification]
+endmodule
+
+module INT-SYMBOLIC-KORE [symbolic, kore, haskell]
+  imports INT-COMMON
+  imports ML-SYNTAX
+  imports private BOOL
+
+  // Definability Conditions
+  rule #Ceil(@I1:Int /Int   @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int %Int   @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int modInt @I2:Int) => {(@I2 =/=Int 0) #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int >>Int  @I2:Int) => {(@I2 >=Int 0)  #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+  rule #Ceil(@I1:Int <<Int  @I2:Int) => {(@I2 >=Int 0)  #Equals true} #And #Ceil(@I1) #And #Ceil(@I2) [simplification]
+endmodule
+
+module INT-KORE [kore, symbolic]
+  imports private K-EQUAL
+  imports private BOOL
+  imports INT-COMMON
+
+  rule I1:Int ==K I2:Int => I1 ==Int I2 [simplification]
+  rule {K1 ==Int K2 #Equals true} => {K1 #Equals K2} [simplification]
+  rule {true #Equals K1 ==Int K2} => {K1 #Equals K2} [simplification]
+  rule {K1 ==Int K2 #Equals false} => #Not({K1 #Equals K2}) [simplification]
+  rule {false #Equals K1 ==Int K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=Int K2 #Equals true} => #Not({K1 #Equals K2}) [simplification]
+  rule {true #Equals K1 =/=Int K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=Int K2 #Equals false} => {K1 #Equals K2} [simplification]
+  rule {false #Equals K1 =/=Int K2} => {K1 #Equals K2} [simplification]
+
+  // Arithmetic Normalization
+  rule I +Int B => B +Int I          [concrete(I), symbolic(B), simplification(51)]
+  rule A -Int I => A +Int (0 -Int I) [concrete(I), symbolic(A), simplification(51)]
+
+  rule (A +Int I2) +Int I3 => A +Int (I2 +Int I3) [concrete(I2, I3), symbolic(A), simplification]
+  rule I1 +Int (B +Int I3) => B +Int (I1 +Int I3) [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 -Int (B +Int I3) => (I1 -Int I3) -Int B [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 +Int (I2 +Int C) => (I1 +Int I2) +Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule I1 +Int (I2 -Int C) => (I1 +Int I2) -Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule (I1 -Int B) +Int I3 => (I1 +Int I3) -Int B [concrete(I1, I3), symbolic(B), simplification]
+  rule I1 -Int (I2 +Int C) => (I1 -Int I2) -Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule I1 -Int (I2 -Int C) => (I1 -Int I2) +Int C [concrete(I1, I2), symbolic(C), simplification]
+  rule (C -Int I2) -Int I3 => C -Int (I2 +Int I3) [concrete(I2, I3), symbolic(C), simplification]
+
+  rule I1 &Int (I2 &Int C) => (I1 &Int I2) &Int C [concrete(I1, I2), symbolic(C), simplification]
+endmodule
+
+module INT
+  imports INT-COMMON
+  imports INT-SYMBOLIC
+  imports INT-KORE
+  imports private K-EQUAL
+  imports private BOOL
+
+  rule bitRangeInt(I::Int, IDX::Int, LEN::Int) => (I >>Int IDX) modInt (1 <<Int LEN)
+
+  rule signExtendBitRangeInt(I::Int, IDX::Int, LEN::Int) => (bitRangeInt(I, IDX, LEN) +Int (1 <<Int (LEN -Int 1))) modInt (1 <<Int LEN) -Int (1 <<Int (LEN -Int 1))
+
+  rule I1:Int divInt I2:Int => (I1 -Int (I1 modInt I2)) /Int I2
+  requires I2 =/=Int 0
+  rule
+    I1:Int modInt I2:Int
+  =>
+    ((I1 %Int absInt(I2)) +Int absInt(I2)) %Int absInt(I2)
+  requires I2 =/=Int 0    [concrete, simplification]
+
+  rule minInt(I1:Int, I2:Int) => I1 requires I1 <=Int I2
+  rule minInt(I1:Int, I2:Int) => I2 requires I1 >=Int I2
+
+  rule I1:Int =/=Int I2:Int => notBool (I1 ==Int I2)
+  rule (I1:Int dividesInt I2:Int) => (I2 %Int I1) ==Int 0
+
+  syntax Int ::= freshInt(Int)    [freshGenerator, function, total, private]
+  rule freshInt(I:Int) => I
+endmodule
+
      +

      IEEE 754 Floating-point Numbers

      +

      Provided here is the syntax of an implementation of arbitrary-precision +floating-point arithmetic in K based on a generalization of the IEEE 754 +standard. This type is hooked to an implementation of floats provided by the +backend.

      +

      The syntax of ordinary floating-point values in K consists of an optional sign +(+ or -) followed by an optional integer part, followed by a decimal point, +followed by an optional fractional part. Either the integer part or the +fractional part must be specified. The mantissa is followed by an optional +exponent part, which consists of an e or E, an optional sign (+ or -), +and an integer. The expoennt is followed by an optional suffix, which can be +either f, F, d, D, or pNxM where N and M are positive integers. +p and x can be either upper or lowercase.

      +

      The value of a floating-point literal is computed as follows: First the +mantissa is read as a rational number. Then it is multiplied by 10 to the +power of the exponent, which is interpreted as an integer, and defaults to +zero if it is not present. Finally, it is rounded to the nearest possible +value in a floating-point type represented like an IEEE754 floating-point type, +with the number of bits of precision and exponent specified by the suffix. +A suffix of f or f represents the IEEE binary32 format. A suffix of d +or D, or no suffix, represents the IEEE binary64 format. A suffix of +pNxM (either upper or lowercase) specifies exactly N bits of precision and +M bits of exponent. The number of bits of precision is assumed to include +any optional 1 that precedes the IEEE 754 mantissa. In other words, p24x8 +is equal to the IEEE binary32 format, and p53x11 is equal to the IEEE +binary64 format.

      +
      k
      module FLOAT-SYNTAX
+  syntax Float [hook(FLOAT.Float)]
+  syntax Float ::= r"([\\+-]?[0-9]+(\\.[0-9]*)?|\\.[0-9]+)([eE][\\+-]?[0-9]+)?([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(1)]
+  syntax Float ::= r"[\\+-]?Infinity([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(3)]
+  syntax Float ::= r"NaN([fFdD]|([pP][0-9]+[xX][0-9]+))?" [token, prec(3)]
+endmodule
+
+module FLOAT
+  imports FLOAT-SYNTAX
+  imports private BOOL
+  imports private INT-SYNTAX
+
      +

      Float precision

      +

      You can retrieve the number of bits of precision in a Float.

      +
      k
        syntax Int ::= precisionFloat(Float) [function, total, hook(FLOAT.precision)]
+
      +

      Float exponent bits

      +

      You can retrieve the number of bits of exponent range in a Float.

      +
      k
        syntax Int ::= exponentBitsFloat(Float) [function, total, hook(FLOAT.exponentBits)]
+
      +

      Float exponent

      +

      You can retrieve the value of the exponent bits of a Float as an integer.

      +
      k
        syntax Int ::= exponentFloat(Float) [function, total, hook(FLOAT.exponent)]
+
      +

      Float sign

      +

      You can retrieve the value of the sign bit of a Float as a boolean. True +means the sign bit is set.

      +
      k
        syntax Bool ::= signFloat(Float)      [function, total, hook(FLOAT.sign)]
+
      +

      Float special values

      +

      You can check whether a Float value is infinite or Not-a-Number.

      +
      k
        syntax Bool ::= isNaN(Float)          [function, total, smt-hook(fp.isNaN), hook(FLOAT.isNaN)]
+                | isInfinite(Float)     [function, total]
+
      +

      Float arithmetic

      +

      You can:

      +
        +
      • Compute the unary negation --Float of a float. --Float X is distinct +from 0.0 -Float X. For example, 0.0 -Float 0.0 is positive zero. +--Float 0.0 is negative zero.
        • +
      • Compute the exponentation ^Float of two floats.
        • +
      • Compute the product *Float, quotient /Float, or remainder %Float of two +floats. The remainder is computed based on rounding the quotient of the two +floats to the nearest integer.
        • +
      • Compute the sum +Float or difference -Float of two floats.
        • +
      +
      k
        syntax Float ::= "--Float" Float             [function, total, smt-hook(fp.neg), hook(FLOAT.neg)]
+                 > Float "^Float" Float        [function, left, latex({#1}^{#2}), hook(FLOAT.pow)]
+                 > left:
+                   Float "*Float" Float        [function, left, smt-hook((fp.mul roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\ast_{\scriptstyle\it Float}}{#2}), hook(FLOAT.mul)]
+                 | Float "/Float" Float        [function, left, smt-hook((fp.div roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\div_{\scriptstyle\it Float}}{#2}), hook(FLOAT.div)]
+                 | Float "%Float" Float        [function, left, smt-hook((fp.rem roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{\%_{\scriptstyle\it Float}}{#2}), hook(FLOAT.rem)]
+                 > left:
+                   Float "+Float" Float        [function, left, smt-hook((fp.add roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{+_{\scriptstyle\it Float}}{#2}), hook(FLOAT.add)]
+                 | Float "-Float" Float        [function, left, smt-hook((fp.sub roundNearestTiesToEven #1 #2)), latex({#1}\mathrel{-_{\scriptstyle\it Float}}{#2}), hook(FLOAT.sub)]
+
      +

      Floating-point mathematics

      +

      You can:

      +
        +
      • Compute the Nth integer root rootFloat of a float.
        • +
      • Compute the absolute value absFloat of a float.
        • +
      • Round a floating-point number to a specified precision and exponent +range (roundFloat). The resulting Float will yield the specified values +when calling precisionFloat and exponentBitsFloat and when performing +further computation.
        • +
      • Round a float to the next lowest floating-point value which is an integer +(floorFloat).
        • +
      • Round a float to the next highest floating-point value which is an integer +(ceilFloat).
        • +
      • Round a float to the next closest floating-point value which is an integer, in +the direction of zero (truncFloat).
        • +
      • Compute the natural exponential expFloat of a float (i.e. e^x).
        • +
      • Compute the natural logarithm logFloat of a float.
        • +
      • Compute the sine sinFloat of a float.
        • +
      • Compute the cosine cosFloat of a float.
        • +
      • Compute the tangent tanFlooat of a float.
        • +
      • Compute the arcsine asinFloat of a float.
        • +
      • Compute the arccosine acosFloat of a float.
        • +
      • Compute the arctangent atanFloat of a float.
        • +
      • Compute the arctangent atan2Float of two floats.
        • +
      • Compute the maximum maxFloat of two floats.
        • +
      • Compute the minimum minFloat of two floats.
        • +
      • Compute the square root sqrtFloat of a float.
        • +
      • Compute the largest finite value expressible in a specified precision and +exponent range (maxValueFloat).
        • +
      • Compute the smallest positive finite value expressible in a specified +precision and exponent range (minValueFloat).
        • +
      +
      k
        syntax Float ::= rootFloat(Float, Int)        [function, hook(FLOAT.root)]
+                 | absFloat(Float)              [function, total, smt-hook(fp.abs), hook(FLOAT.abs)]
+                 | roundFloat(Float, precision: Int, exponentBits: Int)  [function, hook(FLOAT.round)]
+                 | floorFloat(Float)            [function, total, hook(FLOAT.floor)]
+                 | ceilFloat(Float)             [function, total, hook(FLOAT.ceil)]
+                 | truncFloat(Float)            [function, total, hook(FLOAT.trunc)]
+                 | expFloat(Float)              [function, total, hook(FLOAT.exp)]
+                 | logFloat(Float)              [function, hook(FLOAT.log)]
+                 | sinFloat(Float)              [function, total, hook(FLOAT.sin)]
+                 | cosFloat(Float)              [function, total, hook(FLOAT.cos)]
+                 | tanFloat(Float)              [function, hook(FLOAT.tan)]
+                 | asinFloat(Float)             [function, hook(FLOAT.asin)]
+                 | acosFloat(Float)             [function, hook(FLOAT.acos)]
+                 | atanFloat(Float)             [function, total, hook(FLOAT.atan)]
+                 | atan2Float(Float, Float)     [function, hook(FLOAT.atan2)]
+                 | maxFloat(Float, Float)       [function, smt-hook(fp.max), hook(FLOAT.max)]
+                 | minFloat(Float, Float)       [function, smt-hook(fp.min), hook(FLOAT.min)]
+                 | sqrtFloat(Float)             [function]
+                 | maxValueFloat(precision: Int, exponentBits: Int)      [function, hook(FLOAT.maxValue)]
+                 | minValueFloat(precision: Int, exponentBits: Int)      [function, hook(FLOAT.minValue)]
+
      +

      Floating-point comparisons

      +

      Compute whether a float is less than or equasl to, less than, greater than or +equal to, greater than, equal, or unequal to another float. Note that +X ==Float Y and X ==K Y might yield different values. The latter should be +used in cases where you want to compare whether two values of sort Float +contain the same term. The former should be used when you want to implement +the == operator of a programming language. In particular, NaN =/=Float NaN +is true, because NaN compares unequal to all values, including itself, in +IEEE 754 arithmetic. 0.0 ==Float -0.0 is also true.

      +
      k
        syntax Bool ::= Float "<=Float" Float       [function, smt-hook(fp.leq), latex({#1}\mathrel{\leq_{\scriptstyle\it Float}}{#2}), hook(FLOAT.le)]
+                | Float "<Float" Float        [function, smt-hook(fp.lt), latex({#1}\mathrel{<_{\scriptstyle\it Float}}{#2}), hook(FLOAT.lt)]
+                | Float ">=Float" Float       [function, smt-hook(fp.geq), latex({#1}\mathrel{\geq_{\scriptstyle\it Float}}{#2}), hook(FLOAT.ge)]
+                | Float ">Float" Float        [function, smt-hook(fg.gt), latex({#1}\mathrel{>_{\scriptstyle\it Float}}{#2}), hook(FLOAT.gt)]
+                | Float "==Float" Float       [function, comm, smt-hook(fp.eq), latex({#1}\mathrel{==_{\scriptstyle\it Float}}{#2}), hook(FLOAT.eq), klabel(_==Float_)]
+                | Float "=/=Float" Float      [function, comm, smt-hook((not (fp.eq #1 #2))), latex({#1}\mathrel{\neq_{\scriptstyle\it Float}}{#2})]
+
+  rule F1:Float =/=Float F2:Float => notBool (F1 ==Float F2)
+
      +

      Conversion between integer and float

      +

      You can convert an integer to a floating-point number with the specified +precision and exponent range. You can also convert a floating-point number +to the nearest integer. This operation rounds to the nearest integer, but it +also avoids the double-rounding that is present in ceilFloat and floorFloat +if the nearest integer is not representable in the specified floating-point +type.

      +
      k
        syntax Float ::= Int2Float(Int, precision: Int, exponentBits: Int)    [function, latex({\\it{}Int2Float}), hook(FLOAT.int2float)]
+  syntax Int ::= Float2Int(Float)    [function, total, latex({\\it{}Float2Int}), hook(FLOAT.float2int)]
+
      +

      Implementation of Floats

      +

      The remainder of this section consists of an implementation in K of some of the +operators above.

      +
      k
        rule sqrtFloat(F:Float) => rootFloat(F, 2)
+
+  rule isInfinite(F:Float) => F >Float maxValueFloat(precisionFloat(F), exponentBitsFloat(F)) orBool F <Float --Float maxValueFloat(precisionFloat(F), exponentBitsFloat(F))
+
+endmodule
+
      +

      Strings

      +

      Provided here is the syntax of an implementation of Unicode strings in K. This +type is hooked to an implementation of strings provided by the backend. The +implementation is currently incomplete and does not fully support encodings +and code points beyond the initial 256 code points of the Basic Latin and +Latin-1 Supplement blocks. In the future, there may be breaking changes to +the semantics of this module in order to support this functionality.

      +

      The syntax of strings in K is delineated by double quotes. Inside the double +quotes, any character can appear verbatim except double quotes, backslash, +newline, and carriage return. K also supports the following escape sequences:

      +
        +
      • " - the " character
        • +
      • \ - the \ character
        • +
      • \n - newline character
        • +
      • \r - carriage return character
        • +
      • \t - tab character
        • +
      • \f - form feed character
        • +
      • \xFF - \x followed by two hexadecimal characters indicates a code point +between 0x00 and 0xff
        • +
      • \uFFFF - \u followed by four hexadecimal characters indicates a code point +between 0x0000 and 0xffff
        • +
      • \UFFFFFFFF - \U followed by eight hexadecimal characters indicates a code +point between 0x000000 and 0x10ffff
        • +
      +
      k
      module STRING-SYNTAX
+  syntax String [hook(STRING.String)]
+  syntax String ::= r"[\\\"](([^\\\"\\n\\r\\\\])|([\\\\][nrtf\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2})|([\\\\][u][0-9a-fA-F]{4})|([\\\\][U][0-9a-fA-F]{8}))*[\\\"]"      [token]
+endmodule
+
+module STRING-COMMON
+  imports STRING-SYNTAX
+  imports private INT
+  imports private FLOAT-SYNTAX
+  imports private K-EQUAL
+  imports private BOOL
+
      +

      String concatenation

      +

      You can concatenate two strings in O(N) time. For successive concatenation +operations, it may be better to use the STRING-BUFFER module.

      +
      k
        syntax String ::= String "+String" String    [function, total, left, latex({#1}+_{\scriptstyle\it String}{#2}), hook(STRING.concat)]
+
      +

      String length

      +

      You can get the length of a string in O(1) time.

      +
      k
        syntax Int ::= lengthString ( String ) [function, total, hook(STRING.length)]
+
      +

      Character and integer conversion

      +

      You can convert between a character (as represented by a string containing +a single code point) and an integer in O(1) time.

      +
      k
        syntax String ::= chrChar ( Int )      [function, hook(STRING.chr)]
+  syntax Int ::= ordChar ( String )      [function, hook(STRING.ord)]
+
      +

      String substring

      +

      You can compute a substring of a string in O(N) time (where N is the +length of the substring). There are two important facts to note:

      +
        +
      1. the range generated includes the character at startIndex but excludes the +character at endIndex, i.e., the range is [startIndex..endIndex).
        2. +
      2. this function is only defined on valid indices (i.e., it is defined when +startIndex < endIndex and endIndex is less than or equal to the string +length).
        3. +
      +
      k
        syntax String ::= substrString ( String , startIndex: Int , endIndex: Int ) [function, total, hook(STRING.substr)]
+
      + +

      You can find the first (respectively, last) occurrence of a substring, starting +at a certain index, in another string in O(N*M) time. +Returns -1 if the substring is not found.

      +
      k
        syntax Int ::= findString ( haystack: String , needle: String , index: Int )   [function, hook(STRING.find)]
+  syntax Int ::= rfindString ( haystack: String , needle: String , index: Int )  [function, hook(STRING.rfind)]
+
      + +

      You can find the first (respectively, last) occurrence of one of the characters +of the search string, starting at a certain index, in another string in +O(N*M) time.

      +
      k
        syntax Int ::= findChar ( haystack: String , needles: String , index: Int )     [function, hook(STRING.findChar)]
+  syntax Int ::= rfindChar ( haystack: String , needles: String , index: Int )    [function, hook(STRING.rfindChar)]
+
      +

      String and Bool conversion

      +
      k
        syntax String ::= Bool2String(Bool) [function, total]
+  rule Bool2String(true)  => "true"
+  rule Bool2String(false) => "false"
+
      +
      k
        syntax Bool ::= String2Bool(String) [function]
+  rule String2Bool("true")  => true
+  rule String2Bool("false") => false
+
      +

      String and float conversion

      +

      You can convert between a String and a Float. The String will be +represented in the syntax of the Float sort (see the section on the FLOAT +module above for details of that syntax). Which particular string is returned +by Float2String is determined by the backend, but the same Float is +guaranteed to return the same String, and converting that String back to a +Float is guaranteed to return the original Float.

      +

      You can also convert a Float to a string in a particular syntax using the +variant of Float2String with a format. In this case, the resulting string +is one which results directly from passing that format to mpfr_printf. This +functionality may not be supported on backends that do not use Gnu MPFR to +implement floating-point numbers.

      +
      k
        syntax String ::= Float2String ( Float )              [function, total, hook(STRING.float2string)]
+  syntax String ::= Float2String ( Float , format: String )     [function, klabel(FloatFormat), hook(STRING.floatFormat)]
+  syntax Float  ::= String2Float ( String )             [function, hook(STRING.string2float)]
+
      +

      String and integer conversions

      +

      You can convert between a String and an Int. The String will be represented +in the syntax of the INT module (i.e., a nonempty sequence of digits +optionally prefixed by a sign). When converting from an Int to a String, +the sign will not be present unless the integer is negative.

      +

      You can also convert between a String and an Int in a particular radix. +This radix can be anywhere between 2 and 36. For a radix 2 <= N <= 10, the +digits 0 to N-1 will be used. For a radix 11 <= N <= 36, the digits 0 to 9 +and the first N-10 letters of the Latin alphabet will be used. Both uppercase +and lowercase letters are supported by String2Base. Whether the letters +returned by Base2String are upper or lowercase is determined by the backend, +but the backend will consistently choose one or the other.

      +
      k
        syntax Int    ::= String2Int   ( String )             [function, hook(STRING.string2int)]
+  syntax String ::= Int2String   ( Int )                [function, total, hook(STRING.int2string)]
+  syntax String ::= Base2String  ( Int , base: Int )          [function, hook(STRING.base2string)]
+  syntax Int    ::= String2Base  ( String , base: Int )       [function, hook(STRING.string2base)]
+
      +

      String count and replace

      +

      You can replace one, some, or all occurrences of a string within another +string in O(N*M) time. The replaceAll, replace, and replaceFirst methods +are identical, except replaceFirst replaces exactly one ocurrence of the +string, the first occurrence. replace replaces the first times occurrences. +And replaceAll replaces every occurrence.

      +

      You can also count the number of times a string occurs within another string +using countAllOccurrences.

      +
      k
        syntax String ::= "replaceAll" "(" haystack: String "," needle: String "," replacement: String ")"      [function, total, hook(STRING.replaceAll)]
+  syntax String ::= "replace" "(" haystack: String "," needle: String "," replacement: String "," times: Int ")" [function, hook(STRING.replace)]
+  syntax String ::= "replaceFirst" "(" haystack: String "," needle: String "," replacement: String ")"    [function, total, hook(STRING.replaceFirst)]
+  syntax Int ::= "countAllOccurrences" "(" haystack: String "," needle: String ")"            [function, total, hook(STRING.countAllOccurrences)]
+
      +

      String equality and lexicographic comparison

      +

      You can compare whether two strings are equal or unequal, or whether one string +is less than, less than or equal to, greater than, or greater than or equal to +another according to the natural lexicographic ordering of strings.

      +
      k
        syntax Bool ::= String "==String" String  [function, total, comm, hook(STRING.eq)]
+                | String "=/=String" String [function, total, comm, hook(STRING.ne)]
+  rule S1:String =/=String S2:String => notBool (S1 ==String S2)
+
+  syntax Bool ::= String  "<String" String [function, total, hook(STRING.lt)]
+                | String "<=String" String [function, total, hook(STRING.le)]
+                | String  ">String" String [function, total, hook(STRING.gt)]
+                | String ">=String" String [function, total, hook(STRING.ge)]
+
      +

      Implementation of Strings

      +

      What follows is a few String hooks which are deprecated and only are supported +on certain outdated backends of K, as well as an implementation of several +of the above operations in K.

      +
      k
        syntax String ::= categoryChar(String)       [function, hook(STRING.category)]
+                  | directionalityChar(String) [function, hook(STRING.directionality)]
+
+  syntax String ::= "newUUID" [function, hook(STRING.uuid), impure]
+
+  rule S1:String <=String S2:String => notBool (S2 <String S1)
+  rule S1:String >String S2:String => S2 <String S1
+  rule S1:String >=String S2:String => notBool (S1 <String S2)
+
+  rule findChar(S1:String, S2:String, I:Int) => #if findString(S1, substrString(S2, 0, 1), I) ==Int -1 #then findChar(S1, substrString(S2, 1, lengthString(S2)), I) #else #if findChar(S1, substrString(S2, 1, lengthString(S2)), I) ==Int -1 #then findString(S1, substrString(S2, 0, 1), I) #else minInt(findString(S1, substrString(S2, 0, 1), I), findChar(S1, substrString(S2, 1, lengthString(S2)), I)) #fi #fi requires S2 =/=String ""
+  rule findChar(_, "", _) => -1
+  rule rfindChar(S1:String, S2:String, I:Int) => maxInt(rfindString(S1, substrString(S2, 0, 1), I), rfindChar(S1, substrString(S2, 1, lengthString(S2)), I)) requires S2 =/=String ""
+  rule rfindChar(_, "", _) => -1
+
+  rule countAllOccurrences(Source:String, ToCount:String) => 0
+            requires findString(Source, ToCount, 0) <Int 0
+  rule countAllOccurrences(Source:String, ToCount:String) => 1 +Int countAllOccurrences(substrString(Source, findString(Source, ToCount, 0) +Int lengthString(ToCount), lengthString(Source)), ToCount)
+            requires findString(Source, ToCount, 0) >=Int 0
+
+  rule replaceFirst(Source:String, ToReplace:String, Replacement:String) => substrString(Source, 0, findString(Source, ToReplace, 0))
+                +String Replacement +String substrString(Source, findString(Source, ToReplace, 0) +Int lengthString(ToReplace), lengthString(Source))
+                requires findString(Source, ToReplace, 0) >=Int 0
+  rule replaceFirst(Source:String, ToReplace:String, _:String) => Source
+        requires findString(Source, ToReplace, 0) <Int 0
+
+
+  rule replace(Source:String, ToReplace:String, Replacement:String, Count:Int) =>
+       substrString(Source, 0, findString(Source, ToReplace, 0)) +String Replacement +String
+       replace(substrString(Source, findString(Source, ToReplace, 0) +Int lengthString(ToReplace), lengthString(Source)), ToReplace, Replacement, Count -Int 1)
+        requires Count >Int 0
+  rule replace(Source:String, _, _, 0) => Source
+  rule replaceAll(Source:String, ToReplace:String, Replacement:String) => replace(Source, ToReplace, Replacement, countAllOccurrences(Source, ToReplace))
+
+endmodule
+
+module STRING-KORE [kore, symbolic]
+  imports private K-EQUAL
+  imports STRING-COMMON
+
+  rule S1:String ==K S2:String => S1 ==String S2 [simplification]
+
+endmodule
+
+module STRING
+  imports STRING-COMMON
+  imports STRING-KORE
+endmodule
+
      +

      String Buffers

      +

      It is a well known fact that repeated string concatenations are quadratic +in performance whereas use of an efficient mutable representation of arrays +can yield linear performance. We thus provide such a sort, the StringBuffer +sort. Axiomatically, it is implemented below on symbolic backends using the +String module. However, on concrete backends it provides an efficient +implementation of string concatenation. There are three operations:

      +
        +
      • .StringBuffer creates a new StringBuffer with current content equal +to the empty string.
        • +
      • +String takes a StringBuffer and a String and appends the String to +the end of the StringBuffer
        • +
      • StringBuffer2String converts a StringBuffer to a String. This operation +copies the string so that subsequent modifications to the StringBuffer +will not change the value of the String returned by this function.
        • +
      +
      k
      module STRING-BUFFER-IN-K [symbolic]
+  imports private BASIC-K
+  imports STRING
+
+  syntax StringBuffer ::= ".StringBuffer" [function, total]
+  syntax StringBuffer ::= StringBuffer "+String" String [function, total, avoid]
+  syntax StringBuffer ::= String
+  syntax String ::= StringBuffer2String ( StringBuffer ) [function, total]
+
+  rule {SB:String +String S:String}<:StringBuffer => (SB +String S)::String
+  rule .StringBuffer => ""
+  rule StringBuffer2String(S:String) => S
+endmodule
+
+module STRING-BUFFER-HOOKED [concrete]
+  imports private BASIC-K
+  imports STRING
+
+  syntax StringBuffer [hook(BUFFER.StringBuffer)]
+  syntax StringBuffer ::= ".StringBuffer" [function, total, hook(BUFFER.empty), impure]
+  syntax StringBuffer ::= StringBuffer "+String" String [function, total, hook(BUFFER.concat), avoid]
+  syntax String ::= StringBuffer2String ( StringBuffer ) [function, total, hook(BUFFER.toString)]
+endmodule
+
+module STRING-BUFFER
+  imports STRING-BUFFER-HOOKED
+  imports STRING-BUFFER-IN-K
+endmodule
+
      +

      Byte Arrays

      +

      Provided here is the syntax of an implementation of fixed-width arrays of Bytes +in K. This type is hooked to an implementation of bytes provided by the +backend. In concrete backends, this representation is mutable and thus multiple +references can occur to the same Bytes object and when one is modified, the +others are also modified. Care should be taken not to rely on this fact however +as this is not the case in symbolic backends and thus you will experience +divergent behavior unless the Bytes type is used in a manner that preserves +consistency.

      +
      k
      module BYTES-SYNTAX
+  imports private STRING-SYNTAX
+
+  syntax Bytes [hook(BYTES.Bytes)]
+  syntax Bytes ::= r"b[\\\"](([\\x20\\x21\\x23-\\x5B\\x5D-\\x7E])|([\\\\][tnfr\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2}))*[\\\"]"      [token]
+endmodule
+
      +
      k
      module BYTES-STRING-ENCODE [kore, symbolic]
+  imports BYTES-SYNTAX
+
      +

      Encoding/decoding between Bytes and String

      +

      You can encode/decode between Bytes and String using UTF-8, UTF-16LE, UTF-16BE, UTF-32LE, and UTF-32BE

      +
      k
          syntax String ::= decodeBytes ( encoding: String , contents: Bytes ) [function, hook(BYTES.decodeBytes)]
+    syntax Bytes ::= encodeBytes ( encoding: String , contents: String ) [function, hook(BYTES.encodeBytes)]
+endmodule
+
      +
      k
      module BYTES-HOOKED
+  imports STRING-SYNTAX
+  imports BYTES-SYNTAX
+  imports BYTES-STRING-ENCODE
+
      +

      Empty byte array

      +

      The byte array of length zero is represented by .Bytes.

      +
      k
        syntax Bytes ::= ".Bytes" [function, total, hook(BYTES.empty)]
+
      +

      Endianness

      +

      When converting to/from an integer, byte arrays can be treated as either little +endian (ie, least significant byte first) or big endian (ie, most significant +byte first).

      +
      k
        syntax Endianness ::= "LE" [klabel(littleEndianBytes), symbol]
+                      | "BE" [klabel(bigEndianBytes), symbol]
+
      +

      Signedness

      +

      When converting to/from an integer, byte arrays can be treated as either signed +or unsigned.

      +
      k
        syntax Signedness ::= "Signed" [klabel(signedBytes), symbol]
+                      | "Unsigned" [klabel(unsignedBytes), symbol]
+
      +

      Integer and Bytes conversion

      +

      You can convert from a Bytes to an Int. In order to do this, the endianness +and signedness of the Bytes must be provided. The resulting integer is +created by means of interpreting the Bytes as either a twos-complement +representation, or an unsigned representation, of an integer, in the specified +byte order.

      +

      You can also convert from an Int to a Bytes. This comes in two variants. +In the first, the length of the resulting Bytes in bytes is explicitly +specified. If the length is greater than the highest set bit in the magnitude +of the integer, the result is padded with 0 bits if the number is positive +and 1 bits if the number is negative. If the length is less than the highest +bit set in the magnitude of the integer, the most-significant bits of the +integer will be truncated. The endianness of the resulting Bytes object +is as specified.

      +

      In the second variant, both endianness and signedness are specified, and +the resulting Bytes object will be the smallest number of bytes necessary +for the resulting Bytes object to be convertible back to the original integer +via Bytes2Int. In other words, if the highest bit set in the magnitude of the +integer is N, then the byte array will be at least N+1 bits long, rounded up +to the nearest byte.

      +
      k
        syntax Int ::= Bytes2Int(Bytes, Endianness, Signedness) [function, total, hook(BYTES.bytes2int)]
+  syntax Bytes ::= Int2Bytes(length: Int, Int, Endianness) [function, total, hook(BYTES.int2bytes)]
+                 | Int2Bytes(Int, Endianness, Signedness) [function, total, klabel(Int2BytesNoLen)]
+
      +

      String and Bytes conversion

      +

      You can convert between a Bytes and a String in O(N) time. The resulting +value is a copy of the original and will not be affected by subsequent +mutations of the input or output value.

      +
      k
        syntax String ::= Bytes2String(Bytes) [function, total, hook(BYTES.bytes2string)]
+  syntax Bytes ::= String2Bytes(String) [function, total, hook(BYTES.string2bytes)]
+
      +

      Bytes update

      +

      You can set the value of a particular byte in a Bytes object in O(1) time. +The result is #False if value is not in the range [0..255] or if index +is not a valid index (ie, less than zero or greater than or equal to the length +of the Bytes term).

      +
      k
        syntax Bytes ::= Bytes "[" index: Int "<-" value: Int "]" [function, hook(BYTES.update)]
+
      +

      Bytes lookup

      +

      You can get the value of a particular byte in a Bytes object in O(1) time. +The result is #False if index is not a valid index (see above).

      +
      k
        syntax Int ::= Bytes "[" Int "]" [function, hook(BYTES.get)]
+
      +

      Bytes substring

      +

      You can get a new Bytes object containing a range of bytes from the input +Bytes in O(N) time (where N is the length of the substring). The range +of bytes included is [startIndex..endIndex). The resulting Bytes is +a copy and mutations to it do not affect mutations to the original Bytes. +The result is #False if startIndex or endIndex are not valid.

      +
      k
        syntax Bytes ::= substrBytes(Bytes, startIndex: Int, endIndex: Int) [function, hook(BYTES.substr)]
+
      +

      Multiple bytes update

      +

      You can modify a Bytes to return a Bytes which is equal to dest except +the N elements starting at index are replaced with the contents of src in +O(N) time. This does not create a new Bytes object and will instead modify +the original on concrete backends. The result is #False if index + N +is not a valid index.

      +
      k
        syntax Bytes ::= replaceAtBytes(dest: Bytes, index: Int, src: Bytes) [function, hook(BYTES.replaceAt)]
+
      +

      Bytes padding

      +

      You can create a new Bytes object which is at least length bytes long +by taking the input sequence and padding it on the right (respectively, on the +left) with the specified value. This does not create a new Bytes object +if the input is already at least length bytes long, and will instead +return the input unchanged. The result is #False if value is not in the +range [0..255], or if the length is negative.

      +
      k
        syntax Bytes ::= padRightBytes(Bytes, length: Int, value: Int) [function, hook(BYTES.padRight)]
+                 | padLeftBytes(Bytes, length: Int, value: Int) [function, hook(BYTES.padLeft)]
+
      +

      Bytes reverse

      +

      You can reverse a Bytes object in O(N) time. This does not create a new +Bytes object and will instead modify the original on concrete backends.

      +
      k
        syntax Bytes ::= reverseBytes(Bytes) [function, total, hook(BYTES.reverse)]
+
      +

      Bytes length

      +

      You can get the length of a Bytes term in O(1) time.

      +
      k
        syntax Int ::= lengthBytes(Bytes) [function, total, hook(BYTES.length), smtlib(lengthBytes)]
+
      +

      Bytes concatenation

      +

      You can create a new Bytes object by concatenating two Bytes objects +together in O(N) time.

      +
      k
        syntax Bytes ::= Bytes "+Bytes" Bytes [function, total, hook(BYTES.concat), right]
+
+endmodule
+
      +

      Implementation of Bytes

      +

      The remainder of this module consists of an implementation of some of the +operators listed above in K.

      +
      k
      module BYTES-CONCRETE [concrete]
+  imports BYTES-HOOKED
+endmodule
+
+module BYTES-KORE [kore]
+  imports BYTES-HOOKED
+  imports BYTES-SYMBOLIC-CEIL
+endmodule
+
+module BYTES-SYMBOLIC-CEIL [symbolic, kore]
+  imports BYTES-HOOKED
+  imports private INT
+  imports private BOOL
+
+  rule #Ceil(padRightBytes(_, LEN, VAL)) => {(0 <=Int LEN andBool 0 <=Int VAL andBool VAL <Int 256) #Equals true} [simplification]
+  rule #Ceil(padLeftBytes(_, LEN, VAL))  => {(0 <=Int LEN andBool 0 <=Int VAL andBool VAL <Int 256) #Equals true} [simplification]
+endmodule
+
+module BYTES
+  imports BYTES-CONCRETE
+  imports BYTES-KORE
+  imports private INT
+
+  rule Int2Bytes(I::Int, E::Endianness, Unsigned) => Int2Bytes((log2Int(I) +Int 8) /Int 8, I, E)
+    requires I >Int 0
+  rule Int2Bytes(0, _::Endianness, _) => .Bytes
+  rule Int2Bytes(I::Int, E::Endianness, Signed) => Int2Bytes((log2Int(I) +Int 9) /Int 8, I, E)
+    requires I >Int 0
+  rule Int2Bytes(I::Int, E::Endianness, Signed) => Int2Bytes((log2Int(~Int I) +Int 9) /Int 8, I, E)
+    requires I <Int -1
+  rule Int2Bytes(-1, E::Endianness, Signed) => Int2Bytes(1, -1, E)
+endmodule
+
      +

      Program identifiers

      +

      Provided here is an implementation for program identifiers in K. Developers +of semantics for a particular language may wish to use their own implementation +instead of the one provided here if their syntax differs from the syntax +defined below. However, this is provided for convenience for developers who +do not care about the lexical syntax of identifiers.

      +

      Provided are the following pieces of functionality:

      +
        +
      • Id2String - Convert an Id to a String containing its name
        • +
      • String2Id - Convert a String to an Id with the specified name
        • +
      • !X:Id - You can get a fresh identifier distinct from any previous identifier +generated by this syntax.
        • +
      +
      k
      module ID-SYNTAX-PROGRAM-PARSING
+  imports BUILTIN-ID-TOKENS
+  syntax Id ::= r"(?<![A-Za-z0-9\\_])[A-Za-z\\_][A-Za-z0-9\\_]*"     [prec(1), token]
+              | #LowerId                                             [token]
+              | #UpperId                                             [token]
+endmodule
+
+module ID-SYNTAX
+  syntax Id [token]
+endmodule
+
+module ID-COMMON
+  imports ID-SYNTAX
+  imports private STRING
+
+  syntax String ::= Id2String ( Id )    [function, total, hook(STRING.token2string)]
+  syntax Id ::= String2Id (String) [function, total, hook(STRING.string2token)]
+  syntax Id ::= freshId(Int)    [freshGenerator, function, total, private]
+
+  rule freshId(I:Int) => String2Id("_" +String Int2String(I))
+endmodule
+
+module ID
+  imports ID-COMMON
+endmodule
+
      +

      Equality and conditionals

      +

      Provided here are implementations of two important primitives in K:

      +
        +
      • ==K - the equality between two terms. Returns true if they are equal +and false if they are not equal.
        • +
      • #if #then #else #fi - polymorphic conditional function. If the first +argument evaluates to true, the second argument is returned. Otherwise, +the third argument is returned. Note that this does not short-circuit on +symbolic backends.
        • +
      +
      k
      module K-EQUAL-SYNTAX
+  imports private BOOL
+  imports private BASIC-K
+
+  syntax Bool ::= left:
+                  K "==K" K           [function, total, comm, smt-hook(=), hook(KEQUAL.eq), klabel(_==K_), symbol, latex({#1}\mathrel{=_K}{#2}), group(equalEqualK)]
+                | K "=/=K" K          [function, total, comm, smt-hook(distinct), hook(KEQUAL.ne), klabel(_=/=K_), symbol, latex({#1}\mathrel{\neq_K}{#2}), group(notEqualEqualK)]
+
+  syntax priorities equalEqualK notEqualEqualK > boolOperation mlOp
+
+  syntax {Sort} Sort ::= "#if" Bool "#then" Sort "#else" Sort "#fi"   [function, total, smt-hook(ite), hook(KEQUAL.ite)]
+
+endmodule
+
+module K-EQUAL-KORE [kore, symbolic]
+  import private BOOL
+  import K-EQUAL-SYNTAX
+
+  rule K1:Bool ==K K2:Bool => K1 ==Bool K2 [simplification]
+  rule {K1 ==K K2 #Equals true} => {K1 #Equals K2} [simplification]
+  rule {true #Equals K1 ==K K2} => {K1 #Equals K2} [simplification]
+  rule {K1 ==K K2 #Equals false} => #Not({K1 #Equals K2}) [simplification]
+  rule {false #Equals K1 ==K K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=K K2 #Equals true} => #Not({K1 #Equals K2}) [simplification]
+  rule {true #Equals K1 =/=K K2} => #Not({K1 #Equals K2}) [simplification]
+  rule {K1 =/=K K2 #Equals false} => {K1 #Equals K2} [simplification]
+  rule {false #Equals K1 =/=K K2} => {K1 #Equals K2} [simplification]
+
+endmodule
+
+module K-EQUAL
+  import private BOOL
+  import K-EQUAL-SYNTAX
+  import K-EQUAL-KORE
+
+  rule K1:K =/=K K2:K => notBool (K1 ==K K2)
+
+  rule #if C:Bool #then B1::K #else _ #fi => B1 requires C
+  rule #if C:Bool #then _ #else B2::K #fi => B2 requires notBool C
+
+endmodule
+
      +

      Meta operations

      +

      Provided below are a few miscellaneous, mostly deprecated functions in K. +It is not recommended to use any of them directly as they are largely +unsupported in modern K. There are a few exceptions:

      +
        +
      • #getenv - Returns the value of an environment variable
        • +
      • #parseKORE - Takes a String containing a K intermediate representation of +a term such as is returned by kast -o kore and converts it to a term. +This is NOT type-safe. The responsibility is on the user to ensure that the +string they provide is a valid representation of a term of the sort exactly +equal to the sort where the function appears.
        • +
      • #kompiledDirectory - Returns the path to the current compiled K definition +directory.
        • +
      • #unparseKORE = Takes a K term and converts it to a string.
        • +
      +
      k
      module K-REFLECTION
+  imports BASIC-K
+  imports STRING
+
+  syntax K ::= "#configuration" [function, impure, hook(KREFLECTION.configuration)]
+  syntax String ::= #sort(K) [function, hook(KREFLECTION.sort)]
+  syntax KItem ::= #fresh(String)   [function, hook(KREFLECTION.fresh), impure]
+  syntax KItem ::= getKLabel(K)  [function, hook(KREFLECTION.getKLabel)]
+
+  syntax K ::= #getenv(String) [function, impure, hook(KREFLECTION.getenv)]
+
+  syntax String ::= #kompiledDirectory() [function, hook(KREFLECTION.kompiledDir)]
+
+  // meaningful only for the purposes of compilation to a binary, otherwise
+  // undefined
+  syntax List ::= #argv() [function, hook(KREFLECTION.argv)]
+
+  // Takes as input a string and returns a K term
+  syntax {Sort} Sort ::= #parseKORE(String) [function, hook(KREFLECTION.parseKORE)]
+  syntax {Sort} String ::= #unparseKORE(Sort) [function, hook(KREFLECTION.printKORE)]
+  syntax IOError ::= "#noParse" "(" String ")" [klabel(#noParse), symbol]
+
+endmodule
+
      +

      I/O in K

      +

      Concrete execution in K supports I/O operations. This functionality is not +supported during symbolic execution, because symbolic execution must exist +completely free of side-effects, and I/O is an irreducible type of side effect. +However, it is useful in many cases when defining concrete execution to be able +to make reference to I/O operations.

      +

      The design of these I/O operations is based on the POSIX standard, for the most +part. For example, the #read K function maps to the read POSIX function. We +do not at this time have a higher-level API for I/O, but this may be +implemented at some point in the future.

      +

      I/O operations generally return either their result, or an IOError term +corresponding to the errno returned by the underlying system call.

      +
      k
      module K-IO
+  imports private LIST
+  imports private STRING
+  imports private INT
+
      +

      I/O errors

      +

      Aside from EOF, which is returned by #getc if the file is at end-of-file, all +of the below I/O errors correspond to possible values for errno after calling +a library function. If the errno returned is not one of the below errnos +known to K, #unknownIOError is returned along with the integer errno value.

      +
      k
        syntax IOError ::= "#EOF" [klabel(#EOF), symbol] | #unknownIOError(errno: Int) [symbol]
+                   | "#E2BIG" [klabel(#E2BIG), symbol]
+                   | "#EACCES" [klabel(#EACCES), symbol]
+                   | "#EAGAIN" [klabel(#EAGAIN), symbol]
+                   | "#EBADF" [klabel(#EBADF), symbol]
+                   | "#EBUSY" [klabel(#EBUSY), symbol]
+                   | "#ECHILD" [klabel(#ECHILD), symbol]
+                   | "#EDEADLK" [klabel(#EDEADLK), symbol]
+                   | "#EDOM" [klabel(#EDOM), symbol]
+                   | "#EEXIST" [klabel(#EEXIST), symbol]
+                   | "#EFAULT" [klabel(#EFAULT), symbol]
+                   | "#EFBIG" [klabel(#EFBIG), symbol]
+                   | "#EINTR" [klabel(#EINTR), symbol]
+                   | "#EINVAL" [klabel(#EINVAL), symbol]
+                   | "#EIO" [klabel(#EIO), symbol]
+                   | "#EISDIR" [klabel(#EISDIR), symbol]
+                   | "#EMFILE" [klabel(#EMFILE), symbol]
+                   | "#EMLINK" [klabel(#EMLINK), symbol]
+                   | "#ENAMETOOLONG" [klabel(#ENAMETOOLONG), symbol]
+                   | "#ENFILE" [klabel(#ENFILE), symbol]
+                   | "#ENODEV" [klabel(#ENODEV), symbol]
+                   | "#ENOENT" [klabel(#ENOENT), symbol]
+                   | "#ENOEXEC" [klabel(#ENOEXEC), symbol]
+                   | "#ENOLCK" [klabel(#ENOLCK), symbol]
+                   | "#ENOMEM" [klabel(#ENOMEM), symbol]
+                   | "#ENOSPC" [klabel(#ENOSPC), symbol]
+                   | "#ENOSYS" [klabel(#ENOSYS), symbol]
+                   | "#ENOTDIR" [klabel(#ENOTDIR), symbol]
+                   | "#ENOTEMPTY" [klabel(#ENOTEMPTY), symbol]
+                   | "#ENOTTY" [klabel(#ENOTTY), symbol]
+                   | "#ENXIO" [klabel(#ENXIO), symbol]
+                   | "#EPERM" [klabel(#EPERM), symbol]
+                   | "#EPIPE" [klabel(#EPIPE), symbol]
+                   | "#ERANGE" [klabel(#ERANGE), symbol]
+                   | "#EROFS" [klabel(#EROFS), symbol]
+                   | "#ESPIPE" [klabel(#ESPIPE), symbol]
+                   | "#ESRCH" [klabel(#ESRCH), symbol]
+                   | "#EXDEV" [klabel(#EXDEV), symbol]
+                   | "#EWOULDBLOCK" [klabel(#EWOULDBLOCK), symbol]
+                   | "#EINPROGRESS" [klabel(#EINPROGRESS), symbol]
+                   | "#EALREADY" [klabel(#EALREADY), symbol]
+                   | "#ENOTSOCK" [klabel(#ENOTSOCK), symbol]
+                   | "#EDESTADDRREQ" [klabel(#EDESTADDRREQ), symbol]
+                   | "#EMSGSIZE" [klabel(#EMSGSIZE), symbol]
+                   | "#EPROTOTYPE" [klabel(#EPROTOTYPE), symbol]
+                   | "#ENOPROTOOPT" [klabel(#ENOPROTOOPT), symbol]
+                   | "#EPROTONOSUPPORT" [klabel(#EPROTONOSUPPORT), symbol]
+                   | "#ESOCKTNOSUPPORT" [klabel(#ESOCKTNOSUPPORT), symbol]
+                   | "#EOPNOTSUPP" [klabel(#EOPNOTSUPP), symbol]
+                   | "#EPFNOSUPPORT" [klabel(#EPFNOSUPPORT), symbol]
+                   | "#EAFNOSUPPORT" [klabel(#EAFNOSUPPORT), symbol]
+                   | "#EADDRINUSE" [klabel(#EADDRINUSE), symbol]
+                   | "#EADDRNOTAVAIL" [klabel(#EADDRNOTAVAIL), symbol]
+                   | "#ENETDOWN" [klabel(#ENETDOWN), symbol]
+                   | "#ENETUNREACH" [klabel(#ENETUNREACH), symbol]
+                   | "#ENETRESET" [klabel(#ENETRESET), symbol]
+                   | "#ECONNABORTED" [klabel(#ECONNABORTED), symbol]
+                   | "#ECONNRESET" [klabel(#ECONNRESET), symbol]
+                   | "#ENOBUFS" [klabel(#ENOBUFS), symbol]
+                   | "#EISCONN" [klabel(#EISCONN), symbol]
+                   | "#ENOTCONN" [klabel(#ENOTCONN), symbol]
+                   | "#ESHUTDOWN" [klabel(#ESHUTDOWN), symbol]
+                   | "#ETOOMANYREFS" [klabel(#ETOOMANYREFS), symbol]
+                   | "#ETIMEDOUT" [klabel(#ETIMEDOUT), symbol]
+                   | "#ECONNREFUSED" [klabel(#ECONNREFUSED), symbol]
+                   | "#EHOSTDOWN" [klabel(#EHOSTDOWN), symbol]
+                   | "#EHOSTUNREACH" [klabel(#EHOSTUNREACH), symbol]
+                   | "#ELOOP" [klabel(#ELOOP), symbol]
+                   | "#EOVERFLOW" [klabel(#EOVERFLOW), symbol]
+
      +

      I/O result sorts

      +

      Here we see sorts defined to contain either an Int or an IOError, or +either a String or an IOError. These sorts are used to implement the +return sort of functions that may succeed, in which case they return a value, +or may fail, in which case their return value indicates an error and the +error indicated is returned via errno.

      +
      k
        syntax IOInt ::= Int | IOError
+  syntax IOString ::= String | IOError
+
      +

      Opening a file

      +

      You can open a file in K using #open. An optional mode indicates the file +open mode, which can have any value allowed by the fopen function in C. +The returned value is the file descriptor that was opened, or an error.

      +
      k
        syntax IOInt ::= "#open" "(" path: String ")" [function]
+               | "#open" "(" path: String "," mode: String ")" [function, hook(IO.open), impure]
+
+  rule #open(S:String) => #open(S:String, "r+")
+
      +

      Get/set position in file

      +

      You can get the current offset in a file using #tell. You can also seek +to a particular offset using #seek or #seekEnd. #seek is implemented via +a call to lseek with the SEEK_SET whence. #seekEnd is implemented via a +call to lseek with the SEEK_END whence. You can emulate the SEEK_CUR +whence by means of #seek(FD, #tell(FD) +Int Offset).

      +
      k
        syntax IOInt ::= "#tell" "(" fd: Int ")" [function, hook(IO.tell), impure]
+  syntax K ::= "#seek" "(" fd: Int "," index: Int ")" [function, hook(IO.seek), impure]
+             | "#seekEnd" "(" fd: Int "," fromEnd: Int ")" [function, hook(IO.seekEnd), impure]
+
      +

      Read from file

      +

      You can read a single character from a file using #getc. #EOF is returned +if you are at end-of-fie.

      +

      You can also read up to length characters in a file using #read. The +resulting read characters are returned, which may be fewer characters than +requested. A string of zero length being returned indicates end-of-file.

      +
      k
        syntax IOInt ::= "#getc" "(" fd: Int ")"             [function, hook(IO.getc), impure]
+  syntax IOString ::= "#read" "(" fd: Int "," length: Int ")"    [function, hook(IO.read), impure]
+
      +

      Write to file

      +

      You can write a single character to a file using #putc. You can also write +a string to a file using #write. The returned value on success is .K.

      +
      k
        syntax K ::= "#putc" "(" fd: Int "," value: Int ")"      [function, hook(IO.putc), impure]
+             | "#write" "(" fd: Int "," value: String ")" [function, hook(IO.write), impure]
+
      +

      Closing a file

      +

      You can close a file using #close. The returned value on success is .K.

      +
      k
        syntax K ::= "#close" "(" fd: Int ")" [function, hook(IO.close), impure]
+
      +

      Locking/unlocking a file

      +

      You can lock or unlock parts of a file using the #lock and #unlock +functions. The lock starts at the beginning of the file and continues for +endIndex bytes. Note that Unix systems do not actually prevent locked files +from being read and modified; you will have to lock both sides of a concurrent +access to guarantee exclusivity.

      +
      k
        syntax K ::= "#lock" "(" fd: Int "," endIndex: Int ")" [function, hook(IO.lock), impure]
+             | "#unlock" "(" fd: Int "," endIndex: Int ")" [function, hook(IO.unlock), impure]
+
      +

      Networking

      +

      You can accept a connection on a socket using #accept, or shut down the +write end of a socket with #shutdownWrite. Note that facility is not provided +for opening, binding, and listening on sockets. These functions are implemented +in order to support creating stateful request/response servers where the +request loop is implemented using rewriting in K, but the connection +initialization is written in native code and linked into the LLVM backend.

      +
      k
        syntax IOInt ::= "#accept" "(" fd: Int ")" [function, hook(IO.accept), impure]
+  syntax K ::= "#shutdownWrite" "(" fd: Int ")" [function, hook(IO.shutdownWrite), impure]
+
      +

      Time

      +

      You can get the current time in seconds since midnight UTC on January 1, 1970 +using #time.

      +
      k
        syntax Int ::= "#time" "(" ")" [function, hook(IO.time), impure]
+
      +

      Builtin file descriptors

      +

      Provided here are functions that return the file descriptor for standard input, +standard output, and standard error.

      +
      k
        syntax Int ::= "#stdin"   [function, total]
+                | "#stdout" [function, total]
+                | "#stderr" [function, total]
+
+  rule #stdin => 0
+  rule #stdout => 1
+  rule #stderr => 2
+
      +

      Shell access

      +

      You can execute a command using the shell using the #system operator. Care +must be taken to sanitize inputs to this function or security issues may +result. Note that K has no facility for reasoning about logic that happens +outside its process, so any functionality that you wish to be able to formally +reason about in K should not be implemented via the #system operator.

      +
      k
        syntax KItem ::= #system ( String ) [function, hook(IO.system), impure]
+                 | "#systemResult" "(" Int /* exit code */ "," String /* stdout */ "," String /* stderr */ ")" [klabel(#systemResult), symbol]
+
      +

      Temporary files

      +

      You can get a temporary file and open it atomically using the #mkstemp +operator. The resulting file will be closed and deleted when K rewriting ends. +For more info on the argument to #mkstemp, see man mkstemp.

      +
      k
        syntax IOFile ::= #mkstemp(template: String) [function, hook(IO.mkstemp), impure]
+  syntax IOFile ::= IOError
+                  | "#tempFile" "(" path: String "," fd: Int ")" [klabel(#tempFile), symbol]
+
      +

      Deleting a file

      +

      You can delete a file using its absolute or relative path using the #remove +operator. It returns .K on success or an IOError on failure.

      +
      k
        syntax K ::= #remove(path: String) [function, total, hook(IO.remove), impure]
+
      +

      Logging

      +

      You can log information to disk using the #logToFile operator. Semantically, +this operator returns .K. However, it has a side effect that is not reasoned +about which is that value will be written to a uniquely-identified file +containing name in its name. The file is only flushed to disk when rewriting +finishes.

      +
      k
        syntax K ::= #logToFile(name: String, value: String) [function, total, hook(IO.log), impure, returnsUnit, symbol]
+
      +

      Strings can also be logged via the logging mechanisms available to the backend. +On the LLVM backend, this just means logging the text to standard error. On the +Haskell backend, a log message of type InfoUserLog is created with the +specified text.

      +
      k
        syntax K ::= #log(value: String) [function, total, hook(IO.logString), impure, returnsUnit, symbol]
+
      +

      Terms can also be logged to standard error in surface syntax, rather than as +KORE using #trace. This operator has similar semantics to #logToFile (i.e. +it returns .K, but prints as an impure side effect). Note that calling +#trace is equivalent to invoking the kprint tool for the first term that is +logged, which requires re-parsing the underlying K definition. Subsequent calls +do not incur this overhead again; the definition is cached.

      +
      k
        syntax K ::= #trace(value: KItem) [function, total, hook(IO.traceTerm), impure, returnsUnit, symbol]
+             | #traceK(value: K)    [function, total, hook(IO.traceTerm), impure, returnsUnit, symbol]
+
      +

      Implementation of high-level I/O streams in K

      +

      Below is an implementation of the stream="stdin" and stream="stdout" +cell attributes in K. You should not refer to these symbols or modules directly +in your definition. It is provided only so that the K compiler can make use of +it. For more information on how to use this feature, refer to IMP++ in the K +tutorial.

      +
      k
        syntax Stream ::= #buffer(K)
+                  | #istream(Int)
+                  | #parseInput(String, String)
+                  | #ostream(Int)
+
+endmodule
+
+// NOTE: DO NOT DIRECTLY IMPORT *-STREAM MODULES
+// These stream modules will be automatically instantiated and implicitly imported
+// into the main module when `stream` attributes appear in configuration cells.
+// Only `Stream` productions and `[stream]` rules will be imported.
+// The cell name will be replaced with the one of the main configuration.
+
+module STDIN-STREAM
+  imports K-IO
+  imports K-REFLECTION
+  imports LIST
+  imports INT
+  imports BOOL
+
+  configuration <stdin> ListItem(#buffer($STDIN:String)) ListItem($IO:String) ListItem(#istream(#stdin)) </stdin>
+
+  // read one character at a time until we read whitespace
+  rule [stdinGetc]:
+       <stdin>
+       ListItem(#parseInput(_:String, Delimiters:String))
+       ListItem(#buffer(S:String => S +String chrChar({#getc(N)}:>Int)))
+       ListItem("on")
+       ListItem(#istream(N:Int))
+       </stdin>
+    requires findChar(S, Delimiters, 0) ==Int -1 // [stdin]
+       [stream, priority(200)]
+
+  // when we reach whitespace, if it parses create a ListItem
+  rule [stdinParseString]:
+       <stdin>
+       (ListItem(#parseInput("String", Delimiters:String)) => ListItem(S))
+       ListItem(#buffer(S:String => ""))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1 // [stdin]
+       [stream]
+
+  // a hack: handle the case when we read integers without the help of the IO server
+  rule [stdinParseInt]:
+       <stdin>
+       (ListItem(#parseInput("Int", Delimiters:String))
+       => ListItem(String2Int(substrString(S, 0, findChar(S, Delimiters, 0)))))
+       ListItem(#buffer(S:String => substrString(S,findChar(S, Delimiters, 0) +Int 1, lengthString(S))))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1
+       andBool lengthString(S) >Int 1 // [stdin]
+       [stream]
+
+  rule [stdinTrim]:
+       <stdin>
+       ListItem(#parseInput(Sort:String, Delimiters:String))
+       ListItem(#buffer(S:String => substrString(S, 1, lengthString(S))))
+       _:List
+       </stdin>
+    requires findChar(S, Delimiters, 0) =/=Int -1
+       andBool Sort =/=String "String"
+       andBool lengthString(S) <=Int 1 // [stdin]
+       [stream]
+
+  // NOTE: This unblocking rule will be instantiated and inserted carefully
+  // when necessary according to user-defined rules, since otherwise it will
+  // lead to a diverging (i.e., non-terminating) transition system definition.
+  // Currently, it supports only a simple pattern matching on the top of the
+  // input stream cell, e.g.,
+  //   rule <k> read() => V ... </k> <in> ListItem(V:Int) => .List ...  </in>
+  // Non-supported rules that refer to the input stream cell in a sophisticated
+  // way will get stuck in concrete execution mode with real IO enabled (i.e.,
+  // under `--io on` option), while they will still work in symbolic execution
+  // mode or concrete execution mode with real IO disabled (i.e., under `--io
+  // off`, `--search`, or `--debug` options).
+  //
+  // TODO: More patterns need to be supported as well. In that case, we need to
+  // have a way to specify such patterns.
+  rule [stdinUnblock]:
+       <stdin>
+         (.List => ListItem(#parseInput(?Sort:String, ?Delimiters:String)))
+         ListItem(#buffer(_:String))
+         ...
+       </stdin>
+
+  /*
+  syntax Stream ::= "#noIO"
+
+  rule ListItem(#buffer(_))
+       (ListItem(#noIO) ListItem(#istream(_:Int)) => .List) [stdin]
+  */
+
+endmodule
+
+module STDOUT-STREAM
+  imports K-IO
+  imports LIST
+  imports STRING
+
+  configuration <stdout> ListItem(#ostream(#stdout)) ListItem($IO:String) ListItem(#buffer("")) </stdout>
+//configuration <stderr> ListItem(#ostream(#stderr)) ListItem($IO:String) ListItem(#buffer("")) </stderr>
+
+  rule [stdoutBufferFloat]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String Float2String(F)))
+       (ListItem(F:Float) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+  rule [stdoutBufferInt]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String Int2String(I)))
+       (ListItem(I:Int) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+  rule [stdoutBufferString]:
+       <stdout>
+       ListItem(#ostream(_))
+       ListItem(_)
+       ListItem(#buffer(Buffer:String => Buffer +String S))
+       (ListItem(S:String) => .List)
+       _:List
+       </stdout>
+       // [stdout, stderr]
+       [stream, priority(25)]
+
+  // Send first char from the buffer to the server
+  rule [stdoutWrite]:
+       <stdout>
+       ListItem(#ostream(N:Int => {#write(N, S) ~> N:Int}:>Int))
+       ListItem("on")
+       ListItem(#buffer(S:String => ""))
+       _:List
+       </stdout>
+    requires S =/=String "" // [stdout, stderr]
+       [stream, priority(30)]
+
+  /*
+  syntax Stream ::= "#noIO"
+
+  rule ListItem(#buffer(Buffer:String => Buffer +String Float2String(F)))
+       (ListItem(F:Float) => .List)
+       _:List [stdout, stderr]
+  rule ListItem(#buffer(Buffer:String => Buffer +String Int2String(I)))
+       (ListItem(I:Int) => .List)
+       _:List [stdout, stderr]
+  rule ListItem(#buffer(Buffer:String => Buffer +String S))
+       (ListItem(S:String) => .List)
+       _:List [stdout, stderr]
+
+  rule (ListItem(#ostream(_:Int)) ListItem(#noIO) => .List)
+       ListItem(#buffer(_))
+       _:List [stdout, stderr]
+  */
+
+endmodule
+
      +

      Machine Integers

      +

      Provided here is an implementation of arbitrarily large fixed-precision binary +integers in K. This type is hooked to an implementation of integers provided +by the backend, and in particular makes use of native machine integers for +certain sizes of integer. For arbitrary-precision integers, see the INT +module above.

      +

      The syntax of machine integers in K is the same as arbitrary-precision integers +(i.e., an optional sign followed by a sequence of digits) except that machine +integers always end in a suffix pN where N is an integer indicating the +width in bits of the integer. The MInt sort is parametric, and this is +reflected in the literals. For example, the sort of 0p8 is MInt{8}.

      +
      k
      module MINT-SYNTAX
+  /*@\section{Description} The MInt implements machine integers of arbitrary
+   * bit width represented in 2's complement. */
+  syntax {Width} MInt{Width} [hook(MINT.MInt)]
+
+  /*@ Machine integer of bit width and value. */
+  syntax {Width} MInt{Width} ::= r"[\\+-]?[0-9]+[pP][0-9]+" [token, prec(2), hook(MINT.literal)]
+endmodule
+
+module MINT
+  imports MINT-SYNTAX
+  imports private INT
+  imports private BOOL
+
      +

      Bitwidth of MInt

      +

      You can get the number of bits of width in an MInt using bitwidthMInt.

      +
      k
        syntax {Width} Int ::= bitwidthMInt(MInt{Width})   [function, total, hook(MINT.bitwidth)]
+
      +

      Int and MInt conversions

      +

      You can convert from an MInt to an Int using the MInt2Signed and +MInt2Unsigned functions. an MInt does not have a sign; its sign is instead +reflected in how operators interpret its value either as a signed integer or as +an unsigned integer. Thus, you can interpret a MInt as a signed integer witth +MInt2Signed, or as an unsigned integer respectively using MInt2Unsigned.

      +

      You can also convert from an Int to an MInt using Int2MInt. Care must +be given to ensure that the sort context where the Int2MInt operator appears +has the correct bitwidth, as this will influence the width of the resulting +MInt.

      +
      k
        syntax {Width} Int ::= MInt2Signed(MInt{Width})     [function, total, hook(MINT.svalue)]
+                       | MInt2Unsigned(MInt{Width})     [function, total, hook(MINT.uvalue), smt-hook(bv2int)]
+
+  syntax {Width} MInt{Width} ::= Int2MInt(Int) [function, total, hook(MINT.integer), smt-hook(int2bv)]
+
      +

      MInt min and max values

      +

      You can get the minimum and maximum values of a signed or unsigned MInt +with az specified bit width using sminMInt, smaxMInt, uminMInt, and +umaxMInt.

      +
      k
        syntax Int ::= sminMInt(Int)    [function]
+               | smaxMInt(Int)    [function]
+               | uminMInt(Int)    [function]
+               | umaxMInt(Int)    [function]
+  rule sminMInt(N:Int) => 0 -Int (1 <<Int (N -Int 1))
+  rule smaxMInt(N:Int) => (1 <<Int (N -Int 1)) -Int 1
+  rule uminMInt(_:Int) => 0
+  rule umaxMInt(N:Int) => (1 <<Int N) -Int 1
+
      +

      MInt bounds checking

      +

      You can check whether a specified Int will be represented in an MInt +with a specified width without any loss of precision when interpreted as +a signed or unsigned integer using soverflowMInt and uoverflowMInt.

      +
      k
        syntax Bool ::= soverflowMInt(width: Int, Int)   [function]
+                | uoverflowMInt(width: Int, Int)   [function]
+  rule
+    soverflowMInt(N:Int, I:Int)
+  =>
+    I <Int sminMInt(N) orBool I >Int smaxMInt(N)
+  rule
+    uoverflowMInt(N:Int, I:Int)
+  =>
+    I <Int uminMInt(N) orBool I >Int umaxMInt(N)
+
      +

      MInt arithmetic

      +

      You can:

      +
        +
      • Compute the bitwise complement ~MInt of an MInt.
        • +
      • Compute the unary negation --MInt of an MInt.
        • +
      • Compute the product *MInt of two MInts.
        • +
      • Compute the quotient /sMInt of two MInts interpreted as signed integers.
        • +
      • Compute the modulus %sMInt of two MInts interpreted as signed integers.
        • +
      • Compute the quotient /uMInt of two MInts interpreted as unsigned +integers.
        • +
      • Compute the modulus %uMInt of two MInts interpreted as unsigned integers.
        • +
      • Compute the sum +MInt of two MInts.
        • +
      • Compute the difference -MInt of two MInts.
        • +
      • Compute the left shift <<MInt of two MInts. The second MInt is always +interpreted as positive.
        • +
      • Compute the arithmetic right shift >>aMInt of two MInts. The second +MInt is always interpreted as positve.
        • +
      • Compute the logical right shift >>lMInt of two MInts. The second MInt +is always interpreted as positive.
        • +
      • Compute the bitwise and &MInt of two MInts.
        • +
      • Compute the bitwise xor xorMInt of two MInts.
        • +
      • Compute the bitwise inclusive or |MInt of two MInts.
        • +
      +
      k
        syntax {Width} MInt{Width} ::= "~MInt" MInt{Width} [function, total, hook(MINT.not), smt-hook(bvnot)]
+                               | "--MInt" MInt{Width} [function, total, hook(MINT.neg), smt-hook(bvuminus)]
+                               > left:
+                                 MInt{Width} "*MInt" MInt{Width} [function, total, hook(MINT.mul), smt-hook(bvmul)]
+                               | MInt{Width} "/sMInt" MInt{Width} [function, hook(MINT.sdiv), smt-hook(bvsdiv)]
+                               | MInt{Width} "%sMInt" MInt{Width} [function, hook(MINT.srem), smt-hook(bvsrem)]
+                               | MInt{Width} "/uMInt" MInt{Width} [function, hook(MINT.udiv), smt-hook(bvudiv)]
+                               | MInt{Width} "%uMInt" MInt{Width} [function, hook(MINT.urem), smt-hook(bvurem)]
+                               > left:
+                                 MInt{Width} "+MInt" MInt{Width} [function, total, hook(MINT.add), smt-hook(bvadd)]
+                               | MInt{Width} "-MInt" MInt{Width} [function, total, hook(MINT.sub), smt-hook(bvsub)]
+                               > left:
+                                 MInt{Width} "<<MInt" MInt{Width} [function, hook(MINT.shl), smt-hook(bvshl)]
+                               | MInt{Width} ">>aMInt" MInt{Width} [function, hook(MINT.ashr), smt-hook(bvashr)]
+                               | MInt{Width} ">>lMInt" MInt{Width} [function, hook(MINT.lshr), smt-hook(bvlshr)]
+                               > left:
+                                 MInt{Width} "&MInt" MInt{Width} [function, total, hook(MINT.and), smt-hook(bvand)]
+                               > left:
+                                 MInt{Width} "xorMInt" MInt{Width} [function, total, hook(MINT.xor), smt-hook(bvxor)]
+                               > left:
+                                 MInt{Width} "|MInt" MInt{Width} [function, total, hook(MINT.or), smt-hook(bvor)]
+
      +

      MInt comparison

      +

      You can compute whether one MInt is less than, less than or equal to, greater +than, or greater than or equal to another MInt when interpreted as signed +or unsigned integers. You can also compute whether one MInt is equal to or +unequal to another MInt.

      +
      k
        syntax {Width} Bool ::= MInt{Width} "<sMInt" MInt{Width} [function, total, hook(MINT.slt), smt-hook(bvslt)]
+                        | MInt{Width} "<uMInt" MInt{Width} [function, total, hook(MINT.ult), smt-hook(bvult)]
+                        | MInt{Width} "<=sMInt" MInt{Width} [function, total, hook(MINT.sle), smt-hook(bvsle)]
+                        | MInt{Width} "<=uMInt" MInt{Width} [function, total, hook(MINT.ule), smt-hook(bvule)]
+                        | MInt{Width} ">sMInt" MInt{Width} [function, total, hook(MINT.sgt), smt-hook(bvsgt)]
+                        | MInt{Width} ">uMInt" MInt{Width} [function, total, hook(MINT.ugt), smt-hook(bvugt)]
+                        | MInt{Width} ">=sMInt" MInt{Width} [function, total, hook(MINT.sge), smt-hook(bvsge)]
+                        | MInt{Width} ">=uMInt" MInt{Width} [function, total, hook(MINT.uge), smt-hook(bvuge)]
+                        | MInt{Width} "==MInt" MInt{Width} [function, total, hook(MINT.eq), smt-hook(=)]
+                        | MInt{Width} "=/=MInt" MInt{Width} [function, total, hook(MINT.ne), smt-hook(distinct)]
+
      +

      MInt min/max

      +

      You can compute the signed minimum sMinMInt, the signed maximum sMaxMInt, +the unsigned minimum uMinMInt, and the unsigned maximum uMaxMInt of two +MInts.

      +
      k
        syntax {Width} MInt{Width} ::= sMaxMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.smax), smt-hook((ite (bvslt #1 #2) #2 #1))]
+                               | sMinMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.smin), smt-hook((ite (bvslt #1 #2) #1 #2))]
+                               | uMaxMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.umax), smt-hook((ite (bvult #1 #2) #2 #1))]
+                               | uMinMInt(MInt{Width}, MInt{Width}) [function, total, hook(MINT.umin), smt-hook((ite (bvult #1 #2) #1 #2))]
+
      +

      MInt to MInt conversion

      +

      You can convert an MInt of one width to another width with roundMInt. +The resulting MInt will be truncated starting from the most significant bit +if the resulting width is smaller than the input. The resulting MInt will be +zero-extended with the same low-order bits if the resulting width is larger +than the input.

      +
      k
        syntax {Width1, Width2} MInt{Width1} ::= roundMInt(MInt{Width2}) [function, total, hook(MINT.round)]
+
      +
      k
      endmodule
+
      +

      Strategies

      +

      K supports a built-in strategy language that allows you to control how rules +apply. In order to enable it, simply import the STRATEGY module in your +definition. This includes the following basic strategy constructs:

      +
        +
      • ^ Category:#RuleTag - This is a strategy that indicates that you should +apply a rule with the specified category exactly once. By default, all rules +get the regular tag. This can be changed on individual rules with the tag +attribute.
        • +
      • ~ Category:#RuleTag - This is the state the strategy cell will be in +after a rule has applied. In other words, all rules that do not mention the +strategy cell are automatically instrumented so that they rewrite the current +top of the <s> cell from ^ to ~
        • +
      • <s> - This is the strategy cell and contains the current strategy.
        • +
      • #STUCK() - By default, a rule is automatically inserted into the definition +which adds #STUCK() to the top of the <s> cell if no other rules apply +and if it is not already at the top of the <s> cell.
        • +
      +
      k
      module STRATEGY
+    imports ML-SYNTAX
+    imports KVARIABLE-SYNTAX
+    imports private K-EQUAL
+
+    syntax #RuleTag ::= #KVariable
+
+    syntax Strategy ::= #STUCK()    [symbol]
+                      | "^" #RuleTag [symbol, klabel(#applyRule)]
+                      | "~" #RuleTag [symbol, klabel(#appliedRule)]
+
+    configuration <s> $STRATEGY:K </s>
+endmodule
+
+module RULE-TAG-SYNTAX
+    imports BUILTIN-ID-TOKENS
+
+    syntax #RuleTag ::= #LowerId [token]
+endmodule
+
      +

      This is not a complete strategy language. However, it provides several basics. +The user can extend this strategy language into a complete strategy language +in one of three ways. First, they can import the DEFAULT-STRATEGY module, +which provides a very basic strategy that is essentially equivalent to +execution without a strategy. This can be useful if you wish to wholly +manipulate the strategy cell yourself within other rules.

      +
      k
      module DEFAULT-STRATEGY-CONCRETE [concrete]
+    imports STRATEGY
+    imports RULE-TAG-SYNTAX
+    rule ~ regular => ^ regular [anywhere]
+endmodule
+
+module DEFAULT-STRATEGY-SYMBOLIC [symbolic]
+    imports STRATEGY
+    imports RULE-TAG-SYNTAX
+    rule <s> ~ regular => ^ regular ... </s>
+endmodule
+
+module DEFAULT-STRATEGY
+    imports STRATEGY
+    imports DEFAULT-STRATEGY-CONCRETE
+    imports DEFAULT-STRATEGY-SYMBOLIC
+
+    rule initSCell(_) => <s> ^ regular </s> [priority(25)]
+endmodule
+
      +

      The second way you can extend the strategy language is with the +STRATEGY-ABSTRACT module. This provides a slightly more advanced set of +strategies which you can use to compose the basic strategies in the STRATEGY +module into more complex strategies. Note however that the functionality +provided is still relatively basic.

      +
      k
      module STRATEGY-ABSTRACT
+    imports STRATEGY
+
+    syntax #RuleTag ::= "(" #RuleTag ")" [bracket]
+ // ----------------------------------------------
+
+    syntax Strategy ::= ".Strategy"
+                      | "(" Strategy ")" [bracket]
+ // ----------------------------------------------
+    rule <s> .Strategy => . ... </s>
+
+    syntax KItem ::= #catchSTUCK ( Strategy )
+ // -----------------------------------------
+    rule <s> #catchSTUCK(_) => . ... </s>
+    rule <s> #STUCK() ~> (_S:Strategy => .) ... </s>
+    rule <s> #STUCK() ~> #catchSTUCK(S) => S ... </s>
+
+    syntax Strategy ::= Strategy ";" Strategy [left]
+ // ------------------------------------------------
+    rule <s> S:Strategy ; S':Strategy => S ~> S' ... </s>
+
+    syntax #RuleTag ::= #RuleTag "|" #RuleTag [left, klabel(#alternateRule)]
+ // ------------------------------------------------------------------------
+    rule <s> ^ RT:#RuleTag | RT':#RuleTag => ^ RT ~> #catchSTUCK(^ RT') ... </s>
+
+    syntax #RuleTag ::= #RuleTag "*" [klabel(#repeatRule)]
+ // ------------------------------------------------------
+    rule <s> ^ RT:#RuleTag * => ^ RT ~> ^ RT * ~> #catchSTUCK(.Strategy) ... </s>
+endmodule
+
      +

      The third mechanism for extending the strategy language is to define your own +strategy language. No special facility is required in K in order to compose +strategies together, so you are free to write whatever rules that manipulate +the strategy cell you want, and thus whatever more complex strategies you can +compose from the base builtin strategies.

      +
      +
      + + +
      + +
      +
      + + Basic Builtin Types in K + +
      +
      + + Default Modules +
      + + Arrays + + +
      +
      + + Maps + + +
      +
      + + Range Maps + + +
      +
      + + Sets + + +
      +
      + + Lists + + +
      +
      + + Collection Conversions +
      + + Booleans + + +
      +
      + + Integers + + +
      +
      + + IEEE 754 Floating-point Numbers + + +
      +
      + + Strings + + +
      +
      + + String Buffers +
      + + Byte Arrays + + +
      +
      + + Program identifiers +
      + + Equality and conditionals +
      + + Meta operations +
      + + I/O in K + + +
      +
      + + Machine Integers + + +
      +
      + + Strategies +
      +
      +
      + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/ffi/index.html b/k-distribution/include/kframework/builtin/ffi/index.html new file mode 100644 index 00000000000..2c803162447 --- /dev/null +++ b/k-distribution/include/kframework/builtin/ffi/index.html @@ -0,0 +1,619 @@ + + + + + + + + + + + + + + +K Foreign Function Interface | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Foreign Function Interface

      +

      The K Foreign Function Interface (FFI) module provides a way to call native +functions directly from a K semantics using the C ABI. It also provides +utilities for allocating and deallocating byte buffers with static addresses +that are suitable for being passed to native code.

      +

      It is built off of the underlying libffi library +(https://sourceware.org/libffi/) and is subject to some of the same +limitations as that library. Bear in mind, because this library exposes +a number of unsafe C APIs directly, misuse of the library is likely to lead +to memory corruption in your interpreter and can cause segmentation faults or +corrupted term representations that lead to undefined behavior at runtime.

      +
      k
      require "domains.md"
+
+module FFI-SYNTAX
+  imports private LIST
+
      +

      The FFIType sort is used to declare the native C ABI types of operands passed +to the #ffiCall function. These types roughly correspond to the types +declared in ffi.h by libffi.

      +
      k
        syntax FFIType ::= "#void" [klabel(#ffi_void), symbol]
+                  | "#uint8" [klabel(#ffi_uint8), symbol]
+                  | "#sint8" [klabel(#ffi_sint8), symbol]
+                  | "#uint16" [klabel(#ffi_uint16), symbol]
+                  | "#sint16" [klabel(#ffi_sint16), symbol]
+                  | "#uint32" [klabel(#ffi_uint32), symbol]
+                  | "#sint32" [klabel(#ffi_sint32), symbol]
+                  | "#uint64" [klabel(#ffi_uint64), symbol]
+                  | "#sint64" [klabel(#ffi_sint64), symbol]
+                  | "#float" [klabel(#ffi_float), symbol]
+                  | "#double" [klabel(#ffi_double), symbol]
+                  | "#uchar" [klabel(#ffi_uchar), symbol]
+                  | "#schar" [klabel(#ffi_schar), symbol]
+                  | "#ushort" [klabel(#ffi_ushort), symbol]
+                  | "#sshort" [klabel(#ffi_sshort), symbol]
+                  | "#uint" [klabel(#ffi_uint), symbol]
+                  | "#sint" [klabel(#ffi_sint), symbol]
+                  | "#ulong" [klabel(#ffi_ulong), symbol]
+                  | "#slong" [klabel(#ffi_slong), symbol]
+                  | "#longdouble" [klabel(#ffi_longdouble), symbol]
+                  | "#pointer" [klabel(#ffi_pointer), symbol]
+                  | "#complexfloat" [klabel(#ffi_complexfloat), symbol]
+                  | "#complexdouble" [klabel(#ffi_complexdouble), symbol]
+                  | "#complexlongdouble" [klabel(#ffi_complexlongdouble), symbol]
+                  | "#struct" "(" List ")" [klabel(#ffi_struct), symbol]
+endmodule
+
+module FFI
+  imports FFI-SYNTAX
+  imports private BYTES
+  imports private STRING
+  imports private BOOL
+  imports private LIST
+  imports private INT
+
+
      +

      FFI Calls

      +

      The #ffiCall functions are designed to call a native C ABI function and +return a native result. They come in three variants:

      +

      Non-variadic

      +

      In the first variant, #ffiCall(Address, Args, ArgTypes, ReturnType) takes +an integer address of a function (which can be obtained from +#functionAddress), a List of Bytes containing the arguments of the +function, a List of FFITypes containing the types of the parameters of the +function, and an FFIType containing the return type of the function, and +returns the return value of the function as a Bytes.

      +
      k
        syntax Bytes ::= "#ffiCall" "(" Int "," List "," List "," FFIType ")" [function, hook(FFI.call)]
+
      +

      Variadic

      +

      In the second variant, +#ffiCall(Address, Args, FixedTypes, VariadicTypes, ReturnType takes an +integer address of a function, a List of Bytes containing the arguments +of the call, a List of FFITypes containing the types of the fixed +parameters of the function, a List of FFITypes containing the types of the +variadic parameters of the function, and an FFIType containing the return +type of the function, and returns the return value of the function as a +Bytes.

      +
      k
        syntax Bytes ::= "#ffiCall" "(" Int "," List "," List "," List "," FFIType ")" [function, hook(FFI.call_variadic)]
+
      +

      Generic

      +

      In the third variant, +#ffiCall(IsVariadic, Address, Args, ArgTypes, NFixed, ReturnType takes +a boolean indicating whether the function is variadic or not, an integer +address of a function, a List of Bytes containing the arguments of the +call, a List of FFITypes containing the parameter typess of the call +followed by the types of the variadic arguments of the call, if any, an Int +containing how many of the arguments of the call are fixed or not, and an +FFIType containing the return type of the function, and returns the return +value of the function as a Bytes.

      +
      k
        syntax Bytes ::= "#ffiCall" "(" Bool "," Int "," List "," List "," Int "," FFIType ")" [function]
+
+  rule #ffiCall(false, Addr::Int, Args::List, Types::List, _, Ret::FFIType) => #ffiCall(Addr, Args, Types, Ret)
+  rule #ffiCall(true, Addr::Int, Args::List, Types::List, NFixed::Int, Ret::FFIType) => #ffiCall(Addr, Args, range(Types, 0, size(Types) -Int NFixed), range(Types, NFixed, 0), Ret)
+
      +

      Symbol Lookup

      +

      The FFI module provides a mechanism to look up any function symbol and return +that function's address.

      +
      k
        syntax Int ::= "#functionAddress" "(" String ")" [function, hook(FFI.address)]
+
      +

      Direct Memory Management

      +

      Most memory used by the LLVM backend to represent terms is managed +automatically via garbage collection. However, a consequence of this is that +a particular term does not have a fixed address across its entire lifetime +in most cases. Sometimes this is undesirable, especially if you intend for +the address of the memory to be taken by the semantics or if you intend +to pass this memory directly to native code. As a result, the FFI module +exposes the following unsafe APIs for memory management. Note that use of +these APIs leaves the burden of memory management completely on the user, +and thus misuse of these functions can lead to things like use-after-free +and other memory corruption bugs.

      +

      Allocation

      +

      #alloc(Key, Size, Align) will allocate Size bytes with an alignment +requirement of Align (which must be a power of two), and return it as a +Bytes term. The memory is uniquely identified by its key and that key will +be used later to free the memory. The memory is not implicitly freed by garbage +collection; failure to call #free on the memory at a later date can lead to +memory leaks.

      +
      k
        syntax Bytes ::= "#alloc" "(" KItem "," Int "," Int ")" [function, hook(FFI.alloc)]
+
      +

      Addressing

      +

      #addess(B) will return an Int representing the address of the first byte of +B, which must be a Bytes. Unless the Bytes term was allocated by #alloc, +the return value is unspecified and may not be the same across multipl +invocations on the same byte buffer. However, it is guaranteed that memory +allocated by #alloc will have the same address throughout its lifetime.

      +
      k
        syntax Int ::= "#address" "(" Bytes ")" [function, hook(FFI.bytes_address)]
+
      +

      Deallocation

      +

      #free(Key) will free the memory of the Bytes object that was allocated +by a previous call to #alloc. If Key was not used in a previous call to +#alloc, or the memory was already freed, no action is taken. It will generate +undefined behavior if the Bytes term returned by the previous call to +#alloc is still referenced by any other term in the configuration or a +currently evaluating rule. The function returns .K.

      +
      k
        syntax K ::= "#free" "(" KItem ")" [function, hook(FFI.free)]
+
      +

      Reading

      +

      #nativeRead(Addr, Mem) will read native memory at address Addr into Mem, +reading exactly lengthBytes(Mem) bytes. This will generate undefined behavior +if Addr does not point to a readable segment of memory at least +lengthBytes(Mem) bytes long.

      +
      k
        syntax K ::= "#nativeRead" "(" Int "," Bytes ")" [function, hook(FFI.read)]
+
      +

      Writing

      +

      #nativeWrite(Addr, Mem) will write the contents of Mem to native memory at +address Addr. The memory will be read prior to being written, and a write +will only happen if the memory has a different value than the current value of +Mem. This will generate undefined behavior if Addr does not point to a +readable segment of memory at least lengthBytes(Mem) bytes long, or if the +memory at address Addr has a different value than currently contained in +Mem, and the memory in question is not writeable.

      +
      k
        syntax K ::= "#nativeWrite" "(" Int "," Bytes ")" [function, hook(FFI.write)]
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/index.html b/k-distribution/include/kframework/builtin/index.html new file mode 100644 index 00000000000..11bae62ee99 --- /dev/null +++ b/k-distribution/include/kframework/builtin/index.html @@ -0,0 +1,405 @@ + + + + + + + + + + + + + + +K Builtins | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Builtins

      +

      The K Builtins (also referred to as the K Prelude or the K Standard Library) +consists of several files which contain definitions that make working with K +simpler. These files can be found under include/kframework/builtin in your K +installation directory, and can be imported with requires "FILENAME" (without +the path prefix).

      +
        +
      • domains: Basic datatypes which are universally useful.
        • +
      • kast: Representation of K internal data-structures (not to be +included in normal definitions).
        • +
      • prelude: Automatically included into every K definition.
        • +
      • ffi: FFI interface for calling out to native C code from K.
        • +
      • json: JSON datatype and parsers/unparsers for JSON strings.
        • +
      • rat: Rational number representation.
        • +
      • substitution: Hooked implementation of capture-aware +sustitution for K definitions.
        • +
      • unification: Hooked implementation of unification +exposed directly to K definitions.
        • +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/json/index.html b/k-distribution/include/kframework/builtin/json/index.html new file mode 100644 index 00000000000..709ba94406a --- /dev/null +++ b/k-distribution/include/kframework/builtin/json/index.html @@ -0,0 +1,428 @@ + + + + + + + + + + + + + + +Syntax of JSON | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Syntax of JSON

      +

      K provides builtin support for reading/writing to JSON. While the JSON-SYNTAX +module is not precisely the syntax of JSON (utilizing K's syntax for strings, +integers, and floating point numbers rather than the syntax used by JSON), +you can still convert directly to/from the actual syntax of JSON using +the JSON2String and String2JSON hooks.

      +
      k
      module JSON-SYNTAX
+    imports INT-SYNTAX
+    imports STRING-SYNTAX
+    imports BOOL-SYNTAX
+    imports FLOAT-SYNTAX
+
+    syntax JSONs   ::= List{JSON,","}      [klabel(JSONs)      , symbol]
+    syntax JSONKey ::= String
+    syntax JSON    ::= "null"              [klabel(JSONnull)   , symbol]
+                     | String | Int | Float | Bool
+                     | JSONKey ":" JSON    [klabel(JSONEntry)  , symbol]
+                     | "{" JSONs "}"       [klabel(JSONObject) , symbol]
+                     | "[" JSONs "]"       [klabel(JSONList)   , symbol]
+endmodule
+
      +

      Conversion between JSON and String

      +

      Given a string written in valid JSON, you can convert it to the JSON +sort with the String2JSON function. Assuming the user has not extended +the syntax of the JSON sort with their own constructors, any term of sort +JSON can also be converted to a String using the JSON2String function.

      +
      k
      module JSON
+    imports JSON-SYNTAX
+
+    syntax String ::= JSON2String(JSON) [function, hook(JSON.json2string)]
+
+    syntax JSON ::= String2JSON(String) [function, hook(JSON.string2json)]
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/kast/index.html b/k-distribution/include/kframework/builtin/kast/index.html new file mode 100644 index 00000000000..9544f3effd8 --- /dev/null +++ b/k-distribution/include/kframework/builtin/kast/index.html @@ -0,0 +1,966 @@ + + + + + + + + + + + + + + +K Language Features | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Language Features

      +

      Defined below is a series of modules used to parse inner syntax in K (ie, the +contents of rules, configuration declarations, and contexts).

      +

      Much of this file exists in tight correspondence with the K implementation, and +K will not work correctly if it is altered without corresponding changes to the +source code of the K tools.

      +

      Users should only import a few modules from this file. In particular, this +includes SORT-K, BASIC-K, ML-SYNTAX, DEFAULT-LAYOUT, +DEFAULT-CONFIGURATION, and K-AMBIGUITIES. The remaining modules should not +be imported by the user; they are used implicitly by the implementation of K.

      +

      Basic K Sorts

      +

      The SORT-K module declares the K sort, and nothing else.

      +
      k
      module SORT-K
+  syntax K [hook(K.K)]
+endmodule
+
      +

      The BASIC-K module declares the K, KItem, and KConfigVar sorts, and +imports the syntax of matching logic.

      +
      k
      module BASIC-K
+  imports ML-SYNTAX
+  imports SORT-BOOL
+  syntax KItem [hook(K.KItem)]
+  syntax K     ::= KItem
+  syntax KConfigVar [token]
+  syntax KItem ::= KConfigVar
+endmodule
+
      +

      KAST Abstract Syntax

      +

      Below is defined the abstract syntax of concrete terms in K, the KAST syntax. +Users should rarely if ever have to refer to this syntax; in general, it +suffices to use concrete syntax in rules, configuration declarations, contexts, +etc.

      +

      This syntax is used directly by the K implementation, and exists here as a +reference for the syntax of KAST, but it should not be imported directly by +the user.

      +
      k
      module KSTRING
+  syntax KString ::= r"[\\\"](([^\\\"\\n\\r\\\\])|([\\\\][nrtf\\\"\\\\])|([\\\\][x][0-9a-fA-F]{2})|([\\\\][u][0-9a-fA-F]{4})|([\\\\][U][0-9a-fA-F]{8}))*[\\\"]"      [token]
+  // optionally qualified strings, like in Scala "abc", i"abc", r"a*bc", etc.
+endmodule
+
+module BUILTIN-ID-TOKENS
+  syntax #LowerId ::= r"[a-z][a-zA-Z0-9]*"                    [prec(2), token]
+  syntax #UpperId ::= r"[A-Z][a-zA-Z0-9]*"                    [prec(2), token]
+endmodule
+
+module SORT-KBOTT
+  imports SORT-K
+  syntax KBott
+endmodule
+
+module KAST
+  imports BASIC-K
+  imports SORT-KBOTT
+  imports KSTRING
+  imports BUILTIN-ID-TOKENS
+
+  syntax KBott ::= "#token" "(" KString "," KString ")"  [klabel(#KToken), symbol]
+                 | "#klabel" "(" KLabel ")"              [klabel(#WrappedKLabel), symbol]
+                 | KLabel "(" KList ")"                  [klabel(#KApply), symbol]
+  syntax KItem ::= KBott
+
+  syntax KLabel ::= r"`(\\\\`|\\\\\\\\|[^`\\\\\\n\\r])+`" [token]
+                  | #LowerId                                   [token]
+                  | r"(?<![a-zA-Z0-9])[#a-z][a-zA-Z0-9]*"               [token, prec(1)]
+                       // something that doesn't collide with meta-variables
+
+  syntax KList ::= K
+                 | ".KList"          [klabel(#EmptyKList), symbol]
+                 | ".::KList"        [klabel(#EmptyKList), symbol]
+                 | KList "," KList   [klabel(#KList), left, assoc, unit(#EmptyKList), symbol, prefer]
+endmodule
+
+
+// To be used when parsing/pretty-printing ground configurations
+module KSEQ
+  imports KAST
+  imports K-TOP-SORT
+  syntax K ::= ".K"      [klabel(#EmptyK), symbol, unparseAvoid]
+             | "."       [klabel(#EmptyK), symbol]
+             | ".::K"    [klabel(#EmptyK), symbol, unparseAvoid]
+  syntax K ::= K "~>" K  [klabel(#KSequence), left, assoc, unit(#EmptyK), symbol]
+  syntax left #KSequence
+  syntax {Sort} Sort     ::= "(" Sort ")"    [bracket, group(defaultBracket), applyPriority(1)]
+endmodule
+
      +

      Syntax of Matching Logic

      +

      K provides direct access to the symbols of Matching Logic, while giving them +their own concrete syntax distinct from the syntax of the KORE intermediate +representation. These symbols are primarily used during symbolic execution. +The LLVM Backend has relatively little understanding of Matching Logic directly +and use of these symbols directly in rules is likely to cause it to crash. +However, these symbols are necessary when providing lemmas and other types of +logical assistance to proofs and symbolic execution in the Haskell Backend.

      +

      The correspondance between K symbols and KORE symbols is as follows:

      +
        +
      • #Top - \top
        • +
      • #Bottom - \bottom
        • +
      • #Not - \not
        • +
      • #Ceil - \ceil
        • +
      • #Floor - \floor
        • +
      • #Equals - \equals
        • +
      • #And - \and
        • +
      • #Or - \or
        • +
      • #Implies - \implies
        • +
      • #Exists - \exists
        • +
      • #Forall - \forall
        • +
      • #AG - allPathGlobally
        • +
      • #wEF - weakExistsFinally
        • +
      • #wAF - weakAlwaysFinally
        • +
      +
      k
      module ML-SYNTAX [not-lr1]
+  imports SORT-K
+
+  syntax {Sort} Sort ::= "#Top" [klabel(#Top), symbol, group(mlUnary)]
+                       | "#Bottom" [klabel(#Bottom), symbol, group(mlUnary)]
+                       | "#True" [klabel(#Top), symbol, group(mlUnary), unparseAvoid]
+                       | "#False" [klabel(#Bottom), symbol, group(mlUnary), unparseAvoid]
+                       | "#Not" "(" Sort ")" [klabel(#Not), symbol, mlOp, group(mlUnary)]
+
+  syntax {Sort1, Sort2} Sort2 ::= "#Ceil" "(" Sort1 ")" [klabel(#Ceil), symbol, mlOp, group(mlUnary)]
+                                | "#Floor" "(" Sort1 ")" [klabel(#Floor), symbol, mlOp, group(mlUnary)]
+                                | "{" Sort1 "#Equals" Sort1 "}" [klabel(#Equals), symbol, mlOp, group(mlEquals), comm, format(%1%i%n%2%d%n%3%i%n%4%d%n%5)]
+
+  syntax priorities mlUnary > mlEquals > mlAnd
+
+  syntax {Sort} Sort ::= Sort "#And" Sort [klabel(#And), symbol, assoc, left, comm, unit(#Top), mlOp, group(mlAnd), format(%i%1%d%n%2%n%i%3%d)]
+                       > Sort "#Or" Sort [klabel(#Or), symbol, assoc, left, comm, unit(#Bottom), mlOp, format(%i%1%d%n%2%n%i%3%d)]
+                       > Sort "#Implies" Sort [klabel(#Implies), symbol, mlOp, group(mlImplies), format(%i%1%d%n%2%n%i%3%d)]
+
+  syntax priorities mlImplies > mlQuantifier
+
+  syntax {Sort1, Sort2} Sort2 ::= "#Exists" Sort1 "." Sort2 [klabel(#Exists), symbol, mlOp, mlBinder, group(mlQuantifier)]
+                                | "#Forall" Sort1 "." Sort2 [klabel(#Forall), symbol, mlOp, mlBinder, group(mlQuantifier)]
+
+  syntax {Sort} Sort ::= "#AG" "(" Sort ")" [klabel(#AG), symbol, mlOp]
+                       | "#wEF" "(" Sort ")" [klabel(weakExistsFinally), symbol, mlOp]
+                       | "#wAF" "(" Sort ")" [klabel(weakAlwaysFinally), symbol, mlOp]
+endmodule
+
      +

      Variables in K

      +

      Provided below is the syntax of variables in K. There are four types of +variables in K:

      +
        +
      1. Regular variables. These are denoted by variables that begin with an +underscore or a capital letter. These variables match exactly one value +and can be used to refer to it on the right-hand-side.
        2. +
      2. Fresh constants. These are denoted by variables that begin with an !. This +is a convenience syntax which can be used on the right-hand-side only, and +refer to a unique value of the specified sort which is distinct from any +other value that has been generated or will be generated by the !X syntax. +Note that this may not be distinct from values produced via other means.
        3. +
      3. Existential variables. This refers to variables that are existentially +quantified and begin with a ?. They are not required to appear on the +left-hand-side prior to appearing on the right-hand-side, and generally +refer to symbolic quantities that are introduced during rewriting. Refer to +K's documentation for more details.
        4. +
      4. Set variables. These are denoted by variables that begin with a @. +These variables refer to a set of values and are generally used when writing +simplification rules in the Haskell Backend. For more information, refer to +K's documentation.
        5. +
      +

      There is also a fifth type of "variable", although it is not technically a +variable. This refers to configuration variables, which are used to insert +values into the initial configuration that come from outside the semantics. +The most common of these is the $PGM variable, which conventionally contains +the program being executed and is placed in the <k> cell in the configuration +declaration. These "variables" begin with a $ and their values are populated +by the frontend prior to symbolic or concrete execution of a program.

      +
      k
      module KVARIABLE-SYNTAX
+  syntax #KVariable
+endmodule
+
+// To be used when parsing/pretty-printing symbolic configurations
+module KSEQ-SYMBOLIC
+  imports KSEQ
+  imports ML-SYNTAX
+  imports KVARIABLE-SYNTAX
+
+  syntax #KVariable ::= r"(?<![A-Za-z0-9_\\$!\\?@])(\\!|\\?|@)?([A-Z][A-Za-z0-9'_]*|_|_[A-Z][A-Za-z0-9'_]*)"   [token, prec(1)]
+                      | #UpperId                                                          [token]
+  syntax KConfigVar ::= r"(?<![A-Za-z0-9_\\$!\\?@])(\\$)([A-Z][A-Za-z0-9'_]*)"            [token]
+  syntax KBott      ::= #KVariable
+  syntax KBott      ::= KConfigVar
+  syntax KLabel     ::= #KVariable
+endmodule
+
      +

      Syntax of Cells

      +

      While the backend treats cells as regular productions like any other, the +frontend provides a significant amount of convenience notation for dealing with +groups of cells, in order to make writing modular definitions easier. As a +result, we need a syntax for groups of cells and for referring to cells within +rules, configuration declarations, and functions.

      +

      For historical reasons, the Bag sort is used to refer to groups of cells. +This may change in a future release. Users can combine cells in any order +by concatenating them together, and can refer to the absence of any cells with +the .Bag symbol. You can also refer to cells within a function by placing +the cell context symbol, [[ K ]] at the top of a rule, placing a function +symbol inside, and referring to cells afterwards. This implicitly inserts +a reference to the configuration at the time prior to the currently-applied +rule being applied which can be matched on within the function. Functions with +such context cannot be referred to in the initial configuration, because the +prior configuration does not yet exist.

      +
      k
      module KCELLS
+  imports KAST
+
+  syntax Cell
+  syntax Bag ::= Bag Bag  [left, assoc, klabel(#cells), symbol, unit(#cells)]
+               | ".Bag"   [klabel(#cells), symbol]
+               | ".::Bag" [klabel(#cells), symbol]
+               | Cell
+  syntax Bag ::= "(" Bag ")" [bracket]
+  syntax KItem ::= Bag
+  syntax #RuleBody ::= "[" "[" K "]" "]" Bag    [klabel(#withConfig), symbol, avoid]
+  syntax non-assoc #withConfig
+  syntax Bag ::= KBott
+endmodule
+
      +

      Users can also refer to cells in rules. When doing so, an optional ... can +be placed immediately after the start of the cell or immediately before the +end. In a cell whose contents are commutative, these are equivalent to one +another and are also equivalent to placing ... in both places. This means +that what is placed in the cell will be combined with the cell contents' +concatenation operator with an unnamed variable. In other words, you match on +some number of elements in the collection and do not care about the rest of +the collection.

      +

      In a cell whose contents are not commutative, the ... operators correspond +to a variable on the respective side of the contents of the cell that the +... appears. For example, <foo>... L </foo>, if L is a list, means +some number of elements followed by L. Note that not all combinations are +supported. Cells whose contents are sort K can only have ... appear at the +tail of the cell, and cells whose contents are sort List can only have ... +appear on at most one side in a single rule.

      +
      k
      module RULE-CELLS
+  imports KCELLS
+  imports RULE-LISTS
+  // if this module is imported, the parser automatically
+  // generates, for all productions that have the attribute 'cell' or 'maincell',
+  // a production like below:
+  //syntax Cell ::= "<top>" #OptionalDots K #OptionalDots "</top>" [klabel(<top>)]
+
+  syntax #OptionalDots ::= "..." [klabel(#dots), symbol]
+                         | ""    [klabel(#noDots), symbol]
+
+  syntax Int
+  // this production will be added by the compiler to help handle bang variables,
+  // however, it is valuable to put it here because without this production, it
+  // will not exist at the point in time when rules and claims are parsed, and
+  // as a result it makes it very difficult to write proof claims over fragments
+  // of code that exercise rules containing bang variables. We put it here because
+  // this production will "vanish" after parsing finishes and not be picked up
+  // by the compiler, which is the behavior we want in this case since an actual
+  // production will be generated by the compiler later on.
+  syntax GeneratedCounterCell ::= "<generatedCounter>" Int "</generatedCounter>" [cell, klabel(<generatedCounter>), symbol, internal]
+endmodule
+
      +

      Users can also declare cells in a configuration declaration. This generates a +specific set of productions that is used internally to implement the cell. The +most important of these is the cell itself, and attributes on this production +can be specified in an xml-attribute-like syntax.

      +

      You can also use an xml-short-tag-like syntax to compose configuration cells +together which were defined in different modules. However, it is a requirement +that any K definition have at most one fully-composed configuration; thus, all +other configuration declarations must appear composed within another +configuration declaration.

      +
      k
      module CONFIG-CELLS
+  imports KCELLS
+  imports RULE-LISTS
+  syntax #CellName ::= r"[a-zA-Z][a-zA-Z0-9\\-]*"  [token, prec(1)]
+                     | #LowerId            [token]
+                     | #UpperId            [token]
+
+  syntax Cell ::= "<" #CellName #CellProperties ">" K "</" #CellName ">" [klabel(#configCell), symbol]
+  syntax Cell ::= "<" #CellName "/>" [klabel(#externalCell), symbol]
+
+  syntax #CellProperties ::= #CellProperty #CellProperties [klabel(#cellPropertyList), symbol]
+                           | ""                            [klabel(#cellPropertyListTerminator), symbol]
+  syntax #CellProperty ::= #CellName "=" KString           [klabel(#cellProperty), symbol]
+endmodule
+
      +

      Syntax of Rules

      +

      Rules can have an optional requires clause or an ensures clause. For backwards- +compatibility, you can refer to the requires clause with both the requires +and when keywords; The latter, however, is deprecated and may be removed in +a future release.

      +

      The requires clause specifies the preconditions that must be true in order +for the rule to apply. The ensures clause specifies the information which +becomes true after the rule has applied. It is a requirement that information +present in the ensures clause refer to existential variables only.

      +

      When doing concrete execution, you can think of the requires clause as a +side-condition. In other words, even if the rule matches, it will not apply +unless the requires clause, which must be of sort Bool, evaluates to +true.

      +
      k
      module REQUIRES-ENSURES
+  imports BASIC-K
+
+  syntax #RuleBody ::= K
+
+  syntax #RuleContent ::= #RuleBody                                 [klabel("#ruleNoConditions"), symbol]
+                        | #RuleBody "requires" Bool                 [klabel("#ruleRequires"), symbol]
+                        | #RuleBody "when" Bool                     [klabel("#ruleRequires"), symbol]
+                        | #RuleBody "ensures"  Bool                 [klabel("#ruleEnsures"), symbol]
+                        | #RuleBody "requires" Bool "ensures" Bool  [klabel("#ruleRequiresEnsures"), symbol]
+                        | #RuleBody "when" Bool "ensures" Bool      [klabel("#ruleRequiresEnsures"), symbol]
+endmodule
+
      +

      Miscellaneous modules

      +

      The below modules are used in various ways as indicators to the implementation +that certain automatically generated syntax should be created by the parser. +These modules should not be imported directly by the user.

      +
      k
      module K-TOP-SORT
+  imports SORT-KBOTT
+  syntax KItem ::= KBott
+  syntax {Sort} KItem ::= Sort
+endmodule
+
+module K-BOTTOM-SORT
+  imports SORT-KBOTT
+  syntax KItem ::= KBott
+  syntax {Sort} Sort ::= KBott
+endmodule
+
+module K-SORT-LATTICE
+  imports K-TOP-SORT
+  imports K-BOTTOM-SORT
+endmodule
+
+module AUTO-CASTS
+  // if this module is imported, the parser automatically
+  // generates, for all sorts, productions of the form:
+  // Sort  ::= Sort ":Sort"  // semantic cast - force the inner term to be `Sort` or a subsort
+  // Sort  ::= Sort "::Sort" // strict cast - force the inner term to be exactly `Sort`. Useful for disambiguation
+  // Sort ::= "{" Sort "}" "<:Sort" // synonym for strict cast
+  // Sort  ::= "{" K "}"    ":>Sort" // projection cast. Allows any term to be placed in a context that expects `Sort`
+  // this is part of the mechanism that allows concrete user syntax in K
+endmodule
+
+module AUTO-FOLLOW
+  // if this module is imported, the parser automatically
+  // generates a follow restriction for every terminal which is a prefix
+  // of another terminal. This is useful to prevent ambiguities such as:
+  // syntax K ::= "a"
+  // syntax K ::= "b"
+  // syntax K ::= "ab"
+  // syntax K ::= K K
+  // #parse("ab", "K")
+  // In the above example, the terminal "a" is not allowed to be followed by a "b"
+  // because it would turn the terminal into the terminal "ab".
+endmodule
+
+module PROGRAM-LISTS
+  imports SORT-K
+  // if this module is imported, the parser automatically
+  // replaces the default productions for lists:
+  // Es ::= E "," Es [userList("*"), klabel('_,_)]
+  //      | ".Es"    [userList("*"), klabel('.Es)]
+  // into a series of productions more suitable for programs:
+  // Es#Terminator ::= ""      [klabel('.Es)]
+  // Ne#Es ::= E "," Ne#Es     [klabel('_,_)]
+  //         | E Es#Terminator [klabel('_,_)]
+  // Es ::= Ne#Es
+  //      | Es#Terminator      // if the list is *
+endmodule
+
+module RULE-LISTS
+  // if this module is imported, the parser automatically
+  // adds the subsort production to the parsing module only:
+  // Es ::= E        [userList("*")]
+
+endmodule
+
+module RECORD-PRODUCTIONS
+  // if this module is imported, prefix productions of the form
+  // syntax Sort ::= name(Args)
+  // will be able to be parsed with don't-care variables according
+  // to their nonterminal's names
+endmodule
+
+module SORT-PREDICATES
+  // if this module is imported, the Bool sort will be annotated with
+  // syntax Bool ::= isSort(K) [function]
+  // and all sorts will be annotated with
+  // syntax Sort ::= project:Sort(K) [function]
+endmodule
+
      +

      Additional Syntax for K Terms in Rules

      +

      Certain additional features are available when parsing the contents of rules +and contexts. For more information on each of these, refer to K's +documentation.

      +
      k
      module KREWRITE
+  syntax {Sort} Sort ::= Sort "=>" Sort [klabel(#KRewrite), symbol]
+  syntax non-assoc #KRewrite
+  syntax priority #KRewrite > #withConfig
+endmodule
+
+// To be used to parse semantic rules
+module K
+  imports KSEQ-SYMBOLIC
+  imports REQUIRES-ENSURES
+  imports RECORD-PRODUCTIONS
+  imports SORT-PREDICATES
+  imports K-SORT-LATTICE
+  imports AUTO-CASTS
+  imports AUTO-FOLLOW
+  imports KREWRITE
+
+  syntax {Sort} Sort ::= Sort "#as" Sort [klabel(#KAs), symbol]
+  // functions that preserve sorts and can therefore have inner rewrites
+  syntax {Sort} Sort ::= "#fun" "(" Sort ")" "(" Sort ")" [klabel(#fun2), symbol, prefer]
+  // functions that do not preserve sort and therefore cannot have inner rewrites
+  syntax {Sort1, Sort2} Sort1 ::= "#fun" "(" Sort2 "=>" Sort1 ")" "(" Sort2 ")" [klabel(#fun3), symbol]
+
+  syntax {Sort1, Sort2} Sort1 ::= "#let" Sort2 "=" Sort2 "#in" Sort1 [klabel(#let), symbol]
+
+  /*@ Set membership over terms. In addition to equality over
+      concrete patterns, K also supports computing equality
+      between a concrete pattern and a symbolic pattern.
+      This is compiled efficiently down to pattern matching,
+      and can be used by putting a term with unbound variables
+      in the left child of :=K or =/=K. Note that this does not
+      bind variables used on the lhs however (although this may
+      change in the future).*/
+
+  syntax Bool ::= left:
+                  K ":=K" K           [function, total, klabel(_:=K_), symbol, group(equalEqualK)]
+                | K ":/=K" K          [function, total, klabel(_:/=K_), symbol, group(notEqualEqualK)]
+endmodule
+
+// To be used to parse terms in full K
+module K-TERM
+  imports KSEQ-SYMBOLIC
+  imports RECORD-PRODUCTIONS
+  imports SORT-PREDICATES
+  imports K-SORT-LATTICE
+  imports AUTO-CASTS
+  imports AUTO-FOLLOW
+  imports KREWRITE
+endmodule
+
      +

      Layout Information

      +

      When constructing a scanner for use during parsing, often you wish to ignore +certain types of text, such as whitespace and comments. However, the specific +syntax which each language must ignore is a little different from language +to language, and thus you wish to specify it manually. You can do this by +defining productions of the #Layout sort. For more information, refer to +K's documentation. However, this module will be implicitly imported if no +productions are declared of sort #Layout. This module will also be used +for the purposes of parsing K rules. If you wish to declare a language with +no layout productions, simply create a sort declaration for the #Layout sort +in your code (e.g. syntax #Layout).

      +
      k
      module DEFAULT-LAYOUT
+    syntax #Layout ::= r"(\\/\\*([^\\*]|(\\*+([^\\*\\/])))*\\*+\\/)" // C-style multi-line comments
+                     | r"(\\/\\/[^\\n\\r]*)"                         // C-style single-line comments
+                     | r"([\\ \\n\\r\\t])"                           // Whitespace
+endmodule
+
      +

      Default Configuration

      +

      If the user has no configuration declaration in their seamantics, the below +configuration declaration will be implicitly imported.

      +
      k
      module DEFAULT-CONFIGURATION
+  imports BASIC-K
+
+  configuration <k> $PGM:K </k>
+endmodule
+
      +

      Parsing Ambiguous Languages

      +

      On occasion, it may be desirable to parse a language with an ambiguous grammar +when parsing a program, and perform additional semantic analysis at a later +time in order to resolve the ambiguities. A good example of this is as a +substitute for the lexer hack in parsers of the C programming language.

      +

      The following module contains a declaration for ambiguities in K. Usually, +an ambiguous parse is an error. However, when you use the --gen-glr-parser +flag to kast, or the --gen-glr-bison-parser flag to kompile, ambiguities +instead become instances of the below parametric production, which you can use +regular K rules to disambiguate as necessary.

      +
      k
      module K-AMBIGUITIES
+
+  syntax {Sort} Sort ::= amb(Sort, Sort) [symbol]
+
+endmodule
+
      +

      Annotating Parses with Locations

      +

      Another feature of K's Bison parser is the ability to annotate terms parsed +with location information about the file and line where they occurred. For +more information about how to use this, refer to K's documentation. However, +the below module exists to provide a user syntax for the annotations that +are generated by the parser.

      +
      k
      module K-LOCATIONS
+  imports STRING-SYNTAX
+  imports INT-SYNTAX
+
+  // filename, startLine, startCol, endLine, endCol
+  syntax {Sort} Sort ::= #location(Sort, String, Int, Int, Int, Int) [symbol, format(%3)]
+
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/prelude/index.html b/k-distribution/include/kframework/builtin/prelude/index.html new file mode 100644 index 00000000000..75d63852786 --- /dev/null +++ b/k-distribution/include/kframework/builtin/prelude/index.html @@ -0,0 +1,399 @@ + + + + + + + + + + + + + + +K Prelude | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Prelude

      +

      The following files, integral to defining semantics in K, are automatically +required by every definition via this file. This behavior can be disabled +via kompile --no-prelude, however, semantics will likely break unless +they provide their own versions of these files, which are assumed to exist +by the compiler. There are, however, circumstances where passing this flag is +appropriate, such as if you are manually requiring these files in your +definition, if your definition was automatically condensed into a single file +with kompile -E, or if you wish to modify the inner syntax of K by providing +your own version of these files with different syntax.

      +
      k
      require "kast.md"
+require "domains.md"
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/rat/index.html b/k-distribution/include/kframework/builtin/rat/index.html new file mode 100644 index 00000000000..77d65b847d9 --- /dev/null +++ b/k-distribution/include/kframework/builtin/rat/index.html @@ -0,0 +1,620 @@ + + + + + + + + + + + + + + +Rational Numbers in K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Rational Numbers in K

      +

      K provides support for arbitrary-precision rational numbers represented as a +quotient between two integers. The sort representing these values is Rat. +Int is a subsort of Rat, and it is guaranteed that any integer will be +represented as an Int and can be matched as such on the left hand side +of rules. K also supports the usual arithmetic operators over rational numbers.

      +
      k
      module RAT-SYNTAX
+  imports INT-SYNTAX
+  imports private BOOL
+
+  syntax Rat
+
+  syntax Rat ::= Int
+
      +

      Arithmetic

      +

      You can:

      +
        +
      • Raise a rational number to any negative or nonnegative integer.
        • +
      • Multiply or divide two rational numbers to obtain a product or quotient.
        • +
      • Add or subtract two rational numbers to obtain a sum or difference.
        • +
      +
      k
        syntax Rat ::= left:
+                 Rat "^Rat" Int [function, total, klabel(_^Rat_), symbol, smtlib(ratpow), hook(RAT.pow)]
+               > left:
+                 Rat "*Rat" Rat [function, total, klabel(_*Rat_), symbol, left, smtlib(ratmul), hook(RAT.mul)]
+               | Rat "/Rat" Rat [function,             klabel(_/Rat_), symbol, left, smtlib(ratdiv), hook(RAT.div)]
+               > left:
+                 Rat "+Rat" Rat [function, total, klabel(_+Rat_), symbol, left, smtlib(ratadd), hook(RAT.add)]
+               | Rat "-Rat" Rat [function, total, klabel(_-Rat_), symbol, left, smtlib(ratsub), hook(RAT.sub)]
+
      +

      Comparison

      +

      You can determine whether two rational numbers are equal, unequal, or compare +one of less than, less than or equalto, greater than, or greater than or equal +to the other:

      +
      k
        syntax Bool ::= Rat  "==Rat" Rat [function, total, klabel(_==Rat_),  symbol, smtlib(rateq), hook(RAT.eq)]
+                | Rat "=/=Rat" Rat [function, total, klabel(_=/=Rat_), symbol, smtlib(ratne), hook(RAT.ne)]
+                | Rat   ">Rat" Rat [function, total, klabel(_>Rat_),   symbol, smtlib(ratgt), hook(RAT.gt)]
+                | Rat  ">=Rat" Rat [function, total, klabel(_>=Rat_),  symbol, smtlib(ratge), hook(RAT.ge)]
+                | Rat   "<Rat" Rat [function, total, klabel(_<Rat_),   symbol, smtlib(ratlt), hook(RAT.lt)]
+                | Rat  "<=Rat" Rat [function, total, klabel(_<=Rat_),  symbol, smtlib(ratle), hook(RAT.le)]
+
      +

      Min/Max

      +

      You can compute the minimum and maximum of two rational numbers:

      +
      k
        syntax Rat ::= minRat(Rat, Rat) [function, total, klabel(minRat), symbol, smtlib(ratmin), hook(RAT.min)]
+               | maxRat(Rat, Rat) [function, total, klabel(maxRat), symbol, smtlib(ratmax), hook(RAT.max)]
+
      +

      Conversion to Floating Point

      +

      You can convert a rational number to the nearest floating point number that +is representable in a Float of a specified number of precision and exponent +bits:

      +
      k
        syntax Float ::= Rat2Float(Rat, precision: Int, exponentBits: Int) [function]
+endmodule
+
      +

      Implementation of Rational Numbers

      +

      The remainder of this file consists of an implementation in K of the +operations listed above. Users of the RAT module should not use any of the +syntax defined in any of these modules.

      +

      As a point of reference for users, it is worth noting that rational numbers +are normalized to a canonical form by this module,. with the canonical form +bearing the property that it is either an Int, or a pair of integers +I /Rat J such that +I =/=Int 0 andBool J >=Int 2 andBool gcdInt(I, J) ==Int 1 is always true.

      +
      k
      module RAT-COMMON
+  imports RAT-SYNTAX
+
+  // invariant of < I , J >Rat : I =/= 0, J >= 2, and I and J are coprime
+  syntax Rat ::= "<" Int "," Int ">Rat" [format(%2 /Rat %4)]
+endmodule
+
+module RAT-SYMBOLIC [symbolic, kore]
+  imports private RAT-COMMON
+  imports ML-SYNTAX
+  imports private BOOL
+
+  rule
+    #Ceil(@R1:Rat /Rat @R2:Rat)
+  =>
+    {(@R2 =/=Rat 0) #Equals true} #And #Ceil(@R1) #And #Ceil(@R2)
+  [simplification]
+endmodule
+
+module RAT-KORE [kore]
+  imports private RAT-COMMON
+  imports private K-EQUAL
+
+  /*
+   * equalities
+   */
+
+  // NOTE: the two rules below may not work correctly in non-kore backends
+
+  rule R ==Rat S => R ==K S
+
+  rule R =/=Rat S => R =/=K S
+endmodule
+
+module RAT [private]
+  imports private RAT-COMMON
+  imports public RAT-SYMBOLIC
+  imports public RAT-KORE
+  imports public RAT-SYNTAX
+  imports private INT
+  imports private BOOL
+
+  /*
+   * arithmetic
+   */
+
+  rule < I , I' >Rat +Rat < J , J' >Rat => ((I *Int J') +Int (I' *Int J)) /Rat (I' *Int J')
+  rule I:Int         +Rat < J , J' >Rat => ((I *Int J') +Int J) /Rat J'
+  rule < J , J' >Rat +Rat I:Int         => I +Rat < J , J' >Rat
+  rule I:Int         +Rat J:Int         => I +Int J
+
+  rule < I , I' >Rat *Rat < J , J' >Rat => (I *Int J) /Rat (I' *Int J')
+  rule I:Int         *Rat < J , J' >Rat => (I *Int J) /Rat J'
+  rule < J , J' >Rat *Rat I:Int         => I *Rat < J , J' >Rat
+  rule I:Int         *Rat J:Int         => I *Int J
+
+  rule < I , I' >Rat /Rat < J , J' >Rat => (I *Int J') /Rat (I' *Int J)
+  rule I:Int         /Rat < J , J' >Rat => (I *Int J') /Rat J
+  rule < I , I' >Rat /Rat J:Int         => I /Rat (I' *Int J) requires J =/=Int 0
+  rule I:Int         /Rat J:Int         => makeRat(I, J)      requires J =/=Int 0
+
+  // derived
+
+  rule R -Rat S => R +Rat (-1 *Rat S)
+
+  // normalize
+
+  syntax Rat ::= makeRat(Int, Int)      [function]
+               | makeRat(Int, Int, Int) [function]
+
+  rule makeRat(0, J) => 0 requires J =/=Int 0
+
+  rule makeRat(I, J) => makeRat(I, J, gcdInt(I,J)) requires I =/=Int 0 andBool J =/=Int 0
+
+  // makeRat(I, J, D) is defined when I =/= 0, J =/= 0, D > 0, and D = gcd(I,J)
+  rule makeRat(I, J, D) => I /Int D                       requires J ==Int D // implies J > 0 since D > 0
+  rule makeRat(I, J, D) => < I /Int D , J /Int D >Rat     requires J >Int 0 andBool J =/=Int D
+  rule makeRat(I, J, D) => makeRat(0 -Int I, 0 -Int J, D) requires J <Int 0
+
+  // gcdInt(a,b) computes the gcd of |a| and |b|, which is positive.
+  syntax Int ::= gcdInt(Int, Int) [function, public]
+
+  rule gcdInt(A, 0) => A        requires A >Int 0
+  rule gcdInt(A, 0) => 0 -Int A requires A <Int 0
+  rule gcdInt(A, B) => gcdInt(B, A %Int B) requires B =/=Int 0 // since |A %Int B| = |A| %Int |B|
+
+  /*
+   * exponentiation
+   */
+
+  rule _ ^Rat 0 => 1
+  rule 0 ^Rat N => 0 requires N =/=Int 0
+
+  rule < I , J >Rat ^Rat N => powRat(< I , J >Rat, N) requires N >Int 0
+  rule X:Int        ^Rat N => X ^Int N                requires N >Int 0
+
+  rule X ^Rat N => (1 /Rat X) ^Rat (0 -Int N) requires X =/=Rat 0 andBool N <Int 0
+
+  // exponentiation by squaring
+
+  syntax Rat ::= powRat(Rat, Int) [function]
+
+  // powRat(X, N) is defined when X =/= 0 and N > 0
+  rule powRat(X, 1) => X
+  rule powRat(X, N) => powRat(X *Rat X, N /Int 2) requires N >Int 1 andBool N %Int 2  ==Int 0
+  rule powRat(X, N) => powRat(X, N -Int 1) *Rat X requires N >Int 1 andBool N %Int 2 =/=Int 0
+
+  /*
+   * inequalities
+   */
+
+  rule R >Rat S => R -Rat S >Rat 0 requires S =/=Rat 0
+
+  rule < I , _ >Rat >Rat 0 => I >Int 0
+  rule I:Int        >Rat 0 => I >Int 0
+
+  // derived
+
+  rule R >=Rat S => notBool R <Rat S
+
+  rule R <Rat S => S >Rat R
+
+  rule R <=Rat S => S >=Rat R
+
+  rule minRat(R, S) => R requires R <=Rat S
+  rule minRat(R, S) => S requires S <=Rat R
+
+  rule maxRat(R, S) => R requires R >=Rat S
+  rule maxRat(R, S) => S requires S >=Rat R
+
+  syntax Float ::= #Rat2Float(Int, Int, Int, Int) [function, hook(FLOAT.rat2float)]
+  rule Rat2Float(Num:Int, Prec:Int, Exp:Int) => #Rat2Float(Num, 1, Prec, Exp)
+  rule Rat2Float(< Num, Dem >Rat, Prec, Exp) => #Rat2Float(Num, Dem, Prec, Exp)
+
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/include/kframework/builtin/substitution/index.html b/k-distribution/include/kframework/builtin/substitution/index.html new file mode 100644 index 00000000000..4e9c0229513 --- /dev/null +++ b/k-distribution/include/kframework/builtin/substitution/index.html @@ -0,0 +1,484 @@ + + + + + + + + + + + + + + +Capture-Aware Substitution in K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Capture-Aware Substitution in K

      +

      One of the traditional ways in which functional languages are given operational +semantics is via substitution. In particular, you can view a function as +declaring a particular bound variable, the parameter of the function, as well +as the body of the function, within which both bound and free variables can +occur, and implement the process of beta-reduction (one of the axioms of the +lambda calculus) by means of a substitution operator which is aware of the +difference between free variables and bound variables and prevents variable +capture.

      +

      In K this is implemented using two mechanisms: The KVar sort, and the +binder attribute.

      +

      The KVar Sort

      +

      K introduces a new hooked sort, KVar, which the substitution operator +(defined below) understands in a particular way. The syntax of KVar is the +same as for sort Id in DOMAINS, but with a different sort name. Similarly, +some of the same operators are defined over KVar which are defined for Id, +such as conversion from String to KVar and support for the !Var:KVar +syntax.

      +

      A KVar is simply an identifier with special meaning during substitution. +KVars must begin with a letter or underscore, +and can be followed by zero or more letters, numbers, or underscores.

      +
      k
      module KVAR-SYNTAX-PROGRAM-PARSING
+  imports BUILTIN-ID-TOKENS
+
+  syntax KVar ::= r"(?<![A-Za-z0-9\\_])[A-Za-z\\_][A-Za-z0-9\\_]*"     [prec(1), token]
+                | #LowerId                                             [token]
+                | #UpperId                                             [token]
+endmodule
+
+module KVAR-SYNTAX
+  syntax KVar [token, hook(KVAR.KVar)]
+endmodule
+
+module KVAR-COMMON
+  imports KVAR-SYNTAX
+  imports private STRING
+
+  syntax KVar ::= String2KVar (String) [function, total, hook(STRING.string2token)]
+  syntax KVar ::= freshKVar(Int)    [freshGenerator, function, total, private]
+
+  rule freshKVar(I:Int) => String2KVar("_" +String Int2String(I))
+endmodule
+
+module KVAR
+  imports KVAR-COMMON
+endmodule
+
      +

      The binder Attribute

      +

      A production can be given the attribute binder. Such a production must have +at least two nonterminals. The first nonterminal from left to right must be of +sort KVar, and contains the bound variable. The last nonterminal from left +to right contains the term that is bound. For example, I could describe lambdas +in the lambda calculus with the production +syntax Val ::= "lambda" KVar "." Exp [binder].

      +

      Substitution

      +

      K provides a hooked implementation of substitution, currently only implemented +on the Java and LLVM backends. Two variants exist: the first substitutes +a single KVar for a single KItem. The second takes a Map with KVar +keys and KItem values, and substitutes each element in the map atomically.

      +

      Internally, this is implemented in the LLVM backend by a combination of +de Bruijn indices for bound variables and names for free variables. Free +variables are also sometimes given a unique numeric identifier in order to +prevent capture, and the rewriter will automatically assign unique names to +such identifiers when rewriting finishes. The names assigned will always begin +with the original name of the variable and be followed by a unique integer +suffix. However, the names assigned after rewriting finishes might be different +from the names that would be assigned if rewriting were to halt prematurely, +for example due to krun --depth.

      +
      k
      module SUBSTITUTION
+  imports private MAP
+  imports KVAR
+
+  syntax {Sort} Sort ::= Sort "[" KItem "/" KItem "]"  [function, hook(SUBSTITUTION.substOne), impure]
+  syntax {Sort} Sort ::= Sort "[" Map "]"      [function, hook(SUBSTITUTION.substMany), impure]
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/01_installing/index.html b/k-distribution/k-tutorial/1_basic/01_installing/index.html new file mode 100644 index 00000000000..059c8618dbf --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/01_installing/index.html @@ -0,0 +1,473 @@ + + + + + + + + + + + + + + +Lesson 1.1: Setting up a K Environment | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.1: Setting up a K Environment

      +

      The first step to learning K is to install K on your system, and configure your +editor for K development.

      +

      Installing K

      +

      You have two options for how to install K, depending on how you intend to +interact with the K codebase. If you are solely a user of K, and have no +interest in developing or making changes to K, you most likely will want to +install one of our binary releases of K. However, if you are going to be a K +developer, or simply want to build K from source, you should follow the +instructions for a source build of K.

      +

      Installing K from a binary release

      +

      K is developed as a rolling release, with each change to K that passes our +CI infrastructure being deployed on GitHub for download. The latest release of +K can be downloaded here. +This page also contains information on how to install K. It is recommended +that you fully uninstall the old version of K prior to installing the new one, +as K does not maintain entries in package manager databases, with the exception +of Homebrew on MacOS.

      +

      Installing K from source

      +

      You can clone K from GitHub with the following Git command:

      +
      git clone https://github.com/runtimeverification/k --recursive
+
      +

      Instructions on how to build K from source can be found +here.

      +

      Configuring your editor

      +

      K maintains a set of scripts for a variety of text editors, including vim and +emacs, in various states of maintenance. You can download these scripts with +the following Git command:

      +
      git clone https://github.com/kframework/k-editor-support
+
      +

      Because K allows users to define their own grammars for parsing K itself, +not all features of K can be effectively highlighted. However, at the cost of +occasionally highlighting things incorrectly, you can get some pretty good +results in many cases. With that being said, some of the editor scripts in the +above repository are pretty out of date. If you manage to improve them, we +welcome pull requests into the repository.

      +

      Troubleshooting

      +

      If you have problems installing K, we encourage you to reach out to us. If you +follow the above install instructions and run into a problem, you can +Create a bug report on GitHub

      +

      Next lesson

      +

      Once you have set up K on your system to your satisfaction, you can continue to +Lesson 1.2: Basics of Functional K.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/02_basics/index.html b/k-distribution/k-tutorial/1_basic/02_basics/index.html new file mode 100644 index 00000000000..cac4f4a06e4 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/02_basics/index.html @@ -0,0 +1,612 @@ + + + + + + + + + + + + + + +Lesson 1.2: Basics of Functional K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.2: Basics of Functional K

      +

      The purpose of this lesson is to explain the basics of productions and +rules in K. These are two types of K sentences. A K file consists of +one or more requires or modules in K. Each module consists of one or +more imports or sentences. For more information on requires, modules, and +sentences, refer to Lesson 1.5. However, for the time +being, just think of a module as a container for sentences, and don't worry +about requires or imports just yet.

      +

      Our first K program

      +

      To start with, input the following program into your editor as file +lesson-02-a.k:

      +
      k
      module LESSON-02-A
+
+  syntax Color ::= Yellow() | Blue()
+  syntax Fruit ::= Banana() | Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+
      +

      Save this file and then run:

      +
      kompile lesson-02-a.k
+
      +

      kompile is K's compiler. By default, it takes a program or specification +written in K and compiles it into an interpreter for that input. Right now we +are compiling a single file. A set of K files that are compiled together are +called a K definition. We will cover multiple file K definitions later on. +kompile will output a directory containing everything needed to execute +programs and perform proofs using that definition. In this case, kompile will +(by default) create the directory lesson-02-a-kompiled under the current +directory.

      +

      Now, save the following input file in your editor as banana.color in the same +directory as lesson-02-a.k:

      +
      colorOf(Banana())
+
      +

      We can now evaluate this K term by running (from the same directory):

      +
      krun banana.color
+
      +

      krun will use the interpreter generated by the first call to kompile to +execute this program.

      +

      You will get the following output:

      +
      <k>
+  Yellow ( ) ~> .
+</k>
+
      +

      For now, don't worry about the <k>, </k>, or ~> . portions of this +output file.

      +

      You can also execute small programs directly by specifying them on the command +line instead of putting them in a file. For example, the same program above +could also have been executed by running the following command:

      +
      krun -cPGM='colorOf(Banana())'
+
      +

      Now, let's look at what this definition and program did.

      +

      Productions, Constructors, and Functions

      +

      The first thing to realize is that this K definition contains 5 productions. +Productions are introduced with the syntax keyword, followed by a sort, +followed by the operator ::= followed by the definition of one or more +productions themselves, separated by the | operator. There are different +types of productions, but for now we only care about constructors and +functions. Each declaration separated by the | operator is individually +a single production, and the | symbol simply groups together productions that +have the same sort. For example, we could equally have written an identical K +definition lesson-02-b.k like so:

      +
      k
      module LESSON-02-B
+
+  syntax Color ::= Yellow()
+  syntax Color ::= Blue()
+  syntax Fruit ::= Banana()
+  syntax Fruit ::= Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+
      +

      You can try compiling and running lesson-02-b.k to see that it produces the same output as lesson-02-a.k:

      +
      kompile lesson-02-b.k
+krun -cPGM='colorOf(Banana())' --definition 'lesson-02-b-kompiled'
+
      +

      where the --definition attribute points to the directory containing a compiled version of LESSON-02-B. +Even the following definition is equivalent:

      +
      k
      module LESSON-02-C
+
+  syntax Color ::= Yellow()
+                 | Blue()
+                 | colorOf(Fruit) [function]
+  syntax Fruit ::= Banana()
+                 | Blueberry()
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+
+endmodule
+
      +

      Each of these types of productions named above has the same underlying syntax, +but context and attributes are used to distinguish between the different +types. Tokens, brackets, lists, macros, aliases, and anywhere productions will +be covered in a later lesson, but this lesson does introduce us to constructors +and functions. Yellow(), Blue(), Banana(), and Blueberry() are +constructors. You can think of a constructor like a constructor for an +algebraic data type, if you're familiar with a functional language. The data +type itself is the sort that appears on the left of the ::= operator. Sorts +in K consist of uppercase identifiers.

      +

      Constructors can have arguments, but these ones do not. We will cover the +syntax of productions in detail in the next lesson, but for now, you can write +a production with no arguments as an uppercase or lowercase identifier followed +by the () operator.

      +

      A function is distinguished from a constructor by the presence of the +function attribute. Attributes appear in a comma separated list between +square brackets after any sentence, including both productions and rules. +Various attributes with built-in meanings exist in K and will be discussed +throughout the tutorial.

      +

      Exercise

      +

      Use krun to compute the return value of the colorOf function on a +Blueberry().

      +

      Rules, Matching, and Variables

      +

      Functions in K are given definitions using rules. A rule begins with the rule +keyword and contains at least one rewrite operator. The rewrite operator +is represented by the syntax =>. The rewrite operator is one of the built-in +productions in K, and we will discuss in more detail how it can be used in +future lessons, but for now, you can think of a rule as consisting of a +left-hand side and a right-hand side, separated by the rewrite +operator. On the left-hand side is the name of the function and zero or more +patterns corresponding to the parameters of the function. On the right-hand +side is another pattern. The meaning of the rule is relatively simple, having +defined these components. If the function is called with arguments that +match the patterns on the left-hand side, then the return value of the +function is the pattern on the right-hand side.

      +

      For example, in the above example, if the argument of the colorOf function +is Banana(), then the return value of the function is Yellow().

      +

      So far we have introduced that a constructor is a type of pattern in K. We +will introduce more complex patterns in later lessons, but there is one other +type of basic pattern: the variable. A variable, syntactically, consists +of an uppercase identifier. However, unlike a constructor, a variable will +match any pattern with one exception: Two variables with the same name +must match the same pattern.

      +

      Here is a more complex example (lesson-02-d.k):

      +
      k
      module LESSON-02-D
+
+  syntax Container ::= Jar(Fruit)
+  syntax Fruit ::= Apple() | Pear()
+
+  syntax Fruit ::= contentsOfJar(Container) [function]
+
+  rule contentsOfJar(Jar(F)) => F
+
+endmodule
+
      +

      Here we see that Jar is a constructor with a single argument. You can write a +production with multiple arguments by putting the sorts of the arguments in a +comma-separated list inside the parentheses.

      +

      In this example, F is a variable. It will match either Apple() or Pear(). +The return value of the function is created by substituting the matched +values of all of the variables into the variables on the right-hand side of +the rule.

      +

      To demonstrate, compile this definition and execute the following program with +krun:

      +
      contentsOfJar(Jar(Apple()))
+
      +

      You will see when you run it that the program returns Apple(), because that +is the pattern that was matched by F.

      +

      Exercises

      +
        +
      1. Extend the definition in lesson-02-a.k with the addition of blackberries +and kiwis. For simplicity, blackberries are black and kiwis are green. Then +compile your definition and test that your additional fruits are correctly +handled by the colorOf function.
        2. +
      2. Create a new definition which defines an outfit as a multi-argument +constructor consisting of a hat, shirt, pants, and shoes. Define a new sort, +Boolean, with two constructors, true and false. Each of hat, shirt, pants, +and shoes will have a single argument (a color), either black or +white. Then define an outfitMatching function that will return true if all +the pieces of the outfit are the same color. You do not need to define the +case that returns false. Write some tests that your function behaves the way +you expect.
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.3: BNF Syntax and Parser Generation.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/03_parsing/index.html b/k-distribution/k-tutorial/1_basic/03_parsing/index.html new file mode 100644 index 00000000000..987344e93d5 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/03_parsing/index.html @@ -0,0 +1,780 @@ + + + + + + + + + + + + + + +Lesson 1.3: BNF Syntax and Parser Generation | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.3: BNF Syntax and Parser Generation

      +

      The purpose of this lesson is to explain the full syntax and semantics of +productions in K as well as how productions and other syntactic +sentences can be used to define grammars for use parsing both rules as well +as programs.

      +

      K's approach to parsing

      +

      K's grammar is divided into two components: the outer syntax of K and the +inner syntax of K. Outer syntax refers to the parsing of requires, +modules, imports, and sentences in a K definition. Inner syntax +refers to the parsing of rules and programs. Unlike the outer syntax of +K, which is predetermined, much of the inner syntax of K is defined by you, the +developer. When rules or programs are parsed, they are parsed within the +context of a module. Rules are parsed in the context of the module in which +they exist, whereas programs are parsed in the context of the +main syntax module of a K definition. The productions and other syntactic +sentences in a module are used to construct the grammar of the module, which +is then used to perform parsing.

      +

      Basic BNF productions

      +

      To illustrate how this works, we will consider a simple K definition which +defines a relatively basic calculator capable of evaluating Boolean expressions +containing and, or, not, and xor.

      +

      Input the following program into your editor as file lesson-03-a.k:

      +
      k
      module LESSON-03-A
+
+  syntax Boolean ::= "true" | "false"
+                   | "!" Boolean [function]
+                   | Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+
      +

      You will notice that the productions in this file look a little different than +the ones from the previous lesson. In point of fact, K has two different +mechanisms for defining productions. We have previously been focused +exclusively on the first mechanism, where the ::= symbol is followed by an +alphanumeric identifier followed by a comma-separated list of sorts in +parentheses. However, this is merely a special case of a more generic mechanism +for defining the syntax of productions using a variant of +BNF Form.

      +

      For example, in the previous lesson, we had the following set of productions:

      +
      k
      module LESSON-03-B
+  syntax Color ::= Yellow() | Blue()
+  syntax Fruit ::= Banana() | Blueberry()
+  syntax Color ::= colorOf(Fruit) [function]
+endmodule
+
      +

      It turns out that this is equivalent to the following definition which defines +the same grammar, but using BNF notation:

      +
      k
      module LESSON-03-C
+  syntax Color ::= "Yellow" "(" ")" | "Blue" "(" ")"
+  syntax Fruit ::= "Banana" "(" ")" | "Blueberrry" "(" ")"
+  syntax Color ::= "colorOf" "(" Fruit ")" [function]
+endmodule
+
      +

      In this example, the sorts of the argument to the function are unchanged, but +everything else has been wrapped in double quotation marks. This is because +in BNF notation, we distinguish between two types of production items: +terminals and non-terminals. A terminal represents simply a literal +string of characters that is verbatim part of the syntax of that production. +A non-terminal, conversely, represents a sort name, where the syntax of that +production accepts any valid term of that sort at that position.

      +

      This is why, when we wrote the program colorOf(Banana()), krun was able to +execute that program: because it represented a term of sort Color that was +parsed and interpreted by K's interpreter. In other words, krun parses and +interprets terms according to the grammar defined by the developer. It is +automatically converted into an AST of that term, and then the colorOf +function is evaluated using the function rules provided in the definition.

      +

      You can ask yourself: How does K match the strings between the double quotes? +The answer is that K uses Flex to generate a scanner for the grammar. Flex looks +for the longest possible match of a regular expression in the input. If there +are ambiguities between 2 or more regular expressions, it will pick the one with +the highest prec attribute. You can learn more about how Flex matching works +here.

      +

      Bringing us back to the file lesson-03-a.k, we can see that this grammar +has given a simple BNF grammar for expressions over Booleans. We have defined +constructors corresponding to the Boolean values true and false, and functions +corresponding to the Boolean operators for and, or, not, and xor. We have also +given a syntax for each of these functions based on their syntax in the C +programming language. As such, we can now write programs in the simple language +we have defined.

      +

      Input the following program into your editor as and.bool in the same +directory:

      +
      true && false
+
      +

      We cannot interpret this program yet, because we have not given rules defining +the meaning of the && function yet, but we can parse it. To do this, you can +run (from the same directory):

      +
      kast --output kore and.bool
+
      +

      kast is K's just-in-time parser. It will generate a grammar from your K +definition on the fly and use it to parse the program passed on the command +line. The --output flag controls how the resulting AST is represented; don't +worry about the possible values yet, just use kore.

      +

      You ought to get the following AST printed on standard output, minus the +formatting:

      +
      inj{SortBoolean{}, SortKItem{}}(
+  Lbl'UndsAnd-And-UndsUnds'LESSON-03-A'Unds'Boolean'Unds'Boolean'Unds'Boolean{}(
+    Lbltrue'Unds'LESSON-03-A'Unds'Boolean{}(),
+    Lblfalse'Unds'LESSON-03-A'Unds'Boolean{}()
+  )
+)
+
      +

      Don't worry about what exactly this means yet, just understand that it +represents the AST of the program that you just parsed. You ought to be able +to recognize the basic shape of it by seeing the words true, false, and +And in there. This is Kore, the intermediate representation of K, and we +will cover it in detail later.

      +

      Note that you can also tell kast to print the AST in other formats. For a +more direct representation of the original K, while still maintaining the +structure of an AST, you can say kast --output kast and.bool. This will +yield the following output:

      +
      `_&&__LESSON-03-A_Boolean_Boolean_Boolean`(
+  `true_LESSON-03-A_Boolean`(.KList),
+  `false_LESSON-03-A_Boolean`(.KList)
+)
+
      +

      Note how the first output is largely a name-mangled version of the second +output. The one difference is the presence of the inj symbol in the KORE +output. We will talk more about this in later lessons.

      +

      Exercise

      +

      Parse the expression false || true with --output kast. See if you can +predict approximately what the corresponding output would be with +--output kore, then run the command yourself and compare it to your +prediction.

      +

      Ambiguities

      +

      Now let's try a slightly more advanced example. Input the following program +into your editor as and-or.bool:

      +
      true && false || false
+
      +

      When you try and parse this program, you ought to see the following error:

      +
      [Error] Inner Parser: Parsing ambiguity.
+1: syntax Boolean ::= Boolean "||" Boolean [function]
+
+`_||__LESSON-03-A_Boolean_Boolean_Boolean`(`_&&__LESSON-03-A_Boolean_Boolean_Boolean`(`true_LESSON-03-A_Boolean`(.KList),`false_LESSON-03-A_Boolean`(.KList)),`false_LESSON-03-A_Boolean`(.KList))
+2: syntax Boolean ::= Boolean "&&" Boolean [function]
+
+`_&&__LESSON-03-A_Boolean_Boolean_Boolean`(`true_LESSON-03-A_Boolean`(.KList),`_||__LESSON-03-A_Boolean_Boolean_Boolean`(`false_LESSON-03-A_Boolean`(.KList),`false_LESSON-03-A_Boolean`(.KList)))
+        Source(./and-or.bool)
+        Location(1,1,1,23)
+
      +

      This error is saying that kast was unable to parse this program because it is +ambiguous. K's just-in-time parser is a GLL parser, which means it can handle +the full generality of context-free grammars, including those grammars which +are ambiguous. An ambiguous grammar is one where the same string can be parsed +as multiple distinct ASTs. In this example, it can't decide whether it should +be parsed as (true && false) || false or as true && (false || false). As a +result, it reports the error to the user.

      +

      Brackets

      +

      Currently there is no way of resolving this ambiguity, making it impossible +to write complex expressions in this language. This is obviously a problem. +The standard solution in most programming languages to this problem is to +use parentheses to indicate the appropriate grouping. K generalizes this notion +into a type of production called a bracket. A bracket production in K +is any production with the bracket attribute. It is required that such a +production only have a single non-terminal, and the sort of the production +must equal the sort of that non-terminal. However, K does not otherwise +impose restrictions on the grammar the user provides for a bracket. With that +being said, the most common type of bracket is one in which a non-terminal +is surrounded by terminals representing some type of bracket such as +(), [], {}, <>, etc. For example, we can define the most common +type of bracket, the type used by the vast majority of programming languages, +quite simply.

      +

      Consider the following modified definition, which we will save to +lesson-03-d.k:

      +
      k
      module LESSON-03-D
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   | "!" Boolean [function]
+                   | Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+
      +

      In this definition, if the user does not explicitly define parentheses, the +grammar remains ambiguous and K's just-in-time parser will report an error. +However, you are now able to parse more complex programs by means of explicitly +grouping subterms with the bracket we have just defined.

      +

      Consider and-or-left.bool:

      +
      (true && false) || false
+
      +

      Now consider and-or-right.bool:

      +
      true && (false || false)
+
      +

      If you parse these programs with kast, you will once again get a single +unique AST with no error. If you look, you might notice that the bracket itself +does not appear in the AST. In fact, this is a property unique to brackets: +productions with the bracket attribute are not represented in the parsed AST +of a term, and the child of the bracket is folded immediately into the parent +term. This is the reason for the requirement that a bracket production have +a single non-terminal of the same sort as the production itself.

      +

      Exercise

      +

      Write out what you expect the AST to be arising from parsing these two programs +above with --output kast, then parse them yourself and compare them to the +AST you expected. Confirm for yourself that the bracket production does not +appear in the AST.

      +

      Tokens

      +

      So far we have seen how we can define the grammar of a language. However, +the grammar is not the only relevant part of parsing a language. Also relevant +is the lexical syntax of the language. Thus far, we have implicitly been using +K's automatic lexer generation to generate a token in the scanner for each +terminal in our grammar. However, sometimes we wish to define more complex +lexical syntax. For example, consider the case of integers in C: an integer +consists of a decimal, octal, or hexadecimal number followed by an optional +suffix indicating the type of the literal.

      +

      In theory it would be possible to define this syntax via a grammar, but not +only would it be cumbersome and tedious, you would also then have to deal with +an AST generated for the literal which is not convenient to work with.

      +

      Instead of doing this, K allows you to define token productions, where +a production consists of a regular expression followed by the token +attribute, and the resulting AST consists of a typed string containing the +value recognized by the regular expression.

      +

      For example, the builtin integers in K are defined using the following +production:

      +
      .k .exclude
      syntax Int ::= r"[\\+-]?[0-9]+" [token]
+
      +

      Here we can see that we have defined that an integer is an optional sign +followed by a nonzero sequence of digits. The r preceding the terminal +indicates that what appears inside the double quotes is a regular expression, +and the token attribute indicates that terms which parse as this production +should be converted into a token by the parser.

      +

      It is also possible to define tokens that do not use regular expressions. This +can be useful when you wish to declare particular identifiers for use in your +semantics later. For example:

      +
      .k .exclude
      syntax Id ::= "main" [token]
+
      +

      Here, we declare that main is a token of sort Id. Instead of being parsed +as a symbol, it gets parsed as a token, generating a typed string in the AST. +This is useful in a semantics of C because the parser generally does not treat +the main function in C specially; only the semantics treats it specially.

      +

      Of course, languages can have more complex lexical syntax. For example, if we +wish to define the syntax of integers in C, we could use the following +production:

      +
      .k .exclude
      syntax IntConstant ::= r"(([1-9][0-9]*)|(0[0-7]*)|(0[xX][0-9a-fA-F]+))(([uU][lL]?)|([uU]((ll)|(LL)))|([lL][uU]?)|(((ll)|(LL))[uU]?))?" [token]
+
      +

      As you may have noted above, long and complex regular expressions +can be hard to read. They also suffer from the problem that unlike a grammar, +they are not particularly modular.

      +

      We can get around this restriction by declaring explicit regular expressions, +giving them a name, and then referring to them in productions.

      +

      Consider the following (equivalent) way to define the lexical syntax of +integers in C:

      +
      .k .exclude
      syntax IntConstant ::= r"({DecConstant}|{OctConstant}|{HexConstant})({IntSuffix}?)" [token]
+syntax lexical DecConstant = r"{NonzeroDigit}({Digit}*)"
+syntax lexical OctConstant = r"0({OctDigit}*)"
+syntax lexical HexConstant = r"{HexPrefix}({HexDigit}+)"
+syntax lexical HexPrefix = r"0x|0X"
+syntax lexical NonzeroDigit = r"[1-9]"
+syntax lexical Digit = r"[0-9]"
+syntax lexical OctDigit = r"[0-7]"
+syntax lexical HexDigit = r"[0-9a-fA-F]"
+syntax lexical IntSuffix = r"{UnsignedSuffix}({LongSuffix}?)|{UnsignedSuffix}{LongLongSuffix}|{LongSuffix}({UnsignedSuffix}?)|{LongLongSuffix}({UnsignedSuffix}?)"
+syntax lexical UnsignedSuffix = r"[uU]"
+syntax lexical LongSuffix = r"[lL]"
+syntax lexical LongLongSuffix = r"ll|LL"
+
      +

      As you can see, this is rather more verbose, but it has the benefit of both +being much easier to read and understand, and also increased modularity. +Note that we refer to a named regular expression by putting the name in curly +brackets. Note also that only the first sentence actually declares a new piece +of syntax in the language. When the user writes syntax lexical, they are only +declaring a regular expression. To declare an actual piece of syntax in the +grammar, you still must actually declare an explicit token production.

      +

      One final note: K uses Flex to implement +its lexical analysis. As a result, you can refer to the +Flex Manual +for a detailed description of the regular expression syntax supported. Note +that for performance reasons, Flex's regular expressions are actually a regular +language, and thus lack some of the syntactic convenience of modern +"regular expression" libraries. If you need features that are not part of the +syntax of Flex regular expressions, you are encouraged to express them via +a grammar instead.

      +

      Ahead-of-time parser generation

      +

      So far we have been entirely focused on K's support for just-in-time parsing, +where the parser is generated on the fly prior to being used. This benefits +from being faster to generate the parser, but it suffers in performance if you +have to repeatedly parse strings with the same parser. For this reason, it is +generally encouraged that when parsing programs, you use K's ahead-of-time +parser generation. K makes use of +GNU Bison to generate parsers.

      +

      By default, you can enable ahead-of-time parsing via the --gen-bison-parser +flag to kompile. This will make use of Bison's LR(1) parser generator. As +such, if your grammar is not LR(1), it may not parse exactly the same as if +you were to use the just-in-time parser, because Bison will automatically pick +one of the possible branches whenever it encounters a shift-reduce or +reduce-reduce conflict. In this case, you can either modify your grammar to be +LR(1), or you can enable use of Bison's GLR support by instead passing +--gen-glr-bison-parser to kompile. Note that if your grammar is ambiguous, +the ahead-of-time parser will not provide you with particularly readable error +messages at this time.

      +

      If you have a K definition named foo.k, and it generates a directory when +you run kompile called foo-kompiled, you can invoke the ahead-of-time +parser you generated by running foo-kompiled/parser_PGM <file> on a file.

      +

      Exercises

      +
        +
      1. +

        Compile lesson-03-d.k with ahead-of-time parsing enabled. Then compare +how long it takes to run kast --output kore and-or-left.bool with how long it +takes to run lesson-03-d-kompiled/parser_PGM and-or-left.bool. Confirm for +yourself that both produce the same result, but that the latter is faster.

        +
        2. +
      2. +

        Define a simple grammar consisting of integers, brackets, addition, +subtraction, multiplication, division, and unary negation. Integers should be +in decimal form and lexically without a sign, whereas negative numbers can be +represented via unary negation. Ensure that you are able to parse some basic +arithmetic expressions using a generated ahead-of-time parser. Do not worry +about disambiguating the grammar or about writing rules to implement the +operations in this definition.

        +
        3. +
      3. +

        Write a program where the meaning of the arithmetic expression based on +the grammar you defined above is ambiguous, and then write programs that +express each individual intended meaning using brackets.

        +
        4. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.4: Disambiguating Parses.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/04_disambiguation/index.html b/k-distribution/k-tutorial/1_basic/04_disambiguation/index.html new file mode 100644 index 00000000000..661fe7f2830 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/04_disambiguation/index.html @@ -0,0 +1,729 @@ + + + + + + + + + + + + + + +Lesson 1.4: Disambiguating Parses | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.4: Disambiguating Parses

      +

      The purpose of this lesson is to teach how to use K's builtin features for +disambiguation to transform an ambiguous grammar into an unambiguous one that +expresses the intended ASTs.

      +

      Priority blocks

      +

      In practice, very few formal languages outside the domain of natural language +processing are ambiguous. The main reason for this is that parsing unambiguous +languages is asymptotically faster than parsing ambiguous languages. +Programming language designers instead usually use the notions of operator +precedence and associativity to make expression grammars unambiguous. These +mechanisms work by instructing the parser to reject certain ASTs in favor of +others in case of ambiguities; it is often possible to remove all ambiguities +in a grammar with these techniques.

      +

      While it is sometimes possible to explicitly rewrite the grammar to remove +these parses, because K's grammar specification and AST generation are +inextricably linked, this is generally discouraged. Instead, we use the +approach of explicitly expressing the relative precedence of different +operators in different situations in order to resolve the ambiguity.

      +

      For example, in C, && binds tighter in precedence than ||, meaning that +the expression true && false || false has only one valid AST: +(true && false) || false.

      +

      Consider, then, the third iteration on the grammar of this definition +(lesson-04-a.k):

      +
      k
      module LESSON-04-A
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > Boolean "&&" Boolean [function]
+                   > Boolean "^" Boolean [function]
+                   > Boolean "||" Boolean [function]
+
+endmodule
+
      +

      In this example, some of the | symbols separating productions in a single +block have been replaced with >. This serves to describe the +priority groups associated with this block of productions. +The first priority group consists of the atoms of the +language: true, false, and the bracket operator. In general, a priority +group starts either at the ::= or > operator and extends until either the +next > operator or the end of the production block. Thus, we can see that the +second, third, fourth, and fifth priority groups in this grammar all consist +of a single production.

      +

      The meaning of these priority groups becomes apparent when parsing programs: +A symbol with a lesser priority, (i.e., one that binds looser), cannot +appear as the direct child of a symbol with a greater priority (i.e., +one that binds tighter. In this case, the > operator can be seen as a +greater-than operator describing a transitive partial ordering on the +productions in the production block, expressing their relative priority.

      +

      To see this more concretely, let's look again at the program +true && false || false. As noted before, previously this program was +ambiguous because the parser could either choose that && was the child of || +or vice versa. However, because a symbol with lesser priority (i.e., ||) +cannot appear as the direct child of a symbol with greater priority +(i.e., &&), the parser will reject the parse where || is under the +&& operator. As a result, we are left with the unambiguous parse +(true && false) || false. Similarly, true || false && false parses +unambiguously as true || (false && false). Conversely, if the user explicitly +wants the other parse, they can express this using brackets by explicitly +writing true && (false || false). This still parses successfully because the +|| operator is no longer the direct child of the && operator, but is +instead the direct child of the () operator, and the && operator is an +indirect parent, which is not subject to the priority restriction.

      +

      Astute readers, however, will already have noticed what seems to be a +contradiction: we have defined () as also having greater priority than ||. +One would think that this should mean that || cannot appear as a direct +child of (). This is a problem because priority groups are applied to every +possible parse separately. That is to say, even if the term is unambiguous +prior to this disambiguation rule, we still reject that parse if it violates +the rule of priority.

      +

      In fact, however, we do not reject this program as a parse error. Why is that? +Well, the rule for priority is slightly more complex than previously described. +In actual fact, it applies only conditionally. Specifically, it applies in +cases where the child is either the first or last production item in the +parent's production. For example, in the production Bool "&&" Bool, the +first Bool non-terminal is not preceded by any terminals, and the last Bool +non-terminal is not followed by any terminals. As a result of this, we apply +the priority rule to both children of &&. However, in the () operator, +the sole non-terminal is both preceded by and followed by terminals. As a +result, the priority rule is not applied when () is the parent. Because of +this, the program we mentioned above successfully parses.

      +

      Exercise

      +

      Parse the program true && false || false using kast, and confirm that the AST +places || as the top level symbol. Then modify the definition so that you +will get the alternative parse.

      +

      Associativity

      +

      Even having broken the expression grammar into priority blocks, the resulting +grammar is still ambiguous. We can see this if we try to parse the following +program (assoc.bool):

      +
      true && false && false
+
      +

      Priority blocks will not help us here: the problem comes between two parses +where both possible parses have a direct parent and child which is within a +single priority block (in this case, && is in the same block as itself).

      +

      This is where the notion of associativity comes into play. Associativity +applies the following additional rules to parses:

      +
        +
      • a left-associative symbol cannot appear as a direct rightmost child of a +symbol with equal priority;
        • +
      • a right-associative symbol cannot appear as a direct leftmost child of a +symbol with equal priority; and
        • +
      • a non-associative symbol cannot appear as a direct leftmost or rightmost +child of a symbol with equal priority.
        • +
      +

      In C, binary operators are all left-associative, meaning that the expression +true && false && false parses unambiguously as (true && false) && false, +because && cannot appear as the rightmost child of itself.

      +

      Consider, then, the fourth iteration on the grammar of this definition +(lesson-04-b.k):

      +
      k
      module LESSON-04-B
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > left: Boolean "&&" Boolean [function]
+                   > left: Boolean "^" Boolean [function]
+                   > left: Boolean "||" Boolean [function]
+
+endmodule
+
      +

      Here each priority group, immediately after the ::= or > operator, can +be followed by a symbol representing the associativity of that priority group: +either left: for left associativity, right: for right associativity, or +non-assoc: for non-associativity. In this example, each priority group we +apply associativity to has only a single production, but we could equally well +write a priority block with multiple productions and an associativity.

      +

      For example, consider the following, different grammar (lesson-04-c.k):

      +
      k
      module LESSON-04-C
+
+  syntax Boolean ::= "true" | "false"
+                   | "(" Boolean ")" [bracket]
+                   > "!" Boolean [function]
+                   > left:
+                     Boolean "&&" Boolean [function]
+                   | Boolean "^" Boolean [function]
+                   | Boolean "||" Boolean [function]
+
+endmodule
+
      +

      In this example, unlike the one above, &&, ^, and || have the same +priority. However, viewed as a group, the entire group is left associative. +This means that none of &&, ^, and || can appear as the right child of +any of &&, ^, or ||. As a result of this, this grammar is also not +ambiguous. However, it expresses a different grammar, and you are encouraged +to think about what the differences are in practice.

      +

      Exercise

      +

      Parse the program true && false && false yourself, and confirm that the AST +places the rightmost && at the top of the expression. Then modify the +definition to generate the alternative parse.

      +

      Explicit priority and associativity declarations

      +

      Previously we have only considered the case where all of the productions +which you wish to express a priority or associativity relation over are +co-located in the same block of productions. However, in practice this is not +always feasible or desirable, especially as a definition grows in size across +multiple modules.

      +

      As a result of this, K provides a second way of declaring priority and +associativity relations.

      +

      Consider the following grammar, which we will name lesson-04-d.k and which +will express the exact same grammar as lesson-04-b.k

      +
      k
      module LESSON-04-D
+
+  syntax Boolean ::= "true" [group(literal)] | "false" [group(literal)]
+                   | "(" Boolean ")" [group(atom), bracket]
+                   | "!" Boolean [group(not), function]
+                   | Boolean "&&" Boolean [group(and), function]
+                   | Boolean "^" Boolean [group(xor), function]
+                   | Boolean "||" Boolean [group(or), function]
+
+  syntax priorities literal atom > not > and > xor > or
+  syntax left and
+  syntax left xor
+  syntax left or
+endmodule
+
      +

      This introduces a couple of new features of K. First, the group(_) attribute +is used to conceptually group together sets of sentences under a common +user-defined name. For example, literal in the syntax priorities sentence is +used to refer to all the productions marked with the group(literal) attribute, +i.e., true and false. A production can belong to multiple groups using +syntax such as group(myGrp1,myGrp2).

      +

      Once we understand this, it becomes relatively straightforward to understand +the meaning of this grammar. Each syntax priorities sentence defines a +priority relation where > separates different priority groups. Each priority +group is defined by a list of one or more group names, and consists of all +productions which are members of at least one of those named groups.

      +

      In the same way, a syntax left, syntax right, or syntax non-assoc sentence +defines an associativity relation among left-, right-, or non-associative +groups. Specifically, this means that:

      +
      syntax left a b
+
      +

      is different to:

      +
      syntax left a
+syntax left b
+
      +

      As a consequence of this, syntax [left|right|non-assoc] should not be used to +group together labels with different priority.

      +

      Prefer/avoid

      +

      Sometimes priority and associativity prove insufficient to disambiguate a +grammar. In particular, sometimes it is desirable to be able to choose between +two ambiguous parses directly while still not rejecting any parses if the term +parsed is unambiguous. A good example of this is the famous "dangling else" +problem in imperative C-like languages.

      +

      Consider the following definition (lesson-04-E.k):

      +
      k
      module LESSON-04-E
+
+  syntax Exp ::= "true" | "false"
+  syntax Stmt ::= "if" "(" Exp ")" Stmt
+                | "if" "(" Exp ")" Stmt "else" Stmt
+                | "{" "}"
+endmodule
+
      +

      We can write the following program (dangling-else.if):

      +
      if (true) if (false) {} else {}
+
      +

      This is ambiguous because it is unclear whether the else clause is part of +the outer if or the inner if. At first we might try to resolve this with +priorities, saying that the if without an else cannot appear as a child of +the if with an else. However, because the non-terminal in the parent symbol +is both preceded and followed by a terminal, this will not work.

      +

      Instead, we can resolve the ambiguity directly by telling the parser to +"prefer" or "avoid" certain productions when ambiguities arise. For example, +when we parse this program, we see the following ambiguity as an error message:

      +
      [Error] Inner Parser: Parsing ambiguity.
+1: syntax Stmt ::= "if" "(" Exp ")" Stmt
+
+`if(_)__LESSON-04-E_Stmt_Exp_Stmt`(`true_LESSON-04-E_Exp`(.KList),`if(_)_else__LESSON-04-E_Stmt_Exp_Stmt_Stmt`(`false_LESSON-04-E_Exp`(.KList),`;_LESSON-04-E_Stmt`(.KList),`;_LESSON-04-E_Stmt`(.KList)))
+2: syntax Stmt ::= "if" "(" Exp ")" Stmt "else" Stmt
+
+`if(_)_else__LESSON-04-E_Stmt_Exp_Stmt_Stmt`(`true_LESSON-04-E_Exp`(.KList),`if(_)__LESSON-04-E_Stmt_Exp_Stmt`(`false_LESSON-04-E_Exp`(.KList),`;_LESSON-04-E_Stmt`(.KList)),`;_LESSON-04-E_Stmt`(.KList))
+        Source(./dangling-else.if)
+        Location(1,1,1,30)
+
      +

      Roughly, we see that the ambiguity is between an if with an else or an if +without an else. Since we want to pick the first parse, we can tell K to +"avoid" the second parse with the avoid attribute. Consider the following +modified definition (lesson-04-f.k):

      +
      k
      module LESSON-04-F
+
+  syntax Exp ::= "true" | "false"
+  syntax Stmt ::= "if" "(" Exp ")" Stmt
+                | "if" "(" Exp ")" Stmt "else" Stmt [avoid]
+                | "{" "}"
+endmodule
+
      +

      Here we have added the avoid attribute to the else production. As a result, +when an ambiguity occurs and one or more of the possible parses has that symbol +at the top of the ambiguous part of the parse, we remove those parses from +consideration and consider only those remaining. The prefer attribute behaves +similarly, but instead removes all parses which do not have that attribute. +In both cases, no action is taken if the parse is not ambiguous.

      +

      Exercises

      +
        +
      1. +

        Parse the program if (true) if (false) {} else {} using lesson-04-f.k +and confirm that else clause is part of the innermost if statement. Then +modify the definition so that you will get the alternative parse.

        +
        2. +
      2. +

        Modify your solution from Lesson 1.3, Exercise 2 so that unary negation should +bind tighter than multiplication and division, which should bind tighter than +addition and subtraction, and each binary operator should be left associative. +Write these priority and associativity declarations explicitly, and then +try to write them inline.

        +
        3. +
      3. +

        Write a simple grammar containing at least one ambiguity that cannot be +resolved via priority or associativity, and then use the prefer attribute to +resolve that ambiguity.

        +
        4. +
      4. +

        Explain why the following grammar is not labeled ambiguous by the K parser when parsing abb, then make the parser realize the ambiguity.

        +
        5. +
      +
      k
      module EXERCISE4
+
+syntax Expr ::= "a" Expr "b"
+              | "abb"
+              | "b"
+
+endmodule
+
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.5: Modules, Imports, and Requires.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/05_modules/index.html b/k-distribution/k-tutorial/1_basic/05_modules/index.html new file mode 100644 index 00000000000..c33e38db8a1 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/05_modules/index.html @@ -0,0 +1,670 @@ + + + + + + + + + + + + + + +Lesson 1.5: Modules, Imports, and Requires | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.5: Modules, Imports, and Requires

      +

      The purpose of this lesson is to explain how K definitions can be broken into +separate modules and files and how these distinct components combine into a +complete K definition.

      +

      K's outer syntax

      +

      Recall from Lesson 1.3 that K's grammar is broken +into two components: the outer syntax of K and the inner syntax of K. +Outer syntax, as previously mentioned, consists of requires, modules, +imports, and sentences. A K semantics is expressed by the set of +sentences contained in the definition. The scope of what is considered +contained in that definition is determined both by the main semantics +module of a K definition, as well as the requires and imports present +in the file that contains that module.

      +

      Basic module syntax

      +

      The basic unit of grouping sentences in K is the module. A module consists +of a module name, an optional list of attributes, a list of +imports, and a list of sentences.

      +

      A module name consists of one or more groups of letters, numbers, or +underscores, separated by a hyphen. Here are some valid module names: FOO, +FOO-BAR, foo0, foo0_bar-Baz9. Here are some invalid module names: -, +-FOO, BAR-, FOO--BAR. Stylistically, modules names are usually all +uppercase with hyphens separating words, but this is not strictly enforced.

      +

      Some example modules include an empty module:

      +
      k
      module LESSON-05-A
+
+endmodule
+
      +

      A module with some attributes:

      +
      k
      module LESSON-05-B [group(attr1,attr2), private]
+
+endmodule
+
      +

      A module with some sentences:

      +
      k
      module LESSON-05-C
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+  rule not true => false
+  rule not false => true
+endmodule
+
      +

      Imports

      +

      Thus far we have only discussed definitions containing a single module. +Definitions can also contain multiple modules, in which one module imports +others.

      +

      An import in K appears at the top of a module, prior to any sentences. It can +be specified with the imports keyword, followed by a module name.

      +

      For example, here is a simple definition with two modules (lesson-05-d.k):

      +
      k
      module LESSON-05-D-1
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+endmodule
+
+module LESSON-05-D
+  imports LESSON-05-D-1
+
+  rule not true => false
+  rule not false => true
+endmodule
+
      +

      This K definition is equivalent to the definition expressed by the single module +LESSON-05-C. Essentially, by importing a module, we include all of the +sentences in the module being imported into the module that we import from. +There are a few minor differences between importing a module and simply +including its sentences in another module directly, but we will cover these +differences later. Essentially, you can think of modules as a way of +conceptually grouping sentences in a larger K definition.

      +

      Exercise

      +

      Modify lesson-05-d.k to include four modules: one containing the syntax, two +with one rule each that imports the first module, and a final module +LESSON-05-D containing no sentences that imports the second and third module. +Check to make sure the definition still compiles and that you can still evaluate +the not function.

      +

      Parsing in the presence of multiple modules

      +

      As you may have noticed, each module in a definition can express a distinct set +of syntax. When parsing the sentences in a module, we use the syntax +of that module, enriched with the basic syntax of K, in order to parse +rules in that module. For example, the following definition is a parser error +(lesson-05-e.k):

      +
      .k .error
      module LESSON-05-E-1
+  rule not true => false
+  rule not false => true
+endmodule
+
+module LESSON-05-E-2
+  syntax Boolean ::= "true" | "false"
+  syntax Boolean ::= "not" Boolean [function]
+endmodule
+
      +

      This is because the syntax referenced in module LESSON-05-E-1, namely, not, +true, and false, is not imported by that module. You can solve this problem +by simply importing the modules containing the syntax you want to use in your +sentences.

      +

      Main syntax and semantics modules

      +

      When we are compiling a K definition, we need to know where to start. We +designate two specific entry point modules: the main syntax module +and the main semantics module. The main syntax module, as well as all the +modules it imports recursively, are used to create the parser for programs that +you use to parse programs that you execute with krun. The main semantics +module, as well as all the modules it imports recursively, are used to +determine the rules that can be applied at runtime in order to execute a +program. For example, in the above example, if the main semantics module is +module LESSON-05-D-1, then not is an uninterpreted function (i.e., has no +rules associated with it), and the rules in module LESSON-05-D are not +included.

      +

      While you can specify the entry point modules explicitly by passing the +--main-module and --syntax-module flags to kompile, by default, if you +type kompile foo.k, then the main semantics module will be FOO and the +main syntax module will be FOO-SYNTAX.

      +

      Splitting a definition into multiple files

      +

      So far, while we have discussed ways to break definitions into separate +conceptual components (modules), K also provides a mechanism for combining +multiple files into a single K definition, namely, the requires directive.

      +

      In K, the requires keyword has two meanings. The first, the requires +statement, appears at the top of a K file, prior to any module declarations. It +consists of the keyword requires followed by a double-quoted string. The +second meaning of the requires keyword will be covered in a later lesson, +but it is distinguished because the second case occurs only inside modules.

      +

      The string passed to the requires statement contains a filename. When you run +kompile on a file, it will look at all of the requires statements in that +file, look up those files on disk, parse them, and then recursively process all +the requires statements in those files. It then combines all the modules in all +of those files together, and uses them collectively as the set of modules to +which imports statements can refer.

      +

      Putting it all together

      +

      Putting it all together, here is one possible way in which we could break the +definition lesson-02-c.k from Lesson 1.2 into +multiple files and modules:

      +

      colors.k:

      +
      k
      module COLORS
+  syntax Color ::= Yellow()
+                 | Blue()
+endmodule
+
      +

      fruits.k:

      +
      k
      module FRUITS
+  syntax Fruit ::= Banana()
+                 | Blueberry()
+endmodule
+
      +

      colorOf.k:

      +
      .k .exclude
      requires "fruits.k"
+requires "colors.k"
+
+module COLOROF-SYNTAX
+  imports COLORS
+  imports FRUITS
+
+  syntax Color ::= colorOf(Fruit) [function]
+endmodule
+
+module COLOROF
+  imports COLOROF-SYNTAX
+
+  rule colorOf(Banana()) => Yellow()
+  rule colorOf(Blueberry()) => Blue()
+endmodule
+
      +

      You would then compile this definition with kompile colorOf.k and use it the +same way as the original, single-module definition.

      +

      Exercise

      +

      Modify the name of the COLOROF module, and then recompile the definition. +Try to understand why you now get a compiler error. Then, resolve this compiler +error by passing the --main-module and --syntax-module flags to kompile.

      +

      Include path

      +

      One note can be made about how paths are resolved in requires statements.

      +

      By default, the path you specify is allowed to be an absolute or a relative +path. If the path is absolute, that exact file is imported. If the path is +relative, a matching file is looked for within all of the +include directories specified to the compiler. By default, the include +directories include the current working directory, followed by the +include/kframework/builtin directory within your installation of K. You can +also pass one or more directories to kompile via the -I command line flag, +in which case these directories are prepended to the beginning of the list.

      +

      Exercises

      +
        +
      1. +

        Take the solution to Lesson 1.4, Exercise 2 which included the explicit +priority and associativity declarations, and modify the definition so that +the syntax of integers and brackets is in one module, the syntax of addition, +subtraction, and unary negation is in another module, and the syntax of +multiplication and division is in a third module. Make sure you can still parse +the same set of expressions as before. Place priority declarations in the main +module.

        +
        2. +
      2. +

        Modify lesson-02-d.k from Lesson 1.2 so that the rules and syntax are in +separate modules in separate files.

        +
        3. +
      3. +

        Place the file containing the syntax from Exercise 2 in another directory, +then recompile the definition. Observe why a compilation error occurs. Then +fix the compiler error by passing -I to kompile.

        +
        4. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.6: Integers and Booleans.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/06_ints_and_bools/index.html b/k-distribution/k-tutorial/1_basic/06_ints_and_bools/index.html new file mode 100644 index 00000000000..6dc5ea314f6 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/06_ints_and_bools/index.html @@ -0,0 +1,608 @@ + + + + + + + + + + + + + + +Lesson 1.6: Integers and Booleans | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.6: Integers and Booleans

      +

      The purpose of this lesson is to explain the two most basic types of builtin +sorts in K, the Int sort and the Bool sort, representing +arbitrary-precision integers and Boolean algebra.

      +

      Builtin sorts in K

      +

      K provides definitions of some useful sorts in +domains.md, found in the +include/kframework/builtin directory of the K installation. This file is +defined via a +Literate programming +style that we will discuss in a future lesson. We will not cover all of the +sorts found there immediately, however, this lesson discusses some of the +details surrounding integers and Booleans, as well as providing information +about how to look up more detailed knowledge about builtin functions in K's +documentation.

      +

      Booleans in K

      +

      The most basic builtin sort K provides is the Bool sort, representing +Boolean values (i.e., true and false). You have already seen how we were +able to create this type ourselves using K's parsing and disambiguation +features. However, in the vast majority of cases, we prefer instead to import +the version of Boolean algebra defined by K itself. Most simply, you can do +this by importing the module BOOL in your definition. For example +(lesson-06-a.k):

      +
      k
      module LESSON-06-A
+  imports BOOL
+
+  syntax Fruit ::= Blueberry() | Banana()
+  syntax Bool ::= isBlue(Fruit) [function]
+
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+endmodule
+
      +

      Here we have defined a simple predicate, i.e., a function returning a +Boolean value. We are now able to perform the usual Boolean operations of +and, or, and not over these values. For example (lesson-06-b.k):"

      +
      k
      module LESSON-06-B
+  imports BOOL
+
+  syntax Fruit ::= Blueberry() | Banana()
+  syntax Bool ::= isBlue(Fruit) [function]
+
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+
+  syntax Bool ::= isYellow(Fruit) [function]
+                | isBlueOrYellow(Fruit) [function]
+
+  rule isYellow(Banana()) => true
+  rule isYellow(Blueberry()) => false
+
+  rule isBlueOrYellow(F) => isBlue(F) orBool isYellow(F)
+endmodule
+
      +

      In the above example, Boolean inclusive or is performed via the orBool +function, which is defined in the BOOL module. As a matter of convention, +many functions over builtin sorts in K are suffixed with the name of the +primary sort over which those functions are defined. This happens so that the +syntax of K does not (generally) conflict with the syntax of any other +programming language, which would make it harder to define that programming +language in K.

      +

      Exercise

      +

      Write a function isBlueAndNotYellow which computes the appropriate Boolean +expression. If you are unsure what the appropriate syntax is to use, you +can refer to the BOOL module in +domains.md. Add a term of +sort Fruit for which isBlue and isYellow both return true, and test that +the isBlueAndNotYellow function behaves as expected on all three Fruits.

      +

      Syntax Modules

      +

      For most sorts in domains.md, K defines more than one module that can be +imported by users. For example, for the Bool sort, K defines the BOOL +module that has previously already been discussed, but also provides the +BOOL-SYNTAX module. This module, unlike the BOOL module, only declares the +values true and false, but not any of the functions that operate over the +Bool sort. The rationale is that you may want to import this module into the +main syntax module of your definition in some cases, whereas you generally do +not want to do this with the version of the module that includes all the +functions over the Bool sort. For example, if you were defining the semantics +of C++, you might import BOOL-SYNTAX into the syntax module of your +definition, because true and false are part of the grammar of C++, but +you would only import the BOOL module into the main semantics module, because +C++ defines its own syntax for and, or, and not that is different from the +syntax defined in the BOOL module.

      +

      Here, for example, is how we might redefine our Boolean expression calculator +to use the Bool sort while maintaining an idiomatic structure of modules +and imports, for the first time including the rules to calculate the values of +expressions themselves (lesson-06-c.k):

      +
      k
      module LESSON-06-C-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Bool ::= "(" Bool ")" [bracket]
+                > "!" Bool [function]
+                > left:
+                  Bool "&&" Bool [function]
+                | Bool "^" Bool [function]
+                | Bool "||" Bool [function]
+endmodule
+
+module LESSON-06-C
+  imports LESSON-06-C-SYNTAX
+  imports BOOL
+
+  rule ! B => notBool B
+  rule A && B => A andBool B
+  rule A ^ B => A xorBool B
+  rule A || B => A orBool B
+endmodule
+
      +

      Note the encapsulation of syntax: the LESSON-06-C-SYNTAX module contains +exactly the syntax of our Boolean expressions, and no more, whereas any other +syntax needed to implement those functions is in the LESSON-06-C module +instead.

      +

      Exercise

      +

      Add an "implies" function to the above Boolean expression calculator, using the +-> symbol to represent implication. You can look up K's builtin "implies" +function in the BOOL module in domains.md.

      +

      Integers in K

      +

      Unlike most programming languages, where the most basic integer type is a +fixed-precision integer type, the most commonly used integer sort in K is +the Int sort, which represents the mathematical integers, ie, +arbitrary-precision integers.

      +

      K provides three main modules for import when using the Int sort. The first, +containing all the syntax of integers as well as all of the functions over +integers, is the INT module. The second, which provides just the syntax +of integer literals themselves, is the INT-SYNTAX module. However, unlike +most builtin sorts in K, K also provides a third module for the Int sort: +the UNSIGNED-INT-SYNTAX module. This module provides only the syntax of +non-negative integers, i.e., natural numbers. The reasons for this involve +lexical ambiguity. Generally speaking, in most programming languages, -1 is +not a literal, but instead a literal to which the unary negation operator is +applied. K thus provides this module to ease in specifying the syntax of such +languages.

      +

      For detailed information about the functions available over the Int sort, +refer to domains.md. Note again how we append Int to the end of most of the +integer operations to ensure they do not collide with the syntax of other +programming languages.

      +

      Exercises

      +
        +
      1. +

        Extend your solution from Lesson 1.4, Exercise 2 to implement the rules +that define the behavior of addition, subtraction, multiplication, and +division. Do not worry about the case when the user tries to divide by zero +at this time. Use /Int to implement division. Test your new calculator +implementation by executing the arithmetic expressions you wrote as part of +Lesson 1.3, Exercise 2. Check to make sure each computes the value you expected.

        +
        2. +
      2. +

        Combine the Boolean expression calculator from this lesson with your +solution to Exercise 1, and then extend the combined calculator with the <, +<=, >, >=, ==, and != expressions. Write some Boolean expressions +that combine integer and Boolean operations, and test to ensure that these +expressions return the expected truth value.

        +
        3. +
      3. +

        Compute the following expressions using your solution from Exercise 2: +7 / 3, 7 / -3, -7 / 3, -7 / -3. Then replace the /Int function in +your definition with divInt instead, and observe how the value of the above +expressions changes. Why does this occur?

        +
        4. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.7: Side Conditions and Rule Priority.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/07_side_conditions/index.html b/k-distribution/k-tutorial/1_basic/07_side_conditions/index.html new file mode 100644 index 00000000000..9482aa3ac61 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/07_side_conditions/index.html @@ -0,0 +1,621 @@ + + + + + + + + + + + + + + +Lesson 1.7: Side Conditions and Rule Priority | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.7: Side Conditions and Rule Priority

      +

      The purpose of this lesson is to explain how to write conditional rules in K, +and to explain how to control the order in which rules are tried.

      +

      Side Conditions

      +

      So far, all of the rules we have discussed have been unconditional rules. +If the left-hand side of the rule matches the arguments to the function, the +rule applies. However, there is another type of rule, a conditional rule. +A conditional rule consists of a rule body containing the patterns to +match, and a side condition representing a Boolean expression that must +evaluate to true in order for the rule to apply.

      +

      Side conditions in K are introduced via the requires keyword immediately +following the rule body. For example, here is a rule with a side condition +(lesson-07-a.k):

      +
      k
      module LESSON-07-A
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90
+endmodule
+
      +

      In this case, the gradeFromPercentile function takes a single integer +argument. The function evaluates to letter-A if the argument passed is +greater than 90. Note that the side condition is allowed to refer to variables +that appear on the left-hand side of the rule. In the same manner as variables +appearing on the right-hand side, variables that appear in the side condition +evaluate to the value that was matched on the left-hand side. Then the +functions in the side condition are evaluated, which returns a term of sort +Bool. If the term is equal to true, then the rule applies. Bear in mind +that the side condition is only evaluated at all if the patterns on the +left-hand side of the rule match the term being evaluated.

      +

      Exercise

      +

      Write a rule that evaluates gradeFromPercentile to letter-B if the argument +to the function is in the range [80,90). Test that the function correctly +evaluates various numbers between 80 and 100.

      +

      owise Rules

      +

      So far, all the rules we have introduced have had the same priority. What +this means is that K does not necessarily enforce an order in which the rules +are tried. We have only discussed functions so far in K, so it is not +immediately clear why this choice was made, given that a function is not +considered well-defined if multiple rules for evaluating it are capable of +evaluating the same arguments to different results. However, in future lessons +we will discuss other types of rules in K, some of which can be +non-deterministic. What this means is that if more than one rule is capable +of matching, then K will explore both possible rules in parallel, and consider +each of their respective results when executing your program. Don't worry too +much about this right now, but just understand that because of the potential +later for nondeterminism, we don't enforce a total ordering on the order in +which rules are attempted to be applied.

      +

      However, sometimes this is not practical; It can be very convenient to express +that a particular rule applies if no other rules for that function are +applicable. This can be expressed by adding the owise attribute to a rule. +What this means, in practice, is that this rule has lower priority than other +rules, and will only be tried to be applied after all the other, +higher-priority rules have been tried and they have failed.

      +

      For example, in the above exercise, we had to add a side condition containing +two Boolean comparisons to the rule we wrote to handle letter-B grades. +However, in practice this meant that we compare the percentile to 90 twice. We +can more efficiently and more idiomatically write the letter-B case for the +gradeFromPercentile rule using the owise attribute (lesson-07-b.k):

      +
      k
      module LESSON-07-B
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [owise]
+endmodule
+
      +

      This rule is saying, "if all the other rules do not apply, then the grade is a +B if the percentile is greater than or equal to 80." Note here that we use both +a side condition and an owise attribute on the same rule. This is not +required (as we will see later), but it is allowed. What this means is that the +side condition is only tried if the other rules did not apply and the +left-hand side of the rule matched. You can even use more complex matching on +the left-hand side than simply a variable. More generally, you can also have +multiple higher-priority rules, or multiple owise rules. What this means in +practice is that all of the non-owise rules are tried first, in any order, +followed by all the owise rules, in any order.

      +

      Exercise

      +

      The grades D and F correspond to the percentile ranges [60, 70) and [0, 60) +respectively. Write another implementation of gradeFromPercentile which +handles only these cases, and uses the owise attribute to avoid redundant +Boolean comparisons. Test that various percentiles in the range [0, 70) are +evaluated correctly.

      +

      Rule Priority

      +

      As it happens, the owise attribute is a specific case of a more general +concept we call rule priority. In essence, each rule is assigned an integer +priority. Rules are tried in increasing order of priority, starting with a +rule with priority zero, and trying each increasing numerical value +successively.

      +

      By default, a rule is assigned a priority of 50. If the rule has the owise +attribute, it is instead given the priority 200. You can see why this will +cause owise rules to be tried after regular rules.

      +

      However, it is also possible to directly assign a numerical priority to a rule +via the priority attribute. For example, here is an alternative way +we could express the same two rules in the gradeFromPercentile function +(lesson-07-c.k):

      +
      k
      module LESSON-07-C
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90 [priority(50)]
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [priority(200)]
+endmodule
+
      +

      We can, of course, assign a priority equal to any non-negative integer. For +example, here is a more complex example that handles the remaining grades +(lesson-07-d.k):

      +
      k
      module LESSON-07-D
+  imports BOOL
+  imports INT
+
+  syntax Grade ::= "letter-A"
+                 | "letter-B"
+                 | "letter-C"
+                 | "letter-D"
+                 | "letter-F"
+                 | gradeFromPercentile(Int) [function]
+
+  rule gradeFromPercentile(I) => letter-A requires I >=Int 90 [priority(50)]
+  rule gradeFromPercentile(I) => letter-B requires I >=Int 80 [priority(51)]
+  rule gradeFromPercentile(I) => letter-C requires I >=Int 70 [priority(52)]
+  rule gradeFromPercentile(I) => letter-D requires I >=Int 60 [priority(53)]
+  rule gradeFromPercentile(_) => letter-F                     [priority(54)]
+endmodule
+
      +

      Note that we have introduced a new piece of syntax here: _. This is actually +just a variable. However, as a special case, when a variable is named _, it +does not bind a value that can be used on the right-hand side of the rule, or +in a side condition. Effectively, _ is a placeholder variable that means "I +don't care about this term."

      +

      In this example, we have explicitly expressed the order in which the rules of +this function are tried. Since rules are tried in increasing numerical +priority, we first try the rule with priority 50, then 51, then 52, 53, and +finally 54.

      +

      As a final note, remember that if you assign a rule a priority higher than 200, +it will be tried after a rule with the owise attribute, and if you assign +a rule a priority less than 50, it will be tried before a rule with no +explicit priority.

      +

      Exercises

      +
        +
      1. +

        Write a function isEven that returns whether an integer is an even number. +Use two rules and one side condition. The right-hand side of the rules should +be Boolean literals. Refer back to +domains.md for the relevant +integer operations.

        +
        2. +
      2. +

        Modify the calculator application from Lesson 1.6, Exercise 2, so that division +by zero will no longer make krun crash with a "Divison by zero" exception. +Instead, the / function should not match any of its rules if the denominator +is zero.

        +
        3. +
      3. +

        Write your own implementation of ==, <, <=, >, >= for integers and modify your solution from Exercise 2 to use it. +You can use any arithmetic operations in the INT module, but do not use any built-in boolean functions for comparing integers.

        +

        Hint: Use pattern matching and recursive definitions with rule priorities.

        +
        4. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.8: Literate Programming with Markdown.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/08_literate_programming/index.html b/k-distribution/k-tutorial/1_basic/08_literate_programming/index.html new file mode 100644 index 00000000000..406d20b81c2 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/08_literate_programming/index.html @@ -0,0 +1,571 @@ + + + + + + + + + + + + + + +Lesson 1.8: Literate Programming with Markdown | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.8: Literate Programming with Markdown

      +

      The purpose of this lesson is to teach a paradigm for performing literate +programming in K, and explain how this can be used to create K definitions +that are also documentation.

      +

      Markdown and K

      +

      The K tutorial so far has been written in +Markdown. Markdown, +for those not already familiar, is a lightweight plain-text format for styling +text. From this point onward, we assume you are familiar with Markdown and how +to write Markdown code. You can refer to the above link for a tutorial if you +are not already familiar.

      +

      What you may not necessarily realize, however, is that the K tutorial is also +a sequence of K definitions written in the manner of +Literate Programming. +For detailed information about Literate Programming, you can read the linked +Wikipedia article, but the short summary is that literate programming is a way +of intertwining documentation and code together in a manner that allows +executable code to also be, simultaneously, a documented description of that +code.

      +

      K is provided with built-in support for literate programming using Markdown. +By default, if you pass a file with the .md file extension to kompile, it +will look for any code blocks containing k code in that file, extract out +that K code into pure K, and then compile it as if it were a .k file.

      +

      A K code block begins with a line of text containing the keyword ```k, +and ends when it encounters another ``` keyword.

      +

      For example, if you view the markdown source of this document, this is a K +code block:

      +
      k
      module LESSON-08
+  imports INT
+
      +

      Only the code inside K code blocks will actually be sent to the compiler. The +rest, while it may appear in the document when rendered by a markdown viewer, +is essentially a form of code comment.

      +

      When you have multiple K code blocks in a document, K will append each one +together into a single file before passing it off to the outer parser.

      +

      For example, the following code block contains sentences that are part of the +LESSON-08 module that we declared the beginning of above:

      +
      k
        syntax Int ::= Int "+" Int [function]
+  rule I1 + I2 => I1 +Int I2
+
      +

      Exercise

      +

      Compile this file with kompile README.md --main-module LESSON-08. Confirm +that you can use the resulting compiled definition to evaluate the + +function.

      +

      Markdown Selectors

      +

      On occasion, you may want to generate multiple K definitions from a single +Markdown file. You may also wish to include a block of syntax-highlighted K +code that nonetheless does not appear as part of your K definition. It is +possible to accomplish this by means of the built-in support for syntax +highlighting in Markdown. Markdown allows a code block that was begun with +``` to be immediately followed by a string which is used to signify what +programming language the following code is written in. However, this feature +actually allows arbitrary text to appear describing that code block. Markdown +parsers are able to parse this text and render the code block differently +depending on what text appears after the backticks.

      +

      In K, you can use this functionality to specify one or more +Markdown selectors which are used to describe the code block. A Markdown +selector consists of a sequence of characters containing letters, numbers, and +underscores. A code block can be designated with a single selector by appending +the selector immediately following the backticks that open the code block.

      +

      For example, here is a code block with the foo selector:

      +
      foo
      foo bar
+
      +

      Note that this is not K code. By convention, K code should have the k +selector on it. You can express multiple selectors on a code block by putting +them between curly braces and prepending each with the . character. For +example, here is a code block with the foo and k selectors:

      +
      .k .foo
        syntax Int ::= foo(Int) [function]
+  rule foo(0) => 0
+
      +

      Because this code block contains the k Markdown selector, by default it is +included as part of the K definition being compiled.

      +

      Exercise

      +

      Confirm this fact by using krun to evaluate foo(0).

      +

      Markdown Selector Expressions

      +

      By default, as previously stated, K includes in the definition any code block +with the k selector. However, this is merely a specific instance of a general +principle, namely, that K allows you to control which selectors get included +in your K definition. This is done by means of the --md-selector flag to +kompile. This flag accepts a Markdown selector expression, which you +can essentially think of as a kind of Boolean algebra over Markdown selectors. +Each selector becomes an atom, and you can combine these atoms via the &, +|, !, and () operators.

      +

      Here is a grammar, written in K, of the language of Markdown selector +expressions:

      +
      .k .selector
        syntax Selector ::= r"[0-9a-zA-Z_]+" [token]
+  syntax SelectorExp ::= Selector
+                       | "(" SelectorExp ")" [bracket]
+                       > right:
+                         "!" SelectorExp
+                       > right:
+                         SelectorExp "&" SelectorExp
+                       > right:
+                         SelectorExp "|" SelectorExp
+
      +

      Here is a selector expression that selects all the K code blocks in this +definition except the one immediately above:

      +
      k & (! selector)
+
      +

      Addendum

      +

      This code block exists in order to make the above lesson a syntactically valid +K definition. Consider why it is necessary.

      +
      k
      endmodule
+
      +

      Exercises

      +
        +
      1. +

        Compile this lesson with the selector expression k & (! foo) and confirm +that you get a parser error if you try to evaluate the foo function with the +resulting definition.

        +
        2. +
      2. +

        Compile Lesson 1.3 +as a K definition. Identify why it fails to compile. Then pass an appropriate +--md-selector to the compiler in order to make it compile.

        +
        3. +
      3. +

        Modify your calculator application from Lesson 1.7, Exercise 2, to be written +in a literate style. Consider what text might be appropriate to turn the +resulting markdown file into documentation for your calculator.

        +
        4. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.9: Unparsing and the format and color attributes.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/09_unparsing/index.html b/k-distribution/k-tutorial/1_basic/09_unparsing/index.html new file mode 100644 index 00000000000..c6b2f40ba07 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/09_unparsing/index.html @@ -0,0 +1,718 @@ + + + + + + + + + + + + + + +Lesson 1.9: Unparsing and the format and color attributes | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.9: Unparsing and the format and color attributes

      +

      The purpose of this lesson is to teach the user about how terms are +pretty-printed in K, and how the user can make adjustments to the default +settings for how to print specific terms.

      +

      Parsing, Execution, and Unparsing

      +

      When you use krun to interpret a program, the tool passes through three major +phases. In the first, parsing, the program itself is parsed using either kast +or an ahead-of-time parser generated via Bison, and the resulting AST becomes +the input to the interpreter. In the second phase, execution, K evaluates +functions and (as we will discuss in depth later) performs rewrite steps to +iteratively transform the program state. The third and final phase is called +unparsing, because it consists of taking the final state of the application +after the program has been interpreted, and converting it from an AST back into +text that (in theory, anyway) could be parsed back into the same AST that was +the output of the execution phase.

      +

      In practice, parsing is not always precisely reversible. It turns out +(although we are not going to cover exactly why this is here), that +constructing a sound algorithm that takes a grammar and an AST and emits text +that could be parsed via that grammar to the original AST is an +NP-hard problem. As a result, in the interests of avoiding exponential time +algorithms when users rarely care about unparsing being completely sound, we +take certain shortcuts that provide a linear-time algorithm that approximates +a sound solution to the problem while sacrificing the notion that the result +can be parsed into the exact original term in all cases.

      +

      This is a lot of theoretical explanation, but at root, the unparsing process +is fairly simple: it takes a K term that is the output of execution and pretty +prints it according to the syntax defined by the user in their K definition. +This is useful because the original AST is not terribly user-readable, and it +is difficult to visualize the entire term or decipher information about the +final state of the program at a quick glance. Of course, in rare cases, the +pretty-printed configuration loses information of relevance, which is why K +allows you to obtain the original AST on request.

      +

      As an example of all of this, consider the following K definition +(lesson-09-a.k):

      +
      k
      module LESSON-09-A
+  imports BOOL
+
+  syntax Exp ::= "(" Exp ")" [bracket]
+               | Bool
+               > "!" Exp
+               > left:
+                 Exp "&&" Exp
+               | Exp "^" Exp
+               | Exp "||" Exp
+
+  syntax Exp ::= id(Exp) [function]
+  rule id(E) => E
+endmodule
+
      +

      This is similar to the grammar we defined in LESSON-06-C, with the difference +that the Boolean expressions are now constructors of sort Exp and we define a +trivial function over expressions that returns its argument unchanged.

      +

      We can now parse a simple program in this definition and use it to unparse some +Boolean expressions. For example (exp.bool):

      +
      id(true&&false&&!true^(false||true))
+
      +

      Here is a program that is not particularly legible at first glance, because all +extraneous whitespace has been removed. However, if we run krun exp.bool, we +see that the result of the unparser will pretty-print this expression rather +nicely:

      +
      <k>
+  true && false && ! true ^ ( false || true ) ~> .
+</k>
+
      +

      Notably, not only does K insert whitespace where appropriate, it is also smart +enough to insert parentheses where necessary in order to ensure the correct +parse. For example, without those parentheses, the expression above would parse +equivalent to the following one:

      +
      (((true && false) && ! true) ^ false) || true
+
      +

      Indeed, you can confirm this by passing that exact expression to the id +function and evaluating it, then looking at the result of the unparser:

      +
      <k>
+  true && false && ! true ^ false || true ~> .
+</k>
+
      +

      Here, because the meaning of the AST is the same both with and without +parentheses, K does not insert any parentheses when unparsing.

      +

      Exercise

      +

      Modify the grammar of LESSON-09-A above so that the binary operators are +right associative. Try unparsing exp.bool again, and note how the result is +different. Explain the reason for the difference.

      +

      Custom unparsing of terms

      +

      You may have noticed that right now, the unparsing of terms is not terribly +imaginative. All it is doing is taking each child of the term, inserting it +into the non-terminal positions of the production, then printing the production +with a space between each terminal or non-terminal. It is easy to see why this +might not be desirable in some cases. Consider the following K definition +(lesson-09-b.k):

      +
      k
      module LESSON-09-B
+  imports BOOL
+
+  syntax Stmt ::= "{" Stmt "}" | "{" "}"
+                > right:
+                  Stmt Stmt
+                | "if" "(" Bool ")" Stmt
+                | "if" "(" Bool ")" Stmt "else" Stmt [avoid]
+endmodule
+
      +

      This is a statement grammar, simplified to the point of meaninglessness, but +still useful as an object lesson in unparsing. Consider the following program +in this grammar (if.stmt):

      +
      if (true) {
+  if (true) {}
+  if (false) {}
+  if (true) {
+    if (false) {} else {}
+  } else {
+    if (false) {}
+  }
+}
+
      +

      This is how that term would be unparsed if it appeared in the output of krun:

      +
      if ( true ) { if ( true ) { } if ( false ) { } if ( true ) { if ( false ) { } else { } } else { if ( false ) { } } }
+
      +

      This is clearly much less legible than we started with! What are we to do? +Well, K provides an attribute, format, that can be applied to any production, +which controls how that production gets unparsed. You've seen how it gets +unparsed by default, but via this attribute, the developer has complete control +over how the term is printed. Of course, the user can trivially create ways to +print terms that would not parse back into the same term. Sometimes this is +even desirable. But in most cases, what you are interested in is controlling +the line breaking, indentation, and spacing of the production.

      +

      Here is an example of how you might choose to apply the format attribute +to improve how the above term is unparsed (lesson-09-c.k):

      +
      k
      module LESSON-09-C
+  imports BOOL
+
+  syntax Stmt ::= "{" Stmt "}" [format(%1%i%n%2%d%n%3)] | "{" "}" [format(%1%2)]
+                > right:
+                  Stmt Stmt [format(%1%n%2)]
+                | "if" "(" Bool ")" Stmt [format(%1 %2%3%4 %5)]
+                | "if" "(" Bool ")" Stmt "else" Stmt [avoid, format(%1 %2%3%4 %5 %6 %7)]
+endmodule
+
      +

      If we compile this new definition and unparse the same term, this is the +result we get:

      +
      if (true) {
+  if (true) {}
+  if (false) {}
+  if (true) {
+    if (false) {} else {}
+  } else {
+    if (false) {}
+  }
+}
+
      +

      This is the exact same text we started with! By adding the format attributes, +we were able to indent the body of code blocks, adjust the spacing of if +statements, and put each statement on a new line.

      +

      How exactly was this achieved? Well, each time the unparser reaches a term, +it looks at the format attribute of that term. That format attribute is a +mix of characters and format codes. Format codes begin with the % +character. Each character in the format attribute other than a format code is +appended verbatim to the output, and each format code is handled according to +its meaning, transformed (possibly recursively) into a string of text, and +spliced into the output at the position the format code appears in the format +string.

      +

      Provided for reference is a table with a complete list of all valid format +codes, followed by their meaning:

      + + + + + + + + + +
      Format Code Meaning
      n Insert '\n' followed by the current indentation + level
      i Increase the current indentation level by 1
      d Decrease the current indentation level by 1
      c Move to the next color in the list of colors for + this production (see next section)
      r Reset color to the default foreground color for + the terminal (see next section)
      an integer Print a terminal or non-terminal from the + production. The integer is treated as a 1-based + index into the terminals and non-terminals of + the production. +
      +
      If the offset refers to a terminal, move to the + next color in the list of colors for this + production, print the value of that terminal, + then reset the color to the default foreground + color for the terminal. +
      +
      If the offset refers to a regular expression + terminal, it is an error. +
      +
      If the offset refers to a non-terminal, unparse + the corresponding child of the current term + (starting with the current indentation level) + and print the resulting text, then set the + current color and indentation level to the color + and indentation level following unparsing that + term.
      other char Print that character verbatim
      +

      Exercise

      +

      Change the format attributes for LESSON-09-C so that if.stmt will unparse +as follows:

      +
      if (true)
+{
+  if (true)
+  {
+  }
+  if (false)
+  {
+  }
+  if (true)
+  {
+    if (false)
+    {
+    }
+    else
+    {
+    }
+  }
+  else
+  {
+    if (false)
+    {
+    }
+  }
+}
+
      +

      Output coloring

      +

      When the output of unparsing is displayed on a terminal supporting colors, K +is capable of coloring the output, similar to what is possible with a syntax +highlighter. This is achieved via the color and colors attributes.

      +

      Essentially, both the color and colors attributes are used to construct a +list of colors associated with each production, and then the format attribute +is used to control how those colors are used to unparse the term. At its most +basic level, you can set the color attribute to color all the terminals in +the production a certain color, or you can use the colors attribute to +specify a comma-separated list of colors for each terminal in the production. +At a more advanced level, the %c and %r format codes control how the +formatter interacts with the list of colors specified by the colors +attribute. You can essentially think of the color attribute as a way of +specifying that you want all the colors in the list to be the same color.

      +

      Note that the %c and %r format codes are relatively primitive in nature. +The color and colors attributes merely maintain a list of colors, whereas +the %c and %r format codes merely control how to advance through that list +and how individual text is colored.

      +

      It is an error if the colors attribute does not provide all the colors needed +by the terminals and escape codes in the production. %r does not change the +position in the list of colors at all, so the next %c will advance to the +following color.

      +

      As a complete example, here is a variant of LESSON-09-A which colors the +various boolean operators:

      +
      k
      module LESSON-09-D
+  imports BOOL
+
+  syntax Exp ::= "(" Exp ")" [bracket]
+               | Bool
+               > "!" Exp [color(yellow)]
+               > left:
+                 Exp "&&" Exp [color(red)]
+               | Exp "^" Exp [color(blue)]
+               | Exp "||" Exp [color(green)]
+
+  syntax Exp ::= id(Exp) [function]
+  rule id(E) => E
+endmodule
+
      +

      For a complete list of allowed colors, see +here.

      +

      Exercises

      +
        +
      1. +

        Use the color attribute on LESSON-09-C to color the keywords true and +false one color, the keywords if and else another color, and the operators +(, ), {, and } a third color.

        +
        2. +
      2. +

        Use the format, color, and colors attributes to tell the unparser to +style the expression grammar from Lesson 1.8, Exercise 3 according to your own +personal preferences for syntax highlighting and code formatting. You can +view the result of the unparser on a function term without evaluating that +function by means of the command kparse <file> | kore-print -.

        +
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.10: Strings.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/10_strings/index.html b/k-distribution/k-tutorial/1_basic/10_strings/index.html new file mode 100644 index 00000000000..15040a6698b --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/10_strings/index.html @@ -0,0 +1,554 @@ + + + + + + + + + + + + + + +Lesson 1.10: Strings | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.10: Strings

      +

      The purpose of this lesson is to explain how to use the String sort in K to +represent sequences of characters, and explain where to find additional +information about builtin functions over strings.

      +

      The String Sort

      +

      In addition to the Int and Bool sorts covered in +Lesson 1.6, K provides, among others, the +String sort to represent sequences of characters. You can import this +functionality via the STRING-SYNTAX module, which contains the syntax of +string literals in K, and the STRING module, which contains all the functions +that operate over the String type.

      +

      Strings in K are double-quoted. The following list of escape sequences is +supported:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      Escape SequenceMeaning
      \"The literal character "
      \\The literal character \
      \nThe newline character (ASCII code 0x0a)
      \rThe carriage return character (ASCII code 0x0d)
      \tThe tab character (ASCII code 0x09)
      \fThe form feed character (ASCII code 0x0c)
      \x00\x followed by 2 hexadecimal digits indicates a code point between 0x00 and 0xFF
      \u0000\u followed by 4 hexadecimal digits indicates a code point between 0x0000 and 0xFFFF
      \U00000000\U followed by 8 hexadecimal digits indicates a code point between 0x000000 and 0x10FFFF
      +

      Please note that as of the current moment, K's unicode support is not fully +complete, so you may run into errors using code points greater than 0xff.

      +

      As an example, you can construct a string literal containing the following +block of text:

      +
      This is an example block of text.
+Here is a quotation: "Hello world."
+	This line is indented.
+ÁÉÍÓÚ
+
      +

      Like so:

      +
      "This is an example block of text.\nHere is a quotation: \"Hello world.\"\n\tThis line is indented.\n\xc1\xc9\xcd\xd3\xda\n"
+
      +

      Basic String Functions

      +

      The full list of functions provided for the String sort can be found in +domains.md, but here we +describe a few of the more basic ones.

      +

      String concatenation

      +

      The concatenation operator for strings is +String. For example, consider +the following K rule that constructs a string from component parts +(lesson-10.k):

      +
      k
      module LESSON-10
+  imports STRING
+
+  syntax String ::= msg(String) [function]
+  rule msg(S) => "The string you provided: " +String S +String "\nHave a nice day!"
+endmodule
+
      +

      Note that this operator is O(N), so repeated concatenations are inefficient. +For information about efficient string concatenation, refer to +Lesson 2.14.

      +

      String length

      +

      The function to return the length of a string is lengthString. For example, +lengthString("foo") will return 3, and lengthString("") will return 0. +The return value is the length of the string in code points.

      +

      Substring computation

      +

      The function to compute the substring of a string is substrString. It +takes two string indices, starting from 0, and returns the substring within the +range [start..end). It is only defined if end >= start, start >= 0, and +end <= length of string. Here, for example, we return the first 5 characters +of a string:

      +
      substrString(S, 0, 5)
+
      +

      Here we return all but the first 3 characters:

      +
      substrString(S, 3, lengthString(S))
+
      +

      Exercises

      +
        +
      1. Write a function that takes a paragraph of text (i.e., a sequence of +sentences, each ending in a period), and constructs a new (nonsense) sentence +composed of the first word of each sentence, followed by a period. Do not +worry about capitalization or periods within the sentence which do not end the +sentence (e.g. "Dr."). You can assume that all whitespace within the paragraph +are spaces. For more information about the functions over strings required to +implement such a function, refer to domains.md.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.11: Casting Terms.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/11_casts/index.html b/k-distribution/k-tutorial/1_basic/11_casts/index.html new file mode 100644 index 00000000000..f4c7509dd94 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/11_casts/index.html @@ -0,0 +1,592 @@ + + + + + + + + + + + + + + +Lesson 1.11: Casting Terms | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.11: Casting Terms

      +

      The purpose of this lesson is to explain how to use cast expressions in +order to disambiguate terms using sort information. We also explain how the +variable sort inference algorithm works in K, and how to change the default +behavior by casting variables to a particular sort.

      +

      Casting in K

      +

      Sometimes the grammar you write for your rules in K can be a little bit +ambiguous on purpose. While grammars for programming languages may be +unambiguous when considered in their entirety, K allows you to write rules +involving arbitrary fragments of that grammar, and those fragments can +sometimes be ambiguous by themselves, or similar enough to other fragments +of the grammar to trigger ambiguity. As a result, in addition to the tools +covered in Lesson 1.4, K provides one +additional powerful tool for disambiguation: cast expressions.

      +

      K provides three main types of casts: the semantic cast, the strict cast, and +the projection cast. We will cover each of them, and their similarities and +differences, in turn.

      +

      Semantic casts

      +

      The most basic, and most common, type of cast in K is called the +semantic cast. For every sort S declared in a module, K provides the +following (implicit) production for use in sentences:

      +
        syntax S ::= S ":S"
+
      +

      Note that S simply represents the name of the sort. For example, if we +defined a sort Exp, the actual production for that sort would be:

      +
        syntax Exp ::= Exp ":Exp"
+
      +

      At runtime, this expression will not actually exist; it is merely an annotation +to the compiler describing the sort of the term inside the cast. It is telling +the compiler that the term inside the cast must be of sort Exp. For example, +if we had the following grammar:

      +
      k
      module LESSON-11-A
+  imports INT
+
+  syntax Exp ::= Int | Exp "+" Exp
+  syntax Stmt ::= "if" "(" Exp ")" Stmt | "{" "}"
+endmodule
+
      +

      Then we would be able to write 1:Exp, or (1 + 2):Exp, but not {}:Exp.

      +

      You can also restrict the sort that a variable in a rule will match by casting +it. For example, consider the following additional module:

      +
      k
      module LESSON-11-B
+  imports LESSON-11-A
+  imports BOOL
+
+  syntax Term ::= Exp | Stmt
+  syntax Bool ::= isExpression(Term) [function]
+
+  rule isExpression(_E:Exp) => true
+  rule isExpression(_) => false [owise]
+endmodule
+
      +

      Here we have defined a very simple function that decides whether a term is +an expression or a statement. It does this by casting the variable inside the +isExpression rule to sort Exp. As a result, that variable will only match terms +of sort Exp. Thus, isExpression(1) will return true, as will isExpression(1 + 2), but +isExpression({}) will return false.

      +

      Exercise

      +

      Verify this fact for yourself by running isExpression on the above examples. Then +write an isStatement function, and test that it works as expected.

      +

      Strict casts

      +

      On occasion, a semantic cast is not strict enough. It might be that you want +to, for disambiguation purposes, say exactly what sort a term is. For +example, consider the following definition:

      +
      k
      module LESSON-11-C
+  imports INT
+
+  syntax Exp ::= Int | Exp "+" Exp [group(exp)]
+  syntax Exp2 ::= Exp | Exp2 "+" Exp2 [group(exp2)]
+endmodule
+
      +

      This grammar is a little ambiguous and contrived, but it serves to demonstrate +how a semantic cast might be insufficient to disambiguate a term. If we were +to write the term (I1:Int + I2:Int):Exp2, the term would be ambiguous, +because the cast is not sufficiently strict to determine whether you mean +to derive the "+" production in the group exp or the one in the group exp2.

      +

      In this situation, there is a solution: the strict cast. For every sort +S in your grammar, K also defines the following production:

      +
        syntax S ::= S "::S"
+
      +

      This may at first glance seem the same as the previous cast. And indeed, +from the perspective of the grammar and from the perspective of rewriting, +they are in fact identical. However, the second variant has a unique meaning +in the type system of K: namely, the term inside the cast cannot be a +subsort, i.e., a term of another sort S2 such that the production +syntax S ::= S2 exists.

      +

      As a result, if we were to write in the above grammar the term +(I1:Int + I2:Int)::Exp2, then we would know that the second derivation above +should be chosen, whereas if we want the first derivation, we could write +(I1:Int + I2:Int)::Exp.

      +

      Projection casts

      +

      Thus far we have focused entirely on casts which exist solely to inform the +compiler about the sort of terms. However, sometimes when dealing with grammars +containing subsorts, it can be desirable to reason with the subsort production +itself, which injects one sort into another. Remember from above that such +a production looks like syntax S ::= S2. This type of production, called a +subsort production, can be thought of as a type of inheritance involving +constructors. If we have the above production in our grammar, we say that S2 +is a subsort of S, or that any S2 is also an S. K implicitly maintains a +symbol at runtime which keeps track of where such subsortings occur; this +symbol is called an injection.

      +

      Sometimes, when one sort is a subsort of another, it can be the case that +a function returns one sort, but you actually want to cast the result of +calling that function to another sort which is a subsort of the first sort. +This is similar to what happens with inheritance in an object-oriented +language, where you might cast a superclass to a subclass if you know for +sure the object at runtime is in fact an instance of that class.

      +

      K provides something similar for subsorts: the projection cast.

      +

      For each pair of sorts S and S2, K provides the following production:

      +
        syntax S ::= "{" S2 "}" ":>S"
+
      +

      What this means is that you take any term of sort S2 and cast it to sort +S. If the term of sort S2 consists of an injection containing a term of sort +S, then this will return that term. Otherwise, an error occurs and rewriting +fails, returning the projection function which failed to apply. The sort is +not actually checked at compilation time; rather, it is a runtime check +inserted into the code that runs when the rule applies.

      +

      For example, here is a module that makes use of projection casts:

      +
      k
      module LESSON-11-D
+  imports INT
+  imports BOOL
+
+  syntax Exp ::= Int | Bool | Exp "+" Exp | Exp "&&" Exp
+
+  syntax Exp ::= eval(Exp) [function]
+  rule eval(I:Int) => I
+  rule eval(B:Bool) => B
+  rule eval(E1 + E2) => {eval(E1)}:>Int +Int {eval(E2)}:>Int
+  rule eval(E1 && E2) => {eval(E1)}:>Bool andBool {eval(E2)}:>Bool
+endmodule
+
      +

      Here we have defined constructors for a simple expression language over +Booleans and integers, as well as a function eval that evaluates these +expressions to a value. Because that value could be an integer or a Boolean, +we need the casts in the last two rules in order to meet the type signature of ++Int and andBool. Of course, the user can write ill-formed expressions like +1 && true or false + true, but these will cause errors at runtime, because +the projection cast will fail.

      +

      Exercises

      +
        +
      1. +

        Extend the eval function in LESSON-11-D to include Strings and add a . +operator which concatenates them.

        +
        2. +
      2. +

        Modify your solution from Lesson 1.9, Exercise 2 by using an Exp sort to +express the integer and Boolean expressions that it supports, in the same style +as LESSON-11-D. Then write an eval function that evaluates all terms of +sort Exp to either a Bool or an Int.

        +
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.12: Syntactic Lists.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/12_syntactic_lists/index.html b/k-distribution/k-tutorial/1_basic/12_syntactic_lists/index.html new file mode 100644 index 00000000000..f203c6563af --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/12_syntactic_lists/index.html @@ -0,0 +1,577 @@ + + + + + + + + + + + + + + +Lesson 1.12: Syntactic Lists | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.12: Syntactic Lists

      +

      The purpose of this lesson is to explain how K provides support for syntactic +repetition through the use of the List{} and NeList{} constructs, +generally called syntactic lists.

      +

      The List{} construct

      +

      Sometimes, when defining a grammar in K, it is useful to define a syntactic +construct consisting of an arbitrary-length sequence of items. For example, +you might wish to define a function call construct, and need to express a way +of passing arguments to the function. You can in theory simply define these +productions using ordinary constructors, but it can be tricky to get the syntax +exactly right in K without a lot of tedious glue code.

      +

      For this reason, K provides a way of specifying that a non-terminal represents +a syntactic list (lesson-12-a.k):

      +
      k
      module LESSON-12-A-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= List{Int,","}
+endmodule
+
+module LESSON-12-A
+  imports LESSON-12-A-SYNTAX
+endmodule
+
      +

      Note that instead of a sequence of terminals and non-terminals, the right hand +side of the Ints production contains the symbol List followed by two items +in curly braces. The first item is the non-terminal which is the element type +of the list, and the second item is a terminal representing the separator of +the list. As a special case, lists which are separated only by whitespace can +be specified with a separator of "".

      +

      This List{} construct is roughly equivalent to the following definition +(lesson-12-b.k):

      +
      k
      module LESSON-12-B-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= Int "," Ints | ".Ints"
+endmodule
+
+module LESSON-12-B
+  imports LESSON-12-B-SYNTAX
+endmodule
+
      +

      As you can see, the List{} construct represents a cons-list with an element +at the head and another list at the tail. The empty list is represented by +a . followed by the sort of the list.

      +

      However, the List{} construct provides several key syntactic conveniences +over the above definition. First of all, when writing a list in a rule, +explicitly writing the terminator is not always required. For example, consider +the following additional module (lesson-12-c.k):

      +
      k
      module LESSON-12-C
+  imports LESSON-12-A
+  imports INT
+
+  syntax Int ::= sum(Ints) [function]
+  rule sum(I:Int) => I
+  rule sum(I1:Int, I2:Int, Is:Ints) => sum(I1 +Int I2, Is)
+endmodule
+
      +

      Here we see a function that sums together a non-empty list of integers. Note in +particular the first rule. We do not explicitly mention .Ints, but in fact, +the rule in question is equivalent to the following rule:

      +
        rule sum(I:Int, .Ints) => I
+
      +

      The reason for this is that K will automatically insert a list terminator +anywhere a syntactic list is expected, but an element of that list appears +instead. This works even with lists of more than one element:

      +
        rule sum(I1:Int, I2:Int) => I1 +Int I2
+
      +

      This rule is redundant, but here we explicitly match a list of exactly two +elements, because the .Ints is implicitly added after I2.

      +

      Parsing Syntactic Lists in Programs

      +

      An additional syntactic convenience takes place when you want to express a +syntactic list in the input to krun. In this case, K will automatically +transform the grammar in LESSON-12-B-SYNTAX into the following +(lesson-12-d.k):

      +
      k
      module LESSON-12-D
+  imports INT-SYNTAX
+
+  syntax Ints ::= #NonEmptyInts | #IntsTerminator
+  syntax #NonEmptyInts ::= Int "," #NonEmptyInts
+                         | Int #IntsTerminator
+  syntax #IntsTerminator ::= ""
+endmodule
+
      +

      This allows you to express the usual comma-separated list of arguments where +an empty list is represented by the empty string, and you don't have to +explicitly terminate the list. Because of this, we can write the syntax +of function calls in C very easily (lesson-12-e.k):

      +
      k
      module LESSON-12-E
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id | Exp "(" Exps ")"
+  syntax Exps ::= List{Exp,","}
+endmodule
+
      +

      Exercise

      +

      Write a function concat which takes a list of String and concatenates them +all together. Do not worry if the function is O(n^2). +Test your implementation using the syntactic sugar for lists added by the parser.

      +

      Then write some function call expressions using identifiers in C and verify with +kast that the above grammar captures the intended syntax. Make sure to test +with function calls with zero, one, and two or more arguments.

      +

      The NeList{} construct

      +

      One limitation of the List{} construct is that it is always possible to +write a list of zero elements where a List{} is expected. While this is +desirable in a number of cases, it is sometimes not what the grammar expects.

      +

      For example, in C, it is not allowable for an enum definition to have zero +members. In other words, if we were to write the grammar for enumerations like +so (lesson-12-f.k):

      +
      k
      module LESSON-12-F
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id
+
+  syntax EnumSpecifier ::= "enum" Id "{" Ids "}"
+  syntax Ids ::= List{Id,","}
+endmodule
+
      +

      Then we would be syntactically allowed to write enum X {}, which instead, +ought to be a syntax error.

      +

      For this reason, we introduce the additional NeList{} construct. The syntax +is identical to List{}, except with NeList instead of List before the +curly braces. When parsing rules, it behaves identically to the List{} +construct. However, when parsing inputs to krun, the above grammar, if we +replaced syntax Ids ::= List{Id,","} with syntax Ids ::= NeList{Id,","}, +would become equivalent to the following (lesson-12-g.k):

      +
      k
      module LESSON-12-G
+  syntax Id ::= r"[a-zA-Z_][a-zA-Z0-9_]*" [token]
+  syntax Exp ::= Id
+
+  syntax EnumSpecifier ::= "enum" Id "{" Ids "}"
+  syntax Ids ::= Id | Id "," Ids
+endmodule
+
      +

      In other words, only non-empty lists of Id would be allowed.

      +

      Exercises

      +
        +
      1. +

        Modify the sum function in LESSON-12-C so that the Ints sort is an +NeList{}. Verify that calling sum() with no arguments is now a syntax +error.

        +
        2. +
      2. +

        Write a modified sum function with the List construct that can also sum +up an empty list of arguments. In such a case, the sum ought to be 0.

        +
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.13: Basics of K Rewriting.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/13_rewrite_rules/index.html b/k-distribution/k-tutorial/1_basic/13_rewrite_rules/index.html new file mode 100644 index 00000000000..c8595f2fa16 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/13_rewrite_rules/index.html @@ -0,0 +1,696 @@ + + + + + + + + + + + + + + +Lesson 1.13: Basics of K Rewriting | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.13: Basics of K Rewriting

      +

      The purpose of this lesson is to explain how rewrite rules that are not the +definition of a function behave, and how, using these rules, you can construct +a semantics of programs in a programming language in K.

      +

      Recap: Function rules in K

      +

      Recall from Lesson 1.2 that we have, thus far, +introduced two types of productions in K: constructors and functions. +A function is identified by the function attribute placed on the +production. As you may recall, when we write a rule with a function on the +left-hand side of the => operator, we are defining the meaning of that +function for inputs which match the patterns on the left-hand side of the rule. +If the argument to the function match the patterns, then the function is +evaluated to the value constructed by substituting the bindings for the +variables into the right-hand side of the rule.

      +

      Top-level rules

      +

      However, function rules are not the only type of rule permissible in K, nor +even the most frequently used. K also has a concept of a +top-level rewrite rule. The simplest way to ensure that a rule is treated +as a top-level rule is for the left-hand side of the rule to mention one or +more cells. We will cover how cells work and are declared in more detail +in a later lesson, but for now, what you should know is that when we ran krun +in our very first example in Lesson 1.2 and got the following output:

      +
      <k>
+  Yellow ( ) ~> .
+</k>
+
      +

      <k> is a cell, known by convention as the K cell. This cell is available +by default in any definition without needing to be explicitly declared.

      +

      The K cell contains a single term of sort K. K is a predefined sort in K +with two constructors, that can be roughly represented by the following +grammar:

      +
        syntax K ::= KItem "~>" K
+             | "."
+
      +

      As a syntactic convenience, K allows you to treat ~> like it is an +associative list (i.e., as if it were defined as syntax K ::= K "~>" K). +When a definition is compiled, it will automatically transform the rules you +write so that they treat the K sort as a cons-list. Another syntactic +convenience is that, for disambiguation purposes, you can write .K anywhere +you would otherwise write . and the meaning is identical.

      +

      Now, you may notice that the above grammar mentions the sort KItem. This is +another built-in sort in K. For every sort S declared in a definition (with +the exception of K and KItem), K will implicitly insert the following +production:

      +
        syntax KItem ::= S
+
      +

      In other words, every sort is a subsort of the sort KItem, and thus a term +of any sort can be injected as an element of a term of sort K, also called +a K sequence.

      +

      By default, when you krun a program, the AST of the program is inserted as +the sole element of a K sequence into the <k> cell. This explains why we +saw the output we did in Lesson 1.2.

      +

      With these preliminaries in mind, we can now explain how top-level rewrite +rules work in K. Put simply, any rule where there is a cell (such as the K +cell) at the top on the left-hand side will be a top-level rewrite rule. Once +the initial program has been inserted into the K cell, the resulting term, +called the configuration, will be matched against all the top-level +rewrite rules in the definition. If only one rule matches, the substitution +generated by the matching will be applied to the right-hand side of the rule +and the resulting term is rewritten to be the new configuration. Rewriting +proceeds by iteratively applying rules, also called taking steps, until +no top-level rewrite rule can be applied. At this point the configuration +becomes the final configuration and is output by krun.

      +

      If more than one top-level rule applies, by default, K will pick just one +of those rules, apply it, and continue rewriting. However, it is +non-deterministic which rule applies. In theory, it could be any of them. +By passing the --search flag to krun, you are able to tell krun to +explore all possible non-deterministic choices, and generate a complete list of +all possible final configurations reachable by each nondeterminstic choice that +can be made. Note that the --search flag to krun only works if you pass +--enable-search to kompile first.

      +

      Unlike top-level rewrite rules, function rules are not associated with any +particular set of cells in the configuration (although they can contain cells +in their function arguments and return value). While top-level rewrite rules +apply to the entire term being rewritten, function rules apply anywhere a +function application for that function appears, and are immediately rewritten +to their return value in that position.

      +

      Another key distinction between top-level rules and function rules is that +function symbols, i.e., productions with the function attribute, are +mathematical functions rather than constructors. While a constructor is +logically distinct from any other constructor of the same sort, and can be +matched against unconditionally, a function does not necessaraily have the +same restriction unless it happens to be an injective function. Thus, two +function symbols with different arguments may still ultimately produce the +same value and thus compare equal to one another. Due to this, concrete +execution (i.e., all K definitions introduced thus far; see Lesson 1.21) +introduces the restriction that you cannot match on a function symbol on the +left-hand side of a rule, except as the top symbol on the left-hand side of +a function rule. This restriction will be later lifted when we introduce the +Haskell Backend which performs symbolic execution.

      +

      Exercise

      +

      Pass a program containing no functions to krun. You can use a term of sort +Exp from LESSON-11-D. Observe the output and try to understand why you get +the output you do. Then write two rules that rewrite that program to another. +Run krun --search on that program and observe both results. Then add a third +rule that rewrites one of those results again. Test that that rule applies as +well.

      +

      Using top-level rules to evaluate expressions

      +

      Thus far, we have focused primarily on defining functions over constructors +in K. However, now that we have a basic understanding of top-level rules, +it is possible to introduce a rewrite system to our definitions. A rewrite +system is a collection of top-level rewrite rules which performs an organized +transformation of a particular program into a result which expresses the +meaning of that program. For example, we might rewrite an expression in a +programming language into a value representing the result of evaluating that +expression.

      +

      Recall in Lesson 1.11, we wrote a simple grammar of Boolean and integer +expressions that looked roughly like this (lesson-13-a.k):

      +
      k
      module LESSON-13-A
+  imports INT
+
+  syntax Exp ::= Int
+               | Bool
+               | Exp "+" Exp
+               | Exp "&&" Exp
+endmodule
+
      +

      In that lesson, we defined a function eval which evaluated such expressions +to either an integer or Boolean.

      +

      However, it is more idiomatic to evaluate such expressions using top-level +rewrite rules. Here is how one might do so in K (lesson-13-b.k):

      +
      k
      module LESSON-13-B-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Val ::= Int | Bool
+  syntax Exp ::= Val
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-13-B
+  imports LESSON-13-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int ~> K:K </k> => <k> I1 +Int I2 ~> K </k>
+  rule <k> B1:Bool && B2:Bool ~> K:K </k> => <k> B1 andBool B2 ~> K </k>
+
+  syntax KItem ::= freezer1(Val) | freezer2(Exp)
+                 | freezer3(Val) | freezer4(Exp)
+
+  rule <k> E1:Val + E2:Exp ~> K:K </k> => <k> E2 ~> freezer1(E1) ~> K </k> [priority(51)]
+  rule <k> E1:Exp + E2:Exp ~> K:K </k> => <k> E1 ~> freezer2(E2) ~> K </k> [priority(52)]
+  rule <k> E1:Val && E2:Exp ~> K:K </k> => <k> E2 ~> freezer3(E1) ~> K </k> [priority(51)]
+  rule <k> E1:Exp && E2:Exp ~> K:K </k> => <k> E1 ~> freezer4(E2) ~> K </k> [priority(52)]
+
+  rule <k> E2:Val ~> freezer1(E1) ~> K:K </k> => <k> E1 + E2 ~> K </k>
+  rule <k> E1:Val ~> freezer2(E2) ~> K:K </k> => <k> E1 + E2 ~> K </k>
+  rule <k> E2:Val ~> freezer3(E1) ~> K:K </k> => <k> E1 && E2 ~> K </k>
+  rule <k> E1:Val ~> freezer4(E2) ~> K:K </k> => <k> E1 && E2 ~> K </k>
+endmodule
+
      +

      This is of course rather cumbersome currently, but we will soon introduce +syntactic convenience which makes writing definitions of this type considerably +easier. For now, notice that there are roughly 3 types of rules here: the first +matches a K cell in which the first element of the K sequence is an Exp whose +arguments are values, and rewrites the first element of the sequence to the +result of that expression. The second also matches a K cell with an Exp in +the first element of its K sequence, but it matches when one or both arguments +of the Exp are not values, and replaces the first element of the K sequence +with two new elements: one being an argument to evaluate, and the other being +a special constructor called a freezer. Finally, the third matches a K +sequence where a Val is first, and a freezer is second, and replaces them +with a partially evaluated expression.

      +

      This general pattern is what is known as heating an expression, +evaluating its arguments, cooling the arguments into the expression +again, and evaluating the expression itself. By repeatedly performing +this sequence of actions, we can evaluate an entire AST containing a complex +expression down into its resulting value.

      +

      Exercise

      +

      Write an addition expression with integers. Use krun --depth 1 to see the +result of rewriting after applying a single top-level rule. Gradually increase +the value of --depth to see successive states. Observe how this combination +of rules is eventually able to evaluate the entire expression.

      +

      Simplifying the evaluator: Local rewrites and cell ellipses

      +

      As you saw above, the definition we wrote is rather cumbersome. Over the +remainder of Lessons 1.13 and 1.14, we will greatly simplify it. The first step +in doing so is to teach a bit more about the rewrite operator, =>. Thus far, +all the rules we have written look like rule LHS => RHS. However, this is not +the only way the rewrite operator can be used. It is actually possible to place +a constructor or function at the very top of the rule, and place rewrite +operators inside that term. While a rewrite operator cannot appear nested +inside another rewrite operator, by doing this, we can express that some parts +of what we are matching are not changed by the rewrite operator. For +example, consider the following rule from above:

      +
        rule <k> I1:Int + I2:Int ~> K:K </k> => <k> I1 +Int I2 ~> K </k>
+
      +

      We can equivalently write it like following:

      +
        rule <k> (I1:Int + I2:Int => I1 +Int I2) ~> _:K </k>
+
      +

      When you put a rewrite inside a term like this, in essence, you are telling +the rule to only rewrite part of the left-hand side to the right-hand side. +In practice, this is implemented by lifting the rewrite operator to the top of +the rule by means of duplicating the surrounding context.

      +

      There is a way that the above rule can be simplified further, however. K +provides a special syntax for each cell containing a term of sort K, indicating +that we want to match only on some prefix of the K sequence. For example, the +above rule can be simplified further like so:

      +
        rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+
      +

      Here we have placed the symbol ... immediately prior to the </k> which ends +the cell. What this tells the compiler is to take the contents of the cell, +treat it as the prefix of a K sequence, and insert an anonymous variable of +sort K at the end. Thus we can think of ... as a way of saying we +don't care about the part of the K sequence after the beginning, leaving +it unchanged.

      +

      Putting all this together, we can rewrite LESSON-13-B like so +(lesson-13-c.k):

      +
      k
      module LESSON-13-C-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Val ::= Int | Bool
+  syntax Exp ::= Val
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-13-C
+  imports LESSON-13-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax KItem ::= freezer1(Val) | freezer2(Exp)
+                 | freezer3(Val) | freezer4(Exp)
+
+  rule <k> E1:Val + E2:Exp => E2 ~> freezer1(E1) ...</k> [priority(51)]
+  rule <k> E1:Exp + E2:Exp => E1 ~> freezer2(E2) ...</k> [priority(52)]
+  rule <k> E1:Val && E2:Exp => E2 ~> freezer3(E1) ...</k> [priority(51)]
+  rule <k> E1:Exp && E2:Exp => E1 ~> freezer4(E2) ...</k> [priority(52)]
+
+  rule <k> E2:Val ~> freezer1(E1) => E1 + E2 ...</k>
+  rule <k> E1:Val ~> freezer2(E2) => E1 + E2 ...</k>
+  rule <k> E2:Val ~> freezer3(E1) => E1 && E2 ...</k>
+  rule <k> E1:Val ~> freezer4(E2) => E1 && E2 ...</k>
+endmodule
+
      +

      This is still rather cumbersome, but it is already greatly simplified. In the +next lesson, we will see how additional features of K can be used to specify +heating and cooling rules much more compactly.

      +

      Exercises

      +
        +
      1. Modify LESSON-13-C to add rules to evaluate integer subtraction.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.14: Defining Evaluation Order.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/14_evaluation_order/index.html b/k-distribution/k-tutorial/1_basic/14_evaluation_order/index.html new file mode 100644 index 00000000000..62fe08b3207 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/14_evaluation_order/index.html @@ -0,0 +1,707 @@ + + + + + + + + + + + + + + +Lesson 1.14: Defining Evaluation Order | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.14: Defining Evaluation Order

      +

      The purpose of this lesson is to explain how to use the heat and cool +attributes, context and context alias sentences, and the strict and +seqstrict attributes to more compactly express heating and cooling in K, +and to express more advanced evaluation strategies in K.

      +

      The heat and cool attributes

      +

      Thus far, we have been using rule priority and casts to express when to heat +an expression and when to cool it. For example, the rules for heating have +lower priority, so they do not apply if the term could be evaluated instead, +and the rules for heating are expressly written only to apply if the argument +of the expression is a value.

      +

      However, K has built-in support for deciding when to heat and when to cool. +This support comes in the form of the rule attributes heat and cool as +well as the specially named function isKResult.

      +

      Consider the following definition, which is equivalent to LESSON-13-C +(lesson-14-a.k):

      +
      k
      module LESSON-14-A-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-14-A
+  imports LESSON-14-A-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax KItem ::= freezer1(Exp) | freezer2(Exp)
+                 | freezer3(Exp) | freezer4(Exp)
+
+  rule <k> E:Exp + HOLE:Exp => HOLE ~> freezer1(E) ...</k>
+    requires isKResult(E) [heat]
+  rule <k> HOLE:Exp + E:Exp => HOLE ~> freezer2(E) ...</k> [heat]
+  rule <k> E:Exp && HOLE:Exp => HOLE ~> freezer3(E) ...</k>
+    requires isKResult(E) [heat]
+  rule <k> HOLE:Exp && E:Exp => HOLE ~> freezer4(E) ...</k> [heat]
+
+  rule <k> HOLE:Exp ~> freezer1(E) => E + HOLE ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer2(E) => HOLE + E ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer3(E) => E && HOLE ...</k> [cool]
+  rule <k> HOLE:Exp ~> freezer4(E) => HOLE && E ...</k> [cool]
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      We have introduced three major changes to this definition. First, we have +removed the Val sort. We replace it instead with a function isKResult. +The function in question must have the same signature and attributes as seen in +this example. It ought to return true whenever a term should not be heated +(because it is a value) and false when it should be heated (because it is not +a value). We thus also insert isKResult calls in the side condition of two +of the heating rules, where the Val sort was previously used.

      +

      Second, we have removed the rule priorities on the heating rules and the use of +the Val sort on the cooling rules, and replaced them with the heat and +cool attributes. These attributes instruct the compiler that these rules are +heating and cooling rules, and thus should implicitly apply only when certain +terms on the LHS either are or are not a KResult (i.e., isKResult returns +true versus false).

      +

      Third, we have renamed some of the variables in the heating and cooling rules +to the special variable HOLE. Syntactically, HOLE is just a special name +for a variable, but it is treated specially by the compiler. By naming a +variable HOLE, we have informed the compiler which term is being heated +or cooled. The compiler will automatically insert the side condition +requires isKResult(HOLE) to cooling rules and the side condition +requires notBool isKResult(HOLE) to heating rules.

      +

      Exercise

      +

      Modify LESSON-14-A to add rules to evaluate integer subtraction.

      +

      Simplifying further with Contexts

      +

      The above example is still rather cumbersome to write. We must explicitly write +both the heating and the cooling rule separately, even though they are +essentially inverses of one another. It would be nice to instead simply +indicate which terms should be heated and cooled, and what part of them to +operate on.

      +

      To do this, K introduces a new type of sentence, the context. Contexts +begin with the context keyword instead of the rule keyword, and usually +do not contain a rewrite operator.

      +

      Consider the following definition which is equivalent to LESSON-14-A +(lesson-14-b.k):

      +
      k
      module LESSON-14-B-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp
+               > left: Exp "&&" Exp
+endmodule
+
+module LESSON-14-B
+  imports LESSON-14-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  context <k> E:Exp + HOLE:Exp ...</k>
+    requires isKResult(E)
+  context <k> HOLE:Exp + _:Exp ...</k>
+  context <k> E:Exp && HOLE:Exp ...</k>
+    requires isKResult(E)
+  context <k> HOLE:Exp && _:Exp ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      In this example, the heat and cool rules have been removed entirely, as +have been the productions defining the freezers. Don't worry, they still exist +under the hood; the compiler is just generating them automatically. For each +context sentence like above, the compiler generates a #freezer production, +a heat rule, and a cool rule. The generated form is equivalent to the +rules we wrote manually in LESSON-14-A. However, we are now starting to +considerably simplify the definition. Instead of 3 sentences, we just have one.

      +

      context alias sentences and the strict and seqstrict attributes

      +

      Notice that the contexts we included in LESSON-14-B still seem rather +similar in form. For each expression we want to evaluate, we are declaring +one context for each operand of that expression, and they are each rather +similar to one another. We would like to be able to simplify further by +simply annotating each expression production with information about how +it is to be evaluated instead. We can do this with the seqstrict attribute.

      +

      Consider the following definition, once again equivalent to those above +(lesson-14-c.k):

      +
      .k .alias
      module LESSON-14-C-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp [seqstrict(exp; 1, 2)]
+               > left: Exp "&&" Exp [seqstrict(exp; 1, 2)]
+endmodule
+
+module LESSON-14-C
+  imports LESSON-14-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  context alias [exp]: <k> HERE ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      This definition has two important changes from the one above. The first is +that the individual context sentences have been removed and have been +replaced with a single context alias sentence. You may notice that this +sentence begins with an identifier in square brackets followed by a colon. This +syntax is a way of naming individual sentences in K for reference by the tool +or by other sentences. The context alias sentence also has a special variable +HERE.

      +

      The second is that the productions in LESSON-14-C-SYNTAX have been given a +seqstrict attribute. The value of this attribute has two parts. The first +is the name of a context alias sentence. The second is a comma-separated list +of integers. Each integer represents an index of a non-terminal in the +production, counting from 1. For each integer present, the compiler implicitly +generates a new context sentence according to the following rules:

      +
        +
      1. The compiler starts by looking for the context alias sentence named. If +there is more than one, then one context sentence is created per +context alias sentence with that name.
        2. +
      2. For each context created, the variable HERE in the context alias is +substituted with an instance of the production the seqstrict attribute is +attached to. Each child of that production is a variable. The non-terminal +indicated by the integer offset of the seqstrict attribute is given the name +HOLE.
        3. +
      3. For each integer offset prior in the list to the one currently being +processed, the predicate isKResult(E) is conjuncted together and included +as a side condition, where E is the child of the production term with that +offset, starting from 1. For example, if the attribute lists 1, 2, then +the rule generated for the 2 will include isKResult(E1) where E1 is the +first child of the production.
        4. +
      +

      As you can see if you work through the process, the above code will ultimately +generate the same contexts present in LESSON-14-B.

      +

      Finally, note that there are a few minor syntactic conveniences provided by the +seqstrict attribute. First, in the special case of the context alias sentence +being <k> HERE ...</k>, you can omit both the context alias sentence +and the name from the seqstrict attribute.

      +

      Second, if the numbered list of offsets contains every non-terminal in the +production, it can be omitted from the attribute value.

      +

      Thus, we can finally produce the idiomatic K definition for this example +(lesson-14-d.k):

      +
      k
      module LESSON-14-D-SYNTAX
+  imports UNSIGNED-INT-SYNTAX
+  imports BOOL-SYNTAX
+
+  syntax Exp ::= Int
+               | Bool
+               > left: Exp "+" Exp [seqstrict]
+               > left: Exp "&&" Exp [seqstrict]
+endmodule
+
+module LESSON-14-D
+  imports LESSON-14-D-SYNTAX
+  imports INT
+  imports BOOL
+
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+  rule <k> B1:Bool && B2:Bool => B1 andBool B2 ...</k>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      Exercise

      +

      Modify LESSON-14-D to add a production and rule to evaluate integer +subtraction.

      +

      Nondeterministic evaluation order with the strict attribute

      +

      Thus far, we have focused entirely on deterministic evaluation order. However, +not all languages are deterministic in the order they evaluate expressions. +For example, in C, the expression a() + b() + c() is guaranteed to parse +to (a() + b()) + c(), but it is not guaranteed that a will be called before +b before c. In fact, this evaluation order is non-deterministic.

      +

      We can express non-deterministic evaluation orders with the strict attribute. +Its behavior is identical to the seqstrict attribute, except that step 3 in +the above list (with the side condition automatically added) does not take +place. In other words, if we wrote syntax Exp ::= Exp "+" Exp [strict] +instead of syntax Exp ::= Exp "+" Exp [seqstrict], it would generate the +following two contexts instead of the ones found in LESSON-14-B:

      +
        context <k> _:Exp + HOLE:Exp ...</k>
+  context <k> HOLE:Exp + _:Exp ...</k>
+
      +

      As you can see, these contexts will generate heating rules that can both +apply to the same term. As a result, the choice of which heating rule +applies first is non-deterministic, and as we saw in Lesson 1.13, we can +get all possible behaviors by passing --search to krun.

      +

      Exercises

      +
        +
      1. +

        Add integer division to LESSON-14-D. Make division and addition strict +instead of seqstrict, and write a rule evaluating integer division with a +side condition that the denominator is non-zero. Run krun --search on the +program 1 / 0 + 2 / 1 and observe all possible outputs of the program. How +many are there total, and why?

        +
        2. +
      2. +

        Rework your solution from Lesson 1.9, Exercise 2 to evaluate expressions from left to right using the seqstrict attribute.

        +
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.15: Configuration Declarations and Cell Nesting.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/15_configurations/index.html b/k-distribution/k-tutorial/1_basic/15_configurations/index.html new file mode 100644 index 00000000000..49224fa62b8 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/15_configurations/index.html @@ -0,0 +1,663 @@ + + + + + + + + + + + + + + +Lesson 1.15: Configuration Declarations and Cell Nesting | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.15: Configuration Declarations and Cell Nesting

      +

      The purpose of this lesson is to explain how to store additional information +about the state of your interpreter by declaring cells using the +configuration sentence, as well as how to add additional inputs to your +definition.

      +

      Cells and Configuration Declarations

      +

      We have already covered the absolute basics of cells in K by looking at the +<k> cell. As explained in Lesson 1.13, the +<k> cell is available without being explicitly declared. It turns out this is +because, if the user does not explicitly specify a configuration sentence +anywhere in the main module of their definition, the configuration sentence +from the DEFAULT-CONFIGURATION module of +kast.md is imported +automatically. Here is what that sentence looks like:

      +
        configuration <k> $PGM:K </k>
+
      +

      This configuration declaration declares a single cell, the <k> cell. It also +declares that at the start of rewriting, the contents of that cell should be +initialized with the value of the $PGM configuration variable. +Configuration variables function as inputs to krun. These terms are supplied +to krun in the form of ASTs parsed using a particular module. By default, the +$PGM configuration variable uses the main syntax module of the definition.

      +

      The cast on the configuration variable also specifies the sort that is used as +the entry point to the parser, in this case the K sort. It is often +useful to cast to other sorts there as well for better control over the accepted +language. The sort used for the $PGM variable is referred to as the start +symbol. During parsing, the default start symbol K subsumes all user-defined +sorts except for syntactic lists. These are excluded because they will always +produce an ambiguity error when parsing a single element.

      +

      Note that we did not explicitly specify the $PGM configuration variable when +we invoked krun on a file. This is because krun handles the $PGM variable +specially, and allows you to pass the term for that variable via a file passed +as a positional argument to krun. We did, however, specify the PGM name +explicitly when we called krun with the -cPGM command line argument in +Lesson 1.2. This is the other, explicit, way of +specifying an input to krun.

      +

      This explains the most basic use of configuration declarations in K. We can, +however, declare multiple cells and multiple configuration variables. We can +also specify the initial values of cells statically, rather than dynamically +via krun.

      +

      For example, consider the following definition (lesson-15-a.k):

      +
      k
      module LESSON-15-A-SYNTAX
+  imports INT-SYNTAX
+
+  syntax Ints ::= List{Int,","}
+endmodule
+
+module LESSON-15-A
+  imports LESSON-15-A-SYNTAX
+  imports INT
+
+  configuration <k> $PGM:Ints </k>
+                <sum> 0 </sum>
+
+  rule <k> I:Int, Is:Ints => Is ...</k>
+       <sum> SUM:Int => SUM +Int I </sum>
+endmodule
+
      +

      This simple definition takes a list of integers as input and sums them +together. Here we have declared two cells: <k> and <sum>. Unlike <k>, +<sum> does not get initialized via a configuration variable, but instead +is initialized statically with the value 0.

      +

      Note the rule in the second module: we have explicitly specified multiple +cells in a single rule. K will expect each of these cells to match in order for +the rule to apply.

      +

      Here is a second example (lesson-15-b.k):

      +
      k
      module LESSON-15-B-SYNTAX
+  imports INT-SYNTAX
+endmodule
+
+module LESSON-15-B
+  imports LESSON-15-B-SYNTAX
+  imports INT
+  imports BOOL
+
+  configuration <k> . </k>
+                <first> $FIRST:Int </first>
+                <second> $SECOND:Int </second>
+
+  rule <k> . => FIRST >Int SECOND </k>
+       <first> FIRST </first>
+       <second> SECOND </second>
+endmodule
+
      +

      This definition takes two integers as command-line arguments and populates the +<k> cell with a Boolean indicating whether the first integer is greater than +the second. Notice that we have specified no $PGM configuration variable +here. As a result, we cannot invoke krun via the syntax krun $file. +Instead, we must explicitly pass values for each configuration variable via the +-cFIRST and -cSECOND command line flags. For example, if we invoke +krun -cFIRST=0 -cSECOND=1, we will get the value false in the K cell.

      +

      You can also specify both a $PGM configuration variable and other +configuration variables in a single configuration declaration, in which case +you would be able to initialize $PGM with either a positional argument or the +-cPGM command line flag, but the other configuration variables would need +to be explicitly initialized with -c.

      +

      Exercise

      +

      Modify your solution to Lesson 1.14, Exercise 2 to add a new cell with a +configuration variable of sort Bool. This variable should determine whether +the / operator is evaluated using /Int or divInt. Test that by specifying +different values for this variable, you can change the behavior of rounding on +division of negative numbers.

      +

      Cell Nesting

      +

      It is possible to nest cells inside one another. A cell that contains other +cells must contain only other cells, but in doing this, you are able to +create a hierarchical structure to the configuration. Consider the following +definition (lesson-15-c.k), which is equivalent to the one in LESSON-15-B:

      +
      k
      module LESSON-15-C-SYNTAX
+  imports INT-SYNTAX
+endmodule
+
+module LESSON-15-C
+  imports LESSON-15-C-SYNTAX
+  imports INT
+  imports BOOL
+
+  configuration <T>
+                  <k> . </k>
+                  <state>
+                    <first> $FIRST:Int </first>
+                    <second> $SECOND:Int </second>
+                  </state>
+                </T>
+
+  rule <k> . => FIRST >Int SECOND </k>
+       <first> FIRST </first>
+       <second> SECOND </second>
+endmodule
+
      +

      Note that we have added some new cells to the configuration declaration: +the <T> cell wraps the entire configuration, and the <state> cell is +introduced around the <first> and <second> cells.

      +

      However, we have not changed the rule in this definition. This is because of +a concept in K called configuration abstraction. K allows you to specify +any number of cells in a rule (except zero) in any order you want, and K will +compile the rules into a form that matches the structure of the configuration +specified by the configuration declaration.

      +

      Here then, is how this rule would look after the configuration abstraction +has been resolved:

      +
        rule <T>
+         <k> . => FIRST >Int SECOND </k>
+         <state>
+           <first> FIRST </first>
+           <second> SECOND </second>
+         </state>
+       </T>
+
      +

      In other words, K will complete cells to the top of the configuration by +inserting parent cells where appropriate based on the declared structure of +the configuration. This is useful because as a definition evolves, the +configuration may change, but you don't want to have to modify every single +rule each time. Thus, K follows the principle that you should only mention the +cells in a rule that are actually needed in order to accomplish its specific +goal. By following this best practice, you can significantly increase the +modularity of the definition and make it easier to maintain and modify.

      +

      Note that unlike top-level rewrite rules, cells that appear inside function +rules are not necessarily completed to the top of the configuration. They still +participate in cell ccompletion in the sense that you can mention cell +structure loosely inside a function rule and it will be completed into the +correct cell structure specified by the configuration declaration. However, +they do not complete all the way to the top, instead completing only up to +the top-most cell mentioned in the rule.

      +

      For example, if I write the following function rule in the above definition:

      +
        rule doStuff(<first> FIRST </first>) => FIRST
+
      +

      The function will only match on the first cell, rather than the entire +configuration. However, if we had mentioned a parent cell in the rule, it still +would have completed the children of that parent cell as needed to ensure that +the resulting term is well formed.

      +

      Exercise

      +

      Modify your definition from the previous exercise in this lesson to wrap the +two cells you have declared in a top cell <T>. You should not have to change +any other rules in the definition.

      +

      Cell Variables

      +

      Sometimes it is desirable to explicitly match a variable against certain +fragments of the configuration. Because K's configuration is hierarchical, +we can grab subsets of the configuration as if they were just another term. +However, configuration abstraction applies here as well. +In particular, for each cell you specify in a configuration declaration, a +unique sort is assigned for that cell with a single constructor (the cell +itself). The sort name is taken by removing all special characters, +capitalizing the first letter and each letter after a hyphen, and adding the +word Cell at the end. For example, in the above example, the cell sorts are +TCell, KCell, StateCell, FirstCell, and SecondCell. If we had declared +a cell as <first-number>, then the cell sort name would be FirstNumberCell.

      +

      You can explicitly reference a variable of one of these sorts anywhere you +might instead write that cell. For example, consider the following rule:

      +
        rule <k> true => S </k>
+       (S:StateCell => <state>... .Bag ...</state>)
+
      +

      Here we have introduced two new concepts. The first is the variable of sort +StateCell, which matches the entire <state> part of the configuration. The +second is that we have introduced the concept of ... once again. When a cell +contains other cells, it is also possible to specify ... on either the left, +right or both sides of the cell term. Each of these three syntaxes are +equivalent in this case. When they appear on the left-hand side of a rule, they +indicate that we don't care what value any cells not explicitly named might +have. For example, we might write <state>... <first> 0 </first> ...</state> on +the left-hand side of a rule in order to indicate that we want to match the +rule when the <first> cell contains a zero, regardless of what the <second> +cell contains. If we had not included this ellipsis, it would have been a +syntax error, because K would have expected you to provide a value for each of +the child cells.

      +

      However, if, as in the example above, the ... appeared on the right-hand side +of a rule, this instead indicates that the cells not explicitly mentioned under +the cell should be initialized with their default value from the configuration +declaration. In other words, that rule will set the value of <first> and +<second> to zero.

      +

      You may note the presence of the phrase .Bag here. You can think of this as +the empty set of cells. It is used as the child of a cell when you want to +indicate that no cells should be explicitly named. We will cover other uses +of this term in later lessons.

      +

      Exercises

      +
        +
      1. Modify the definition from the previous exercise in this lesson so that the +Boolean cell you created is initialized to false. Then add a production +syntax Stmt ::= Bool ";" Exp, and a rule that uses this Stmt to set the +value of the Boolean flag. Then add another production +syntax Stmt ::= "reset" ";" Exp which sets the value of the Boolean flag back +to its default value via a ... on the right-hand side. You will need to add +an additional cell around the Boolean cell to make this work.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.16: Maps, Semantic Lists, and Sets.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/16_collections/index.html b/k-distribution/k-tutorial/1_basic/16_collections/index.html new file mode 100644 index 00000000000..65aabdc5e18 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/16_collections/index.html @@ -0,0 +1,742 @@ + + + + + + + + + + + + + + +Lesson 1.16: Maps, Semantic Lists, and Sets | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.16: Maps, Semantic Lists, and Sets

      +

      The purpose of this lesson is to explain how to use the data structure sorts +provided by K: maps, lists, and sets.

      +

      Maps

      +

      The most frequently used type of data structure in K is the map. The sort +provided by K for this purpose is the Map sort, and it is provided in +domains.md in the MAP +module. This type is not (currently) polymorphic. All Map terms are maps that +map terms of sort KItem to other terms of sort KItem. A KItem can contain +any sort except a K sequence. If you need to store such a term in a +map, you can always use a wrapper such as syntax KItem ::= kseq(K).

      +

      A Map pattern consists of zero or more map elements (as represented by the +symbol syntax Map ::= KItem "|->" KItem), mixed in any order, separated by +whitespace, with zero or one variables of sort Map. The empty map is +represented by .Map. If all of the bindings for the variables in the keys +of the map can be deterministically chosen, these patterns can be matched in +O(1) time. If they cannot, then each map element that cannot be +deterministically constructed contributes a single dimension of polynomial +time to the cost of the matching. In other words, a single such element is +linear, two are quadratic, three are cubic, etc.

      +

      Patterns like the above are the only type of Map pattern that can appear +on the left-hand-side of a rule. In other words, you are not allowed to write +a Map pattern on the left-hand-side with more than one variable of sort Map +in it. You are, however, allowed to write such patterns on the right-hand-side +of a rule. You can also write a function pattern in the key of a map element +so long as all the variables in the function pattern can be deterministically +chosen.

      +

      Note the meaning of matching on a Map pattern: a map pattern with no +variables of sort Map will match if the map being matched has exactly as +many bindings as |-> symbols in the pattern. It will then match if each +binding in the map pattern matches exactly one distinct binding in the map +being matched. A map pattern with one Map variable will also match any map +that contains such a map as a subset. The variable of sort Map will be bound +to whatever bindings are left over (.Map if there are no bindings left over).

      +

      Here is an example of a simple definition that implements a very basic +variable declaration semantics using a Map to store the value of variables +(lesson-16-a.k):

      +
      k
      module LESSON-16-A-SYNTAX
+  imports INT-SYNTAX
+  imports ID-SYNTAX
+
+  syntax Exp ::= Id | Int
+  syntax Decl ::= "int" Id "=" Exp ";" [strict(2)]
+  syntax Pgm ::= List{Decl,""}
+endmodule
+
+module LESSON-16-A
+  imports LESSON-16-A-SYNTAX
+  imports BOOL
+
+  configuration <T>
+                  <k> $PGM:Pgm </k>
+                  <state> .Map </state>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // variable declaration
+  rule <k> int X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+
+  // variable lookup
+  rule <k> X:Id => I ...</k>
+       <state>... X |-> I ...</state>
+
+  syntax Bool ::= isKResult(K) [symbol, function]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      There are several new features in this definition. First, note we import +the module ID-SYNTAX. This module is defined in domains.md and provides a +basic syntax for identifiers. We are using the Id sort provided by this +module in this definition to implement the names of program variables. This +syntax is only imported when parsing programs, not when parsing rules. Later in +this lesson we will see how to reference specific concrete identifiers in a +rule.

      +

      Second, we introduce a single new function over the Map sort. This function, +which is represented by the symbol +syntax Map ::= Map "[" KItem "<-" KItem "]", represents the map update +operation. Other functions over the Map sort can be found in domains.md.

      +

      Finally, we have used the ... syntax on a cell containing a Map. In this +case, the meaning of <state>... Pattern ...</state>, +<state>... Pattern </state>, and <state> Pattern ...</state> are the same: +it is equivalent to writing <state> (Pattern) _:Map </state>.

      +

      Consider the following program (a.decl):

      +
      int x = 0;
+int y = 1;
+int a = x;
+
      +

      If we run this program with krun, we will get the following result:

      +
      <T>
+  <k>
+    .
+  </k>
+  <state>
+    a |-> 0
+    x |-> 0
+    y |-> 1
+  </state>
+</T>
+
      +

      Note that krun has automatically sorted the collection for you. This doesn't +happen at runtime, so you still get the performance of a hash map, but it will +help make the output more readable.

      +

      Exercise

      +

      Create a sort Stmt that is a subsort of Decl. Create a production of sort +Stmt for variable assignment in addition to the variable declaration +production. Feel free to use the syntax syntax Stmt ::= Id "=" Exp ";". Write +a rule that implements variable assignment using a map update function. Then +write the same rule using a map pattern. Test your implementations with some +programs to ensure they behave as expected.

      +

      Semantic Lists

      +

      In a previous lesson, we explained how to represent lists in the AST of a +program. However, this is not the only context where lists can be used. We also +frequently use lists in the configuration of an interpreter in order to +represent certain types of program state. For this purpose, it is generally +useful to have an associative-list sort, rather than the cons-list sorts +provided in Lesson 1.12.

      +

      The type provided by K for this purpose is the List sort, and it is also +provided in domains.md, in the LIST module. This type is also not +(currently) polymorphic. Like Map, all List terms are lists of terms of the +KItem sort.

      +

      A List pattern in K consists of zero or more list elements (as represented by +the ListItem symbol), followed by zero or one variables of sort List, +followed by zero or more list elements. An empty list is represented by +.List. These patterns can be matched in O(log(N)) time. This is the only +type of List pattern that can appear on the left-hand-side of a rule. In +other words, you are not allowed to write a List pattern on the +left-hand-side with more than one variable of sort List in it. You are, +however, allowed to write such patterns on the right-hand-side of a rule.

      +

      Note the meaning of matching on a List pattern: a list pattern with no +variables of sort List will match if the list being matched has exactly as +many elements as ListItem symbols in the pattern. It will then match if each +element in sequence matches the pattern contained in the ListItem symbol. A +list pattern with one variable of sort List operates the same way, except +that it can match any list with at least as many elements as ListItem +symbols, so long as the prefix and suffix of the list match the patterns inside +the ListItem symbols. The variable of sort List will be bound to whatever +elements are left over (.List if there are no elements left over).

      +

      The ... syntax is allowed on cells containing lists as well. In this case, +the meaning of <cell>... Pattern </cell> is the same as +<cell> _:List (Pattern) </cell>, the meaning of <cell> Pattern ...</cell> +is the same as <cell> (Pattern) _:List</cell>. Because list patterns with +multiple variables of sort List are not allowed, it is an error to write +<cell>... Pattern ...</cell>.

      +

      Here is an example of a simple definition that implements a very basic +function-call semantics using a List as a function stack (lesson-16-b.k):

      +
      k
      module LESSON-16-B-SYNTAX
+  imports INT-SYNTAX
+  imports ID-SYNTAX
+
+  syntax Exp ::= Id "(" ")" | Int
+  syntax Stmt ::= "return" Exp ";" [strict]
+  syntax Decl ::= "fun" Id "(" ")" "{" Stmt "}"
+  syntax Pgm ::= List{Decl,""}
+  syntax Id ::= "main" [token]
+endmodule
+
+module LESSON-16-B
+  imports LESSON-16-B-SYNTAX
+  imports BOOL
+  imports LIST
+
+  configuration <T>
+                  <k> $PGM:Pgm ~> main () </k>
+                  <functions> .Map </functions>
+                  <fstack> .List </fstack>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // function definitions
+  rule <k> fun X:Id () { S } => . ...</k>
+       <functions>... .Map => X |-> S ...</functions>
+
+  // function call
+  syntax KItem ::= stackFrame(K)
+  rule <k> X:Id () ~> K => S </k>
+       <functions>... X |-> S ...</functions>
+       <fstack> .List => ListItem(stackFrame(K)) ...</fstack>
+
+  // return statement
+  rule <k> return I:Int ; ~> _ => I ~> K </k>
+       <fstack> ListItem(stackFrame(K)) => .List ...</fstack>
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      Notice that we have declared the production syntax Id ::= "main" [token]. +Since we use the ID-SYNTAX module, this declaration is necessary in order to +be able to refer to the main identifier directly in the configuration +declaration. Our <k> cell now contains a K sequence initially: first we +process all the declarations in the program, then we call the main function.

      +

      Consider the following program (foo.func):

      +
      fun foo() { return 5; }
+fun main() { return foo(); }
+
      +

      When we krun this program, we should get the following output:

      +
      <T>
+  <k>
+    5 ~> .
+  </k>
+  <functions>
+    foo |-> return 5 ;
+    main |-> return foo ( ) ;
+  </functions>
+  <fstack>
+    .List
+  </fstack>
+</T>
+
      +

      Note that we have successfully put on the <k> cell the value returned by the +main function.

      +

      Exercise

      +

      Add a term of sort Id to the stackFrame operator to keep track of the +name of the function in that stack frame. Then write a function +syntax String ::= printStackTrace(List) that takes the contents of the +<fstack> cell and pretty prints the current stack trace. You can concatenate +strings with +String in the STRING module in domains.md, and you can +convert an Id to a String with the Id2String function in the ID module. +Test this function by creating a new expression that returns the current stack +trace as a string. Make sure to update isKResult and the Exp sort as +appropriate to allow strings as values.

      +

      Sets

      +

      The final primary data structure sort in K is a set, i.e., an idempotent +unordered collection where elements are deduplicated. The sort provided by K +for this purpose is the Set sort and it is provided in domains.md in the +SET module. Like maps and lists, this type is not (currently) polymorphic. +Like Map and List, all Set terms are sets of terms of the KItem sort.

      +

      A Set pattern has the exact same restrictions as a Map pattern, except that +its elements are treated like keys, and there are no values. It has the same +performance characteristics as well. However, syntactically it is more similar +to the List sort: An empty Set is represented by .Set, but a set element +is represented by the SetItem symbol.

      +

      Matching behaves similarly to the Map sort: a set pattern with no variables +of sort Set will match if the set has exactly as many bindings as SetItem +symbols, and if each element pattern matches one distinct element in the set. +A set with a variable of sort Set also matches any superset of such a set. +As with map, the elements left over will be bound to the Set variable (or +.Set if no elements are left over).

      +

      Like Map, the ... syntax on a set is syntactic sugar for an anonymous +variable of sort Set.

      +

      Here is an example of a simple modification to LESSON-16-A which uses a Set +to ensure that variables are never declared more than once. In practice, you +would likely just use the in_keys symbol over maps to test for this, but +it's still useful as an example of sets in practice:

      +
      k
      module LESSON-16-C-SYNTAX
+  imports LESSON-16-A-SYNTAX
+endmodule
+
+module LESSON-16-C
+  imports LESSON-16-C-SYNTAX
+  imports BOOL
+  imports SET
+
+  configuration <T>
+                  <k> $PGM:Pgm </k>
+                  <state> .Map </state>
+                  <declared> .Set </declared>
+                </T>
+
+  // declaration sequence
+  rule <k> D:Decl P:Pgm => D ~> P ...</k>
+  rule <k> .Pgm => . ...</k>
+
+  // variable declaration
+  rule <k> int X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+       <declared> D => D SetItem(X) </declared>
+    requires notBool X in D
+
+  // variable lookup
+  rule <k> X:Id => I ...</k>
+       <state>... X |-> I ...</state>
+       <declared>... SetItem(X) ...</declared>
+
+  syntax Bool ::= isKResult(K) [symbol, function]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      Now if we krun a program containing duplicate declarations, it will get +stuck on the declaration.

      +

      Exercises

      +
        +
      1. Modify your solution to Lesson 1.14, Exercise 2 and introduce the sorts +Decls, Decl, and Stmt which include variable and function declaration +(without function parameters), and return and assignment statements, as well +as call expressions. Use List and Map to implement these operators, making +sure to consider the interactions between components, such as saving and +restoring the environment of variables at each call site. Don't worry about +local function definitions or global variables for now. Make sure to test the +resulting interpreter.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.17: Cell Multiplicity and Cell Collections.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/17_cell_multiplicity/index.html b/k-distribution/k-tutorial/1_basic/17_cell_multiplicity/index.html new file mode 100644 index 00000000000..8c178e0dade --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/17_cell_multiplicity/index.html @@ -0,0 +1,577 @@ + + + + + + + + + + + + + + +Lesson 1.17: Cell Multiplicity and Cell Collections | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.17: Cell Multiplicity and Cell Collections

      +

      The purpose of this lesson is to explain how you can create optional cells +and cells that repeat multiple times in a configuration using a feature called +cell multiplicity.

      +

      Cell Multiplicity

      +

      K allows you to specify attributes for cell productions as part of the syntax +of configuration declarations. Unlike regular productions, which use the [] +syntax for attributes, configuration cells use an XML-like attribute syntax:

      +
      configuration <k color="red"> $PGM:K </k>
+
      +

      This configuration declaration gives the <k> cell the color red during +unparsing using the color attribute as discussed in +Lesson 1.9.

      +

      However, in addition to the usual attributes for productions, there are some +other attributes that can be applied to cells with special meaning. One such +attribute is the multiplicity attribute. By default, each cell that is +declared occurs exactly once in every configuration term. However, using the +multiplicity attribute, this default behavior can be changed. There are two +values that this attribute can have: ? and *.

      +

      Optional cells

      +

      The first cell multiplicity we will discuss is ?. Similar to a regular +expression language, this attribute tells the compiler that this cell can +appear 0 or 1 times in the configuration. In other words, it is an +optional cell. By default, K does not create optional cells in the initial +configuration, unless that optional cell has a configuration variable inside +it. However, it is possible to override the default behavior and create that +cell initially by adding the additional cell attribute initial="".

      +

      K uses the .Bag symbol to represent the absence of any cells in a particular +rule. Consider the following module:

      +
      k
      module LESSON-17-A
+  imports INT
+
+  configuration <k> $PGM:K </k>
+                <optional multiplicity="?"> 0 </optional>
+
+  syntax KItem ::= "init" | "destroy"
+
+  rule <k> init => . ...</k>
+       (.Bag => <optional> 0 </optional>)
+  rule <k> destroy => . ...</k>
+       (<optional> _ </optional> => .Bag)
+
+endmodule
+
      +

      In this definition, when the init symbol is executed, the <optional> cell +is added to the configuration, and when the destroy symbol is executed, it +is removed. Any rule that matches on that cell will only match if that cell is +present in the configuration.

      +

      Exercise

      +

      Create a simple definition with a Stmts sort that is a List{Stmt,""} and +a Stmt sort with the constructors +syntax Stmt ::= "enable" | "increment" | "decrement" | "disable". The +configuration should have an optional cell that contains an integer that +is created with the enable command, destroyed with the disable command, +and its value is incremented or decremented by the increment and decrement +command.

      +

      Cell collections

      +

      The second type of cell multiplicity we will discuss is *. Simlar to a +regular expression language, this attribute tells the compiler that this cell +can appear 0 or more times in the configuration. In other words, it is a +cell collection. Cells with multiplicity * must be the only child of +their parent cell. As a convention, the inner cell is usually named with the +singular form of what it contains, and the outer cell with the plural form, for +example, "thread" and "threads".

      +

      All cell collections are required to have the type attribute set to either +Set or Map. A Set cell collection is represented as a set and behaves +internally the same as the Set sort, although it actually declares a new +sort. A Map cell collection is represented as a Map in which the first +subcell of the cell collection is the key and the remaining cells are the +value.

      +

      For example, consider the following module:

      +
      k
      module LESSON-17-B
+  imports INT
+  imports BOOL
+  imports ID-SYNTAX
+
+  syntax Stmt ::= Id "=" Exp ";" [strict(2)]
+                | "return" Exp ";" [strict]
+  syntax Stmts ::= List{Stmt,""}
+  syntax Exp ::= Id
+               | Int
+               | Exp "+" Exp [seqstrict]
+               | "spawn" "{" Stmts "}"
+               | "join" Exp ";" [strict]
+
+  configuration <threads>
+                  <thread multiplicity="*" type="Map">
+                    <id> 0 </id>
+                    <k> $PGM:K </k>
+                  </thread>
+                </threads>
+                <state> .Map </state>
+                <next-id> 1 </next-id>
+
+  rule <k> X:Id => I:Int ...</k>
+       <state>... X |-> I ...</state>
+  rule <k> X:Id = I:Int ; => . ...</k>
+       <state> STATE => STATE [ X <- I ] </state>
+  rule <k> S:Stmt Ss:Stmts => S ~> Ss ...</k>
+  rule <k> I1:Int + I2:Int => I1 +Int I2 ...</k>
+
+  rule <thread>...
+         <k> spawn { Ss } => NEXTID ...</k>
+       ...</thread>
+       <next-id> NEXTID => NEXTID +Int 1 </next-id>
+       (.Bag =>
+       <thread>
+         <id> NEXTID </id>
+         <k> Ss </k>
+       </thread>)
+
+  rule <thread>...
+         <k> join ID:Int ; => I ...</k>
+       ...</thread>
+       (<thread>
+         <id> ID </id>
+         <k> return I:Int ; ...</k>
+       </thread> => .Bag)
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_) => false [owise]
+endmodule
+
      +

      This module implements a very basic fork/join semantics. The spawn expression +spawns a new thread to execute a sequence of statements and returns a thread +id, and the join statement waits until a thread executes return and then +returns the return value of the thread.

      +

      Note something quite novel here: the <k> cell is inside a cell of +multiplicity *. Since the <k> cell is just a regular cell (mostly), this +is perfectly allowable. Rules that don't mention a specific thread are +automatically completed to match any thread.

      +

      When you execute programs in this language, the cells in the cell collection +get sorted and printed like any other collection, but they still display like +cells. Rules in this language also benefit from all the structural power of +cells, allowing you to omit cells you don't care about or complete the +configuration automatically. This allows you to have the power of cells while +still being a collection under the hood.

      +

      Exercises

      +
        +
      1. Modify the solution from Lesson 1.16, Exercise 1 so that the cell you use to +keep track of functions in a Map is now a cell collection. Run some programs +and compare how they get unparsed before and after this change.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.18: Term Equality and the Ternary Operator.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/18_equality_and_conditionals/index.html b/k-distribution/k-tutorial/1_basic/18_equality_and_conditionals/index.html new file mode 100644 index 00000000000..7a2d5955991 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/18_equality_and_conditionals/index.html @@ -0,0 +1,483 @@ + + + + + + + + + + + + + + +Lesson 1.18: Term Equality and the Ternary Operator | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.18: Term Equality and the Ternary Operator

      +

      The purpose of this lesson is to introduce how to compare equality of terms in +K, and how to put conditional expressions directly into the right-hand side of +rules.

      +

      Term Equality

      +

      One major way you can compare whether two terms are equal in K is to simply +match both terms with a variable with the same name. This will only succeed +in matching if the two terms are equal structurally. However, sometimes this +is impractical, and it is useful to have access to a way to actually compare +whether two terms in K are equal. The operator for this is found in +domains.md in the K-EQUAL +module. The operator is ==K and takes two terms of sort K and returns a +Bool. It returns true if they are equal. This includes equality over builtin +types such as Map and Set where equality is not purely structural in +nature. However, it does not include any notion of semantic equality over +user-defined syntax. The inverse symbol for inequality is =/=K.

      +

      Ternary Operator

      +

      One way to introduce conditional logic in K is to have two separate rules, +each with a side condition (or one rule with a side condition and another with +the owise attribute). However, sometimes it is useful to explicitly write +a conditional expression directly in the right-hand side of a rule. For this +purpose, K defines one more operator in the K-EQUAL module, which corresponds +to the usual ternary operator found in many languages. Here is an example of its +usage (lesson-18.k):

      +
      k
      module LESSON-18
+  imports INT
+  imports BOOL
+  imports K-EQUAL
+
+  syntax Exp ::= Int | Bool | "if" "(" Exp ")" Exp "else" Exp [strict(1)]
+
+  syntax Bool ::= isKResult(K) [function, symbol]
+  rule isKResult(_:Int) => true
+  rule isKResult(_:Bool) => true
+
+  rule if (B:Bool) E1:Exp else E2:Exp => #if B #then E1 #else E2 #fi
+endmodule
+
      +

      Note the symbol on the right-hand side of the final rule. This symbol is +polymorphic: B must be of sort Bool, but E1 and E2 could have been +any sort so long as both were of the same sort, and the sort of the entire +expression becomes equal to that sort. K supports polymorphic built-in +operators, but does not yet allow users to write their own polymorphic +productions.

      +

      The behavior of this function is to evaluate the Boolean expression to a +Boolean, then pick one of the two children and return it based on whether the +Boolean is true or false. Please note that it is not a good idea to use this +symbol in cases where one or both of the children is potentially undefined +(for example, an integer expression that divides by zero). While the default +implementation is smart enough to only evaluate the branch that happens to be +picked, this will not be true when we begin to do program verification. If +you need short circuiting behavior, it is better to use a side condition.

      +

      Exercises

      +
        +
      1. +

        Write a function in K that takes two terms of sort K and returns an +Int: the Int should be 0 if the terms are equal and 1 if the terms are +unequal.

        +
        2. +
      2. +

        Modify your solution to Lesson 1.16, Exercise 1 and introduce an if +Stmt to the syntax of the language, then implement it using the #if symbol. +Make sure to write tests for the resulting interpreter.

        +
        3. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.19: Debugging with GDB.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/19_debugging/index.html b/k-distribution/k-tutorial/1_basic/19_debugging/index.html new file mode 100644 index 00000000000..3fb07bd2d53 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/19_debugging/index.html @@ -0,0 +1,912 @@ + + + + + + + + + + + + + + +Lesson 1.19: Debugging with GDB or LLDB | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.19: Debugging with GDB or LLDB

      +

      The purpose of this lesson is to teach how to debug your K interpreter using +the K-language support provided in GDB or +LLDB.

      +

      Caveats

      +

      This lesson has been written with GDB support on Linux in mind. Unfortunately, +on macOS, GDB has limited support. To address this, we have introduced early +experimental support for debugging with LLDB on macOS. In some cases, the +features supported by LLDB are slightly different to those supported by GDB; the +tutorial text will make this clear where necessary. If you use a macOS with an +LLVM version older than 15, you may need to upgrade it to use the LLDB +correctly. If you encounter an issue on either operating system, please open an +issue against the K repository.

      +

      Getting started

      +

      On Linux, you will need GDB in order to complete this lesson. If you do not +already have GDB installed, then do so. Steps to install GDB are outlined in +this GDB Tutorial.

      +

      On macOS, LLDB should already have been installed with K's build dependencies +(whether you have built K from source, or installed it using kup or Homebrew).

      +

      The first thing neccessary in order to debug a K interpreter is to build the +interpreter with full debugging support enabled. This can be done relatively +simply. First, run kompile with the command line flag --enable-llvm-debug. +The resulting compiled K definition will be ready to support debugging.

      +

      Once you have a compiled K definition and a program you wish to debug, you can +start the debugger by passing the --debugger flag to krun. This will +automatically load the program you are executing into GDB and drop you into a +GDB shell ready to start executing the program.

      +

      As an example, consider the following K definition (lesson-19-a.k):

      +
      k
      module LESSON-19-A
+  imports INT
+
+  rule I => I +Int 1
+    requires I <Int 100
+endmodule
+
      +

      If we compile this definition with kompile lesson-19-a.k --enable-llvm-debug, +and run the program 0 in the debugger with krun -cPGM=0 --debugger, we will +see the following output (roughly, and depending on which platform you are +using):

      +

      GDB / Linux

      +
      GNU gdb (Ubuntu 9.2-0ubuntu1~20.04) 9.2
+Copyright (C) 2020 Free Software Foundation, Inc.
+License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>
+This is free software: you are free to change and redistribute it.
+There is NO WARRANTY, to the extent permitted by law.
+Type "show copying" and "show warranty" for details.
+This GDB was configured as "x86_64-linux-gnu".
+Type "show configuration" for configuration details.
+For bug reporting instructions, please see:
+<http://www.gnu.org/software/gdb/bugs/>.
+Find the GDB manual and other documentation resources online at:
+    <http://www.gnu.org/software/gdb/documentation/>.
+
+For help, type "help".
+Type "apropos word" to search for commands related to "word"...
+Reading symbols from ./lesson-19-a-kompiled/interpreter...
+warning: File "/home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter" auto-loading has been declined by your `auto-load safe-path' set to "$debugdir:$datadir/auto-load".
+To enable execution of this file add
+        add-auto-load-safe-path /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter
+line to your configuration file "/home/dwightguth/.gdbinit".
+To completely disable this security protection add
+        set auto-load safe-path /
+line to your configuration file "/home/dwightguth/.gdbinit".
+For more information about this security protection see the
+"Auto-loading safe path" section in the GDB manual.  E.g., run from the shell:
+        info "(gdb)Auto-loading safe path"
+(gdb)
+
      +

      To make full advantage of the GDB features of K, you should follow the first +command listed in this output message and add the corresponding +add-auto-load-safe-path command to your ~/.gdbinit file as prompted. +Please note that the path will be different on your machine than the one +listed above. Adding directories to the "load safe path" effectively tells GDB +to trust those directories. All content under a given directory will be recursively +trusted, so if you want to avoid having to add paths to the "load safe path" every +time you kompile a different K definition, then you can just trust a minimal +directory containing all your kompiled files; however, do not choose a top-level directory containing arbitrary files as this amounts to trusting arbitrary files and is a security risk. More info on the load safe path +can be found here.

      +

      LLDB / macOS

      +
      (lldb) target create "./lesson-19-a-kompiled/interpreter"
+warning: 'interpreter' contains a debug script. To run this script in this debug session:
+
+    command script import "/Users/brucecollie/code/scratch/lesson-19-a-kompiled/interpreter.dSYM/Contents/Resources/Python/interpreter.py"
+
+To run all discovered debug scripts in this session:
+
+    settings set target.load-script-from-symbol-file true
+
+Current executable set to '/Users/brucecollie/code/scratch/lesson-19-a-kompiled/interpreter' (x86_64).
+(lldb) settings set -- target.run-args  ".krun-2023-03-20-11-22-46-TcYt9ffhb2/tmp.in.RupiLwHNfn" "-1" ".krun-2023-03-20-11-22-46-TcYt9ffhb2/result.kore"
+(lldb) 
+
      +

      LLDB applies slightly different security policies to GDB. To load K's debugging +scripts for this session only, you can run the command script import line at +the LLDB prompt. The loaded scripts will not persist across debugging sessions +if you do this. It is also possible to configure LLDB to automatically load the +K scripts when an interpreter is started in LLDB; doing so requires a slightly +less broad permission than GDB.

      +

      On macOS, the .dSYM directory that contains debugging symbols for an +executable can also contain Python scripts in Contents/Resources/Python. If +there is a Python script with a name matching the name of the current executable +(here, interpreter and interpreter.py), it will be automatically loaded if +the target.load-script-from-symbol-file setting is set). You can therefore add +the settings set command to your ~/.lldbinit without enabling full arbitrary +code execution, but you should be aware of the paths from which code can be +executed if you do so.

      +

      Basic commands

      +
      +

      LLDB Note: the k start and k step commands are currently not +implemented in the K LLDB scripts. To work around this limitation temporarily, +you can run process launch --stop-at-entry instead of k start. To emulate +k step, first run rbreak k_step once, then continue instead of each k step. We hope to address these limitations soon.

      +
      +

      The most basic commands you can execute in the K GDB session are to run your +program or to step through it. The first can be accomplished using GDB's +built-in run command. This will automatically start the program and begin +executing it. It will continue until the program aborts or finishes, or the +debugger is interrupted with Ctrl-C.

      +

      Sometimes you want finer-grained control over how you proceed through the +program you are debugging. To step through the rule applications in your +program, you can use the k start and k step GDB commands.

      +

      k start is similar to the built-in start command in that it starts the +program and then immediately breaks before doing any work. However, unlike +the start command which will break immediately after the main method of +a program is executed, the K start program will initialize the rewriter, +evaluate the initial configuration, and break immediately prior to applying +any rewrite steps.

      +

      In the example above, here is what we see when we run the k start command:

      +
      Temporary breakpoint 1 at 0x239210
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-a-kompiled/interpreter .krun-2021-08-13-14-10-50-sMwBkbRicw/tmp.in.01aQt85TaA -1 .krun-2021-08-13-14-10-50-sMwBkbRicw/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Temporary breakpoint 1, 0x0000000000239210 in main ()
+0x0000000000231890 in step (subject=<k>
+  0 ~> .
+</k>)
+(gdb)
+
      +

      As you can see, we are stopped at the step function in the interpreter. +This function is responsible for taking top-level rewrite steps. The subject +parameter to this function is the current K configuration.

      +

      We can step through K rewrite steps one at a time by running the k step +command. By default, this takes a single rewrite step (including any function +rule applications that are part of that step).

      +

      Here is what we see when we run that command:

      +
      Continuing.
+
+Temporary breakpoint -22, 0x0000000000231890 in step (subject=<k>
+  1 ~> .
+</k>)
+(gdb)
+
      +

      As we can see, we have taken a single rewrite step. We can also pass a number +to the k step command which indicates the number of rewrite steps to take.

      +

      Here is what we see if we run k step 10:

      +
      Continuing.
+
+Temporary breakpoint -23, 0x0000000000231890 in step (subject=<k>
+  11 ~> .
+</k>)
+(gdb)
+
      +

      As we can see, ten rewrite steps were taken.

      +

      Breakpoints

      +

      The next important step in debugging an application in GDB is to be able to +set breakpoints. Generally speaking, there are three types of breakpoints we +are interested in a K semantics: Setting a breakpoint when a particular +function is called, setting a breakpoint when a particular rule is applied, +and setting a breakpoint when a side condition of a rule is evaluated.

      +

      The easiest way to do the first two things is to set a breakpoint on the +line of code containing the function or rule.

      +

      For example, consider the following K definition (lesson-19-b.k):

      +
      k
      module LESSON-19-B
+  imports BOOL
+
+  syntax Bool ::= isBlue(Fruit) [function]
+  syntax Fruit ::= Blueberry() | Banana()
+  rule isBlue(Blueberry()) => true
+  rule isBlue(Banana()) => false
+
+  rule F:Fruit => isBlue(F)
+endmodule
+
      +

      Once this program has been compiled for debugging, we can run the program +Blueberry(). We can then set a breakpoint that stops when the isBlue +function is called with the following command in GDB:

      +
      break lesson-19-b.k:4
+
      +

      Similarly, in LLDB, run:

      +
      breakpoint set --file lesson-19-b.k --line 4
+
      +

      Here is what we see if we set this breakpoint and then run the interpreter:

      +
      (gdb) break lesson-19-b.k:4
+Breakpoint 1 at 0x231040: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k, line 4.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-20-27-vXOQmV6lwS/tmp.in.fga98yqXlc -1 .krun-2021-08-13-14-20-27-vXOQmV6lwS/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit (_1=Blueberry ( )) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:4
+4         syntax Bool ::= isBlue(Fruit) [function]
+(gdb)
+
      +
      (lldb) breakpoint set --file lesson-19-b.k --line 4
+Breakpoint 1: where = interpreter`LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit + 20 at lesson-19-b.k:4:19, address = 0x0000000100003ff4
+(lldb) run
+Process 50546 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50546 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100003ff4 interpreter`LblisBlue'LParUndsRParUnds'LESSON-19-B'Unds'Bool'Unds'Fruit(_1=Blueberry ( )) at lesson-19-b.k:4:19
+   1   	module LESSON-19-B
+   2   	  imports BOOL
+   3   	
+-> 4   	  syntax Bool ::= isBlue(Fruit) [function]
+   5   	  syntax Fruit ::= Blueberry() | Banana()
+   6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+(lldb)
+
      +

      As we can see, we have stopped at the point where we are evaluating that +function. The value _1 that is a parameter to that function shows the +value passed to the function by the caller.

      +

      We can also break when the isBlue(Blueberry()) => true rule applies by simply +changing the line number to the line number of that rule:

      +
      (gdb) break lesson-19-b.k:6
+Breakpoint 1 at 0x2af710: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-32-36-7kD0ic7XwD/tmp.in.8JNH5Qtmow -1 .krun-2021-08-13-14-32-36-7kD0ic7XwD/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, apply_rule_138 () at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:6
+6         rule isBlue(Blueberry()) => true
+(gdb)
+
      +
      (lldb) breakpoint set --file lesson-19-b.k --line 6
+Breakpoint 1: where = interpreter`apply_rule_140 at lesson-19-b.k:6:8, address = 0x0000000100004620
+(lldb) run
+Process 50681 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50681 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100004620 interpreter`apply_rule_140 at lesson-19-b.k:6:8
+   3   	
+   4   	  syntax Bool ::= isBlue(Fruit) [function]
+   5   	  syntax Fruit ::= Blueberry() | Banana()
+-> 6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+   8   	
+   9   	  rule F:Fruit => isBlue(F)
+(lldb) 
+
      +

      We can also do the same with a top-level rule:

      +
      (gdb) break lesson-19-b.k:9
+Breakpoint 1 at 0x2aefa0: lesson-19-b.k:9. (2 locations)
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b-kompiled/interpreter .krun-2021-08-13-14-33-13-9fC8Sz4aO3/tmp.in.jih1vtxSiQ -1 .krun-2021-08-13-14-33-13-9fC8Sz4aO3/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, apply_rule_107 (Var'Unds'DotVar0=<generatedCounter>
+  0
+</generatedCounter>, Var'Unds'DotVar1=., VarF=Blueberry ( )) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-b.k:9
+9         rule F:Fruit => isBlue(F)
+(gdb)
+
      +
      (lldb) breakpoint set --file lesson-19-b.k --line 9
+Breakpoint 1: 2 locations.
+(lldb) run
+Process 50798 launched: '/Users/brucecollie/code/scratch/lesson-19-b-kompiled/interpreter' (x86_64)
+Process 50798 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x0000000100003f2e interpreter`apply_rule_109(Var'Unds'DotVar0=<generatedCounter>
+  0
+</generatedCounter>, Var'Unds'DotVar1=., VarF=Blueberry ( )) at lesson-19-b.k:9:8
+   6   	  rule isBlue(Blueberry()) => true
+   7   	  rule isBlue(Banana()) => false
+   8   	
+-> 9   	  rule F:Fruit => isBlue(F)
+   10  	endmodule
+(lldb)  
+
      +

      Unlike the function rule above, we see several parameters to this function. +These are the substitution that was matched for the function. Variables only +appear in this substitution if they are actually used on the right-hand side +of the rule.

      +

      Advanced breakpoints

      +

      Sometimes it is inconvenient to set the breakpoint based on a line number.

      +

      It is also possible to set a breakpoint based on the rule label of a particular +rule. Consider the following definition (lesson-19-c.k):

      +
      k
      module LESSON-19-C
+  imports INT
+  imports BOOL
+
+  syntax Bool ::= isEven(Int) [function]
+  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+
+endmodule
+
      +

      We will run the program isEven(4). We can set a breakpoint for when a rule +applies by means of the MODULE-NAME.label.rhs syntax:

      +
      (gdb) break LESSON-19-C.isEven.rhs
+Breakpoint 1 at 0x2afda0: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c-kompiled/interpreter .krun-2021-08-13-14-40-29-LNNT8YEZ61/tmp.in.ZG93vWCGGC -1 .krun-2021-08-13-14-40-29-LNNT8YEZ61/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LESSON-19-C.isEven.rhs () at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+6         rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+(gdb)
+
      +
      (lldb) breakpoint set --name LESSON-19-C.isEven.rhs
+Breakpoint 1: where = interpreter`LESSON-19-C.isEven.rhs at lesson-19-c.k:6:18, address = 0x00000001000038e0
+(lldb) run
+Process 51205 launched: '/Users/brucecollie/code/scratch/lesson-19-c-kompiled/interpreter' (x86_64)
+Process 51205 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x00000001000038e0 interpreter`LESSON-19-C.isEven.rhs at lesson-19-c.k:6:18
+   3   	  imports BOOL
+   4   	
+   5   	  syntax Bool ::= isEven(Int) [function]
+-> 6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8   	
+   9   	endmodule
+(lldb) 
+
      +

      We can also set a breakpoint for when a rule's side condition is evaluated +by means of the MODULE-NAME.label.sc syntax:

      +
      (gdb) break LESSON-19-C.isEven.sc
+Breakpoint 1 at 0x2afd70: file /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k, line 6.
+(gdb) run
+Starting program: /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c-kompiled/interpreter .krun-2021-08-13-14-41-48-1BoGfJRbYc/tmp.in.kg4F8cwfCe -1 .krun-2021-08-13-14-41-48-1BoGfJRbYc/result.kore
+[Thread debugging using libthread_db enabled]
+Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
+
+Breakpoint 1, LESSON-19-C.isEven.sc (VarI=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+6         rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+(gdb) finish
+Run till exit from #0  LESSON-19-C.isEven.sc (VarI=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:6
+0x00000000002b2662 in LblisEven'LParUndsRParUnds'LESSON-19-C'Unds'Bool'Unds'Int (_1=4) at /home/dwightguth/kframework-5.0.0/k-distribution/k-tutorial/1_basic/19_debugging/lesson-19-c.k:5
+5         syntax Bool ::= isEven(Int) [function]
+Value returned is $1 = true
+(gdb)
+
      +
      (lldb) breakpoint set --name LESSON-19-C.isEven.sc
+Breakpoint 1: where = interpreter`LESSON-19-C.isEven.sc + 1 at lesson-19-c.k:6:18, address = 0x00000001000038c1
+(lldb) run
+Process 52530 launched: '/Users/brucecollie/code/scratch/lesson-19-c-kompiled/interpreter' (x86_64)
+Process 52530 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = breakpoint 1.1
+    frame #0: 0x00000001000038c1 interpreter`LESSON-19-C.isEven.sc(VarI=0x0000000101800088) at lesson-19-c.k:6:18
+   3   	  imports BOOL
+   4   	
+   5   	  syntax Bool ::= isEven(Int) [function]
+-> 6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8   	
+   9   	endmodule
+(lldb) finish
+Process 52649 stopped
+* thread #1, queue = 'com.apple.main-thread', stop reason = step out
+Return value: (bool) $0 = true
+
+    frame #0: 0x00000001000069e5 interpreter`LblisEven'LParUndsRParUnds'LESSON-19-C'Unds'Bool'Unds'Int(_1=0x0000000101800088) at lesson-19-c.k:5:19
+   2   	  imports INT
+   3   	  imports BOOL
+   4   	
+-> 5   	  syntax Bool ::= isEven(Int) [function]
+   6   	  rule [isEven]: isEven(I) => true requires I %Int 2 ==Int 0
+   7   	  rule [isOdd]: isEven(I) => false requires I %Int 2 =/=Int 0
+   8
+(lldb)
+
      +

      Here we have used the built-in command finish to tell us whether the side +condition returned true or not. Note that once again, we see the substitution +that was matched from the left-hand side. Like before, a variable will only +appear here if it is used in the side condition.

      +

      Debugging rule matching

      +

      Sometimes it is useful to try to determine why a particular rule did or did +not apply. K provides some basic debugging commands which make it easier +to determine this.

      +

      Consider the following K definition (lesson-19-d.k):

      +
      k
      module LESSON-19-D
+
+  syntax Foo ::= foo(Bar)
+  syntax Bar ::= bar(Baz) | bar2(Baz)
+  syntax Baz ::= baz() | baz2()
+
+  rule [baz]: foo(bar(baz())) => .K
+
+endmodule
+
      +

      Suppose we try to run the program foo(bar(baz2())). It is obvious from this +example why the rule in this definition will not apply. However, in practice, +such cases are not always obvious. You might look at a rule and not immediately +spot why it didn't apply on a particular term. For this reason, it can be +useful to get the debugger to provide a log about how it tried to match that +term. You can do this with the k match command. If you are stopped after +having run k start or k step, you can obtain this log for any rule after +any step by running the command k match MODULE.label subject for a particular +top-level rule label.

      +

      For example, with the baz rule above, we get the following output:

      +
      (gdb) k match LESSON-19-D.baz subject
+Subject:
+baz2 ( )
+does not match pattern:
+baz ( )
+
      +
      (lldb) k match LESSON-19-D.baz subject
+Subject:
+baz2 ( )
+does not match pattern:
+baz ( )
+
      +

      As we can see, it provided the exact subterm which did not match against the +rule, as well as the particular subpattern it ought to have matched against.

      +

      This command does not actually take any rewrite steps. In the event that +matching actually succeeds, you will still need to run the k step command +to advance to the next step.

      +

      Final notes

      +

      In addition to the functionality provided above, you have the full power of +GDB or LLDB at your disposal when debugging. Some features are not particularly +well-adapted to K code and may require more advanced knowledge of the +term representation or implementation to use effectively, but anything that +can be done in GDB or LLDB can in theory be done using this debugging functionality. +We suggest you refer to the +GDB Documentation or +LLDB Tutorial if you +want to try to do something and are unsure as to how.

      +

      Exercises

      +
        +
      1. Compile your solution to Lesson 1.18, Exercise 2 with debugging support +enabled and step through several programs you have previously used to test. +Then set a breakpoint on the isKResult function and observe the state of the +interpreter when stopped at that breakpoint. Set a breakpoint on the rule for +addition and run a program that causes it to be stopped at that breakpoint. +Finally, step through the program until the addition symbol is at the top +of the K cell, and then use the k match command to report the reason why +the subtraction rule does not apply. You may need to modify the definition +to insert some rule labels.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.20: K Backends and the Haskell Backend.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/20_backends/index.html b/k-distribution/k-tutorial/1_basic/20_backends/index.html new file mode 100644 index 00000000000..9dae9b9c010 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/20_backends/index.html @@ -0,0 +1,503 @@ + + + + + + + + + + + + + + +Lesson 1.20: K Backends and the Haskell Backend | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.20: K Backends and the Haskell Backend

      +

      The purpose of this lesson is to teach about the multiple backends of K, +in particular the Haskell Backend which is the complement of the backend we +have been using so far.

      +

      K Backends

      +

      Thus far, we have not discussed the distinction between the K frontend and +the K backends at all. We have simply assumed that if you run kompile on a +K definition, there will be a compiler backend that will allow you to execute +the K definition you have compiled.

      +

      K actually has multiple different backends. The one we have been using so far +implicitly, the default backend, is called the LLVM Backend. It is +designed to support efficient, optimized concrete execution and search. It +does this by compiling your K definition to LLVM bitcode and then using LLVM +to generate machine code for it that is compiled and linked and executed. +However, K is a formal methods toolkit at the end of the day, and the primary +goal many people have when defining a programming language in K is to +ultimately be able to perform more advanced verification on programs in their +programming language.

      +

      It is for this purpose that K also provides the Haskell Backend, so called +because it is implemented in Haskell. While we will cover the features of the +Haskell Backend in more detail in the next two lessons, the important thing to +understand is that it is a separate backend which is optimized for more formal +reasoning about programming languages. While it is capable of performing +concrete execution, it does not do so as efficiently as the LLVM Backend. +In exchange, it provides more advanced features.

      +

      Choosing a backend

      +

      You can choose which backend to use to compile a K definition by means of the +--backend flag to kompile. By default, if you do not specify this flag, it +is equivalent to if you had specified --backend llvm. However, to use the +Haskell Backend instead, you can simply say kompile --backend haskell on a +particular K definition.

      +

      As an example, here is a simple K definition that we have seen before in the +previous lesson (lesson-20.k):

      +
      k
      module LESSON-20
+  imports INT
+
+  rule I => I +Int 1
+    requires I <Int 100
+endmodule
+
      +

      Previously we compiled this definition using the LLVM Backend, but if we +instead execute the command kompile lesson-20.k --backend haskell, we +will get an interpreter for this K definition that is implemented in Haskell +instead. Unlike the default LLVM Backend, the Haskell Backend is not a +compiler per se. It does not generate new Haskell code corresponding to your +programming language and then compile and execute it. Instead, it is an +interpreter which reads the generated IR from kompile and implements in +Haskell an interpreter that is capable of interpreting any K definition.

      +

      Note that on arm64 macOS (Apple Silicon), there is a known issue with the Compact +library that causes crashes in the Haskell backend. Pass the additional flag +--no-haskell-binary to kompile to resolve this. +This flag is also needed when using krun.

      +

      Exercise

      +

      Try running the program 0 in this K definition on the Haskell Backend and +compare the final configuration to what you would get compiling the same +definition with the LLVM Backend.

      +

      Legacy backends

      +

      As a quick note, K does provide one other backend, which exists primarily as +legacy code which should be considered deprecated. This is the +Java Backend. The Java Backend is essentially a precursor to the Haskell +Backend. We will not cover this backend in any detail since it is deprecated, +but we still mention it here for the purposes of understanding.

      +

      Exercises

      +
        +
      1. Compile your solution to Lesson 1.18, Exercise 2 with the Haskell Backend +and execute some programs. Compare the resulting configurations with the +output of the same program on the LLVM Backend. Note that if you are getting +different behaviors on the Haskell backend, you might have some luck debugging +by passing --search to krun when using the LLVM backend.
        2. +
      +

      Next lesson

      +

      Once you have completed the above exercises, you can continue to +Lesson 1.21: Unification and Symbolic Execution.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/21_symbolic_execution/index.html b/k-distribution/k-tutorial/1_basic/21_symbolic_execution/index.html new file mode 100644 index 00000000000..a7f0bfb2468 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/21_symbolic_execution/index.html @@ -0,0 +1,583 @@ + + + + + + + + + + + + + + +Lesson 1.21: Unification and Symbolic Execution | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.21: Unification and Symbolic Execution

      +

      The purpose of this lesson is to teach the basic concepts of symbolic execution +in order to introduce the unique capabilities of the Haskell Backend at a +conceptual level.

      +

      Symbolic Execution

      +

      Thus far, all of the programs we have run using K have been concrete +configurations. What this means is that the configuration we use to initialize +the K rewrite engine is concrete; in other words, contains no logical +variables. The LLVM Backend is a concrete execution engine, meaning that +it is only capable of rewriting concrete configurations.

      +

      By contrast, the Haskell Backend performs symbolic execution, which is +capable of rewriting any configuration, including those where parts of the +configuration are symbolic, ie, contain variables or uninterpreted +functions.

      +

      Unification

      +

      Previously, we have introduced the concept that K rewrite rules operate by +means of pattern matching: the current configuration being rewritten is pattern +matched against the left-hand side of the rewrite rule, and the substitution +is used in order to construct a new term from the right-hand side. In symbolic +execution, we use +unification +instead of pattern matching. To summarize, unification behaves akin to a +two-way pattern matching where both the configuration and the left-hand side +of the rule can contain variables, and the algorithm generates a +most general unifier containing substitutions for the variables in both +which will make both terms equal.

      +

      Feasibility

      +

      Unification by itself cannot completely solve the problem of symbolic +execution. One task symbolic execution must perform is to identify whether +a particular symbolic term is feasible, that is to say, that there actually +exists a concrete instantiation of that term such that all the logical +constraints on that term can actually be satisfied. The Haskell Backend +delegates this task to Z3, an +SMT solver. +This solver is used to periodically trim configurations that are determined +to be mathematically infeasible.

      +

      Symbolic terms

      +

      The final component of symbolic execution consists of the task of introducing +symbolic terms into the configuration. This can be done one of two different +ways. First, the term being passed to krun can actually be symbolic. This +is less frequently used because it requires the user to construct an AST +that contains variables, something which our current parsing capabilities are +not well-equipped to do. The second, more common, way of introducing symbolic +terms into a configuration consists of writing rules where there exists an +existentially qualified variable on the right-hand side of the rule that does +not exist on the left-hand side of the rule.

      +

      In order to prevent users from writing such rules by accident, K requires +that such variables begin with the ? prefix. For example, here is a rule +that rewrites a constructor foo to a symbolic integer:

      +
      rule <k> foo => ?X:Int ...</k>
+
      +

      When this rule applies, a fresh variable is introduced to the configuration, which +then is unified against the rules that might apply in order to symbolically +execute that configuration.

      +

      ensures clauses

      +

      We also introduce here a new feature of K rules that applies when a rule +has this type of variable on the right-hand side: the ensures clause. +An ensures clause is similar to a requires clause and can appear after +a rule body, or after a requires clause. The ensures clause is used to +introduce constraints that might apply to the variable that was introduced by +that rule. For example, we could write the rule above with the additional +constraint that the symbolic integer that was introduced must be less than +five, by means of the following rule:

      +
      rule <k> foo => ?X:Int ...</k> ensures ?X <Int 5
+
      +

      Putting it all together

      +

      Putting all these pieces together, it is possible to use the Haskell Backend +to perform symbolic reasoning about a particular K module, determining all the +possible states that can be reached by a symbolic configuration.

      +

      For example, consider the following K definition (lesson-21.k):

      +
      k
      module LESSON-21
+    imports INT
+
+    rule <k> 0 => ?X:Int ... </k> ensures ?X =/=Int 0
+    rule <k> X:Int => 5  ... </k> requires X >=Int 10
+endmodule
+
      +

      When we symbolically execute the program 0, we get the following output +from the Haskell Backend:

      +
          <k>
+      5 ~> .
+    </k>
+  #And
+    {
+      true
+    #Equals
+      ?X:Int >=Int 10
+    }
+  #And
+    #Not ( {
+      ?X:Int
+    #Equals
+      0
+    } )
+#Or
+    <k>
+      ?X:Int ~> .
+    </k>
+  #And
+    #Not ( {
+      true
+    #Equals
+      ?X:Int >=Int 10
+    } )
+  #And
+    #Not ( {
+      ?X:Int
+    #Equals
+      0
+    } )
+
      +

      Note some new symbols introduced by this configuration: #And, #Or, and +#Equals. While andBool, orBool, and ==K represent functions of sort +Bool, #And, #Or, and #Equals are matching logic connectives. We +will discuss matching logic in more detail later in the tutorial, but the basic +idea is that these symbols represent Boolean operators over the domain of +configurations and constraints, as opposed to over the Bool sort.

      +

      Notice that the configuration listed above is a disjunction of conjunctions. +This is the most common form of output that can be produced by the Haskell +Backend. In this case, each conjunction consists of a configuration and a set +of constraints. What this conjunction describes, essentially, is a +configuration and a set of information that was derived to be true while +rewriting that configuration.

      +

      Similar to how we saw --search in a previous lesson, the reason we have +multiple disjuncts is because there are multiple possible output states +for this program, depending on whether or not the second rule applied. In the +first case, we see that ?X is greater than or equal to 10, so the second rule +applied, rewriting the symbolic integer to the concrete integer 5. In the +second case, we see that the second rule did not apply because ?X is less +than 10. Moreover, because of the ensures clause on the first rule, we know +that ?X is not zero, therefore the first rule will not apply a second time. +If we had omitted this constraint, we would have ended up infinitely applying +the first rule, leading to krun not terminating.

      +

      In the next lesson, we will cover how symbolic execution forms the backbone +of deductive program verification in K and how we can use K to prove programs +correct against a specification.

      +

      Exercises

      +
        +
      1. Create another rule in LESSON-21 that rewrites odd integers greater than +ten to a symbolic even integer less than 10 and greater than 0. This rule will +now apply nondeterministically along with the existing rules. Predict what the +resulting output configuration will be from rewriting 0 after adding this +rule. Then run the program and see whether your prediction is correct.
        2. +
      +

      Once you have completed the above exercises, you can continue to +Lesson 1.22: Basics of Deductive Program Verification using K.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/22_proofs/index.html b/k-distribution/k-tutorial/1_basic/22_proofs/index.html new file mode 100644 index 00000000000..cc4331f0400 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/22_proofs/index.html @@ -0,0 +1,819 @@ + + + + + + + + + + + + + + +Lesson 1.22: Basics of Deductive Program Verification using K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 1.22: Basics of Deductive Program Verification using K

      +

      In this lesson, you will familiarize yourself with the basics of using K for +deductive program verification.

      +

      1. Setup: Simple Programming Language with Function Calls

      +

      We base this lesson on a simple programming language with functions, +assignment, if conditionals, and while loops. Take your time to study its +formalization below (lesson-22.k):

      +
      module LESSON-22-SYNTAX
+    imports INT-SYNTAX
+    imports BOOL-SYNTAX
+    imports ID-SYNTAX
+
+    syntax Exp ::= IExp | BExp
+
+    syntax IExp ::= Id | Int
+
+    syntax KResult ::= Int | Bool | Ints
+
+    // Take this sort structure:
+    //
+    //     IExp
+    //    /    \
+    // Int      Id
+    //
+    // Through the List{_, ","} functor.
+    // Must add a `Bot`, for a common subsort for the empty list.
+
+    syntax Bot
+    syntax Bots ::= List{Bot, ","} [klabel(exps)]
+    syntax Ints ::= List{Int, ","} [klabel(exps)]
+                  | Bots
+    syntax Ids  ::= List{Id, ","}  [klabel(exps)]
+                  | Bots
+    syntax Exps ::= List{Exp, ","} [klabel(exps), seqstrict]
+                  | Ids | Ints
+
+    syntax IExp ::= "(" IExp ")" [bracket]
+                  | IExp "+" IExp [seqstrict]
+                  | IExp "-" IExp [seqstrict]
+                  > IExp "*" IExp [seqstrict]
+                  | IExp "/" IExp [seqstrict]
+                  > IExp "^" IExp [seqstrict]
+                  | Id "(" Exps ")" [strict(2)]
+
+    syntax BExp ::= Bool
+
+    syntax BExp ::= "(" BExp ")" [bracket]
+                  | IExp "<=" IExp [seqstrict]
+                  | IExp "<"  IExp [seqstrict]
+                  | IExp ">=" IExp [seqstrict]
+                  | IExp ">"  IExp [seqstrict]
+                  | IExp "==" IExp [seqstrict]
+                  | IExp "!=" IExp [seqstrict]
+
+    syntax BExp ::= BExp "&&" BExp
+                  | BExp "||" BExp
+
+    syntax Stmt ::=
+         Id "=" IExp ";" [strict(2)]                        // Assignment
+       | Stmt Stmt [left]                                   // Sequence
+       | Block                                              // Block
+       | "if" "(" BExp ")" Block "else" Block [strict(1)]   // If conditional
+       | "while" "(" BExp ")" Block                         // While loop
+       | "return" IExp ";"                    [seqstrict]   // Return statement
+       | "def" Id "(" Ids ")" Block                         // Function definition
+
+    syntax Block ::=
+         "{" Stmt "}"    // Block with statement
+       | "{" "}"         // Empty block
+endmodule
+
+module LESSON-22
+    imports INT
+    imports BOOL
+    imports LIST
+    imports MAP
+    imports LESSON-22-SYNTAX
+
+    configuration
+      <k> $PGM:Stmt </k>
+      <store> .Map </store>
+      <funcs> .Map </funcs>
+      <stack> .List </stack>
+
+ // -----------------------------------------------
+    rule <k> I1 + I2 => I1 +Int I2 ... </k>
+    rule <k> I1 - I2 => I1 -Int I2 ... </k>
+    rule <k> I1 * I2 => I1 *Int I2 ... </k>
+    rule <k> I1 / I2 => I1 /Int I2 ... </k>
+    rule <k> I1 ^ I2 => I1 ^Int I2 ... </k>
+
+    rule <k> I:Id => STORE[I] ... </k>
+         <store> STORE </store>
+
+ // ------------------------------------------------
+    rule <k> I1 <= I2 => I1  <=Int I2 ... </k>
+    rule <k> I1  < I2 => I1   <Int I2 ... </k>
+    rule <k> I1 >= I2 => I1  >=Int I2 ... </k>
+    rule <k> I1  > I2 => I1   >Int I2 ... </k>
+    rule <k> I1 == I2 => I1  ==Int I2 ... </k>
+    rule <k> I1 != I2 => I1 =/=Int I2 ... </k>
+
+    rule <k> B1 && B2 => B1 andBool B2 ... </k>
+    rule <k> B1 || B2 => B1  orBool B2 ... </k>
+
+    rule <k> S1:Stmt S2:Stmt => S1 ~> S2 ... </k>
+
+    rule <k> ID = I:Int ; => . ... </k>
+         <store> STORE => STORE [ ID <- I ] </store>
+
+    rule <k> { S } => S ... </k>
+    rule <k> {   } => . ... </k>
+
+    rule <k> if (true)   THEN else _ELSE => THEN ... </k>
+    rule <k> if (false) _THEN else  ELSE => ELSE ... </k>
+
+    rule <k> while ( BE ) BODY => if ( BE ) { BODY while ( BE ) BODY } else { } ... </k>
+
+    rule <k> def FNAME ( ARGS ) BODY => . ... </k>
+         <funcs> FS => FS [ FNAME <- def FNAME ( ARGS ) BODY ] </funcs>
+
+    rule <k> FNAME ( IS:Ints ) ~> CONT => #makeBindings(ARGS, IS) ~> BODY </k>
+         <funcs> ... FNAME |-> def FNAME ( ARGS ) BODY ... </funcs>
+         <store> STORE => .Map </store>
+         <stack> .List => ListItem(state(CONT, STORE)) ... </stack>
+
+    rule <k> return I:Int ; ~> _ => I ~> CONT </k>
+         <stack> ListItem(state(CONT, STORE)) => .List ... </stack>
+         <store> _ => STORE </store>
+
+    rule <k> return I:Int ; ~> . => I </k>
+         <stack> .List </stack>
+
+    syntax KItem ::= #makeBindings(Ids, Ints)
+                   | state(continuation: K, store: Map)
+ // ----------------------------------------------------
+    rule <k> #makeBindings(.Ids, .Ints) => . ... </k>
+    rule <k> #makeBindings((I:Id, IDS => IDS), (IN:Int, INTS => INTS)) ... </k>
+         <store> STORE => STORE [ I <- IN ] </store>
+endmodule
+
      +

      Next, compile this example using kompile lesson-22.k --backend haskell. If +your processor is an Apple Silicon processor, add the --no-haskell-binary +flag if the compilation fails.

      +

      2. Setup: Proof Environment

      +

      Next, take the following snippet of K code and save it in lesson-22-spec.k. +This is a skeleton of the proof environment, and we will complete it as the +lesson progresses.

      +
      requires "lesson-22.k"
+requires "domains.md"
+
+module LESSON-22-SPEC-SYNTAX
+    imports LESSON-22-SYNTAX
+
+endmodule
+
+module VERIFICATION
+    imports K-EQUAL
+    imports LESSON-22-SPEC-SYNTAX
+    imports LESSON-22
+    imports MAP-SYMBOLIC
+
+endmodule
+
+module LESSON-22-SPEC
+    imports VERIFICATION
+
+endmodule
+
      +

      3. Claims

      +
        +
      1. The first claim we will ask K to prove is that 3 + 4, in fact, equals 7. +Claims are stated using the claim keyword, followed by the claim +statement:
        2. +
      +
      claim <k> 3 + 4 => 7 ... </k>
+
      +

      Add this claim to the LESSON-22-SPEC module and run the K prover using the +command kprove lesson-22-spec.k. You should get back the output #Top, +which denotes the Matching Logic equivalent of true and means, in this +context, that all claims have been proven correctly.

      +
        +
      1. The second claim reasons about the if statement that has a concrete condition:
        2. +
      +
      claim <k> if ( 3 + 4 == 7 ) {
+            $a = 1 ;
+            } else {
+            $a = 2 ;
+            }
+        => . ... </k>
+        <store> STORE => STORE [ $a <- 1 ] </store>
+
      +

      stating that the given program terminates (=> .), and when it does, the value +of the variable $a is set to 1, meaning that the execution will have taken +the then branch. Add this claim to the LESSON-22-SPEC module, but also add

      +
      syntax Id ::= "$a" [token]
+
      +

      to the LESSON-22-SPEC-SYNTAX module in order to declare $a as a token so +that it can be used as a program variable. Re-run the K prover, which should +again return #Top.

      +
        +
      1. Our third claim demonstrates how to reason about both branches of an if +statement at the same time:
        2. +
      +
      claim <k> $a = A:Int ; $b = B:Int ;
+          if ($a < $b) {
+            $c = $b ;
+          } else {
+            $c = $a ;
+          }
+        => . ... </k>
+        <store> STORE => STORE [ $a <- A ] [ $b <- B ] [ $c <- ?C:Int ] </store>
+    ensures (?C ==Int A) orBool (?C ==Int B)
+
      +

      The program in question first assigns symbolic integers A and B to program +variables $a and $b, respectively, and then executes the given if +statement, which has a symbolic condition (A < B), updating the value of the +program variable $c in both branches. The specification we give states that +the if statement terminates, with $a and $b updated, respectively, to A +and B, and $c updated to some symbolic integer value ?C. Via the +ensures clause, which is used to specify additional constraints that hold +after execution, we also state that this existentially quantified ?C equals +either A or B.

      +

      Add the productions declaring $b and $c as tokens to the +LESSON-22-SPEC-SYNTAX module, the claim to the LESSON-22-SPEC module, run +the K prover again, and observe the output, which should not be #Top this +time. This means that K was not able to prove the claim, and we now need to +understand why. We do so by examining the output, which should look as follows:

      +
          (InfoReachability) while checking the implication:
+    The configuration's term unifies with the destination's term,
+    but the implication check between the conditions has failed.
+
+  #Not (
+    #Exists ?C . {
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- ?C:Int ]
+      #Equals
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    }
+  #And
+    {
+      true
+    #Equals
+      ?C ==Int A orBool ?C ==Int B
+    }
+  )
+#And
+  <generatedTop>
+    <k>
+      _DotVar1
+    </k>
+    <store>
+      STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    </store>
+    <funcs>
+      _Gen3
+    </funcs>
+    <stack>
+      _Gen5
+    </stack>
+  </generatedTop>
+#And
+  {
+    true
+  #Equals
+    A <Int B
+  }
+
      +

      This output starts with a message telling us at which point the proof failed, +followed by the final state, which consists of three parts: some negative +Matching Logic (ML) constraints, the final configuration (<generatedTop> ... </generatedTop>), and some positive ML constraints. Generally speaking, +these positive and the negative constraints could arise from various sources, +such as (but not limited to) branches taken by the execution +(e.g. { true #Equals A <Int B } or #Not ( { true #Equals A <Int B } ) ), +or ensures constraints.

      +

      First, we examine the message:

      +
      (InfoReachability) while checking the implication:
+The configuration's term unifies with the destination's term,
+but the implication check between the conditions has failed.
+
      +

      which tells us that the structure of the final configuration is as expected, +but that some of the associated constraints cannot be proven. We next look at +the final configuration, in which the relevant item is the <store> ... </store> cell, because it is the only one that we are reasoning about. By +inspecting its contents:

      +
      STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+
      +

      we see that we should be within the constraints of the ensures, since the +value of $c in the store equals B in this branch. We next examine the +negative and positive constraints of the output and, more often than not, the +goal is to instruct K how to use the information from the final configuration +and the positive constraints to falsify one of the negative constraints. This +is done through simplifications.

      +

      So, the positive constraint that we have is

      +
      { true #Equals A <Int B }
+
      +

      meaning that A <Int B holds. Given the analysed program, this tells us that +we are in the then branch of the if. The negative constraint is

      +
        #Not (
+    #Exists ?C . {
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- ?C:Int ]
+      #Equals
+        STORE [ $a <- A:Int ] [ $b <- B:Int ] [ $c <- B:Int ]
+    }
+  #And
+    { true #Equals ?C ==Int A orBool ?C ==Int B }
+  )
+
      +

      and we observe, from the first equality, that the existential ?C should be +instantiated with B. This would make both branches of the #And true, +falsifying the outside #Not. We just need to show K how to conclude that +?C ==Int B. We do so by introducing the following simplification into the +VERIFICATION module:

      +
      rule { M:Map [ K <- V ] #Equals M [ K <- V' ] } => { V #Equals V' } [simplification]
+
      +

      which formalizes our internal understanding of ?C ==Int B. The rule states +that when we update the same key in the same map with two values, and the +resulting maps are equal, then the two values must be equal as well. The +[simplification] attribute indicates to K to use this rule to simplify the +state when trying to prove claims. Like function rules, simplification rules +do not complete to the top of the configuration, but instead apply anywhere +their left-hand-side matches. Re-run the K prover, which should now return +#Top, indicating that K was able to use the simplification and prove the +required claims.

      +
        +
      1. Next, we show how to state and prove properties of while loops. In +particular, we consider the following loop
        2. +
      +
      claim
+    <k>
+        while ( 0 < $n ) {
+            $s = $s + $n;
+            $n = $n - 1;
+            } => . ...
+    </k>
+    <store>
+        $s |-> (S:Int => S +Int ((N +Int 1) *Int N /Int 2))
+        $n |-> (N:Int => 0)
+    </store>
+    requires N >=Int 0
+
      +

      which adds the sum of the first $n integers to $s, assuming the value of $n +is non-negative to begin with. This is reflected in the store by stating that, +after the execution of the loop, the original value of $s (which is set to +equal some symbolic integer S) is incremented by ((N +Int 1) *Int N /Int 2), and the value of $n always equals 0. Add $n and $s as tokens in +the LESSON-22-SPEC-SYNTAX module, the above claim to the LESSON-22-SPEC +module, and run the K prover, which should return #Top.

      +
        +
      1. Finally, our last claim is about a program that uses function calls:
        2. +
      +
      claim
+    <k>
+        def $sum($n, .Ids) {
+            $s = 0 ;
+            while (0 < $n) {
+                $s = $s + $n;
+                $n = $n - 1;
+            }
+            return $s;
+        }
+
+        $s = $sum(N:Int, .Ints);
+    => . ... </k>
+    <funcs> .Map => ?_ </funcs>
+    <store> $s |-> (_ => ((N +Int 1) *Int N /Int 2)) </store>
+    <stack> .List </stack>
+    requires N >=Int 0
+
      +

      Essentially, we have wrapped the while loop from claim 3.4 into a function +$sum, and then called that function with a symbolic integer N, storing the +return value in the variable $s. The specification states that this program +ends up storing the sum of the first N integers in the variable $n. Add $sum +to the LESSON-22-SPEC-SYNTAX module, the above claim to the +LESSON-22-SPEC module, and run the K prover, which should again return +#Top.

      +

      Exercises

      +
        +
      1. +

        Change the condition of the if statement in part 3.2 to take the else +branch and adjust the claim so that the proof passes.

        +
        2. +
      2. +

        The post-condition of the specification in part 3.3 loses some information. +In particular, the value of ?C is in fact the maximum of A and B. +Prove the same claim as in 3.2, but with the post-condition ensures (?C ==Int maxInt(A, B)). For this, you will need to extend the VERIFICATION +module with two simplifications that capture the meaning of maxInt(A:Int, B:Int). Keep in mind that any rewriting rule can be used as a +simplification; in particular, that simplifications can have requires +clauses.

        +
        3. +
      3. +

        Following the pattern shown in part 3.4, assuming a non-negative initial +value of $b, specify and verify the following while loop:

        +
        4. +
      +
      while ( 0 < $b ) {
+    $a = $a + $c;
+    $b = $b - 1;
+    $c = $c - 1;
+}
+
      +

      Hint: You will not need additional simplifications---once you've got the +specification right, the proof will go through.

      +
        +
      1. Write an arbitrary yet not-too-complex function (or several functions +interacting with each other), and try to specify and verify it (them) in K.
        2. +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/1_basic/index.html b/k-distribution/k-tutorial/1_basic/index.html new file mode 100644 index 00000000000..9b60d6a1bf9 --- /dev/null +++ b/k-distribution/k-tutorial/1_basic/index.html @@ -0,0 +1,401 @@ + + + + + + + + + + + + + + +Section 1: Basic K Concepts | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Section 1: Basic K Concepts

      +

      The goal of this first section of the K tutorial is to teach the basic +principles of K to someone with no prior experience with K as a programming +language. However, this is not written with the intended audience of someone +who is a complete beginner to programming. We are assuming that the reader +has a firm grounding in computer science broadly, as well as that they have +experience writing code in functional programming languages before.

      +

      By the end of this section, the reader ought to be able to write specifications +of simple languages in K, use these specifications to generate a fast +interpreter for their programming language, as well as write basic deductive +program verification proofs over programs in their language. This should give +them the theoretical grounding they need to begin expanding their knowledge +of K in Section 2: Intermediate K Concepts.

      +

      To begin this section, refer to +Lesson 1.1: Setting up a K Environment.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/01_macros/index.html b/k-distribution/k-tutorial/2_intermediate/01_macros/index.html new file mode 100644 index 00000000000..77082d4599f --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/01_macros/index.html @@ -0,0 +1,567 @@ + + + + + + + + + + + + + + +Lesson 2.1: Macros, Aliases, and Anywhere Rules | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.1: Macros, Aliases, and Anywhere Rules

      +

      The purpose of this lesson is to explain the behavior of the macro, +macro-rec, alias, and alias-rec production attributes, as well as the +anywhere rule attribute. These attributes control the meaning of how rules +associated with them are applied.

      +

      Macros

      +

      Thus far in the K tutorial, we have described three different types of rules:

      +
        +
      1. Top-level rewrite rules, which rewrite a configuration composed of cells to +another configuration;
        2. +
      2. Function rules, which define the behavior of a function written over +arbitrary input and output types; and
        3. +
      3. Simplification rules, which describe ways in which the symbolic execution +engine ought to simplify terms containing symbolic values.
        4. +
      +

      This lesson introduces three more types of rules, the first of which are +macros. A production is a macro if it has the macro attribute, and all +rules whose top symbol on the left hand side is a macro are macro rules +which define the behavior of the macro. Like function rules and simplification +rules, macro rules do not participate in cell completion. However, unlike +function rules and simplification rules, macro rules are applied statically +before rewriting begins, and the macro symbol is expected to no longer appear +in the initial configuration for rewriting once all macros in that +configuration are rewritten.

      +

      The rationale behind macros is they allow you to define one piece of syntax +in terms of another piece of syntax without any runtime overhead associated +with the cost of rewriting one to the other. This process is a common one in +programming language design and specification and is referred to as +desugaring; The syntax that is transformed is typically also referred to as +syntactic sugar for another type of syntax. For example, in a language with +if statements and curly braces, you could write the following fragment +(lesson-01.k):

      +
      k
      module LESSON-01
+  imports BOOL
+
+  syntax Stmt ::= "if" "(" Exp ")" Stmt             [macro]
+                | "if" "(" Exp ")" Stmt "else" Stmt
+                | "{" Stmts "}"
+  syntax Stmts ::= List{Stmt,""}
+  syntax Exp ::= Bool
+
+  rule if ( E ) S => if ( E ) S else { .Stmts }
+endmodule
+
      +

      In this example, we see that an if statement without an else clause is +defined in terms of one with an else clause. As a result, we would only +need to give a single rule for how to rewrite if statements, rather than +two separate rules for two types of if statements. This is a common pattern +for dealing with program syntax that contains an optional component to it.

      +

      It is worth noting that by default, macros are not applied recursively. To be +more precise, by default a macro that arises as a result of the expansion of +the same macro is not rewritten further. This is primarily to simplify the +macro expansion process and reduce the risk that improperly defined macros will +lead to non-terminating behavior.

      +

      It is possible, however, to tell K to expand a macro recursively. To do this, +simply replace the macro attribute with the macro-rec attribute. Note that +K does not do any kind of checking to ensure termination here, so it is +important that rules be defined correctly to always terminate, otherwise the +macro expansion phase will run forever. Fortunately, in practice it is very +simple to ensure this property for most of the types of macros that are +typically used in real-world semantics.

      +

      Exercise

      +

      Using a Nat sort containing the constructors 0 and S (i.e., a +Peano-style axiomatization of the +natural numbers where S(N) = N + 1, S(S(N)) = N + 2, etc), write a macro +that will compute the sum of two numbers.

      +

      Aliases

      +

      NOTE: This lesson introduces the concept of "aliases", which are a variant +of macros. While similar, this is different from the concept of "aliases" in +matching logic, which is introduced in Lesson 2.16.

      +

      Macros can be very useful in helping you define a programming language. +However, they can be disruptive while pretty printing a configuration. For +example, you might write a set of macros that transforms the code the user +wrote into equivalent code that is slightly harder to read. This can make it +more difficult to understand the code when it is pretty printed as part of the +output of rewriting.

      +

      K defines a relatively straightforward but novel solution to this problem, +which is known as a K alias. An alias in K is very similar to a macro, +with the exception that the rewrite rule will also be applied backwards +during the pretty-printing process.

      +

      It is very simple to make a production be an alias instead of a macro: simply +use the alias or alias-rec attributes instead of the macro or macro-rec +attributes. For example, if the example involving if statements above was +declared using an alias instead of a macro, the Stmt term if (E) {} else {} +would be pretty-printed as if (E) {}. This is because during pretty-printing, +the term participates in another macro-expansion pass. However, this macro +expansion step will only apply rules with the alias or alias-rec attribute, +and, critically, it will reverse the rule by treating the left-hand side as if +it were the right-hand side, and vice versa.

      +

      This can be very useful to allow you to define one construct in terms of +another while still being able to pretty-print the result as if it were +the original term in question. This can be especially useful for applications +of K where we are taking the output of rewriting and attempting to use it as +a code fragment that we then execute, such as with test generation.

      +

      Exercise

      +

      Modify LESSON-01 above to use an alias instead of a macro and experiment +with how various terms are pretty-printed by invoking krun on them.

      +

      anywhere rules

      +

      The last type of rule introduced in this lesson is the anywhere rule. An +anywhere rule is specified by adding the anywhere attribute to a rule. Such a +rule is similar to a function rule in that it does not participate in cell +completion, and will apply anywhere that the left-hand-side matches in the +configuration, but distinct in that the symbol in question can still be matched +against in the left-hand side of other rules, even during concrete rewriting. +The reasoning behind this is that instead of the symbol in question being a +constructor, it is a constructor modulo the axioms defined with the +anywhere attribute. Essentially, the rules with the anywhere attribute will +apply as soon as they appear in the right-hand side of a rule being applied, +but the symbol in question will still be treated as a symbol that can be +matched on if it is not completely removed by those rules.

      +

      This can be useful in certain cases to allow you to define transformations over +particular pieces of syntax while still generally giving those pieces of syntax +another meaning when the anywhere rule does not apply. For example, the ISO C +standard defines the semantics of *&x as exactly equal to x, with no +reading or writing of memory taking place, and the K semantics of C implements +this functionality using an anywhere rule that is applied at compilation time.

      +

      NOTE: the anywhere attribute is only implemented on the LLVM backend +currently. Attempting to use it in a semantics that is compiled with the +Haskell backend will result in an error being reported by the compiler. This +should be remembered when using this attribute, as it may not be suitable for +a segment of a semantics which is intended to be symbolically executed.

      +

      Exercises

      +
        +
      1. Write a version of the calculator from Lesson 1.14 Exercise 1, which uses +the same syntax for evaluating expressions, but defines its arithmetic logic +using anywhere rules rather than top-level rewrite rules.
        2. +
      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/02_fresh_constants/index.html b/k-distribution/k-tutorial/2_intermediate/02_fresh_constants/index.html new file mode 100644 index 00000000000..3e69716d3ad --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/02_fresh_constants/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.2: Fresh Constants | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.2: Fresh Constants

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/03_klabels/index.html b/k-distribution/k-tutorial/2_intermediate/03_klabels/index.html new file mode 100644 index 00000000000..8e5f9e8f279 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/03_klabels/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.3: KLabels and Abstract Syntax | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.3: KLabels and Abstract Syntax

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/04_overloading/index.html b/k-distribution/k-tutorial/2_intermediate/04_overloading/index.html new file mode 100644 index 00000000000..1c75aa455bc --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/04_overloading/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.4: Overloaded Symbols | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.4: Overloaded Symbols

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/05_matching_logic/index.html b/k-distribution/k-tutorial/2_intermediate/05_matching_logic/index.html new file mode 100644 index 00000000000..36cef934f47 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/05_matching_logic/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.5: Matching Logic Connectives and #Or Patterns | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.5: Matching Logic Connectives and #Or Patterns

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/06_function_context/index.html b/k-distribution/k-tutorial/2_intermediate/06_function_context/index.html new file mode 100644 index 00000000000..804b00f1242 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/06_function_context/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.6: Function Context | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.6: Function Context

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.7: Record Productions and Named Nonterminals

      +

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      +

      Click here to return to the Table of Contents for Section 2.

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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.8: #fun and #let

      +

      Return to Top

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      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.9: #as Patterns

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.10: The Matching Operators, :=K and :/=K

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/11_evaluation_order/index.html b/k-distribution/k-tutorial/2_intermediate/11_evaluation_order/index.html new file mode 100644 index 00000000000..f8a0380b720 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/11_evaluation_order/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.11: Uncommon Evaluation Order Concepts | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.11: Uncommon Evaluation Order Concepts

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/12_floats_and_machine_ints/index.html b/k-distribution/k-tutorial/2_intermediate/12_floats_and_machine_ints/index.html new file mode 100644 index 00000000000..38fc97bfa75 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/12_floats_and_machine_ints/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.12: IEEE 754 Floating Point and Fixed Width Integers | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.12: IEEE 754 Floating Point and Fixed Width Integers

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/13_substitution/index.html b/k-distribution/k-tutorial/2_intermediate/13_substitution/index.html new file mode 100644 index 00000000000..3ae969e0e87 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/13_substitution/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.13: Alpha-renaming-aware Substitution | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.13: Alpha-renaming-aware Substitution

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/14_io/index.html b/k-distribution/k-tutorial/2_intermediate/14_io/index.html new file mode 100644 index 00000000000..98564c080de --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/14_io/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.14: File I/O | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.14: File I/O

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/15_string_buffers_and_bytes/index.html b/k-distribution/k-tutorial/2_intermediate/15_string_buffers_and_bytes/index.html new file mode 100644 index 00000000000..32d034e941d --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/15_string_buffers_and_bytes/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.15: String Buffers and Byte Sequences | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.15: String Buffers and Byte Sequences

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/16_kore/index.html b/k-distribution/k-tutorial/2_intermediate/16_kore/index.html new file mode 100644 index 00000000000..9f3bb4aa9a7 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/16_kore/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.16: The Intermediate Language of K, KORE | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.16: The Intermediate Language of K, KORE

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/17_debugging_proofs/index.html b/k-distribution/k-tutorial/2_intermediate/17_debugging_proofs/index.html new file mode 100644 index 00000000000..87d572e93c5 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/17_debugging_proofs/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Lesson 2.17: Debugging Proofs using the Haskell Backend REPL | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 2.17: Debugging Proofs using the Haskell Backend REPL

      +

      Return to Top

      +

      Click here to return to the Table of Contents for Section 2.

      +
      +
      + + +
      + + + +
      +
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      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/2_intermediate/index.html b/k-distribution/k-tutorial/2_intermediate/index.html new file mode 100644 index 00000000000..5a36a6c0b15 --- /dev/null +++ b/k-distribution/k-tutorial/2_intermediate/index.html @@ -0,0 +1,430 @@ + + + + + + + + + + + + + + +Section 2: Intermediate K Concepts | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Section 2: Intermediate K Concepts

      +

      The goal of this second section is to supplement a beginning developer's +knowledge of K after they have gained a basic understanding of K. Each lesson +in this section can be completed independently in order to learn about a +particular facet of the K language. The lessons are written to provide basic +understanding of less commonly-used features of K to someone who is still +learning K. For more complete references of these features, the reader ought to +consult the User Manual.

      +

      The reader ought to be able to complete lessons in this section as needed in +order to learn about specific features of interest, but if desired, can also +complete the entire section in one go. Someone who has completed this entire +section ought to be able to read and understand most K specifications, as well +as write their own specifications of some complexity, and use them to perform +most common K-related tasks. They can then read about specific lessons in +Section 3: Advanced K Concepts if they want to +learn more.

      +

      Table of Contents

      +
        +
      1. Macros, Aliases, and Anywhere Rules
        2. +
      2. Fresh Constants
        3. +
      3. KLabels and Abstract Syntax
        4. +
      4. Overloaded Symbols
        5. +
      5. Matching Logic Connectives and #Or Patterns
        6. +
      6. Function Context
        7. +
      7. Record Productions and Named Nonterminals
        8. +
      8. #fun and #let
        9. +
      9. #as patterns
        10. +
      10. The Matching Operators, :=K and :/=K
        11. +
      11. Uncommon Evaluation Order Concepts
        12. +
      12. IEEE 754 Floating Point and Fixed Width Integers
        13. +
      13. Alpha-renaming-aware Substitution
        14. +
      14. File I/O
        15. +
      15. String Buffers and Byte Sequences
        16. +
      16. The Intermediate Language of K, KORE
        17. +
      17. Debugging Proofs using the Haskell Backend REPL
        18. +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/1_parsing/1_layout/index.html b/k-distribution/k-tutorial/3_advanced/1_parsing/1_layout/index.html new file mode 100644 index 00000000000..d719822711b --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/1_parsing/1_layout/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.1.1: Using #Layout to define the syntax of comments and whitespace | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.1.1: Using #Layout to define the syntax of comments and whitespace

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/1_parsing/2_ambiguities/index.html b/k-distribution/k-tutorial/3_advanced/1_parsing/2_ambiguities/index.html new file mode 100644 index 00000000000..aca9187d778 --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/1_parsing/2_ambiguities/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.1.2: Using amb for parsing context-sensitive languages | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.1.2: Using amb for parsing context-sensitive languages

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/1_parsing/3_locations/index.html b/k-distribution/k-tutorial/3_advanced/1_parsing/3_locations/index.html new file mode 100644 index 00000000000..4beb3ba5853 --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/1_parsing/3_locations/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.1.3: Using #location to annotate terms with file, line, and column information | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.1.3: Using #location to annotate terms with file, line, and column information

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/2_libraries/1_json/index.html b/k-distribution/k-tutorial/3_advanced/2_libraries/1_json/index.html new file mode 100644 index 00000000000..c292da0c09d --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/2_libraries/1_json/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.2.1: The JSON Module | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.2.1: The JSON Module

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/2_libraries/2_rat/index.html b/k-distribution/k-tutorial/3_advanced/2_libraries/2_rat/index.html new file mode 100644 index 00000000000..79a4749ac47 --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/2_libraries/2_rat/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.2.2: The RAT Module | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.2.2: The RAT Module

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/2_libraries/3_ffi/index.html b/k-distribution/k-tutorial/3_advanced/2_libraries/3_ffi/index.html new file mode 100644 index 00000000000..0d65f55f5ed --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/2_libraries/3_ffi/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.2.3: The FFI Module | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.2.3: The FFI Module

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/3_extending_k/1_custom_hooks/index.html b/k-distribution/k-tutorial/3_advanced/3_extending_k/1_custom_hooks/index.html new file mode 100644 index 00000000000..bf0cc885d23 --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/3_extending_k/1_custom_hooks/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lesson 3.3.1: Extending K by adding new builtin functions | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lesson 3.3.1: Extending K by adding new builtin functions

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/3_extending_k/2_scripting_k/index.html b/k-distribution/k-tutorial/3_advanced/3_extending_k/2_scripting_k/index.html new file mode 100644 index 00000000000..ed4c9e3f3c5 --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/3_extending_k/2_scripting_k/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Lessonm 3.3.2: Scripting K to provide language-specific tooling | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Lessonm 3.3.2: Scripting K to provide language-specific tooling

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/3_advanced/index.html b/k-distribution/k-tutorial/3_advanced/index.html new file mode 100644 index 00000000000..32680d065cc --- /dev/null +++ b/k-distribution/k-tutorial/3_advanced/index.html @@ -0,0 +1,387 @@ + + + + + + + + + + + + + + +Section 3: Advanced K Concepts | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Section 3: Advanced K Concepts

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/k-tutorial/index.html b/k-distribution/k-tutorial/index.html new file mode 100644 index 00000000000..f5042b5c225 --- /dev/null +++ b/k-distribution/k-tutorial/index.html @@ -0,0 +1,398 @@ + + + + + + + + + + + + + + +K Tutorial | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Tutorial

      +

      The purpose of this series of lessons is to teach developers how to program in +K. While the primary use of K is in the specification of operational semantics +of programming languages, this tutorial is agnostic on how the knowledge of K +is used. For a more detailed tutorial explaining the basic principles of +programming language design, refer to the +K PL Tutorial. Note that that tutorial is somewhat +out of date presently.

      +

      This K tutorial is a work in progress. Many lessons are currently simply +placeholders for future content.

      +

      To start the K tutorial, begin with +Section 1: Basic Programming in K.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/index.html new file mode 100644 index 00000000000..68dc84ca002 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/index.html @@ -0,0 +1,404 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Part 1: Defining LAMBDA

      +

      Here you will learn how to define a very simple language in K and the basics +of how to use the K tool. The language is a variant of call-by-value lambda +calculus and its definition is based on substitution. Specifically, you will +learn the following:

      +
        +
      • How to define a module.
        • +
      • How to define a language syntax.
        • +
      • How to use the defined syntax to parse programs.
        • +
      • How to import predefined modules.
        • +
      • How to define evaluation strategies using strictness attributes.
        • +
      • How to define semantic rules.
        • +
      • How the predefined generic substitution works.
        • +
      • How to generate PDF and HTML documentation from ASCII definitions.
        • +
      • How to include builtins (integers and Booleans) into your language.
        • +
      • How to define derived language constructs.
        • +
      +

      This folder contains several lessons, each adding new features to LAMBDA.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/NOTES/index.html new file mode 100644 index 00000000000..2eec042b8d7 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/NOTES/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      We now support the following line to the syntax module:

      +
      syntax priority lambda_._ > __  // exact syntax subject to change
+
      +

      This will allow for fewer parentheses in programs.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/index.html new file mode 100644 index 00000000000..f59c2c206f4 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_1/index.html @@ -0,0 +1,471 @@ + + + + + + + + + + + + + + +Syntax Modules and Basic K Commands | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Syntax Modules and Basic K Commands

      +

      Here we define our first K module, which contains the initial syntax of the +LAMBDA language, and learn how to use the basic K commands.

      +

      Let us create an empty working folder, and open a terminal window +(to the left) and an editor window (to the right). We will edit our K +definition in the right window in a file called lambda.k, and will call +the K tool commands in the left window.

      +

      Let us start by defining a K module, containing the syntax of LAMBDA.

      +

      K modules are introduced with the keywords module ... endmodule.

      +

      The keyword syntax adds new productions to the syntax grammar, using a +BNF-like notation.

      +

      Terminals are enclosed in double-quotes, like strings.

      +

      You can define multiple productions for the same non-terminal in the same +syntax declaration using the | separator.

      +

      Productions can have attributes, which are enclosed in square brackets.

      +

      The attribute left tells the parser that we want the lambda application to be +left associative. For example, a b c d will then parse as (((a b) c) d).

      +

      The attribute bracket tells the parser to not generate a node for the +parenthesis production in the abstract syntax trees associated to programs. +In other words, we want to allow parentheses to be used for grouping, but we +do not want to bother to give them their obvious (ignore) semantics.

      +

      In our variant of lambda calculus defined here, identifiers and lambda +abstractions are meant to be irreducible, that is, are meant to be values. +However, so far Val is just another non-terminal, just like Exp, +without any semantic meaning. It will get a semantic meaning later.

      +

      After we are done typing our definition in the file lambda.k, we can kompile +it with the command:

      +
      kompile lambda.k
+
      +

      If we get no errors then a parser has been generated. This parser will be +called from now on by default by the krun tool. To see whether and how the +parser works, we are going to write some LAMBDA programs and store them in +files with the extension .lambda.

      +

      Let us create a file identity.lambda, which contains the identity lambda +abstraction:

      +
      lambda x . x
+
      +

      Now let us call krun on identity.lambda:

      +
      krun identity.lambda
+
      +

      Make sure you call the krun command from the folder containing your language +definition (otherwise type krun --help to learn how to pass a language +definition as a parameter to krun). The krun command produces the output:

      +
      <k>
+  lambda x . x
+</k>
+
      +

      If you see such an output it means that your program has been parsed (and then +pretty printed) correctly. If you want to see the internal abstract syntax +tree (AST) representation of the parsed program, which we call the K AST, then +type kast in the command instead of krun:

      +
      kast identity.lambda
+
      +

      You should normally never need to see this internal representation in your +K definitions, so do not get scared (yes, it is ugly for humans, but it is +very convenient for tools).

      +

      Note that krun placed the program in a <k> ... </k> cell. In K, computations +happen only in cells. If you do not define a configuration in your definition, +like we did here, then a configuration will be created automatically for you +which contains only one cell, the default k cell, which holds the program.

      +

      Next, let us create a file free-variable-capture.lambda, which contains an +expression which, in order to execute correctly in a substitution-based +semantics of LAMBDA, the substitution operation needs to avoid +variable-capture:

      +
      a (((lambda x.lambda y.x) y) z)
+
      +

      Next, file closed-variable-capture.lambda shows an expression which also +requires a capture-free substitution, but this expression is closed (that is, +it has no free variables) and all its bound variables are distinct (I believe +this is the smallest such expression):

      +
      (lambda z.(z z)) (lambda x.lambda y.(x y))
+
      +

      Finally, the file omega.lambda contains the classic omega combinator +(or closed expression), which is the smallest expression which loops forever +(not now, but after we define the semantics of LAMBDA):

      +
      (lambda x.(x x)) (lambda x.(x x))
+
      +

      Feel free to define and parse several other LAMBDA programs to get a feel for +how the parser works. Parse also some incorrect programs, to see how the +parser generates error messages.

      +

      In the next lesson we will see how to define semantic rules that iteratively +rewrite expressions over the defined syntax until they evaluate to a result. +This way, we obtain our first programming language defined using K.

      +

      Go to Lesson 2, LAMBDA: Module Importing, Rules, Variables

      +

      MOVIE (out of date) [4'07"]

      +
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      + + +
      + + + +
      +
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      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2.5/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2.5/NOTES/index.html new file mode 100644 index 00000000000..7c56461eb04 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2.5/NOTES/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      This folder has been added after the original tutorial was made +and after the videos were recorded. Eventually we will renumber +the lessons and redo the videos. A README.md file is also needed +here.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/NOTES/index.html new file mode 100644 index 00000000000..7bf341bd799 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/NOTES/index.html @@ -0,0 +1,385 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Substitution has been reimplemented in the meanwhile, where the fresh +variables are resolved locally. So there is no global counter for +fresh variables anymore as shown in the video, and fewer variable +renamings take place.

      +

      When calling krun on the programs in lesson_1, a different path is +shown than in the README.md.

      +

      Marking the beta-reduction rule with [anywhere] will give us the +conventional lambda-calculus. A new lesson has been added, 2.5, +showing that. The README.md file should be changed at the end to +point to lesson 2.5.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/index.html new file mode 100644 index 00000000000..6feefd30f91 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_2/index.html @@ -0,0 +1,453 @@ + + + + + + + + + + + + + + +Module Importing, Rules, Variables | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Module Importing, Rules, Variables

      +

      We here learn how to include a predefined module (SUBSTITUTION), how to +use it to define a K rule (the characteristic rule of lambda calculus), +and how to make proper use of variables in rules.

      +

      Let us continue our lambda.k definition started in the previous lesson.

      +

      The requires keyword takes a .k file containing language features that +are needed for the current definition, which can be found in the +k-distribution/include/kframework/builtin folder. Thus, the command

      +
      require "substitution.k"
+
      +

      says that the subsequent definition of LAMBDA needs the generic substitution, +which is predefined in file substitution.k under the folder +k-distribution/include/kframework/builtin. Note that substitution can be defined itself in K, +although it uses advanced features that we have not discussed yet in this +tutorial, so it may not be easy to understand now.

      +

      Using the imports keyword, we can now modify LAMBDA to import the module +SUBSTITUTION, which is defined in the required substitution.k file.

      +

      Now we have all the substitution machinery available for our definition. +However, since our substitution is generic, it cannot know which language +constructs bind variables, and what counts as a variable; however, this +information is critical in order to correctly solve the variable capture +problem. Thus, you have to tell the substitution that your lambda construct +is meant to be a binder, and that your Id terms should be treated as variables +for substitution. The former is done using the attribute binder. +By default, binder binds all the variables occurring anywhere in the first +argument of the corresponding syntactic construct within its other arguments; +you can configure which arguments are bound where, but that will be discussed +in subsequent lectures. To tell K which terms are meant to act as variables +for binding and substitution, we have to explicitly subsort the desired syntactic +categories to the builtin KVariable sort.

      +

      Now we are ready to define our first K rule. Rules are introduced with the +keyword rule and make use of the rewrite symbol, =>. In our case, +the rule defines the so-called lambda calculus beta-reduction, which +makes use of substitution in its right-hand side, as shown in lambda.k.

      +

      By convention, variables that appear in rules start with a capital letter +(the current implementation of the K tool may even enforce that).

      +

      Variables may be explicitly tagged with their syntactic category (also called +sort). If tagged, the matching term will be checked at run-time for +membership to the claimed sort. If not tagged, then no check will be made. +The former is safer, but involves the generation of a side condition to the +rule, so the resulting definition may execute slightly slower overall.

      +

      In our rule in lambda.k we tagged all variables with their sorts, so we chose +the safest path. Only the V variable really needs to be tagged there, +because we can prove (using other means, not the K tool, as the K tool is not +yet concerned with proving) that the first two variables will always have the +claimed sorts whenever we execute any expression that parses within our +original grammar.

      +

      Let us compile the definition and then run some programs. For example,

      +
      krun closed-variable-capture.lambda
+
      +

      yields the output

      +
      <k>
+  lambda y . ((lambda x . (lambda y . (x  y))) y)
+</k> 
+
      +

      Notice that only certain programs reduce (some even yield non-termination, +such as omega.lambda), while others do not. For example, +free-variable-capture.lambda does not reduce its second argument expression +to y, as we would expect. This is because the K rewrite rules between syntactic +terms do not apply anywhere they match. They only apply where they have been +given permission to apply by means of appropriate evaluation strategies of language +constructs, which is done using strictness attributes, evaluation contexts, +heating/cooling rules, etc., as discussed in the next lessons.

      +

      The next lesson will show how to add LAMBDA the desired evaluation strategies +using strictness attributes.

      +

      Go to Lesson 3, LAMBDA: Evaluation Strategies using Strictness

      +

      MOVIE (out of date) [4'03"]

      +
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      + + +
      + + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/NOTES/index.html new file mode 100644 index 00000000000..1238d2cdc85 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/NOTES/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      When we say "previous lesson" we refer to lesson 2. This will need to change +when we incorporate lesson 2.5 properly.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/index.html new file mode 100644 index 00000000000..1a1c441f4f0 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_3/index.html @@ -0,0 +1,418 @@ + + + + + + + + + + + + + + +Evaluation Strategies using Strictness | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Evaluation Strategies using Strictness

      +

      Here we learn how to use the K strict attribute to define desired evaluation +strategies. We will also learn how to tell K which terms are already +evaluated, so it does not attempt to evaluate them anymore and treats them +internally as results of computations.

      +

      Recall from the previous lecture that the LAMBDA program +free-variable-capture.lambda was stuck, because K was not given permission +to evaluate the arguments of the lambda application construct.

      +

      You can use the attribute strict to tell K that the corresponding construct +has a strict evaluation strategy, that is, that its arguments need to be +evaluated before the semantics of the construct applies. The order of +argument evaluation is purposely unspecified when using strict, and indeed +the K tool allows us to detect all possible non-deterministic behaviors that +result from such intended underspecification of evaluation strategies. We will +learn how to do that when we define the IMP language later in this tutorial; +we will also learn how to enforce a particular order of evaluation.

      +

      In order for the above strictness declaration to work effectively and +efficiently, we need to tell the K tool which expressions are meant to be +results of computations, so that it will not attempt to evaluate them anymore. +One way to do it is to make Val a syntactic subcategory of the builtin +KResult syntactic category. Since we use the same K parser to also parse +the semantics, we use the same syntax keyword to define additional syntax +needed exclusively for the semantics (like KResults). See lambda.k.

      +

      Compile again and then run some programs. They should all work as expected. +In particular, free-variable-capture.lambda now evaluates to a y.

      +

      We now got a complete and working semantic definition of call-by-value +lambda-calculus. While theoretically correct, our definition is not +easy to use and disseminate. In the next lessons we will learn how to +generate formatted documentation for LAMBDA and how to extend LAMBDA +in order to write human readable and interesting programs.

      +

      Go to Lesson 4, LAMBDA: Generating Documentation; Latex Attributes.

      +

      MOVIE (out of date) [2'20"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/NOTES/index.html new file mode 100644 index 00000000000..3d2e15d9aa1 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/NOTES/index.html @@ -0,0 +1,376 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      README.md refers to Lesson 9. This will need to be updated.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/index.html new file mode 100644 index 00000000000..632e29bbc47 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_4/index.html @@ -0,0 +1,412 @@ + + + + + + + + + + + + + + +Generating Documentation; Latex Attributes | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Generating Documentation; Latex Attributes

      +

      In this lesson we learn how to generate formatted documentation from K +language definitions. We also learn how to use Latex attributes to control +the formatting of language constructs, particularly of ones which have a +mathematical flavor and we want to display accordingly.

      +

      To enhance readability, we may want to replace the keyword lambda by the +mathematical lambda symbol in the generated documentation. We can control +the way we display language constructs in the generated documentation +by associating them Latex attributes.

      +

      This is actually quite easy. All we have to do is to associate a latex +attribute to the production defining the construct in question, following +the Latex syntax for defining new commands (or macros).

      +

      In our case, we associate the attribute latex(\lambda{#1}.{#2}) to the +production declaring the lambda abstraction (recall that in Latex, #n refers +to the n-th argument of the defined new command).

      +

      We will later see, in Lesson 9, that we can add arbitrarily complex Latex +comments and headers to our language definitions, which give us maximum +flexibility in formatting our language definitions.

      +

      Now we have a simple programming language, with a nice documentation. However, +it is not easy to write interesting programs in this language. Almost all +programming languages build upon existing data-types and libraries. The K +tool provides a few of these (and you can add more).

      +

      In the next lesson we show how we can add builtin integers and Booleans to +LAMBDA, so we can start to evaluate meaningful expressions.

      +

      Go to Lesson 5, LAMBDA: Adding Builtins; Side Conditions.

      +

      MOVIE (out of date) [3'13"]

      +
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      + + +
      + + + +
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      + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/NOTES/index.html new file mode 100644 index 00000000000..06345c30c59 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/NOTES/index.html @@ -0,0 +1,381 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The builtins have changed, they are now generic for all backends.

      +

      Talk about sort inference for variables, for example from I1 +Int I2 +we infer the sort of I1 and I2 is Int.

      +

      Check the entire tutorial for instances where we give the sort of a +variable but we don't have to. Many of those are artifacts since we were +not able to infer sorts that well.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/index.html new file mode 100644 index 00000000000..6cd1d7dee4f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_5/index.html @@ -0,0 +1,465 @@ + + + + + + + + + + + + + + +Adding Builtins; Side Conditions | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Adding Builtins; Side Conditions

      +

      We have already added the builtin identifiers (sort Id) to LAMBDA expressions, +but those had no operations on them. In this lesson we add integers and +Booleans to LAMBDA, and extend the builtin operations on them into +corresponding operations on LAMBDA expressions. We will also learn how to add +side conditions to rules, to limit the number of instances where they can +apply.

      +

      The K tool provides several builtins, which are automatically included in all +definitions. These can be used in the languages that we define, typically by +including them in the desired syntactic categories. You can also define your +own builtins in case the provided ones are not suitable for your language +(e.g., the provided builtin integers and operations on them are arbitrary +precision).

      +

      For example, to add integers and Booleans as values to our LAMBDA, we have to +add the productions

      +
      syntax Val ::= Int | Bool
+
      +

      Int and Bool are the nonterminals that correspond to these builtins.

      +

      To make use of these builtins, we have to add some arithmetic operation +constructs to our language. We prefer to use the conventional infix notation +for these, and the usual precedences (i.e., multiplication and division bind +tighter than addition, which binds tighter than relational operators). +Inspired from SDF, we use > instead of +| to state that all the previous constructs bind tighter than all the +subsequent ones. See lambda.k.

      +

      The only thing left is to link the LAMBDA arithmetic operations to the +corresponding builtin operations, when their arguments are evaluated. +This can be easily done using trivial rewrite rules, as shown in lambda.k. +In general, the K tool attempts to uniformly add the corresponding builtin +name as a suffix to all the operations over builtins. For example, the +addition over integers is an infix operation named +Int.

      +

      Compile the new lambda.k definition and evaluate some simple arithmetic +expressions. For example, if arithmetic.lambda is (1+2*3)/4 <= 1, then

      +
      krun arithmetic.lambda
+
      +

      yields, as expected, true. Note that the parser took the desired operation +precedence into account.

      +

      Let us now try to evaluate an expression which performs a wrong computation, +namely a division by zero. Consider the expression arithmetic-div-zero.lambda +which is 1/(2/3). Since division is strict and 2/3 evaluates to 0, this +expression reduces to 1/0, which further reduces to 1 /Int 0 by the rule for +division, which is now stuck (with the current back-end to the K tool).

      +

      In fact, depending upon the back-end that we use to execute K definitions and +in particular to evaluate expressions over builtins, 1 /Int 0 can evaluate to +anything. It just happens that the current back-end keeps it as an +irreducible term. Other K back-ends may reduce it to an explicit error +element, or issue a segmentation fault followed by a core dump, or throw an +exception, etc.

      +

      To avoid requesting the back-end to perform an illegal operation, we may use a +side condition in the rule of division, to make sure it only applies when the +denominator is non-zero.

      +

      Like in other operational formalisms, the role of the K side +conditions is to filter the number of instances of the rule. The notion +of a side condition comes from logics, where a sharp distinction is made +between a side condition (cheap) and a premise (expensive). Premises are +usually resolved using further (expensive) logical derivations, while side +conditions are simple (cheap) conditions over the rule meta-variables within +the underlying mathematical domains (which in K can be extended by the user, +as we will see in future lessons). Regarded as a logic, K derives rewrite +rules from other rewrite rules; therefore, the K side conditions cannot +contain other rewrites in them (using =>). This contrasts other rewrite +engines, for example Maude, which +allow conditional rules with rewrites in conditions. +The rationale behind this deliberate restriction in K is twofold:

      +
        +
      • On the one hand, general conditional rules require a complex, and thus slower +rewrite engine, which starts recursive (sometimes exhaustive) rewrite sessions +to resolve the rewrites in conditions. In contrast, the side conditions in K +can be evaluated efficiently by back-ends, for example by evaluating builtin +expressions and/or by calling builtin functions.
        • +
      • On the other hand, the semantic definitional philosophy of K is that rule +premises are unnecessary, so there is no need to provide support for them.
        • +
      +

      Having builtin arithmetic is useful, but writing programs with just lambda +and arithmetic constructs is still a pain. In the next two lessons we will +add conditional (if_then_else) and binding (let and letrec) constructs, +which will allow us to write nicer programs.

      +

      Go to Lesson 6, LAMBDA: Selective Strictness; Anonymous Variables.

      +

      MOVIE (out of date) [4'52"]

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_6/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_6/index.html new file mode 100644 index 00000000000..dce9ab0bb35 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_6/index.html @@ -0,0 +1,421 @@ + + + + + + + + + + + + + + +Selective Strictness; Anonymous Variables | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Selective Strictness; Anonymous Variables

      +

      We here show how to define selective strictness of language constructs, +that is, how to state that certain language constructs are strict only +in some arguments. We also show how to use anonymous variables.

      +

      We next define a conditional if construct, which takes three arguments, +evaluates only the first one, and then reduces to either the second or the +third, depending on whether the first one evaluated to true or to false.

      +

      K allows to define selective strictness using the same strict attribute, +but passing it a list of numbers. The numbers correspond to the arguments +in which we want the defined construct to be strict. In our case,

      +
      syntax Exp ::= "if" Exp "then" Exp "else" Exp   [strict(1)]
+
      +

      states that the conditional construct is strict in the first argument.

      +

      We can now assume that its first argument will eventually reduce to a value, so +we only write the following two semantic rules:

      +
      rule if true  then E else _ => E
+rule if false then _ else E => E
+
      +

      Thus, we assume that the first argument evaluates to either true or false.

      +

      Note the use of the anonymous variable _. We use such variables purely for +structural reasons, to state that something is there but we don't care what. +An anonymous variable is therefore completely equivalent to a normal variable +which is unsorted and different from all the other variables in the rule. If +you use _ multiple times in a rule, they will all be considered distinct.

      +

      Compile lambda.k and write and execute some interesting expressions making +use of the conditional construct. For example, the expression

      +
      if 2<=1 then 3/0 else 10
+
      +

      evaluates to 10 and will never evaluate 3/0, thus avoiding an unwanted +division-by-zero.

      +

      In the next lesson we will introduce two new language constructs, called +let and letrec and conventionally found in functional programming +languages, which will allow us to already write interesting LAMBDA programs.

      +

      Go to Lesson 7, LAMBDA: Derived Constructs; Extending Predefined Syntax.

      +

      MOVIE (out of date) [2'14"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/NOTES/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/NOTES/index.html new file mode 100644 index 00000000000..66206dd2dfc --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/NOTES/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      In more recent definitions, we prefer to make some [macro] rules. +Macros apply statically, before the program is executed, thus +increasing the execution performance. The let and letrec constructs +here could be made into [macro].

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/index.html new file mode 100644 index 00000000000..b03207cde21 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_7/index.html @@ -0,0 +1,459 @@ + + + + + + + + + + + + + + +Derived Constructs, Extending Predefined Syntax | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Derived Constructs, Extending Predefined Syntax

      +

      In this lesson we will learn how to define derived language constructs, that +is, ones whose semantics is defined completely in terms of other language +constructs. We will also learn how to add new constructs to predefined +syntactic categories.

      +

      When defining a language, we often want certain language constructs to be +defined in terms of other constructs. For example, a let-binding construct +of the form

      +
      let x = e in e'
+
      +

      is nothing but syntactic sugar for

      +
      (lambda x . e') e
+
      +

      This can be easily achieved with a rule, as shown in lambda.k.

      +

      Compile lambda.k and write some programs using let binders.

      +

      For example, consider a lets.lambda program which takes arithmetic.lambda +and replaces each integer by a let-bound variable. It should evaluate to +true, just like the original arithmetic.lambda.

      +

      Let us now consider a more interesting program, namely one that calculates the +factorial of 10:

      +
      let f = lambda x . (
+        (lambda t . lambda x . (t t x))
+        (lambda f . lambda x . (if x <= 1 then 1 else (x * (f f (x + -1)))))
+        x
+      )
+in (f 10)
+
      +

      This program follows a common technique to define fixed points in untyped +lambda calculus, based on passing a function to itself.

      +

      We may not like to define fixed-points following the approach above, because +it requires global changes in the body of the function meant to be recursive, +basically to pass it to itself (f f in our case above). The approach below +isolates the fixed-point aspect of the function in a so-called fixed-point +combinator, which we call fix below, and then apply it to the function +defining the body of the factorial, without any changes to it:

      +
      let fix = lambda f . (
+          (lambda x . (f (lambda y . (x x y))))
+          (lambda x . (f (lambda y . (x x y))))
+        )
+in let f = fix (lambda f . lambda x .
+                (if x <= 1 then 1 else (x * (f (x + -1)))))
+   in (f 10)
+
      +

      Although the above techniques are interesting and powerful (indeed, untyped +lambda calculus is in fact Turing complete), programmers will probably not +like to write programs this way.

      +

      We can easily define a more complex derived construct, called letrec and +conventionally encountered in functional programming languages, whose semantics +captures the fixed-point idea above. In order to keep its definition simple +and intuitive, we define a simplified variant of letrec, namely one which only +allows to define one recursive one-argument function. See lambda.k.

      +

      There are two interesting observations here.

      +

      First, note that we have already in-lined the definition of the fix +combinator in the definition of the factorial, to save one application of the +beta reduction rule (and the involved substitution steps). We could have +in-lined the definition of the remaining let, too, but we believe that the +current definition is easier to read.

      +

      Second, note that we extended the predefined Id syntactic category with two +new constants, $x and $y. The predefined identifiers cannot start with +$, so programs that will be executed with this semantics cannot possibly +contain the identifiers xandx andy. In other words, by adding them to Id they +become indirectly reserved for the semantics. This is indeed desirable, +because any possible uses of xinthebodyofthefunctiondefinedusing‘letrec‘wouldbecapturedbythe‘lambdax in the body of the function defined +using `letrec` would be captured by the `lambdaxdeclaration in the definition ofletrec`.

      +

      Using letrec, we can now write the factorial program as elegantly as it can +be written in a functional language:

      +
      letrec f x = if x <= 1 then 1 else (x * (f (x + -1)))
+in (f 10)
+
      +

      In the next lesson we will discuss an alternative definition of letrec, based +on another binder, mu, specifically designed to define fixed points.

      +

      Go to Lesson 8, LAMBDA: Multiple Binding Constructs.

      +

      MOVIE (out of date) [5'10"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/SK-combinators/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/SK-combinators/index.html new file mode 100644 index 00000000000..a5043ae62bd --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/SK-combinators/index.html @@ -0,0 +1,415 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define LAMBDA using the S/K combinators instead of substitution. +You new definition will not require the substitution.k module anymore, +and will not use environments (discussed in future lectures), either.

      +

      Recall that the S and K combinators are defined as follows:

      +
      K E1 E2 = E1
+S E1 E2 E3 = E1 E3 (E2 E3)
+
      +

      where the application is that of LAMBDA (left associative binary operation), +and that the lambda construct can be desugared to combinators using the +following simple rules:

      +
      lambda X . X = S K K
+lambda X . Y = K Y    when Y is a name different from X
+lambda X . (E1 E2) = S (lambda X . E1) (lambda X . E2)
+lambda X . B = K B    when B is any constant, including S or K
+
      +

      To distinguish the S and K combinators from K variables and make them +more visible, we prefer to write them as SS and KK instead of S and K.

      +

      If defined correctly and completely, all the tests should pass when you call +ktest on the provided config.xml file. The tests include all the programs +previously executed using LAMBDA (lesson_8), plus the additional program of +the mu-derived exercise, plus a few more simple programs given with this +exercise to help you better test your definition and nail down the notation.

      +

      The syntax of the new LAMBDA should be the same as before, although +mu needs to be desugared as in the mu-desugared exercise (using a macro). +The tricky part is how to deal with the builtin operations. For example, +lambda x . if x then y else z cannot be transformed into combinators as is, +but it can if we assume a builtin conditional function constant, say cond, +and desugar if_then_else_ to it. Then this expression becomes +lambda x . (((cond x) y) z), which we know how to transform. The drawback +of this cond constant approach is that it may induce non-termination +in recursive programs, but that appears to not be a problem in our examples.

      +

      You will have to do the same for all builtin functions, and you will have +to make sure that you define your values correctly! In our previous +definition we were able to say that lambda x . e was a value, but now that +is not possible anymore, because the lambda construct will be eliminated. +Instead, you will have to explicitly say it using the isVal membership +predicate that all the expressions that involve builtin functions and +yield functions are values; for example, isVal(cond V:Val) => true and +isVal(cond V1:Val V2:Val) => true need to be added, but obviously not +isVal(cond V1:Val V2:Val V3:Val) => true.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/mu-derived/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/mu-derived/index.html new file mode 100644 index 00000000000..9320f48df0b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/exercises/mu-derived/index.html @@ -0,0 +1,380 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The current K LAMBDA semantics of mu (in Lesson 8) is based on +substitution, and then letrec is defined as a derived operation using +mu. Give mu a different semantics, as a derived construct by +translation into other LAMBDA constructs, like we defined letrec in +Lesson 7.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/index.html new file mode 100644 index 00000000000..03e306960b6 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_8/index.html @@ -0,0 +1,432 @@ + + + + + + + + + + + + + + +Multiple Binding Constructs | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Multiple Binding Constructs

      +

      Here we learn how multiple language constructs that bind variables can +coexist. We will also learn about or recall another famous binder besides +lambda, namely mu, which can be used to elegantly define all kinds of +interesting fixed-point constructs.

      +

      The mu binder has the same syntax as lambda, except that it replaces +lambda with mu.

      +

      Since mu is a binder, in order for substitution to know how to deal with +variable capture in the presence of mu, we have to tell it that mu is a +binding construct, same like lambda. We take advantage of being there and +also add mu its desired latex attribute.

      +

      The intuition for

      +
      mu x . e
+
      +

      is that it reduces to e, but each free occurrence of x in e behaves +like a pointer that points back to mu x . e.

      +

      With that in mind, let us postpone the definition of mu and instead redefine +letrec F X = E in E' as a derived construct, assuming mu available. The +idea is to simply regard F as a fixed-point of the function

      +
      lambda X . E
+
      +

      that is, to first calculate

      +
      mu F . lambda X . E
+
      +

      and then to evaluate E' where F is bound to this fixed-point:

      +
      let F = mu F . lambda X . E in E'
+
      +

      This new definition of letrec may still look a bit tricky, particularly +because F is bound twice, but it is much simpler and cleaner than our +previous definition. Moreover, now it is done in a type-safe manner +(this aspect goes beyond our objective in this tutorial).

      +

      Let us now define the semantic rule of mu.

      +

      The semantics of mu is actually disarmingly simple. We just have to +substitute mu X . E for each free occurrence of X in E:

      +
      mu X . E => E[(mu X . E) / X]
+
      +

      Compile lambda.k and execute some recursive programs. They should be now +several times faster. Write a few more recursive programs, for example ones +for calculating the Ackermann function, for calculating the number of moves +needed to solve the Hanoi tower problem, etc.

      +

      We have defined our first programming language in K, which allows us to +write interesting functional programs. In the next lesson we will learn how +to fully document our language definition, in order to disseminate it, to ship +it to colleagues or friends, to publish it, to teach it, and so on.

      +

      Go to Lesson 9, LAMBDA: A Complete and Commented Definition.

      +

      MOVIE (out of date) [2'40"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/index.html new file mode 100644 index 00000000000..a031f9017af --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/index.html @@ -0,0 +1,445 @@ + + + + + + + + + + + + + + +A Complete and Documented K Definition | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      A Complete and Documented K Definition

      +

      In this lesson you will learn how to add formal comments to your K definition, +in order to nicely document it. The generated document can be then used for +various purposes: to ease understanding the K definition, to publish it, +to send it to others, etc.

      +

      The K tool allows a literate programming style, where the executable +language definition can be documented by means of annotations. One such +annotation is the latex(_) annotation, where you can specify how to format +the given production when producing Latex output via the --output latex +option to krun, kast, and kprove.

      +

      There are three types of comments, which we discuss next.

      +

      Ordinary comments

      +

      These use // or /* ... */, like in various programming languages. These +comments are completely ignored.

      +

      Document annotations

      +

      Use the @ symbol right after // or /* in order for the comment to be +considered an annotation and thus be processed by the K tool when it +generates documentation.

      +

      As an example, we can go ahead and add such an annotation at the beginning +of the LAMBDA module, explaining how we define the syntax of this language.

      +

      Header annotations

      +

      Use the ! symbol right after // or /* if you want the comment to be +considered a header annotation, that is, one which goes before +\begin{document} in the generated Latex. You typically need header +annotations to include macros, or to define a title, etc.

      +

      As an example, let us set a Latex length and then add a title and an +author to this K definition.

      +

      Compile the documentation and take a look at the results. Notice the title.

      +

      Feel free to now add lots of annotations to lambda.k.

      +

      Then compile and check the result. Depending on your PDF viewer, you +may also see a nice click-able table of contents, with all the sections +of your document. This could be quite convenient when you define large +languages, because it helps you jump to any part of the semantics.

      +

      Tutorial 1 is now complete. The next tutorial will take us through the +definition of a simple imperative language and will expose us to more +feature of the K framework and the K tool.

      +

      MOVIE (out of date) [6'07"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/lambda/index.html b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/lambda/index.html new file mode 100644 index 00000000000..8564a6c6743 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/1_lambda/lesson_9/lambda/index.html @@ -0,0 +1,510 @@ + + + + + + + + + + + + + + +Tutorial 1 --- LAMBDA | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K code can be nested inside Markdown using annotated code blocks. +Use the tag k to tell the compiler which blocks to select.

      +

      Inside .k files, C/Java-like comments are available.

      +
      k
      // Single line comment
+/* Multiline
+comments */
+
      +

      Tutorial 1 --- LAMBDA

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Abstract

      +

      This file defines a simple functional language in K, called LAMBDA, +using a substitution style. The explicit objective here is to teach some +K concepts and how they work in the K tool, and not to teach +λ-calculus or to argue for one definitional style against another +(e.g., some may prefer environment/closure-based definitions of such +languages).

      +

      Note that the subsequent definition is so simple, that it hardly shows any +of the strengths of K. Perhaps the most interesting K aspect it shows is +that substitution can be defined fully generically, and then used to give +semantics to various constructs in various languages.

      +

      Note: +K follows the +literate programming +approach. The various semantic features defined in a K +module can be reordered at will and can be commented using normal +comments like in C/C++/Java. +While comments are useful in general, they can annoy the expert user +of K. To turn them off, you can do one of the following (unless you +want to remove them manually):
      +(1) Use an editor which can +hide or highlight Markdown and conventional C-like comments; or
      +(2) Run kompile --debug <def>. Inside ./.kompiled-xxx/.md2.k/ you will find +all the K code extracted from the markdown files as used for compilation.

      +

      Substitution

      +

      We need the predefined substitution module, so we require it with the command +below. Then we should make sure we import its module called SUBSTITUTION +in our LAMBDA module below.

      +
      k
      require "substitution.md"
+
+module LAMBDA-SYNTAX
+  imports DOMAINS-SYNTAX
+  imports KVAR-SYNTAX
+
      +

      Basic Call-by-value λ-Calculus Syntax

      +

      We first define the syntax of conventional call-by-value λ-calculus, making +sure we declare the lambda abstraction construct to be a binder, the +lambda application to be strict, and the parentheses used for grouping as +a bracket.

      +

      Note: +Syntax in K is defined using the familiar BNF notation, with +terminals enclosed in quotes and nonterminals starting with capital +letters. K actually extends BNF with several attributes, which will be +described in this tutorial.

      +

      Note: +The strict constructs can evaluate their arguments in any (fully +interleaved) order.

      +

      The initial syntax of our λ-calculus:

      +
      k
        syntax Val ::= KVar
+               | "lambda" KVar "." Exp  [binder, latex(\lambda{#1}.{#2})]
+  syntax Exp ::= Val
+               | Exp Exp              [left, strict]
+               | "(" Exp ")"          [bracket]
+
      +

      Integer and Boolean Builtins Syntax

      +

      The LAMBDA arithmetic and Boolean expression constructs are simply rewritten +to their builtin counterparts once their arguments are evaluated. +The annotated operators in the right-hand side of the rules below are +builtin and come with the corresponding builtin sort. Note that the +variables appearing in these rules have integer sort. That means that these +rules will only be applied after the arguments of the arithmetic constructs +are fully evaluated to K results; this will happen thanks to their strictness +attributes declared as annotations to their syntax declarations (below).

      +
      k
        syntax Val ::= Int | Bool
+  syntax Exp ::= "-" Int
+               > Exp "*" Exp          [strict, left]
+               | Exp "/" Exp          [strict]
+               > Exp "+" Exp          [strict, left]
+               > Exp "<=" Exp         [strict]
+
      +

      Conditional Syntax

      +

      Note that the if construct is strict only in its first argument.

      +
      k
        syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+
      +

      Let Binder

      +

      The let binder is a derived construct, because it can be defined using λ.

      +
      k
        syntax Exp ::= "let" KVar "=" Exp "in" Exp [macro]
+  rule let X = E in E':Exp => (lambda X . E') E
+
      +

      Letrec Binder

      +

      We prefer a definition based on the μ construct. Note that μ is not +really necessary, but it makes the definition of letrec easier to understand +and faster to execute.

      +
      k
        syntax Exp ::= "letrec" KVar KVar "=" Exp "in" Exp [macro]
+               | "mu" KVar "." Exp                   [binder, latex(\mu{#1}.{#2})]
+  rule letrec F:KVar X:KVar = E in E' => let F = mu F . lambda X . E in E'
+endmodule
+
      +

      LAMBDA module

      +
      k
      module LAMBDA
+  imports LAMBDA-SYNTAX
+  imports SUBSTITUTION
+  imports DOMAINS
+
+  syntax KResult ::= Val
+
      +

      β-reduction

      +
      k
        rule (lambda X:KVar . E:Exp) V:Val => E[V / X]
+
      +

      Integer Builtins

      +
      k
        rule - I => 0 -Int I
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2  requires I2 =/=Int 0
+  rule I1 + I2 => I1 +Int I2
+  rule I1 <= I2 => I1 <=Int I2
+
      +

      Conditional

      +
      k
        rule if true  then E else _ => E
+  rule if false then _ else E => E
+
      +

      Mu

      +
      k
        rule mu X . E => E[(mu X . E) / X]
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/NOTES/index.html b/k-distribution/pl-tutorial/1_k/2_imp/NOTES/index.html new file mode 100644 index 00000000000..49748e7f56b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/NOTES/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +
        +
      • Revise the change of S1 S2 into S1:Stmt S2:Stmt, if needed; only S2 +really needs to be sorted.
        • +
      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/index.html b/k-distribution/pl-tutorial/1_k/2_imp/index.html new file mode 100644 index 00000000000..2cbc79223c1 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/index.html @@ -0,0 +1,403 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Part 2: Defining IMP

      +

      Here you will learn how to define a very simple imperative language in K +and the basics of how to work with configurations, cells, and computations. +Specifically, you will learn the following:

      +
        +
      • How to define languages using multiple modules.
        • +
      • How to define sequentially strict syntactic constructs.
        • +
      • How to use K's syntactic lists.
        • +
      • How to define, initialize and configure configurations.
        • +
      • How the language syntax is swallowed by the builtin K syntactic category.
        • +
      • The additional syntax of the K syntactic category.
        • +
      • How the strictness annotations are automatically desugared into rules.
        • +
      • The first steps of the configuration abstraction mechanism.
        • +
      +

      Like in the previous tutorial, this folder contains several lessons, each +adding new features to IMP. Do them in order. Also, make sure you completed +and understood the previous tutorial.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_1/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_1/index.html new file mode 100644 index 00000000000..2abb0407a55 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_1/index.html @@ -0,0 +1,541 @@ + + + + + + + + + + + + + + +Defining a More Complex Syntax | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Defining a More Complex Syntax

      +

      Here we learn how to define a more complex language syntax than LAMBDA's, +namely the C-like syntax of IMP. Also, we will learn how to define languages +using multiple modules, because we are going to separate IMP's syntax from +its semantics using modules. Finally, we will also learn how to use K's +builtin support for syntactic lists.

      +

      The K tool provides modules for grouping language features. In general, we +can organize our languages in arbitrarily complex module structures. +While there are no rigid requirements or even guidelines for how to group +language features in modules, we often separate the language syntax from the +language semantics in different modules.

      +

      In our case here, we start by defining two modules, IMP-SYNTAX and IMP, and +import the first in the second, using the keyword imports. As their names +suggest, we will place all IMP's syntax definition in IMP-SYNTAX and all its +semantics in IMP.

      +

      Note, however, that K does no more than simply includes all the +contents of the imported module in the one which imports it (making sure +that everything is only kept once, even if you import it multiple times). +In other words, there is currently nothing fancy in K tool's module system.

      +

      IMP has six syntactic categories, as shown in imp.k: AExp for arithmetic +expressions, BExp for Boolean expressions, Block for blocks, Stmt for +statements, Pgm for programs and Ids for comma-separated lists of +identifiers. Blocks are special statements, whose role is to syntactically +constrain the conditional statement and the while loop statement to only +take blocks as branches and body, respectively.

      +

      There is nothing special about arithmetic and Boolean expressions. They +are given the expected strictness attributes, except for <= and &&, +for demonstration purposes.

      +

      The <= is defined to be seqstrict, which means that it evaluates its +arguments in order, from left-to-right (recall that the strict operators +can evaluate their arguments in any, fully interleaved, orders). Like +strict, the seqstrict annotation can also be configured; for example, one +can specify in which arguments and in what order. By default, seqstrict +refers to all the arguments, in their left-to-right order. In our case here, +it is equivalent with seqstrict(1 2).

      +

      The && is only strict in its first argument, because we will give it a +short-circuited semantics (its second argument will only be evaluated when +the first evaluates to true). Recall the K tool also allows us to associate +LaTex attributes to constructs, telling the document generator how to display +them. For example, we associate <= the attribute latex({#1}\leq{#2}), +which makes it be displayed ≤\leq everywhere in the generated LaTex +documentation.

      +

      In this tutorial we take the freedom to associate the various constructs +parsing precedences that we have already tested and we know work well, so that +we can focus on the semantics here instead of syntax. In practice, though, +you typically need to experiment with precedences until you obtain the desired +parser.

      +

      Blocks are defined using curly brackets, and they can either be empty or +hold a statement.

      +

      Nothing special about the IMP statements. Note that ; is an assignment +statement terminator, not a statement separator. Note also that blocks are +special statements.

      +

      An IMP program declares a comma-separated list of variables using the keyword +int like in C, followed by a semicolon ;, followed by a statement. +Syntactically, the idea here is that we can wrap any IMP program within a +main(){...} function and get a valid C program. IMP does not allow variable +declarations anywhere else except through this construct, at the top-level of +the program. Other languages provided with the K distribution (see, e.g., the +IMP++ language also discussed in this tutorial) remove this top-level program +construct of IMP and add instead variable declaration as a statement construct, +which can be used anywhere in the program, not only at the top level.

      +

      Note how we defined the comma-separated list of identifiers using +List{Id,","}. The K tool provides builtin support for generic syntactic +lists. In general,

      +
      syntax B ::= List{A,T}
+
      +

      declares a new non-terminal, B, corresponding to T-separated sequences of +elements of A, where A is a non-terminal and T is a terminal. These +lists can also be empty, that is, IMP programs declaring no variable are also +allowed (e.g., int; {} is a valid IMP program). To instantiate and use +the K builtin lists, you should alias each instance with a (typically fresh) +non-terminal in your syntax, like we do with the Ids nonterminal.

      +

      Like with other K features, there are ways to configure the syntactic lists, +but we do not discuss them here.

      +

      Recall from Tutorial 1 (LAMBDA) that in order for strictness to work well +we also need to tell K which computations are meant to be results. We do +this as well now, in the module IMP: integers and Booleans are K results.

      +

      Kompile imp.k and test the generated parser by running some programs. +Since IMP is a fragment of C, you may want to select the C mode in your +editor when writing these programs. This will also give your the feel that +you are writing programs in a real programming language.

      +

      For example, here is sum.imp, which sums in sum all numbers up to n:

      +
      int n, sum;
+n = 100;
+sum=0;
+while (!(n <= 0)) {
+  sum = sum + n;
+  n = n + -1;
+}
+
      +

      Now krun it and see how it looks parsed in the default k cell.

      +

      The program collatz.imp tests the Collatz conjecture for all numbers up to +m and accumulates the total number of steps in s:

      +
      int m, n, q, r, s;
+m = 10;
+while (!(m<=2)) {
+  n = m;
+  m = m + -1;
+  while (!(n<=1)) {
+    s = s+1;
+    q = n/2;
+    r = q+q+1;
+    if (r<=n) {
+      n = n+n+n+1;         // n becomes 3*n+1 if odd
+    } else {n=q;}          //        of   n/2 if even
+  }
+}
+
      +

      Finally, program primes.imp counts in s all the prime numbers up to m:

      +
      int i, m, n, q, r, s, t, x, y, z;
+m = 10;  n = 2;
+while (n <= m) {
+  // checking primality of n and writing t to 1 or 0
+  i = 2;  q = n/i;  t = 1;
+  while (i<=q && 1<=t) {
+    x = i;
+    y = q;
+    // fast multiplication (base 2) algorithm
+    z = 0;
+    while (!(x <= 0)) {
+      q = x/2;
+      r = q+q+1;
+      if (r <= x) { z = z+y; } else {}
+      x = q;
+      y = y+y;
+    } // end fast multiplication
+    if (n <= z) { t = 0; } else { i = i+1;  q = n/i; }
+  } // end checking primality
+  if (1 <= t) { s = s+1; } else {}
+  n = n+1;
+}
+
      +

      All the programs above will run once we define the semantics of IMP. If you +want to execute them now, wrap them in a main(){...} function and compile +them and run them with your favorite C compiler.

      +

      Before we move to the K semantics of IMP, we would like to make some +clarifications regarding the K builtin parser, kast. Although it is quite +powerful, you should not expect magic from it! While the K parser can parse +many non-trivial languages (see, for example, the KOOL language in +pl-tutorial/2_languages) in the K distribution), it was +never meant to be a substitute for real parsers. We often call the syntax +defined in K the syntax of the semantics, to highlight the fact that its +role is to serve as a convenient notation when writing the semantics, not +necessarily as a means to define concrete syntax of arbitrarily complex +programming languages. See the KERNELC language for an example on how to connect an external parser for concrete syntax to +the K tool.

      +

      The above being said, we strongly encourage you to strive to make the +builtin parser work with your desired language syntax! Do not give up +simply because you don't want to deal with syntactic problems. On the +contrary, fight for your syntax! If you really cannot define your desired +syntax because of tool limitations, we would like to know. Please tell us.

      +

      Until now we have only seen default configurations. In the next lesson we +will learn how to define a K custom configuration.

      +

      Go to Lesson 2, IMP: Defining a Configuration.

      +

      MOVIE (out of date) [09'15"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_2/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_2/index.html new file mode 100644 index 00000000000..2198bd2d169 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_2/index.html @@ -0,0 +1,453 @@ + + + + + + + + + + + + + + +Defining a Configuration | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Defining a Configuration

      +

      Here we learn how to define a configuration in K. We also learn how to +initialize and how to display it.

      +

      As explained in the overview presentation on K, configurations are quite +important, because all semantic rules match and apply on them. +Moreover, they are the backbone of configuration abstraction, which allows +you to only mention the relevant cells in each semantic rule, the rest of +the configuration context being inferred automatically. The importance of +configuration abstraction will become clear when we define more complex +languages (even in IMP++). IMP does not really need it. K configurations +are constructed making use of cells, which are labeled and can be arbitrarily +nested.

      +

      Configurations are defined with the keyword configuration. Cells are +defined using an XML-ish notation stating clearly where the cell starts +and where it ends.

      +

      While not enforced by the tool, we typically like to put the entire +configuration in a top-level cell, called T. So let's define it:

      +
      configuration <T>...</T>
+
      +

      Cells can have other cells inside. In our case of IMP, we need a cell to +hold the remaining program, cell which we typically call k, and a cell to +hold the program state. Let us add them:

      +
      configuration <T> <k>...</k> <state>...</state> </T>
+
      +

      K allows us to also specify how to initialize a configuration at the same +time with declaring the configuration. All we have to do is to fill in +the contents of the cells with some terms. The syntactic categories of +those terms will also indirectly define the types of the corresponding +cells.

      +

      For example, we want the k cell to initially hold the program that is passed +to krun. K provides a builtin configuration variable, called $PGM, which +is specifically designed for this purpose: krun will place its program there +(after it parses it, or course). The K tool allows users to define their own +configuration variables, too, which can be used to develop custom +initializations of program configurations with the help of krun; this can be +quite useful when defining complex languages, but we do not discuss it in +this tutorial.

      +
      configuration <T> <k> $PGM </k> <state>...</state>  </T>
+
      +

      Moreover, we want the program to be a proper Pgm term (because we do not +want to allow krun to take fragments of programs, for example, statements). +Therefore, we tag $PGM with the desired syntactic category, Pgm:

      +
      configuration <T> <k> $PGM:Pgm </k> <state>...</state>  </T>
+
      +

      Like for other variable tags in K, a run-time check will be performed and the +semantics will get stuck if the passed term is not a well-formed program.

      +

      We next tell K that the state cell should be initialized with the empty map:

      +
      configuration <T> <k> $PGM:Pgm </k> <state> .Map </state>  </T>
+
      +

      Recall that in K . stands for nothing. However, since there are various +types of nothing, to avoid confusion we can suffix the . with its desired +type. K has several builtin data-types, including lists, sets, bags, and +maps. .Map is the empty map.

      +

      Kompile imp.k and run several programs to see how the configuration is +initialized as desired.

      +

      When configurations get large, and they do when defining large programming +languages, you may want to color the cells in order to more easily distinguish +them. This can be easily achieved using the color cell attribute, following +again an XML-ish style:

      +
      configuration <T color="yellow">
+                <k color="green"> $PGM:Pgm </k>
+                <state color="red"> .Map </state>
+              </T>
+
      +

      In the next lesson we will learn how to write rules that involve cells.

      +

      Go to Lesson 3, IMP: Computations, Results, Strictness; Rules Involving Cells.

      +

      MOVIE (out of date) [04'21"]

      +
      +
      + + +
      + + + +
      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_3/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_3/index.html new file mode 100644 index 00000000000..10734f4823e --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_3/index.html @@ -0,0 +1,591 @@ + + + + + + + + + + + + + + +Computations, Results, Strictness; Rules Involving Cells | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Computations, Results, Strictness; Rules Involving Cells

      +

      In this lesson we will learn about the syntactic category K of computations, +about how strictness attributes are in fact syntactic sugar for rewrite rules +over computations, and why it is important to tell the tool which +computations are results. We will also see a K rule that involves cells.

      +

      K Computations

      +

      Computation structures, or more simply computations, extend the abstract +syntax of your language with a list structure using ~> (read followed +by or and then, and written ↷\curvearrowright in Latex) as a separator. +K provides a distinguished sort, K, for computations. The extension of the +abstract syntax of your language into computations is done automatically by +the K tool when you declare constructs using the syntax keyword, so the K +semantic rules can uniformly operate only on terms of sort K. The intuition +for computation structures of the form

      +
      t1 ~> t2 ~> ... ~> tn
+
      +

      is that the listed tasks are to be processed in order. The initial +computation typically contains the original program as its sole task, but +rules can then modify it into task sequences, as seen shortly.

      +

      Strictness in Theory

      +

      The strictness attributes, used as annotations to language constructs, +actually correspond to rules over computations. For example, the +strict(2) attribute of the assignment statement corresponds to the +following two opposite rules (X ranges over Id and A over AExp):

      +
      X=A; => A ~> X=[];
+A ~> X=[]; => X=A;
+
      +

      The first rule pulls A from the syntactic context X=A; and schedules it +for processing. The second rule plugs A back into its context. +Inspired from the chemical abstract machine, we call rules of the first +type above heating rules and rules of the second type cooling rules. +Similar rules are generated for other arguments in which operations are +strict. Iterative applications of heating rules eventually bring to the +top of the computation atomic tasks, such as a variable lookup, or a +builtin operation, which then make computational progress by means of other +rules. Once progress is made, cooling rules can iteratively plug the result +back into context, so that heating rules can pick another candidate for +reduction, and so on and so forth.

      +

      When operations are strict only in some of their arguments, the corresponding +positions of the arguments in which they are strict are explicitly enumerated +in the argument of the strict attribute, e.g., strict(2) like above, or +strict(2 3) for an operation strict in its second and third arguments, etc. +If an operation is simply declared strict then it means that it is strict +in all its arguments. For example, the strictness of addition yields:

      +
      A1+A2 => A1 ~> []+A2
+A1 ~> []+A2 => A1+A2
+A1+A2 => A2 ~> A1+[]
+A2 ~> A1+[] => A1+A2
+
      +

      It can be seen that such heating/cooling rules can easily lead to +non-determinism, since the same term may be heated many different ways; +these different evaluation orders may lead to different behaviors in some +languages (not in IMP, because its expressions do not have side effects, +but we will experiment with non-determinism in its successor, IMP++).

      +

      A similar desugaring applies to sequential strictness, declared with the +keyword seqstrict. While the order of arguments of strict is irrelevant, +it matters in the case of seqstrict: they are to be evaluated in the +specified order; if no arguments are given, then they are assumed by default +to be evaluated from left-to-right. For example, the default heating/cooling +rules associated to the sequentially strict <= construct above are +(A1, A2 range over AExp and I1 over Int):

      +
      A1<=A2 => A1 ~> []<=A2
+A1 ~> []<=A2 => A1<=A2
+I1<=A2 => A2 ~> I1<=[]
+A2 ~> I1<=[] => I1<=A2
+
      +

      In other words, A2 is only heated/cooled after A1 is already evaluated.

      +

      While the heating/cooling rules give us a nice and uniform means to define +all the various allowable ways in which a program can evaluate, all based +on rewriting, the fact that they are reversible comes with a serious practical +problem: they make the K definitions unexecutable, because they lead to +non-termination.

      +

      Strictness in Practice; K Results

      +

      To break the reversibility of the theoretical heating/cooling rules, and, +moreover, to efficiently execute K definitions, the current implementation of +the K tool relies on users giving explicit definitions of their languages' +results.

      +

      The K tool provides a predicate isKResult, which is automatically defined +as we add syntactic constructs to KResult (in fact the K tool defines such +predicates for all syntactic categories, which are used, for example, as +rule side conditions to check user-declared variable memberships, such as +V:Val stating that V belongs to Val).

      +

      The kompile tool, depending upon what it is requested to do, changes the +reversible heating/cooling rules corresponding to evaluation strategy +definitions (e.g., those corresponding to strictness attributes) to avoid +non-termination. For example, when one is interested in obtaining an +executable model of the language (which is the default compilation mode of +kompile), then heating is performed only when the to-be-pulled syntactic +fragment is not a result, and the corresponding cooling only when the +to-be-plugged fragment is a result. In this case, e.g., the heating/cooling +rules for assignment are modified as follows:

      +
      X=A; => A ~> X=[];  requires notBool isKResult(A)
+A ~> X=[]; => X=A;  requires isKResult(A)
+
      +

      Note that non-termination of heating/cooling is avoided now. The only thing +lost is the number of possible behaviors that a program can manifest, but +this is irrelevant when all we want is one behavior.

      +

      As will be discussed in the IMP++ tutorial, the heating/cooling rules are +modified differently by kompile when we are interested in other aspects +of the language definition, such us, for example, in a search-able model that +comprises all program behaviors. This latter model is obviously more general +from a theoretical perspective, but, in practice, it is also slower to execute. +The kompile tool strives to give you the best model of the language for the +task you are interested in.

      +

      Can't Results be Inferred Automatically?

      +

      This is a long story, but the short answer is: No!. Maybe in some cases +it is possible, but we prefer to not attempt it in the K tool. For example, +you most likely do not want any stuck computation to count as a result, +since some of them can happen simply because you forgot a semantic rule that +could have further reduce it! Besides, in our experience with defining large +languages, it is quite useful to take your time and think of what the results +of your language's computations are. This fact in itself may help you improve +your overall language design. We typically do it at the same time with +defining the evaluation strategies of our languages. Although in theory K +could infer the results of your language as the stuck computations, based on +the above we have deliberately decided to not provide this feature, in spite +of requests from some users. So you currently do have to explicitly define +your K results if you want to effectively use the K tool. Note, however, that +theoretical definitions, not meant to be executed, need not worry about +defining results (that's because in theory semantic rules apply modulo the +reversible heating/cooling rules, so results are not necessary).

      +

      A K Rule Involving Cells

      +

      All our K rules so far in the tutorial were of the form

      +
      rule left => right requires condition
+
      +

      where left and right were syntactic, or more generally computation, terms.

      +

      Here is our first K rule explicitly involving cells:

      +
      rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
+
      +

      Recall that the k cell holds computations, which are sequences of tasks +separated by ~>. Also, the state cell holds a map, which is a set of +bindings, each binding being a pair of computations (currently, the +K builtin data-structures, like maps, are untyped; or, said differently, +they are all over the type of computations, K).

      +

      Therefore, the two cells mentioned in the rule above hold collections +of things, ordered or not. The ...s, which we also call cell frames, +stand for more stuff there, which we do not care about.

      +

      The rewrite relation => is allowed in K to appear anywhere in a term, its +meaning being that the corresponding subterm is rewritten as indicated in the +shown context. We say that K's rewriting is local.

      +

      The rule above says that if the identifier X is the first task in the k +cell, and if X is bound to I somewhere in the state, then X rewrites +to I locally in the k cell. Therefore, IMP variables need to be already +declared when looked up.

      +

      Of course, the K rule above can be translated into an ordinary rewrite rule +of the form

      +
      rule <k> X ~> Rest </k> <state> Before (X |-> I) After </state>
+  => <k> I ~> Rest </k> <state> Before (X |-> I) After </state>
+
      +

      Besides being more verbose and thus tedious to write, this ordinary rule +is also more error-prone; for example, we may forget the Rest variable +in the right-hand-side, etc. Moreover, the concurrent semantics of K +allows for its rules to be interpreted as concurrent transactions, where +the context is the read-only component of the transaction, while the +subterms which are rewritten are read/write component of the transaction; +thus, K rule instances can apply concurrently if they only overlap +on read-only parts, while they cannot if regarded as ordinary rewrite logic +rules. Note: our current implementation of the K tool is not concurrent, +so K rules are in fact desugared as normal rewrite rules in the K tool.

      +

      Kompile imp.k using a documentation option and check out how the K rule +looks in the generated document. The ... frames are displayed as cell +tears, metaphorically implying that those parts of the cells that we +do not care about are torn away. The rewrite relation is replaced by a +horizontal line: specifically, the subterm which rewrites, X, is +underlined, and its replacement is written underneath the line.

      +

      In the next lesson we define the complete K semantics of IMP and +run the programs we parsed in the first lesson.

      +

      Go to Lesson 4, IMP: Configuration Abstraction, Part 1; Types of Rules.

      +

      MOVIE (out of date) [10'30"]

      +
      +
      + + +
      + + + +
      +
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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Modify IMP so that the K followed by arrow, ~>, does not explicitly +occur in the definition (it currently occurs in the semantics of +sequential composition).

      +

      Hint: make sequential composition strict(1) or seqstrict, and have +statements reduce to {} instead of .; and don't forget to make +{} a KResult (you may need a new syntactic category for that, which +only includes {} and is included in KResult).

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/exercises/uninitialized-variables/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/exercises/uninitialized-variables/index.html new file mode 100644 index 00000000000..de913f3cf9f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/exercises/uninitialized-variables/index.html @@ -0,0 +1,381 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Modify the K definition of IMP to not automatically initialize +variables to 0. Instead, declared variables should stay uninitialized +until assigned a value, and the execution should get stuck when an +uninitialized variable is looked up. Specifically, you should add a +new undefined construct of sort K, and initialize all the declared +variables with it.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/index.html new file mode 100644 index 00000000000..7474edf2271 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_4/index.html @@ -0,0 +1,497 @@ + + + + + + + + + + + + + + +Configuration Abstraction, Part 1; Types of Rules | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Configuration Abstraction, Part 1; Types of Rules

      +

      Here we will complete the K definition of IMP and, while doing so, we will +learn the very first step of what we call configuration abstraction.

      +

      The IMP Semantic Rules

      +

      Let us add the remaining rules, in the order in which the language constructs +were defined in IMP-SYNTAX.

      +

      The rules for the arithmetic and Boolean constructs are self-explanatory. +Note, however, that K will infer the correct sorts of all the variables in +these rules, because they appear as arguments of the builtin operations +(_+Int_, etc.). Moreover, the inferred sorts will be enforced dynamically. +Indeed, we do not want to apply the rule for addition, for example, when the +two arguments are not integers. In the rules for &&, although we prefer to +not do it here for simplicity, we could have eliminated the dynamic check by +replacing B (and similarly for _) with B:K. Indeed, it can be shown +that whenever any of these rules apply, B (or _) is a BExp anyway. +That's because there is no rule that can touch such a B (or _); this +will become clearer shortly, when we discuss the first step of configuration +abstraction. Therefore, since we know that B will be a BExp anyway, we +could save the time it takes to check its sort; such times may look minor, +but they accumulate, so some designers may prefer to avoid run-time checks +whenever possible.

      +

      The block rules are trivial. However, the rule for non-empty blocks is +semantically correct only because we do not have local variable declarations +in IMP. We will have to change this rule in IMP++.

      +

      The assignment rule has two =>: one in the k cell dissolving the +assignment statement, and the other in the state cell updating the value of +the assigned variable. Note that the one in the state is surrounded by +parentheses: (_ => I). That is because => is greedy: it matches as much +as it can to the left and to the right, until it reaches the cell boundaries +(closed or open). If you want to limit its scope, or for clarity, you can use +parentheses like here.

      +

      The rule for sequential composition simply desugars S1 S2 into S1 ~> S2. +Indeed, the two have exactly the same semantics. Note that statements +evaluate to nothing (.), so once S1 is processed in S1 ~> S2, then the +next task is automatically S2, without wasting any step for the transition.

      +

      The rules for the conditional and while statements are clear. One thing to +keep in mind now is that the while unrolling rule will not apply +indefinitely in the positive branch of the resulting conditional, because +of K's configuration abstraction, which will be discussed shortly.

      +

      An IMP program declares a set of variables and then executes a +statement in the state obtained after initializing all those variables +to 0. The rules for programs initialize the declared variables one by one, +checking also that there are no duplicates. We check for duplicates only for +demonstration purposes, to illustrate the keys predefined operation that +returns the set of keys of a map, and the set membership operation in. +In practice, we typically define a static type checker for our language, +which we execute before the semantics and reject inappropriate programs.

      +

      The use of the .Ids in the second rule is not necessary. We could have +written int; S instead of int .Ids; S and the K tool would parse it and +kompile the definition correctly, because it uses the same parser used for +parsing programs also to parse the semantics. However, we typically prefer to +explicitly write the nothing values in the semantics, for clarity; +the parser has been extended to accept these. Note that the first rule +matches the entire k cell, because int_;_ is the top-level program +construct in IMP, so there is nothing following it in the computation cell. +The anonymous variable stands for the second argument of this top-level program +construct, not for the rest of the computation. The second rule could have +also been put in a complete k cell, but we preferred not to, for simplicity.

      +

      Our IMP semantics is now complete, but there are a few more things that we +need to understand and do.

      +

      Configuration Abstraction, Part 1

      +

      First, let us briefly discuss the very first step of configuration abstraction. +In K, all semantic rules are in fact rules between configurations. As soon +explained in the IMP++ tutorial, the declared configuration cell structure is +used to automatically complete the missing configuration parts in rules. +However, many rules do not involve any cells, being rules between syntactic +terms (of sort K); for example, we had only three rules involving cells in our +IMP semantics. In this case, the k cell will be added automatically and the +actual rewrite will happen on top of the enclosed computation. For example, +the rule for the while loop is automatically translated into the following:

      +
      rule <k> while (B) S => if (B) {S while (B) S} else {} ...</k>
+
      +

      Since the first task in computations is what needs to be done next, the +intuition for this rule completion is that the syntactic transition +only happens when the term to rewrite is ready for processing. This explains, +for example, why the while loop unrolling does not indefinitely apply in the +positive branch of the conditional: the inner while loop is not ready for +evaluation yet. We call this rule completion process, as well as other +similar ones, configuration abstraction. That is because the incomplete +rule abstracts away the configuration structure, thus being easier to read. +As seen soon when we define IMP++, configuration abstraction is not only a +user convenience; it actually significantly increases the modularity of our +definitions. The k-cell-completion is only the very first step, though.

      +

      If you really want certain rewrites over syntactic terms to apply +anywhere they match, then you should tag the rule with the attribute +anywhere, which was discussed in Tutorial 1, Lesson 2.5.

      +

      Kompile and then krun the programs that you only parsed in Lesson 1. They +should all execute as expected. The state cell shows the final state +of the program. The k cell shows the final code contents, which should be +empty whenever the IMP program executes correctly.

      +

      Kompile also with the documentation option and take a look at the generated +documentation. The assignment rule should particularly be of interest, +because it contains two local rewrites.

      +

      In the next lesson we comment the IMP definition and conclude this tutorial.

      +

      Go to Lesson 5, IMP: Completing and Documenting IMP.

      +

      MOVIE (out of date) [09'16"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/imp/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/imp/index.html new file mode 100644 index 00000000000..d4b722f451b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/imp/index.html @@ -0,0 +1,547 @@ + + + + + + + + + + + + + + +IMP | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      IMP

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Abstract

      +

      This is the K semantic definition of the classic IMP language. +IMP is considered a folklore language, without an official inventor, +and has been used in many textbooks and papers, often with slight +syntactic variations and often without being called IMP. It includes +the most basic imperative language constructs, namely basic constructs +for arithmetic and Boolean expressions, and variable assignment, +conditional, while loop and sequential composition constructs for statements.

      +
      k
      module IMP-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      This module defines the syntax of IMP. +Note that <= is sequentially strict, and && is strict only in its first +argument, because we want to give it a short-circuit semantics.

      +
      k
        syntax AExp  ::= Int | Id
+                 | "-" Int                    [format(%1%2)]
+                 | AExp "/" AExp              [left, strict, color(pink)]
+                 | "(" AExp ")"               [bracket]
+                 > AExp "+" AExp              [left, strict, color(pink)]
+  syntax BExp  ::= Bool
+                 | AExp "<=" AExp             [seqstrict, latex({#1}\leq{#2}), color(pink)]
+                 | "!" BExp                   [strict, color(pink)]
+                 | "(" BExp ")"               [bracket]
+                 > BExp "&&" BExp             [left, strict(1), color(pink)]
+  syntax Block ::= "{" "}"
+                 | "{" Stmt "}"               [format(%1%i%n%2%d%n%3)]
+  syntax Stmt  ::= Block
+                 | Id "=" AExp ";"            [strict(2), color(pink), format(%1 %2 %3%4)]
+                 | "if" "(" BExp ")"
+                   Block "else" Block         [strict(1), colors(yellow, white, white, yellow), format(%1 %2%3%4 %5 %6 %7)]
+                 | "while" "(" BExp ")" Block [colors(yellow,white,white), format(%1 %2%3%4 %5)]
+                 > Stmt Stmt                  [left, format(%1%n%2)]
+
      +

      An IMP program declares a set of variables and then executes a +statement in the state obtained after initializing all those variables +to 0. K provides builtin support for generic syntactic lists: +List{Nonterminal,terminal} stands for terminal-separated lists of Nonterminal elements.

      +
      k
        syntax Pgm ::= "int" Ids ";" Stmt           [format(%1 %2%3%n%4), colors(yellow,pink)]
+  syntax Ids ::= List{Id,","}                 [format(%1%2 %3)]
+endmodule
+
      +

      We are done with the definition of IMP's syntax. Make sure +that you write and parse several interesting programs before you move to the +semantics.

      +
      k
      module IMP
+  imports IMP-SYNTAX
+  imports DOMAINS
+
      +

      Semantics

      +

      This module defines the semantics of IMP. +Before you start adding semantic rules to a K definition, you need to +define the basic semantic infrastructure consisting of definitions for +results and the configuration.

      +

      Values and results

      +

      IMP only has two types of values, or results of computations: integers +and Booleans. We here use the K builtin variants for both of them.

      +
      k
        syntax KResult ::= Int | Bool
+
      +

      Configuration

      +

      The configuration of IMP is trivial: it only contains two cells, one +for the computation and another for the state. For good encapsulation +and clarity, we place the two cells inside another cell, the top cell +which is labeled T.

      +
      k
        configuration <T color="yellow">
+                  <k color="green"> $PGM:Pgm </k>
+                  <state color="red"> .Map </state>
+                </T>
+
      +

      The configuration variable PGM tells the K tool where to +place the program. More precisely, the command +krun program parses the program and places the resulting +K abstract syntax tree in the k cell before invoking the +semantic rules described in the sequel. The . in the +state cell, written .Map in ASCII in the +imp.md file, is K's way to say nothing. Technically, it +is a constant which is the unit, or identity, of all maps in K +(similar dot units exist for other K structures, such as lists, sets, +multi-sets, etc.).

      +

      Arithmetic expressions

      +

      The K semantics of each arithmetic construct is defined below.

      +

      Variable lookup

      +

      A program variable X is looked up in the state by matching a binding +of the form X |-> I in the state cell. If such a binding does not +exist, then the rewriting process will get stuck. Thus our semantics of +IMP disallows uses of uninitialized variables. Note that the variable +to be looked up is the first task in the k cell (the cell is +closed to the left and torn to the right), while the binding can be +anywhere in the state cell (the cell is torn at both sides).

      +
      k
        rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
+
      +

      Arithmetic operators

      +

      There is nothing special about these, but recall that K's configuration +abstraction mechanism is at work here! That means that the rewrites in the +rules below all happen at the beginning of the k cell.

      +
      k
        rule I1 / I2 => I1 /Int I2  requires I2 =/=Int 0
+  rule I1 + I2 => I1 +Int I2
+  rule - I1 => 0 -Int I1
+
      +

      Boolean expressions

      +

      The rules below are straightforward. Note the short-circuited semantics +of &&; this is the reason we annotated the syntax of +&& with the K attribute strict(1) instead of strict.

      +
      k
        rule I1 <= I2 => I1 <=Int I2
+  rule ! T => notBool T
+  rule true && B => B
+  rule false && _ => false
+
      +

      Blocks and Statements

      +

      There is one rule per statement construct except for the conditional, +which needs two rules.

      +

      Blocks

      +

      The empty block {} is simply dissolved. The . below is the +unit of the computation list structure K, that is, the empty task. +Similarly, the non-empty blocks are dissolved and replaced by their statement +contents, thus effectively giving them a bracket semantics; we can afford to +do this only because we have no block-local variable declarations yet in IMP.

      +
      k
        rule {} => .
+  rule {S} => S
+
      +

      Assignment

      +

      The assigned variable is updated in the state. The variable is expected +to be declared, otherwise the semantics will get stuck. At the same time, +the assignment is dissolved.

      +
      k
        rule <k> X = I:Int; => . ...</k> <state>... X |-> (_ => I) ...</state>
+
      +

      Sequential composition

      +

      Sequential composition is simply structurally translated into K's +builtin task sequentialization operation.

      +
      k
        rule S1:Stmt S2:Stmt => S1 ~> S2
+
      +

      Conditional

      +

      The conditional statement has two semantic cases, corresponding to +when its condition evaluates to true or to false. +Recall that the conditional was annotated with the attribute +strict(1) in the syntax module above, so only its first +argument is allowed to be evaluated.

      +
      k
        rule if (true)  S else _ => S
+  rule if (false) _ else S => S
+
      +

      While loop

      +

      We give the semantics of the while loop by unrolling.

      +
      k
        rule while (B) S => if (B) {S while (B) S} else {}
+
      +

      Programs

      +

      The semantics of an IMP program is that its body statement is executed +in a state initializing all its global variables to 0. Since K's +syntactic lists are internally interpreted as cons-lists (i.e., lists +constructed with a head element followed by a tail list), we need to +distinguish two cases, one when the list has at least one element and +another when the list is empty. In the first case we initialize the +variable to 0 in the state, but only when the variable is not already +declared (all variables are global and distinct in IMP).

      +
      k
        rule <k> int (X,Xs => Xs);_ </k> <state> Rho:Map (.Map => X|->0) </state>
+    requires notBool (X in keys(Rho))
+  rule int .Ids; S => S
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/index.html b/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/index.html new file mode 100644 index 00000000000..5a5337c634c --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/2_imp/lesson_5/index.html @@ -0,0 +1,400 @@ + + + + + + + + + + + + + + +Completing and Documenting IMP | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Completing and Documenting IMP

      +

      We here learn no new concepts, but it is a good moment to take a break +and contemplate what we learned so far.

      +

      Let us add lots of formal annotations to imp.k.

      +

      Once we are done with the annotations, we kompile with the documentation +option and then take a look at the produced document. We often call these +documents language posters. Depending on how much information you add to +these language posters, they can serve as standalone, formal presentations +of your languages. For example, you can print them as large posters and +post them on the wall, or in poster sessions at conferences.

      +

      This completes our second tutorial. The next tutorials will teach us more +features of the K framework, such as how to define languages with complex +control constructs (like callcc), languages which are concurrent, and so on.

      +

      MOVIE (out of date) [03'45"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/index.html new file mode 100644 index 00000000000..a80bb3ab5b5 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/index.html @@ -0,0 +1,399 @@ + + + + + + + + + + + + + + +Part 3: Defining LAMBDA++ | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Part 3: Defining LAMBDA++

      +

      Here you will learn how to define language constructs which abruptly change +the execution control flow, and how to define language semantics following +and environment/store style. Specifically, you will learn the following:

      +
        +
      • How to define constructs like callcc, which allow you to take snapshots of +program executions and to go back in time at any moment.
        • +
      • How to define languages in an environment/store style.
        • +
      • Some basic notions about the use of closures and closure-like semantic +structures to save and restore execution environments.
        • +
      • Some basic intuitions about reusing existing semantics in new languages, +as well as some of the pitfalls in doing so.
        • +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/NOTES/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/NOTES/index.html new file mode 100644 index 00000000000..fd2c2c41793 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/NOTES/index.html @@ -0,0 +1,378 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Maybe we should change the name of calCC, as it is not a good idea to have +two constructs with different semantics but names which cannot be distinguished +easily.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/callCC/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/callCC/index.html new file mode 100644 index 00000000000..0d7ef72fa5f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/callCC/index.html @@ -0,0 +1,378 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define a variant of callcc, say callCC, which never returns to the +current context unless a value is specifically passed to its argument +continuation. Follow a substitution-based style.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-call-CC-to-callcc/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-call-CC-to-callcc/index.html new file mode 100644 index 00000000000..0762d8e4e9f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-call-CC-to-callcc/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define callcc in terms of callCC, where callCC is explained in the +callCC exercise under LAMBDA++, Lesson 1. Follow a substitution-based style.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-callcc-to-call-CC/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-callcc-to-call-CC/index.html new file mode 100644 index 00000000000..47164a88cc9 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/exercises/from-callcc-to-call-CC/index.html @@ -0,0 +1,381 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define callCC in terms of callcc, where callCC is explained in the +callCC exercise under LAMBDA++, Lesson 1. Follow a substitution-based style.

      +

      To facilitate testing, call the main module CALLCC (see tests/config.xml). +For example, you can define a module CALLCC which imports the previous +definition of callcc and adds the definition of callCC in terms of +callcc.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/index.html new file mode 100644 index 00000000000..392f6dfb2b0 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_1/index.html @@ -0,0 +1,486 @@ + + + + + + + + + + + + + + +Abrupt Changes of Control | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Abrupt Changes of Control

      +

      Here we add call-with-current-continuation (callcc) to the definition of +LAMBDA completed in Tutorial 1, and call the resulting language LAMBDA++. +While doing so, we will learn how to define language constructs that +abruptly change the execution control flow.

      +

      Take over the lambda.k definition from Lesson 8 in Part 1 of this Tutorial, +which is the complete definition of the LAMBDA language, but without the +comments.

      +

      callcc is a good example for studying the capabilities of a framework to +support abrupt changes of control, because it is one of the most +control-intensive language constructs known. Scheme is probably the first +programming language that incorporated the callcc construct, although +similar constructs have been recently included in many other languages in +one form or another.

      +

      Here is a quick description: callcc e passes the remaining computation +context, packaged as a function k, to e (which is expected to be a function); +if during its evaluation e passes any value to k, then the current +execution context is discarded and replaced by the one encoded by k and +the value is passed to it; if e evaluates normally to some value v and +passes nothing to k in the process, then v is returned as a result of +callcc e and the execution continues normally. For example, we want the +program callcc-jump.lambda:

      +
      (callcc (lambda k . ((k 5) + 2))) + 10
+
      +

      to evaluate to 15, not 17! Indeed, the computation context [] + 10 is +passed to callcc's argument, which then sends it a 5, so the computation +resumes to 5 + 10. On the other hand, the program callcc-not-jump.lambda

      +
      (callcc (lambda k . (5 + 2))) + 10
+
      +

      evaluates to 17.

      +

      If you like playing games, you can metaphorically think of callcc e as +saving your game state in a file and passing it to your friend e. +Then e can decide at some moment to drop everything she was doing, load +your game and continue to play it from where you were.

      +

      The behavior of many popular control-changing constructs can be obtained +using callcc. The program callcc-return.lambda shows, for example, how to +obtain the behavior of a return statement, which exits the current execution +context inside a function and returns a value to the caller's context:

      +
      letrec f x = callcc (lambda return . (
+  f (if (x <= 0) then ((return 1) / 0) else 2)
+))
+in (f -3)
+
      +

      This should evaluate to 1, in spite of the recursive call to f +and of the division by zero! Note that return is nothing but a variable +name, but one which is bound to the current continuation at the beginning of +the function execution. As soon as 1 is passed to return, the computation +jumps back in time to where callcc was defined! Change -3 to 3 and the +program will loop forever.

      +

      callcc is quite a powerful and beautiful language construct, although one +which is admittedly hard to give semantics to in some frameworks. +But not in K :) Here is the entire K syntax and semantics of callcc:

      +
      syntax Exp ::= "callcc" Exp  [strict]
+syntax Val ::= cc(K)
+rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+rule <k> cc(K) V ~> _ =>  V ~> K </k>
+
      +

      Let us first discuss the annotated syntax. We declared callcc strict, +because its argument may not necessarily be a function yet, so it may need +to be evaluated. As explained above, we need to encode the remaining +computation somehow and pass it to callcc's argument. More specifically, +since LAMBDA is call-by-value, we have to encode the remaining computation as +a value. We do not want to simply subsort computations to Val, because there +are computations which we do not want to be values. A simple solution to +achieve our goal here is to introduce a new value construct, say cc (from +current-continuation), which holds any computation.

      +

      Note that, inspired from SDF, +K allows you to define the syntax of helping semantic operations, like cc, +more compactly. Typically, we do not need a fancy syntax for such operators; +all we need is a name, followed by open parenthesis, followed by a +comma-separated list of arguments, followed by closed parenthesis. If this +is the syntax that you want for a particular construct, then K allows you to +drop all the quotes surrounding the terminals, as we did above for cc.

      +

      The semantic rules do exactly what the English semantics of callcc says. +Note that here, unlike in our definition of LAMBDA in Tutorial 1, we had +to mention the cell <k/> in our rules. This is because we need to make sure +that we match the entire remaining computation, not only a fragment of it! +For example, if we replace the two rules above with

      +
      rule (callcc V:Val => V cc(K)) ~> K
+rule cc(K) V ~> _ =>  V ~> K
+
      +

      then we get a callcc which is allowed to non-deterministically pick a +prefix of the remaining computation and pass it to its argument, and then +when invoked within its argument, a non-deterministic prefix of the new +computation is discarded and replaced by the saved one. Wow, that would +be quite a language! Would you like to write programs in it? :)

      +

      Consequently, in K we can abruptly change the execution control flow of a +program by simply changing the contents of the <k/> cell. This is one of +the advantages of having an explicit representation of the execution context, +like in K or in reduction semantics with evaluation contexts. Constructs like +callcc are very hard and non-elegant to define in frameworks such as SOS, +because those implicitly represent the execution context as proof context, +and the latter cannot be easily changed.

      +

      Now that we know how to handle cells in configurations and use them in rules, +in the next lesson we take a fresh look at LAMBDA and define it using +an environment-based style, which avoids the complexity of substitution +(e.g., having to deal with variable capture) and is closer in spirit to how +functional languages are implemented.

      +

      Go to Lesson 2, LAMBDA++: Semantic (Non-Syntactic) Computation Items.

      +

      MOVIE (out of date) [6'28"]

      +
      +
      + + +
      + + + +
      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_2/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_2/index.html new file mode 100644 index 00000000000..df3f27be15f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_2/index.html @@ -0,0 +1,535 @@ + + + + + + + + + + + + + + +Semantic (Non-Syntactic) Computation Items | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Semantic (Non-Syntactic) Computation Items

      +

      In this lesson we start another semantic definition of LAMBDA++, which +follows a style based on environments instead of substitution. In terms of +K, we will learn how easy it is to add new items to the syntactic category +of computations K, even ones which do not have a syntactic nature.

      +

      An environment binds variable names of interest to locations where their +values are stored. The idea of environment-based definitions is to maintain +a global store mapping locations to values, and then have environments +available when we evaluate expressions telling where the variables are +located in the store. Since LAMBDA++ is a relatively simple language, we +only need to maintain one global environment. Following a similar style +like in IMP, we place all cells into a top cell T:

      +
      configuration <T>
+                <k> $PGM:Exp </k>
+                <env> .Map </env>
+                <store> .Map </store>
+              </T>
+
      +

      Recall that $PGM is where the program is placed by krun after parsing. So +the program execution starts with an empty environment and an empty store.

      +

      In environment-based definitions of lambda-calculi, lambda abstractions +evaluate to so-called closures:

      +
      rule <k> lambda X:Id . E => closure(Rho,X,E) ...</k>
+     <env> Rho </env>
+
      +

      A closure is like a lambda abstraction, but it also holds the environment +in which it was declared. This way, when invoked, a closure knows where to +find in the store the values of all the variables that its body expression +refers to. We will define the lookup rule shortly.

      +

      Therefore, unlike in the substitution-based definitions of LAMBDA and +LAMBDA++, neither the lambda abstractions nor the identifiers are values +anymore here, because they both evaluate further: lambda abstractions to +closures and identifiers to their values in the store. In fact, the only +values at this moment are the closures, and they are purely semantic entities, +which cannot be used explicitly in programs. That's why we modified the +original syntax of the language to include no Val syntactic category +anymore, and that's why we need to add closures as values now; same like +before, we add a Val syntactic category which is subsorted +to KResult. In general, whenever you have any strictness attributes, +your should also define some K results.

      +

      Invoking a closure is a bit more involved than the substitution-based +beta-reduction: we need to switch to the closure's environment, then create a +new, or fresh, binding for the closure's parameter to the value passed to the +closure, then evaluate the closure's body, and then switch back to the +caller's environment, which needs to be stored somewhere in the meanwhile. +We can do all these with one rule:

      +
      rule <k> closure(Rho,X,E) V:Val => E ~> Rho' ...</k>
+     <env> Rho' => Rho[X <- !N] </env>
+     <store>... .Map => (!N:Int |-> V) ...</store>
+
      +

      Therefore, we atomically do all the following:

      +
        +
      • switch the computation to the closure's body, E, followed by a +caller-environment-recovery task Rho' (note that Rho' is the +current environment),
        • +
      • generate a fresh location !N (the ! is important, we discuss it below), +bind X to !N in closure's environment and switch the current environment +Rho' to that one,
        • +
      • write the value passed to the closure, V, at location !N.
        • +
      +

      This was the most complex K rule we've seen so far in the tutorial. Note, +however, that this one rule achieves a lot. It is, in fact, quite compact +considering how much it does. Note also that everything that this K rule +mentions is needed also conceptually in order to achieve this task, so it +is minimal from that point of view. That would not be the case if we +used, instead, a conventional rewrite rule, because we would have had to +mention the remaining store, say Sigma, in both sides of the rule, to say +it stays unchanged. Here we just use ....

      +

      The declaration of the fresh variable above, !N, is new and needs +some explanation. First, note that !N appears only in the right-hand-side +terms in the rule, that is, it is not matched when the rule is applied. +Instead, a fresh Nat element is generated each time the rule is applied. +In K, we can define syntactic categories which have the capability to +generate fresh elements like above, using unbound variables whose name starts +with a !. The details of how to do that are beyond the scope of this +tutorial (see Tutorial 6). All we need to know here is that an arbitrary +fresh element of that syntactic category is generated each time the rule +is applied. We cannot rely on the particular name or value of the generated +element, because that can change with the next version of the K tool, or +even from execution to execution with the same version. All you can rely +on is that each newly generated element is distinct from the previously +generated elements for the same syntactic category.

      +

      Unlike in the substitution-based definition, we now also need a lookup rule:

      +
      rule <k> X => V ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> V ...</store>
+
      +

      This rule speaks for itself: replace X by the value V located in the store +at X's location N in the current environment.

      +

      The only thing left to define is the auxiliary environment-recovery operation:

      +

      rule _:Val ~> (Rho => .) ... _ => Rho

      +

      When the item preceding the environment recovery task Rho in the +computation becomes a value, replace the current environment with Rho +and dissolve Rho from the computation.

      +

      Let us kompile and ... fail:

      +
      kompile lambda
+
      +

      gives a parsing error saying that V:Val does not fit there in the closure +invocation rule. That's because Val and Exp are currently completely +disconnected, so K rightfully complains that we want to apply a value to +another one, because application was defined to work with expressions, not +values. What we forgot here was to state that Exp includes Val:

      +
      syntax Exp ::= Val
+
      +

      Now everything works, but it is a good time to reflect a bit.

      +

      So we added closures, which are inherently semantic entities, to the syntax +of expressions. Does that mean that we can now write LAMBDA programs with +closures in them? Interestingly, with our current definition of LAMBDA, +which purposely did not follow the nice organization of IMP into syntax and +semantic modules, and with K's default parser, kast, you can. But you are +not supposed to speculate this! In fact, if you use an external parser, that +parser will reject programs with explicit closures. Also, if we split the +LAMBDA definition into two modules, one called LAMBDA-SYNTAX containing +exclusively the desired program syntax and one called LAMBDA importing the +former and defining the syntax of the auxiliary operations and the semantics, +then even K's default parser will reject programs using auxiliary syntactic +constructs.

      +

      Indeed, when you kompile a language, say lang.k, the tool will by default +attempt to find a module LANG-SYNTAX and generate the program parser from +that. If it cannot find it, then it will use the module LANG instead. There +are also ways to tell kompile precisely which syntax module you want to use +for the program parser if you don't like the default convention. +See kompile --help.

      +

      Another insightful thought to reflect upon, is the relationship between your +language's values and other syntactic categories. It is often the case that +values form a subset of the original language syntax, like in IMP (Part 2 of +the tutorial), but sometimes that is not true, like in our case here. When +that happens, in order for the semantics to be given smoothly and uniformly +using the original syntax, you need to extend your language's original +syntactic categories with the new values. The same holds true in other +semantic approaches, not only in K, even in ones which are considered purely +syntactic. As it should be clear by now, K does not enforce you to use a +purely syntactic style in your definitions; nevertheless, K does allow you to +develop purely syntactic definitions, like LAMBDA in Part 1 of the tutorial, +if you prefer those.

      +

      krun some programs, such as those provided in Lesson 1 of the LAMBDA +tutorial (Part 1). Note the closures, both as results in the <k/> cell, +and as values in the store. Also, since variables are not values anymore, +expressions that contain free variables may get stuck with one of those on +top of their computation. See, for example, free-variable-capture.lambda, +which gets stuck on z, because z is free, so it cannot evaluate it. +If you want, you can go ahead and manually provide a configuration with +z mapped to some location in the environment and that location mapped to +some value in the store, and then you can also execute this program. The +program omega.lambda should still loop.

      +

      Although we completely changed the definitional style of LAMBDA, the semantics +of the other constructs do not need to change, as seen in the next lesson.

      +

      Go to Lesson 3, LAMBDA++: Reusing Existing Semantics.

      +

      MOVIE (out of date) [8'02"]

      +
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      + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/NOTES/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/NOTES/index.html new file mode 100644 index 00000000000..14c7a76ca6b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/NOTES/index.html @@ -0,0 +1,393 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +
        +
      • Cut-and-paste is a poor-man's approach to reuse.
        • +
      +

      Indeed, it is. A better way to reuse, which requires a bit of planning ahead, +is to put each feature in its own module. Then you can simply include the +modules containing the features you want to reuse. Our point in this lesson +was that such reuse is possible, not to teach the best way to do it in +practice. Good methodologies on how to use a technology are equally important.

      +
        +
      • Do we need an env/store split? Couldn't we just work with a state?
        • +
      +

      Since in our language so far we never change the value of a variable, it +happens to be OK to only keep a state. That is, to collapse env/store into +state, then embed the state in closures and restore the state instead of the +environment. However, this simplistic approach breaks as soon as we add +references to our language, because functions can then modify the environment +in which they were declared, so we would have to carry over those changes when +returning from function invocations, which would be quite difficult.

      +
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      + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/index.html new file mode 100644 index 00000000000..8bcd08f5e2f --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_3/index.html @@ -0,0 +1,408 @@ + + + + + + + + + + + + + + +Reusing Existing Semantics | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Reusing Existing Semantics

      +

      In this lesson we will learn that, in some cases, we can reuse existing +semantics of language features without having to make any change!

      +

      Although the definitional style of the basic LAMBDA language changed quite +radically in our previous lesson, compared to its original definition in +Part 1 of the tutorial, we fortunately can reuse a large portion of the +previous definition. For example, let us just cut-and-paste the rest of the +definition from Lesson 7 in Part 1 of the tutorial.

      +

      Let us kompile and krun all the remaining programs from Part 1 of the +tutorial. Everything should work fine, although the store contains lots of +garbage. Garbage collection is an interesting topic, but we do not do it +here. Nevertheless, much of this garbage is caused by the intricate use of +the fixed-point combinator to define recursion. In a future lesson in this +tutorial we will see that a different, environment-based definition of +fixed-points will allocate much less memory.

      +

      One interesting question at this stage is: how do we know when we can reuse +an existing semantics of a language feature? Well, I'm afraid the answer is: +we don't. In the next lesson we will learn how reuse can fail for quite subtle +reasons, which are impossible to detect statically (and some non-experts may +fail to even detect them at all).

      +

      Go to Lesson 4, LAMBDA++: Do Not Reuse Blindly!.

      +

      MOVIE (out of date) [3'21"]

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/NOTES/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/NOTES/index.html new file mode 100644 index 00000000000..ea9854f5b3a --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/NOTES/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      With the current version of the tool (as of Sept 12, 2013), the +callcc-env1.lambda program actually evaluates to 4, as expected. +But the comments in the README are still valid, because it could just as +well evaluate to 3. For example, just replace ...+x with x+..., and it +should evaluate to 3 now.

      +

      Also, the first "fix" suggested in the READMEm to make "+" seqstrict, only +works for that particular program. It does not fix the problem if we change +the program as indicated above. In that case "+" it would need to be +seqstrict(2,1).

      +

      Also, callcc-env2.lambda evaluates to 3 instead of 4, because of the +particular order in which the strictness of the application operation is +applied. If you make application seqstrict(2,1) then you get 4.

      +

      Dec 06, 2014: Looks like we should discuss the --search and --transition +options before this lesson, and then kompile the definition with option +--transition = computational and krun it with --search.

      +

      The README.md says "One is to make + seqstrict in the semantics, to +enforce its evaluation from left-to-right. Do it and then run the program +above again;". Then it continues and says "The problem is now the +non-deterministic evaluation strategy of the function application construct". +Grigore will add this as an exercise, asking reader to fix this +non-determinism. Then ask them to propose another example where you still get +non-determinism; can they?

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/index.html new file mode 100644 index 00000000000..0fe48805234 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_4/index.html @@ -0,0 +1,469 @@ + + + + + + + + + + + + + + +Do Not Reuse Blindly! | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Do Not Reuse Blindly!

      +

      It may be tempting to base your decision to reuse an existing semantics of +a language feature solely on syntactic considerations; for example, to reuse +whenever the parser does not complain. As seen in this lesson, this could +be quite risky.

      +

      Let's try (and fail) to reuse the definition of callcc from Lesson 1:

      +
      syntax Exp ::= "callcc" Exp  [strict]
+syntax Val ::= cc(K)
+rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+rule <k> cc(K) V ~> _ =>  V ~> K </k>
+
      +

      The callcc examples that we tried in Lesson 1 work, so it may look it works.

      +

      However, the problem is that cc(K) should also include an environment, +and that environment should also be restored when cc(K) is invoked. +Let's try to illustrate this bug with callcc-env1.lambda

      +
      let x = 1 in
+  ((callcc lambda k . (let x = 2 in (k x))) + x)
+
      +

      where the second argument of +, x, should be bound to the top x, which +is 1. However, since callcc does not restore the environment, that x +should be looked up in the wrong, callcc-inner environment, so we should see +the overall result 4.

      +

      Hm, we get the right result, 3 ... (Note: you may get 4, depending on +your version of K and platform; but both 3 and 4 are possible results, as +explained below and seen in the tests). How can we get 3? Well, recall that ++ is strict, which means that it can evaluate its arguments in any order. +It just happened that in the execution that took place above its second +argument was evaluated first, to 1, and then the callcc was evaluated, but +its cc value K had already included the 1 instead of x ... In Part 4 of +the tutorial we will see how to explore all the non-deterministic behaviors of +a program; we could use that feature of K to debug semantics, too. +For example, in this case, we could search for all behaviors of this program +and we would indeed get two possible value results: 3 and 4.

      +

      One may think that the problem is the non-deterministic evaluation order +of +, and thus that all we need to do is to enforce a deterministic order +in which the arguments of + are evaluated. Let us follow this path to +see what happens. There are two simple ways to make the evaluation order +of +'s arguments deterministic. One is to make + seqstrict in the +semantics, to enforce its evaluation from left-to-right. Do it and then +run the program above again; you should get only one behavior for the +program above, 4, which therefore shows that copying-and-pasting our old +definition of callcc was incorrect. However, as seen shortly, that only +fixed the problem for the particular example above, but not in general. +Another conventional approach to enforce the desired evaluation order is to +modify the program to enforce the left-to-right evaluation order using let +binders, as we do in callcc-env2.lambda:

      +
      let x = 1 in
+  let a = callcc lambda k . (let x = 2 in (k x)) in
+    let b = x in
+      (a + b)
+
      +

      With your installation of K you may get the "expected" result 4 when you +execute this program, so it may look like our non-deterministic problem is +fixed. Unfortunately, it is not. Using the K tool to search for all the +behaviors in the program above reveals that the final result 3 is still +possible. Moreover, both the 3 and the 4 behaviors are possible regardless +of whether + is declared to be seqstrict or just strict. How is that +possible? The problem is now the non-deterministic evaluation strategy of +the function application construct. Indeed, recall that the semantics of +the let-in construct is defined by desugaring to lambda application:

      +
      rule let X = E in E' => (lambda X . E') E
+
      +

      With this, the program above eventually reduces to

      +
      (lambda a . ((lambda b . a + b) x))
+(callcc lambda k . (let x = 2 in (k x)))
+
      +

      in an environment where x is 1. If the first expression evaluates first, +then it does so to a closure in which x is bound to a location holding 1, +so when applied later on to the x inside the argument of callcc (which is +2), it will correctly lookup x in its enclosed environment and thus the +program will evaluate to 3. On the other hand, if the second expression +evaluates first, then the cc value will freeze the first expression as is, +breaking the relationship between its x and the current environment in which +it is bound to 1, being inadvertently captured by the environment of the +let-in construct inside the callcc and thus making the entire expression +evaluate to 4.

      +

      So the morale is: Do not reuse blindly. Think!

      +

      In the next lesson we fix the environment-based semantics of callcc by having +cc also wrap an environment, besides a computation. We will also give a more +direct semantics to recursion, based on environments instead of fixed-point +combinators.

      +

      Go to Lesson 5, LAMBDA++: More Semantic Computation Items.

      +

      MOVIE (out of date) [3'37"]

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_5/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_5/index.html new file mode 100644 index 00000000000..a2050411434 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_5/index.html @@ -0,0 +1,444 @@ + + + + + + + + + + + + + + +More Semantic Computation Items | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      More Semantic Computation Items

      +

      In this lesson we see more examples of semantic (i.e., non-syntactic) +computational items, and how useful they can be. Specifically, we fix the +environment-based definition of callcc and give an environment-based +definition of the mu construct for recursion.

      +

      Let us first fix callcc. As discussed in Lesson 4, the problem that we +noticed there was that we only recovered the computation, but not the +environment, when a value was passed to the current continuation. This is +quite easy to fix: we modify cc to take both an environment and a +computation, and its rules to take a snapshot of the current environment with +it, and to recover it at invocation time:

      +
      syntax Val ::= cc(Map,K)
+rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k> <env> Rho </env>
+rule <k> cc(Rho,K) V:Val ~> _ =>  V ~> K </k> <env> _ => Rho </env>
+
      +

      Let us kompile and make sure it works with the callcc-env2.lambda program, +which should evaluate to 3, not to 4.

      +

      Note that the cc value, which can be used as a computation item in the <k/> +cell, is now quite semantic in nature, pretty much the same as the closures.

      +

      Let us next add one more closure-like semantic computational item, for mu. +But before that, let us reuse the semantics of letrec in terms of mu that +was defined in Lesson 8 of Part 1 of the tutorial on LAMBDA:

      +
      syntax Exp ::= "letrec" Id Id "=" Exp "in" Exp [macro]
+             | "mu" Id "." Exp                 [latex(\mu{#1}.{#2})]
+rule letrec F:Id X = E in E' => let F = mu F . lambda X . E in E'
+
      +

      We removed the binder annotation of mu, because it is not necessary +anymore (since we do not work with substitutions anymore).

      +

      To save the number of locations needed to evaluate mu X . E, let us replace +it with a special closure which already binds X to a fresh location holding +the closure itself:

      +
      syntax Exp ::= muclosure(Map,Exp)
+
+rule <k> mu X . E => muclosure(Rho[X <- !N], E) ...</k>
+     <env> Rho </env>
+     <store>... .Map => (!N:Int |-> muclosure(Rho[X <- !N], E)) ...</store>
+
      +

      Since each time mu X . E is encountered during the evaluation it needs to +evaluate E, we conclude that muclosure cannot be a value. We can declare +it as either an expression or as a computation. Let's go with the former.

      +

      Finally, here is the rule unrolling the muclosure:

      +

      rule muclosure(Rho,E) => E ~> Rho' ... + Rho' => Rho

      +

      Note that the current environment Rho' needs to be saved before and +restored after E is executed, because the fixed point may be invoked +from a context with a completely different environment from the one +in which mu X . E was declared.

      +

      We are done. Let us now kompile and krun factorial-letrec.lambda from +Lesson 7 in Part 1 of the tutorial on LAMBDA. Recall that in the previous +lesson this program generated a lot of garbage into the store, due to the +need to allocate space for the arguments of all those lambda abstractions +needed to run the fixed-point combinator. Now we need much fewer locations, +essentially only locations for the argument of the factorial function, one at +each recursive call. Anyway, much better than before.

      +

      In the next lesson we wrap up the environment definition of LAMBDA++ and +generate its documentation.

      +

      Go to Lesson 6, LAMBDA++: Wrapping Up and Documenting LAMBDA++.

      +

      MOVIE (out of date) [5'19"]

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      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Maybe we should change the name of calCC, as it is not a good idea to have +two constructs with different semantics but names which cannot be distinguished +easily.

      +
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      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/callCC/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/callCC/index.html new file mode 100644 index 00000000000..34615fcc8cc --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/callCC/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      (see similar exercise in Lesson 1, with substitution instead of environments)

      +

      Define a variant of callcc, say callCC, which never returns to the +current context unless a value is specifically passed to its argument +continuation. Follow an environment-based style.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-call-CC-to-callcc/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-call-CC-to-callcc/index.html new file mode 100644 index 00000000000..25615a52aa8 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-call-CC-to-callcc/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define callcc in terms of callCC, where callCC is explained in the +callCC exercise under LAMBDA++, Lesson 1. Follow an environment-based style.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-callcc-to-call-CC/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-callcc-to-call-CC/index.html new file mode 100644 index 00000000000..d9f81dc2f6d --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/exercises/from-callcc-to-call-CC/index.html @@ -0,0 +1,378 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Define callCC in terms of callcc, where callCC is explained in the +callCC exercise under LAMBDA++, Lesson 1. Follow an environment-based +style.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/index.html new file mode 100644 index 00000000000..c1db1d4838d --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/index.html @@ -0,0 +1,394 @@ + + + + + + + + + + + + + + +Wrapping Up and Documenting LAMBDA++ | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Wrapping Up and Documenting LAMBDA++

      +

      In this lesson we wrap up and nicely document LAMBDA++. In doing so, we also +take the freedom to reorganize the semantics a bit, to make it look better.

      +

      See the lambda.k file, which is self-explanatory.

      +

      Part 3 of the tutorial is now complete. Part 4 will teach you more features +of the K framework, in particular how to exhaustively explore the behaviors +of non-deterministic or concurrent programs.

      +

      MOVIE (out of date) [6'23"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/lambda/index.html b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/lambda/index.html new file mode 100644 index 00000000000..4bed87f99a3 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/3_lambda++/lesson_6/lambda/index.html @@ -0,0 +1,542 @@ + + + + + + + + + + + + + + +Tutorial 3--- LAMBDA++ | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Tutorial 3--- LAMBDA++

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Abstract

      +

      This file contains an environment-based K semantic definition of LAMBDA++, an +extension of the LAMBDA language (defined in Tutorial 1) with a callcc +construct. The objective here is to further disseminate some of the features +of the K framework, in particular to illustrate how popular environment-based +and closure-based semantics can be defined in K.

      +

      For notational/kompilation/krun simplicity and to avoid OS errors, we continue +to write LAMBDA and lambda as names for modules and program extensions, +respectively, in the sequel.

      +

      To restrict the default program parser invoked by krun, namely kast, +to only parse proper LAMBDA++ programs no matter what other syntactic +constructs we add to Exp later on in the semantics, we put the actual program +syntax in a module with the suffix -SYNTAX. This issue was discussed in more +detail in Lesson 2 of this tutorial. In short, the parser generated by kompile +to be used by kast will be by default built only based on the syntax in this +module. Type kompile --help to see how to tell the parser which syntax to use.

      +
      k
      module LAMBDA-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      We move all the LAMBDA++ syntax here.

      +
      k
        syntax Val ::= Int | Bool
+  syntax Exp ::= Val
+// Basic lambda-calculus syntax
+               | Id
+               | "lambda" Id "." Exp  [latex(\lambda{#1}.{#2})]
+               | Exp Exp              [strict, left]
+               | "(" Exp ")"          [bracket]
+// Arithmetic
+               > "-" Int
+               | Exp "*" Exp          [strict, left]
+               | Exp "/" Exp          [strict]
+               > Exp "+" Exp          [strict, left]
+               > Exp "<=" Exp         [strict]
+// Other functional constructs
+  syntax Exp ::= "if" Exp "then" Exp "else" Exp  [strict(1)] // Conditional
+               | "let" Id "=" Exp "in" Exp [macro]           // Let binder
+               | "letrec" Id Id "=" Exp "in" Exp [macro]     // Letrec
+               | "mu" Id "." Exp      [latex(\mu{#1}.{#2})]  // Mu
+               | "callcc" Exp  [strict]                      // Callcc
+
      +

      One thing you may want to do, now that the entire syntax is in one +place, is to play with precedences. This way, you can make kompile +generate the parser you want for your programs, so that you won't have to +put lots of parentheses in your programs.

      +
      k
      endmodule
+
+
+module LAMBDA
+  imports LAMBDA-SYNTAX
+  imports DOMAINS
+
      +

      Semantics

      +

      The next module contains the semantics of all the LAMBDA++ constructs, +in the order in which their syntax was declared above.

      +

      The K Results

      +

      We should not forget to define the results of our computations. +Here is a rule of thumb: whenever you have any strictness attributes, your +should also define some K results. Or even simpler: always define your +results! (unless you define a theoretical semantics, for analysis but not +for execution purposes, you will need to define your results)

      +
      k
        syntax KResult ::= Val
+
      +

      Configuration

      +

      Since LAMBDA++ is such a simple language, its configuration is minimal +for an environment-based semantics: it only contains the k cell, +an environment cell, and a store cell. An environment binds variable names +to locations, and a store binds locations to values.

      +
      k
        configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <env color="blue"> .Map </env>
+                  <store color="red"> .Map </store>
+                </T>
+
      +

      Recall that $PGM is where the program is placed by krun after parsing.

      +

      Closures

      +

      In environment-based definitions of lambda-calculi, λ-abstractions +evaluate to closures. A closure is like a λ-abstraction, +but it also holds the environment in which it was declared. This way, when +invoked, a closure knows where to find in the store the values of all the +variables that its body expression refers to. +To invoke a closure, we need to switch to closure's environment, then create +a new binding for closure's parameter, then evaluate the closure's body, and +then switch back to caller's environment.

      +
      k
        syntax Val ::= closure(Map,Id,Exp)
+    [latex(\textsf{closure}_\lambda({#1},{#2},{#3}))]
+
+  rule <k> lambda X:Id . E => closure(Rho,X,E) ...</k>
+       <env> Rho </env>
+  rule <k> closure(Rho,X,E) V:Val => E ~> Rho' ...</k>
+       <env> Rho' => Rho[X <- !N] </env>
+       <store>... .Map => (!N:Int |-> V) ...</store>
+  rule <k> X => V ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> V ...</store>
+
      +

      Environment Recovery

      +

      The environment-recovery computation item defined below is useful in many +semantics, like it was above. It is so useful, that there are discussions +in the K team to add it to the set of pre-defined K features.

      +
      k
        rule <k> _:Val ~> (Rho => .) ...</k> <env> _ => Rho </env>
+
      +

      Arithmetic Constructs

      +

      Not much to say here. They have exactly the same semantics as in LAMBDA and +IMP. Note that we let it in programmer's hands to check that the denominator +of a division is different from zero. If a division-by-zero is issued, then +completely non-deterministic result can happen depending upon what back-end +one uses for the K tool. Currently, Maude is used and Maude gets stuck +with a term of the form I /Int 0, but one should not rely on that. +If you want to catch division-by-zero in the semantics, instead of letting +the back-end do whatever it wants, you should add a side condition to the +division rule.

      +
      k
        rule - I => 0 -Int I
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2
+  rule I1 + I2 => I1 +Int I2
+  rule I1 <= I2 => I1 <=Int I2
+
      +

      Conditional

      +
      k
        rule if true  then E else _ => E
+  rule if false then _ else E => E
+
      +

      Let Binder

      +
      k
        rule let X = E in E':Exp => (lambda X . E') E
+
      +

      Letrec Binder

      +

      We define letrec in term of mu, whose semantics is below.

      +
      k
        rule letrec F:Id X = E in E' => let F = mu F . lambda X . E in E'
+
      +

      Mu

      +

      To save the number of locations needed to evaluate μ X . E, we replace it +with a special closure which binds X to a fresh location holding the closure +itself. This has the same effect as binding X to a reference that points +back to the fixed-point.

      +
      k
        syntax Exp ::= muclosure(Map,Exp)
+    [latex(\textsf{closure}_\mu({#1},{#2}))]
+  rule <k> mu X . E => muclosure(Rho[X <- !N], E) ...</k>
+       <env> Rho </env>
+       <store>... .Map => (!N:Int |-> muclosure(Rho[X <- !N], E)) ...</store>
+  rule <k> muclosure(Rho,E) => E ~> Rho' ...</k>
+       <env> Rho' => Rho </env>
+
      +

      Callcc

      +

      For callcc, we need to create a new closure-like value which +wraps both the remaining computation, and the environment in which it is +supposed to be executed. Forget the environment, and you get a wrong +callcc.

      +
      k
        syntax Val ::= cc(Map,K)
+    [latex(\textsf{closure}_{\texttt{callcc}}({#1},{#2}))]
+  rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k> <env> Rho </env>
+  rule <k> cc(Rho,K) V:Val ~> _ =>  V ~> K </k> <env> _ => Rho </env>
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/index.html new file mode 100644 index 00000000000..10e631d6b9b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/index.html @@ -0,0 +1,408 @@ + + + + + + + + + + + + + + +Part 4: Defining IMP++ | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Part 4: Defining IMP++

      +

      IMP++ extends IMP, which was discussed in Part 2 of this tutorial, with several +new syntactic constructs. Also, some existing syntax is generalized, which +requires non-modular changes of the existing IMP semantics. For example, +global variable declarations become local declarations and can occur +anywhere a statement can occur. In this tutorial we will learn the following:

      +
        +
      • That (and how) existing syntax/semantics may change as a language evolves.
        • +
      • How to refine configurations as a language evolves.
        • +
      • How to define and use fresh elements of desired sorts.
        • +
      • How to tag syntactic constructs and rules, and how to use such tags +with the superheat/supercool/transition options of kompile.
        • +
      • How the search option of krun works.
        • +
      • How to stream cells holding semantic lists to the standard input/output, +and thus obtain interactive interpreters for the defined languages.
        • +
      • How to delete, save and restore cell contents.
        • +
      • How to add/delete cells dynamically.
        • +
      • More details on how the configuration abstraction mechanism works.
        • +
      +

      Like in the previous tutorials, this folder contains several lessons, each +adding new features to IMP++. Do them in order and make sure you completed +and understood the previous tutorials.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/NOTES/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/NOTES/index.html new file mode 100644 index 00000000000..2d40d7dbd25 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/NOTES/index.html @@ -0,0 +1,381 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Add an exercise somewhere with a print which first evaluates all its arguments +and THEN prints them. The idea is to define print to be strict and to +make the AExps list construct seqstrict, so lists of arithmetic +expressions get evaluated from left-to-right whenever they reach the top of +the <k/> cell (replace seqstrict with strict if you want expressions in +a list to evaluate non-deterministically and interleaved).

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/index.html new file mode 100644 index 00000000000..0569e181dfe --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_1/index.html @@ -0,0 +1,465 @@ + + + + + + + + + + + + + + +Extending/Changing an Existing Language Syntax | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Extending/Changing an Existing Language Syntax

      +

      Here we learn how to extend the syntax of an existing language, both with +new syntactic constructs and with more general uses of existing constructs. +The latter, in particular, requires changes of the existing semantics.

      +

      Consider the IMP language, as defined in Lesson 4 of Part 2 of the tutorial.

      +

      Let us first add the new syntactic constructs, with their precedences:

      +
        +
      • variable increment, ++, which increments an integer variable and +evaluates to the new value;
        • +
      • read, which reads and evaluates to a new integer from the input buffer;
        • +
      • print, which takes a comma-separated list of arithmetic expressions and +evaluates and prints each of them in order, from left to right, to the +output buffer; we therefore define a new list syntactic category, AExps, +which we pass as an argument to print; note we do not want to declare +print to be strict, because we do not want to first evaluate the +arguments and then print them (for example, if the second argument performs +an illegal operation, say division by zero, we still want to print the first +argument); we also go ahead and add strings as arithmetic expressions, +because we intend print to also take strings, in order to print nice +messages to the user;
        • +
      • halt, which abruptly terminates the program; and
        • +
      • spawn, which takes a statement and creates a new concurrent thread +executing it and sharing its environment with the parent thread.
        • +
      +

      Also, we want to allow local variable declarations, which can appear anywhere +a statement can appear. Their scope ranges from the place they are defined +until the end of the current block, and they can shadow previous declarations, +both inside and outside the current block. The simplest way to define the +syntax of the new variable declarations is as ordinary statements, at the same +time removing the previous Pgm syntactic category and its construct. +Programs are now just statements.

      +

      We are now done with adding the new syntax and modifying the old one. +Note that the old syntax was modified in a way which makes the previous IMP +programs still parse, but this time as statements. Let us then modify +the configuration variable $PGM to have the sort Stmt instead of Pgm, +and let us try to run the old IMP programs, for example sum.imp.

      +

      Note that they actually get stuck with the global declaration on the top +of their computations. This is because variable declarations are now treated +like any statements, in particular, the sequential composition rule applies. +This makes the old IMP rule for global variable declarations not match anymore. +We can easily fix it by replacing the anonymous variable _, which matched +the program's statement that now turned into the remaining computation in +the <k/> cell, with the cell frame variable ..., which matches the +remaining computation. Similarly, we have to change the rule for the case +where there are no variables left to declare into one that dissolves itself.

      +

      We can now run all the previous IMP programs, in spite of the fact that +our IMP++ semantics is incomplete and, more interestingly, in spite of the +fact that our current semantics of blocks is incorrect in what regards the +semantics of local variable declarations (note that the old IMP programs do +not declare block-local variables, which is why they still run correctly).

      +

      Let us also write some proper IMP++ programs, which we would like to execute +once we give semantics to the new constructs.

      +

      div.imp is a program manifesting non-deterministic behaviors due to the +desired non-deterministic evaluation strategy of division and the fact that +expressions will have side effects once we add variable increment. We will +be able to see all the different behaviors of this program. Challenge: can +you identify the behavior where the program performs a division-by-zero?

      +

      If we run div.imp now, it will get stuck with the variable increment +construct on top of the computation cell. Once we give it a semantics, +div.imp will execute completely (all the other constructs in div.imp +already have their semantics defined as part of IMP).

      +

      Note that some people prefer to define all their semantics in a by need +style, that is, they first write and parse lots of programs, and then they +add semantics to each language construct on which any of the programs gets +stuck, and so on and so forth until they can run all the programs.

      +

      io.imp is a program which exercises the input/output capabilities of the +language: reads two integers and prints three strings and an integer. +Note that the variable declaration is not the first statement anymore.

      +

      sum-io.imp is an interactive variant of the sum program.

      +

      spawn.imp is a program which dynamically creates two threads that interact +with the main thread via the shared variable x. Lots of behaviors will be +seen here once we give spawn the right semantics.

      +

      Finally, locals.imp tests whether variable shadowing/unshadowing works well.

      +

      In the next lesson we will prepare the configuration for the new constructs, +and will see what it takes to adapt the semantics to the new configuration. +Specifically, we will split the state cell into an environment cell and a +store cell, like in LAMBDA++ in Part 3 of the tutorial.

      +

      Go to Lesson 2, IMP++: Configuration Refinement; Freshness.

      +

      MOVIE (out of date) [07'47"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/NOTES/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/NOTES/index.html new file mode 100644 index 00000000000..13564c72ac8 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/NOTES/index.html @@ -0,0 +1,376 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      update discussion on fresh; it has already been explained in lambda++

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/index.html new file mode 100644 index 00000000000..45498ed4262 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_2/index.html @@ -0,0 +1,447 @@ + + + + + + + + + + + + + + +Configuration Refinement; Freshness | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Configuration Refinement; Freshness

      +

      To prepare for the semantics of threads and local variables, in this lesson we +split the state cell into an environment and a store. The environment and +the store will be similar to those in the definition of LAMBDA++ in Part +3 of the Tutorial. This configuration refinement will require us to change +some of IMP's rules, namely those that used the state.

      +

      To split the state map, which binds program variables to values, into an +environment mapping program variables to locations and a store mapping +locations to values, we replace in the configuration declaration the cell

      +
      <state color="red"> .Map </state>
+
      +

      with two cells

      +
      <env color="LightSkyBlue"> .Map </env>
+<store color="red"> .Map </store>
+
      +

      Structurally speaking, this split of a cell into other cells is a major +semantic change, which, unfortunately, requires us to revisit the existing +rules that used the state cell. One could, of course, argue that we could +have avoided this problem if we had followed from the very beginning the +good-practice style to work with an environment and a store, instead of a +monolithic state. While that is a valid argument, highlighting the fact that +modularity is not only a feature of the framework alone, but one should also +follow good practices to achieve it, it is also true that if all we wanted +in Part 2 of the tutorial was to define IMP as is, then the split of the state +in an environment and a store is unnecessary and not really justified.

      +

      The first rule which used a state cell is the lookup rule:

      +
      rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
+
      +

      We modify it as follows:

      +
      rule <k> X:Id => I ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> I ...</store>
+
      +

      So we first match the location N of X in the environment, then the value +I at location N in the store, and finally we rewrite X to I into the +computation. This rule also shows an instance of a more complex +multiset matching, where two variables (X and N) are matched each twice.

      +

      The assignment rule is modified quite similarly.

      +

      The variable declaration rule is trickier, though, because we need to allocate +a fresh location in the store and bind the newly declared variable to it. +This is quite similar to the way we allocated space for variables in +the environment-based definition of LAMBDA++ in Part 3 of the tutorial.

      +
      rule <k> int (X,Xs => Xs); ...</k>
+     <env> Rho => Rho[X <- !N:Int] </env>
+     <store>... .Map => !N |-> 0 ...</store>
+
      +

      Note the use of the fresh (!N) variable notation above. Recall from +the LAMBDA++ tutorial that each time the rule with fresh (!) variables is +applied, fresh elements of corresponding sorts are generated for the fresh +variables, distinct from all the previously generated elements; also, we +cannot and should not assume anything about the particular element that is +being generated, except that it is different from the previous ones.

      +

      kompile and krun sum.imp to see how the fresh locations have been +generated and used. There were two fresh locations needed, for the two +variables. Note also that a cell holding the counter has been added to the +configuration.

      +

      In the next lesson we will add the semantics of variable increment, and see +how that yields non-deterministic behaviors in programs and how to explore +those behaviors using the K tool.

      +

      Go to Lesson 3, IMP++: Tagging; Superheat/Supercool Kompilation Options.

      +

      MOVIE (out of date) [04'06"]

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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/NOTES/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/NOTES/index.html new file mode 100644 index 00000000000..998b5516065 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/NOTES/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      We eliminated the superheat/supercool optimization. Now we only need to use +the transition option. So the video is out of synch now.

      +
      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/index.html new file mode 100644 index 00000000000..128e91a0b64 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_3/index.html @@ -0,0 +1,492 @@ + + + + + + + + + + + + + + +Tagging; Transition Kompilation Option | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
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      + + +
      + +
      +

      Tagging; Transition Kompilation Option

      +

      In this lesson we add the semantics of variable increment. In doing so, we +learn how to tag syntactic constructs and rules and then use such tags to +instruct the kompile tool to generate the desired language model that is +amenable for exhaustive analysis.

      +

      The variable increment rule is self-explanatory:

      +
      rule <k> ++X => I +Int 1 ...</k>
+     <env>... X |-> N ...</env>
+     <store>... N |-> (I => I +Int 1) ...</store>
+
      +

      We can now run programs like our div.imp program introduced in Lesson 1. +Do it.

      +

      The addition of increment makes the evaluation of expressions have side +effects. That, in combination with the non-determinism allowed by the +strictness attributes in how expression constructs evaluate their +arguments, makes expressions in particular and programs in general have +non-deterministic behaviors. One possible execution of the div.imp program +assigns 1 to y's location, for example, but this program manifests several +other behaviors, too.

      +

      To see all the (final-state) behaviors that a program can have, you can call +the krun tool with the option --search. For example:

      +
      krun div.imp --search
+
      +

      Oops, we see only one solution, the same as when we ran it without search.

      +

      Here is what happens. krun can only explore as much of the transition +system associated to a program as kompile allowed the generated language +model to yield. Since most of the K users are interested in language models +that execute efficiently, that is, in faster interpreters for the defined +languages, by default kompile optimizes the generated language model for +execution. In particular, it inserts no backtracking markers, which krun +uses when called with the --search option in order to systematically generate +the entire transition system associated to a program. This is why krun +showed us only one solution when run with the --search option on div.imp.

      +

      We next explain how to tell kompile what kind of language model we are +interested in for analysis purposes. When you experiment with non-determinism +in a language semantics, you should keep it in mind that the --transition +option of kompile allows you to configure what counts as a transition in +your language model. We here only discuss transitions due to the +non-deterministic evaluation strategies of language constructs, but we will +see in future lectures (see Lesson 6 of IMP++, where we add concurrency) that +we can also have transitions due to non-deterministic applications of rewrite +rules.

      +

      If you want to explore the entire behavior space due to non-deterministic +evaluation strategies, then you should include all the language constructs +in the --transition option. This may sound like the obvious thing to +always do, but as soon as you do it you soon realize that it is way too much +in practice when you deal with large languages or programs. There are simply +too many program behaviors to consider, and krun will likely hang +on you or crush. For example, a small ten-statement program where each +statement uses one strict expression construct already has 1000+ behaviors for +krun to explore! Driven by practical needs of its users, the K tool +therefore allows you to finely tune the generated language models using the +--transition option.

      +

      To state which constructs are to be considered to generate transitions in the +generated language model, and for other reasons, too, the K tool allows you to +tag any production and any rule by putting the tag in brackets. You can associate +multiple tags to the same construct or rule, and more than one construct or +rule can have the same tag. As an example, let us tag the division construct +with division, the lookup rule with lookup and the increment rule with +increment. The tags of the rules are not needed in this lesson, we do it only +to demonstrate that rules can also be tagged.

      +

      The least intrusive way to enforce our current language to explore the +entire space of behaviors due to the strictness of division is to kompile it +with the following option:

      +
      kompile imp.k --transition "division"
+
      +

      It is interesting to note that the lookup and increment rules are the only +two rules which can trigger non-deterministic behaviors for division, because +no other rule but these two can ever apply while a division operation is +heated. Previous versions of K allowed you to also specify which rules could +trigger non-deterministic behaviors of operator evaluation strategies, +but that option was rarely used and is not available anymore.

      +

      Note that it is highly non-trivial to say precisely whether a strict language +construct may yield non-deterministic behaviors. For example, division's +strictness would yield no non-determinism if the language had no side effects. +It is even harder to say so for a particular program. Consequently, our K +implementation makes no attempt to automatically detect which operations +should be tagged as transitions. Instead, it provides the functionality to +let you decide it.

      +

      Now the command

      +
      krun div.imp --search
+
      +

      shows us all five behaviors of this program. Interestingly, one +of the five behaviors yields a division by zero!

      +

      The --transition option can be quite useful when you experiment with your +language designs or when you formally analyze programs for certain kinds of +errors. Please let us know if you ever need more finer-grained control over +the non-determinism of your language models.

      +

      Before we conclude this lesson, we'd like to let you know one trick, which +you will hopefully not overuse: you can tag elements in your K definition with +kompile option names, and those elements will be automatically included in +their corresponding options. For example, if you tag the division production +with transition then the command

      +
      kompile imp
+
      +

      is completely equivalent to the previous kompile command.

      +

      Please use this default behavior with caution, or even better, try to avoid +using it! You may be tempted to add the transition tag to lots of elements +and then forget about them; your language models will then be increasingly slower +when you execute them and you may wonder why ... This convention is typically +convenient when you want to quickly experiment with non-determinism and do not +want to bother inventing tag names and calling kompile with options.

      +

      In the next lesson we add input/output to our language and learn how to +generate a model of it which behaves like an interactive interpreter!

      +

      Go to Lesson 4, IMP++: Semantic Lists; Input/Output Streaming.

      +

      MOVIE (out of date) [06'56"]

      +
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      + + + +
      +
      +
      + + +
      + +
      +

      Make sure cells have the same indentation, which should use normal +spaces, not tabs. Tabs look differently in different editors.

      +

      The tests here include all the imp and imp++ programs, but of course +the imp ones do not display any output, so their .out files are empty. +But this way we at least make sure we test that these programs +do not fail/crash and that nothing is output, so it is better that what +we used to have in K3.6.

      +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_4/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_4/index.html new file mode 100644 index 00000000000..0d342321a7c --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_4/index.html @@ -0,0 +1,484 @@ + + + + + + + + + + + + + + +Semantic Lists; Input/Output Streaming | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
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      +

      Semantic Lists; Input/Output Streaming

      +

      In this lesson we add semantics to the read and print IMP++ constructs. +In doing so, we also learn how to use semantic lists and how to connect +cells holding semantic lists to the standard input and standard output. +This allows us to turn the K semantics into an interactive interpreter.

      +

      We start by adding two new cells to the configuration,

      +
      <in color="magenta"> .List </in>
+<out color="Orchid"> .List </out>
+
      +

      each holding a semantic list, initially empty. Semantic lists are +space-separated sequences of items, each item being a term of the form +ListItem(t), where t is a term of sort K. Recall that the semantic maps, +which we use for states, environments, stores, etc., are sets of pairs +t1 |-> t2, where t1 and t2 are terms of sort K. The ListItem wrapper +is currently needed, to avoid parsing ambiguities.

      +

      Since we want the print statement to also print strings, we need to tell +K that strings are results. To make it more interesting, let us also overload +the + symbol on arithmetic expressions to also take strings and, as a +result, to concatenate them. Since + is already strict, we only need to add +a rule reducing the IMP addition of strings to the builtin operation +String +which concatenates two strings.

      +

      The semantics of read is immediate: read and consumes the first integer item +from the <in/> cell; note that our read only reads integer values (it gets +stuck if the first item in the <in/> cell is not an integer).

      +

      The semantics of print is a bit trickier. Recall that print takes an +arbitrary number of arithmetic expression arguments, and evaluates and outputs +each of them in order, from left to right. For example, +print("Hello", 3/0, "Bye"); outputs "Hello" and then gets stuck on the +illegal division by zero operation. In other words, we do not want it to +first evaluate all its arguments and then print them, because that would miss +outputting potentially valuable information. So the first step is to evaluate +the first argument of print. In some sense, what we'd like to say is that +print has the evaluation strategy strict(1). However, strictness +attributes only work with individual language constructs, while what we need +is an evaluation strategy that involves two constructs: print and the list +(comma) construct of AExps. If we naively associate print the strict(1) +evaluation strategy then its first and unique argument, an AExps list, will +be scheduled for evaluation and the execution will get stuck because we have +no rules for evaluating AExps terms. If we make the list construct of +AExps strict then we get the wrong semantics for print which first +evaluates all its arguments and then outputs them. The correct way to +tell K that print should evaluate only its first argument is by using a +context declaration:

      +
      context print(HOLE:AExp, _);
+
      +

      Note the HOLE of sort AExp above. Contexts allow us to define finer-grain +evaluation strategies than the strictness attributes, involving potentially +more than one language construct, like above. The HOLE indicates the +argument which is requested to be evaluated. For example, the strict +attribute of division corresponds to two contexts:

      +
      context HOLE / _
+context _ / HOLE
+
      +

      In their full generality, contexts can be any terms with precisely one +occurrence of a HOLE, and with arbitrary side conditions on any variables +occurring in the context term as well as on the HOLE. See Part 6 of the +tutorial for more examples.

      +

      Once evaluated, the first argument of print is expected to become either an +integer or a string. Since we want to print both integers and string values, +to avoid writing two rules, one for each type of value, we instead add a new +syntactic category, Printable, which is the union of integers and strings.

      +

      Let us kompile and krun the io.imp program discussed in Lesson 1. As +expected, it gets stuck with a read construct on top of the computation and +with an empty <in/> cell. To run it, we need to provide some items in the +<in/> cell, so that the rule of read can match. Let us add

      +
      <in> ListItem(3) ListItem(5) ListItem(7) </in>
+
      +

      Now, if we krun io.imp, we can see that its execution completes normally +(the <k/> cell is empty), that the first two items have been removed by the +two read constructs from the <in/> cell, and that the desired strings and +numbers have been placed into the <out/> cell.

      +

      Cells holding semantic lists can be connected to the standard input and +standard output buffers, and krun knows how to handle these appropriately. +Let us connect the <in/> cell to the standard input using the cell attribute +stream="stdin" and the <out/> cell to the standard output with the +attribute stream="sdtout". A cell connected to the standard input will +take its items from the standard input and block the rewriting process when +an input is needed until an item is available in the standard input buffer. +A cell connected to the standard output buffer will send all its items, in +order, to the standard output.

      +

      Let us kompile and krun io.imp again. It prints the message and then +waits for your input numbers. Type in two numbers, then press <Enter>. +A message with their sum is then printed, followed by the final configuration. +If you do not want to see the final configuration, and thus obtain a realistic +interpreter for our language, then call krun with the option --output none:

      +
      krun io.imp --output none
+
      +

      Let us now krun our interactive sum program, which continuously reads numbers +from the console and prints the sum of numbers up to them:

      +
      krun sum-io.imp
+
      +

      Try a few numbers, then 0. Note that the program terminated, but with junk +in the <k/> cell, essentially with a halt statement on its top. Of course, +because halt has been reached and it has no semantics yet.

      +

      In the next lesson we give the semantics of halt and also fix the semantics +of blocks with local variable declarations.

      +

      Go to Lesson 5, IMP++: Deleting, Saving and Restoring Cell Contents.

      +

      MOVIE (out of date) [05'21"]

      +
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      Deleting, Saving and Restoring Cell Contents

      +

      In this lesson we will see how easily we can delete, save and/or restore +contents of cells in order to achieve the desired semantics of language +constructs that involve abrupt changes of control or environments. We have +seen similar or related K features in the LAMBDA++ language in Part 3 of the +tutorial.

      +

      Let us start by adding semantics to the halt statement. As its name says, +what we want is to abruptly terminate the execution of the program. Moreover, +we want the program configuration to look as if the program terminated +normally, with an empty computation cell. The simplest way to achieve that is +to simply empty the computation cell when halt is encountered:

      +
      rule <k> halt; ~> _ => . </k>
+
      +

      It is important to mention the entire <k/> cell here, with both its membranes +closed, to make sure that its entire contents is discarded. Note the +anonymous variable, which matches the rest of the computation.

      +

      kompile and krun sum-io.imp. Note that unlike in Lesson 4, the program +terminates with an empty computation cell now.

      +

      As mentioned earlier, the semantics of blocks that was inherited from IMP is +wrong. Program locals.imp shows it very clearly: the environments are not +correctly restored at block exits. One way to fix the problem is to take +a snapshot of the current environment when a block is entered and save it +somewhere, and then to restore it when the block is left. There are many +ways to do this, which you can explore on your own: for example you can add +a new list cell for this task where to push/pop the environment snapshots in +a stack style; or you can use the existing environment cell for this purpose, +but then you need to change the variable access rules to search through the +stacked environments for the variable.

      +

      My preferred solution is to follow a style similar to how we saved/restored +LAMBDA++ environments in Part 3 of the Tutorial, namely to use the already +existing <k/> cell for such operations. More specifically, we place a +reminder item in the computation whenever we need to take a snapshot of +some cell contents; the item simply consists of the entire contents of the cell. +Then, when the reminder item is reached, we restore the contents of the cell:

      +
      rule <k> {S} => S ~> Rho ...</k> <env> Rho </env>
+
      +

      The only thing left now is to give the definition of environment restore:

      +
      rule <k> Rho => . ...</k> <env> _ => Rho </env>
+
      +

      Done. kompile and krun locals.imp. Everything should work correctly now. +Note that the rule above is different from the one we had for LAMBDA++ in +Part 3 of the tutorial, in that here there is no value preceding the environment +restoration item in the computation; that's because IMP++ statements, +unlike LAMBDA++'s expressions, evaluate to nothing (.).

      +

      In the next lesson we will give semantics to the spawn S construct, which +dynamically creates a concurrent shared-memory thread executing statement S.

      +

      Go to Lesson 6, IMP++: Adding/Deleting Cells Dynamically; Configuration Abstraction, Part 2.

      +

      MOVIE (out of date) [04'30"]

      +
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      +

      Adding/Deleting Cells Dynamically; Configuration Abstraction, Part 2

      +

      In this lesson we add dynamic thread creation and termination to IMP, and +while doing so we learn how to define and use configurations whose structure +can evolve dynamically.

      +

      Recall that the intended semantics of spawn S is to spawn a new concurrent +thread that executes S. The new thread is being passed at creation time +its parent's environment, so it can share with its parent the memory +locations that its parent had access to at creation time. No other locations +can be shared, and no other memory sharing mechanism is available. +The parent and the child threads can evolve unrestricted, in particular they +can change their environments by declaring new variables or shadowing existing +ones, can create other threads, and so on.

      +

      The above suggests that each thread should have its own computation and its +own environment. This can be elegantly achieved if we group the <k/> and +<env/> cells in a <thread/> cell in the configuration. Since at any given +moment during the execution of a program there could be zero, one or more +instances of such a <thread/> cell in the configuration, it is a good idea +to declare the <thread/> cell with multiplicity * (i.e., zero, one or more):

      +
      <thread multiplicity="*" color="blue">
+  <k color="green"> $PGM:Stmt </k>
+  <env color="LightSkyBlue"> .Map </env>
+</thread>
+
      +

      This multiplicity declaration is not necessary, but it is a good idea to do +it for several reasons:

      +
        +
      1. it may help the configuration abstraction process, +which may in turn significantly increase the compactness and modularity of +your subsequent rules;
        2. +
      2. it may help various analysis and execution tools, +for example static analyzers to give you error messages when you create cells +where you should not, or K compilers to improve performance by starting +actual concurrent hardware threads or processes corresponding to each cell +instance; and
        3. +
      3. it may help you better understand and control the dynamics +of your configuration, and thus your overall semantics.
        4. +
      +

      For good encapsulation, I also prefer to put all thread cells into one cell, +<threads/>. This is technically unnecessary, though; to convince yourself +that this is indeed the case, you can remove this cell once we are done with +the semantics and everything will work without having to make any changes.

      +

      Before we continue, let us kompile an krun some programs that used to +work, say sum-io.imp. In spite of the relatively radical configuration +reorganization, those programs execute just fine! How is that possible? +In particular, why do rules like the lookup and assignment still work, +unchanged, in spite of the fact that the <k/> and <env/> cells are not at +the same level with the <store/> cell in the configuration anymore?

      +

      Welcome to configuration abstraction, part 2. Recall that the role of +configuration abstraction is to allow you to only write the relevant +information in each rule, and have the compiler fill-in the obvious and boring +details. According to the configuration that we declared for our new +language, there is only one reasonable way to complete rules like the lookup, +namely to place the <k/> and </env> cells inside a <thread/> cell, +inside a <threads/> cell:

      +
      rule <threads>...
+       <thread>...
+         <k> X:Id => I ...</k>
+         <env>... X |-> N ...</env>
+       ...</thread>
+     ...<threads/>
+     <store>... N |-> I ...</store>  [lookup]
+
      +

      This is the most direct, compact and local way to complete the configuration +context of the lookup rule. If for some reason you wanted here to match the +<k/> cell of one thread and the <env/> cell of another thread, then you +would need to explicitly tell K so, by mentioning the two thread cells, +for example:

      +
      rule <thread>...
+         <k> X:Id => I ...</k>
+     ...</thread>
+     <thread>...
+         <env>... X |-> N ...</env>
+     ...</thread>
+     <store>... N |-> I ...</store>  [lookup]
+
      +

      By default, K completes rules in a greedy style. Think this way: what is the +minimal number of changes to my rule to make it fit the declared +configuration? That's what the K tool will do.

      +

      Configuration abstraction is technically unnecessary, but once you start +using it and get a feel for how it works, it will become your best friend. +It allows you to focus on the essentials of your semantics, and at the same +time gives you flexibility in changing the configuration later on without +having to touch the rules. For example, it allows you to remove the +<threads/> cell from the configuration, if you don't like it, without +having to touch any rule.

      +

      We are now ready to give the semantics of spawn:

      +
      rule <k> spawn S => . ...</k> <env> Rho </env>
+     (. => <thread>... <k> S </k> <env> Rho </env> ...</thread>)
+
      +

      Note configuration abstraction at work, again. Taking into account +the declared configuration, and in particular the multiplicity information +* in the <thread/> cell, the only reasonable way to complete the rule +above is to wrap the <k/> and <env/> cells on the first line within a +<thread/> cell, and to fill-in the ...s in the child thread with the +default contents of the other subcells in <thread/>. In this case there +are no other cells, so we can get rid of those ...s, but that would +decrease the modularity of this rule: indeed, we may later on add other +cells within <thread/> as the language evolves, for example a function +or an exception stack, etc.

      +

      In theory, we should be able to write the rule above even more compactly +and modularly, namely as

      +
      rule <k> spawn S => . ...</k> <env> Rho </env>
+     (. => <k> S </k> <env> Rho </env>)
+
      +

      Unfortunately, this currently does not work in the K tool, due to some +known limitations of our current configuration abstraction algorithm. +This latter rule would be more modular, because it would not even depend +on the cell name thread. For example, we may later decide to change +thread into agent, and we would not have to touch this rule. +We hope this current limitation will be eliminated soon.

      +

      Once a thread terminates, its computation cell becomes empty. When that +happens, we can go ahead and remove the useless thread cell:

      +
      rule <thread>... <k> . </k> ...</thread> => .
+
      +

      Let's see what we've got. kompile and krun spawn.imp. +Note the following:

      +
        +
      • The <threads/> cell is empty, so all threads terminated normally;
        • +
      • The value printed is different from the value in the store; the store value +is not even the one obtained if the threads executed sequentially.
        • +
      +

      Therefore, interesting behaviors may happen; we would like to see them all!

      +

      Based on prior experience with krun's search option, we would hope that

      +
      krun spawn.imp --search
+
      +

      shows all the behaviors. However, the above does not work, for two reasons.

      +

      First, spawn.imp is an interactive program, which reads a number from the +standard input. When analyzing programs exhaustively using the search option, +krun has to disable the streaming capabilities (just think about it and you +will realize why). The best you can do in terms of interactivity with search +is to pipe some input to krun: krun will flush the standard input buffer +into the cells connected to it when creating the initial configuration (will +do that no matter whether you run it with or without the --search option). +For example:

      +
      echo 23 | krun spawn.imp --search
+
      +

      puts 23 in the standard input buffer, which is then transferred in the +<in/> cell as a list item, and then the exhaustive search procedure is +invoked.

      +

      Second, even after piping some input, the spawn.imp program still manifests +only one behavior, which does not seem right. There should be many more.

      +

      As explained in Lesson 3, by default kompile optimizes the generated +language model for execution. In particular, it does not insert any +backtracking markers where transition attempts should be made, so krun +lacks the information it needs to exhaustively search the generated language +model. Like we did in Lesson 3 with the language constructs, we also have +to explicitly tell kompile which rules should be considered as actual +transitions. A theoretically correct but practically unfeasible approach +to search all possible behaviors is to consider all rules as transitions. +Even more than with the non-deterministic strictness of language constructs +in Lesson 3, such a naive solution would make the number of behaviors, and +thus krun, explode. Remember that a two-thread program with 150 statements +each manifests more behaviors than particles in the known universe! +Consequently, unless your multi-threaded programs are very small, you will +most likely want to control which rules should be considered transitions and +which should not.

      +

      A good rule of thumb is to include as transitions only those rules which +compete for behaviors. That is, those rules which may yield a different +behavior if we choose to apply them when other rules match as well. +The rule for addition, for example, is a clear example of a rule which +should not be a transition: indeed, 3+7 will rewrite to 10 now and also +later. On the other hand, the lookup rule should be a transition. Indeed, +if we delay the lookup of variable x, then other threads may write x in the +meanwhile (with an increment or an assignment rule) and thus yield a +different behavior.

      +

      Let us discuss and tag those rules which should be transitions: lookup and +increment need to be transitions and we already tagged them in Lesson 3; +the read rule needs to also be a transition, because it may complete with +other instances of itself in other threads; assignment needs to also be a +transition, and so should be the first rule for print.

      +

      Let us now kompile with the transition option set as desired:

      +
      kompile imp --transition "lookup increment assignment read print"
+
      +

      Now echo 23 | krun spawn.imp --search gives us all 12 behaviors of the +spawn.imp program.

      +

      Like for non-deterministically strict operations which can be tagged as +transitions, it is highly non-trivial to say precisely which rules need +to be transitions. So krun makes no attempt to automatically detect it. +Instead, it provides the functionality to let you decide it.

      +

      We currently have no mechanism for thread synchronization. In the next lesson +we add a join statement, which allows a thread to wait until another completes.

      +

      Go to Lesson 7, IMP++: Everything Changes: Syntax, Configuration, Semantics.

      +

      MOVIE (out of date) [11'40"]

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      Everything Changes: Syntax, Configuration, Semantics

      +

      In this lesson we add thread joining, one of the simplest thread +synchronization mechanisms. In doing so, we need to add unique ids +to threads in the configuration, and to modify the syntax to allow spawn +to return the id of the newly created thread. This gives us an opportunity +to make several other small syntactic and semantics changes to the language, +which make it more powerful or more compact at a rather low cost.

      +

      Before we start, let us first copy and modify the previous spawn.imp program +from Lesson 1 to make use of thread joining. Recall from Lesson 6 that in some +runs of this program the main thread completed before the child threads, +printing a possibly undesired value of x. What we want now is to assign +unique ids to the two spawned threads, and then to modify the main thread to +join the two child threads before printing. To avoid adding a new type to +the language, let's assume that thread ids are integer numbers. So we declare +two integers, t1 and t2, and assign them the two spawn commands. In order +for this to parse, we will have to change the syntax of spawn to be an +arithmetic expression construct instead of a statement. Once we do that, +we have a slight syntactic annoyance: we need to put two consecutive ; +after the spawn assignment, one for the assignment statement inside the spawn, +and another for the outer assignment. To avoid the two consecutive semicolons, +we can syntactically enforce spawn to take a block as argument, instead of a +statement. Now it looks better. The new spawn.imp program is still +non-deterministic, because the two threads can execute in any order and even +continue to have a data-race on the shared variable x, but we should see fewer +behaviors when we use the join statements. If we want to fully synchronize +this program, we can have the second thread start with a join(t1) statement. +Then we should only see one behavior for this program.

      +

      Let us now modify the language semantics. First, we move the spawn +construct from statements to expressions, and make it take a block. +Second, we add one more sub-cell to the thread cell in the configuration, +<id/>, to hold the unique identifier of the thread. We want the main +thread to have id 0, so we initialize this cell with 0. Third, we modify +the spawn rule to generate a fresh integer identifier, which is put in the +<id/> cell of the child thread and returned as a result of spawn in the +parent thread. Fourth, let us add the join statement to the language, +both syntactically and semantically. So in order for the join(T) statement +to execute, thread T must have its computation empty. However, in order +for this to work we have to get rid of the thread termination cleanup rule. +Indeed, we need to store somewhere the information that thread T terminated; +the simplest way to do it is to not remove the terminated threads. Feel free +to experiment with other possibilities, too, here. For example, you may add +another cell, <done/>, in which you can store all the thread ids of the +terminated and garbage-collected threads.

      +

      Let us now kompile imp.k and convince ourselves that the new spawn.imp +with join statements indeed has fewer behaviors than its variant without +join statements. Also, let us convince ourselves that the fully synchronized +variant of it indeed has only one behavior.

      +

      Note that now spawn, like variable increment, makes the evaluation of +expressions to have side effects. Many programming languages in fact allow +expressions to be evaluated only for their side effects, and not for their +value. This is typically done by simply adding a ; after the expression +and thus turning it into a statement. For example, ++x;. Let as also +allow arithmetic expressions in our language to be used as statements, by +simply adding the production AExp ";" to Stmt, with evaluation strategy +strict and with the expected semantics discarding the value of the AExp.

      +

      Another simple change in syntax and semantics which gives our language more +power, is to remove the ; from the syntax of variable assignments and to make +them expression instead of statement constructs. This change, combined with +the previous one, will still allow us to parse all the programs that we could +parse before, but will also allow us to parse more programs. For example, we +can now do sequence assignments like in C: x = y = z = 0. The semantics +of assignment now has to return the assigned value also to the computation, +because we want the assignment expression to evaluate to the assigned value.

      +

      Let us also make another change, but this time one which only makes the +definition more compact. Instead of defining statement sequential +composition as a binary construct for statements, let us define a new +syntactic construct, Stmts, as whitespace-separated lists of Stmt. This +allows us to get rid of the empty blocks, because we can change the syntax of +blocks to {Stmts} and Stmts also allows the empty sequence of statements. +However, we do have to make sure that .Stmts dissolves.

      +

      In general, unless you are defining a well-established programming language, +it is quite likely that your definitions will suffer lots of changes like the +ones seen in this lecture. You add a new construct, which suggests changes +in the existing syntax making in fact your language parse more programs, +which then requires corresponding changes in the semantics, and so on. +Also, compact definitions are desirable in general, because they are easier +to read and easier to change if needed later.

      +

      In the next lesson we wrap up and document the definition of IMP++.

      +

      Go to Lesson 8, IMP++: Wrapping up Larger Languages.

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      IMP++

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Abstract

      +

      This is the K semantic definition of the IMP++ language. +IMP++ extends the IMP language with the features listed below. We +strongly recommend you to first familiarize yourself with the IMP +language and its K definition in Tutorial 2 before proceeding.

      +

      Strings and concatenation of strings. Strings are useful +for the print statement, which is discussed below. For +string concatenation, we use the same + construct that we use +for addition (so we overload it).

      +

      Variable increment. We only add a pre-increment construct: +++x increments variable x and evaluates to the +incremented value. Variable increment makes the evaluation of +expressions have side effects, and thus makes the evaluation strategies +of the various language constructs have an influence on the set +of possible program behaviors.

      +

      Input and output. IMP++ adds a read() expression +construct which reads an integer number and evaluates to it, and +a variadic (i.e., it has an arbitrary number of arguments) statement +construct print(e1,e2,...,en) which evaluates its arguments +and then outputs their values. Note that the K tool allows to +connect the input and output cells to the standard input and output +buffers, this way compiling the language definition into an +interactive interpreter.

      +

      Abrupt termination. The halt statement simply halts +the program. The K tool shows the resulting configuration, as if the +program terminated normally. We therefore assume that an external +observer does not care whether the program terminates normally or +abruptly, same like with exit statements in conventional +programming languages like C.

      +

      Dynamic threads. The expression construct spawn s +starts a new concurrent thread that executes statement s, +which is expected to be a block, and evaluates immediately to a fresh +thread identifier that is also assigned to the newly created thread. +The new thread is given at creation time the environment of its +parent, so it can access all its parent's variables. This allows for +the parent thread, and the child thread to communicate; it also allows +for races and "unexpected" behaviors, so be careful. +For thread synchronization, IMP++ provides a thread join statement +construct join t;, where t evaluates to a thread +identifier, which stalls the current thread until thread t +completes its computation. For simplicity, we here assume a +sequentially consistent shared memory model. To experiment with other +memory models, see the definition of KERNELC.

      +

      Blocks and local variables. IMP++ allows blocks enclosed by +curly brackets. Also, IMP's global variable declaration construct is +generalized to be used anywhere as a statement, not only at the +beginning of the program. As expected, the scope of the declared +variables is from their declaration point till the end of the most +nested enclosing block.

      +

      What You Will Learn Here

      +
        +
      • How to define a less trivial language in K, as explained above.
        • +
      • How to use the superheat and supercool +options of the K tool kompile to exhaustively explore the +non-determinism due to underspecified evaluation strategies.
        • +
      • How to use the transition option of the K tool to +exhaustively explore the non-determinism due to concurrency.
        • +
      • How to connect certain cells in the configuration to the +standard input and standard output, and thus turn the krun +tool into an interactive interpreter for the defined language.
        • +
      • How to exhaustively search for the non-deterministic behaviors +of a program using the search option of krun.
        • +
      +
      k
      module IMP-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      IMP++ adds several syntactic constructs to IMP. Also, since the +variable declaration construct is generalized to be used anywhere a +statement can be used, not only at the beginning of the program, we +need to remove the previous global variable declaration of IMP and +instead add a variable declaration statement construct

      +

      We do not re-discuss the constructs which are taken over from IMP, +except when their syntax has been subtly modified (such as, for +example, the syntax of the previous "statement" assignment which +is now obtained by composing the new assignment expression, and the +new expression statement constructs); go the last lesson of +Tutorial 2 if you are interested in IMP's constructs. For execution +purposes, we tag the addition and division operations as members of the +addition and division groups. These groups have no theoretical significance, +in that they do not affect the semantics of the language in any way. They only +have practical relevance, specific to our implementation of the K tool. +Specifically, we can tell the K tool (using its superheat and supercool +options) that we want to exhaustively explore all the non-deterministic +behaviors (due to strictness) of these language constructs. For performance +reasons, by default the K tool chooses an arbitrary but fixed order to +evaluate the arguments of the strict language constructs, thus possibly losing +behaviors due to missed interleavings. This aspect was irrelevant in IMP, +because its expressions had no side effects, but it becomes relevant in IMP++.

      +

      The syntax of the IMP++ constructs is self-explanatory. Note that assignment +is now an expression construct. Also, print is variadic, taking a +list of expressions as argument. It is also strict, which means that the +entire list of expressions, that is, each expression in the list, will be +evaluated. Note also that we have now defined sequential composition +of statements as a whitespace-separated list of statements, aliased with +the nonterminal Stmts, and block as such a (possibly empty) sequence +of statements surrounded by curly brackets.

      +
      k
        syntax AExp  ::= Int | String | Id
+                 | "++" Id
+                 | "read" "(" ")"
+                 | "-" AExp                   [strict]
+                 | "(" AExp ")"               [bracket]
+                 > AExp "/" AExp              [left, strict, group(division)]
+                 > AExp "+" AExp              [left, strict]
+                 > "spawn" Block
+                 > Id "=" AExp                [strict(2)]
+  syntax BExp  ::= Bool
+                 | AExp "<=" AExp             [seqstrict, latex({#1}\leq{#2})]
+                 | "!" BExp                   [strict]
+                 | "(" BExp ")"               [bracket]
+                 > BExp "&&" BExp             [left, strict(1)]
+  syntax Block ::= "{" Stmts "}"
+  syntax Stmt  ::= Block
+                 | AExp ";"                   [strict]
+                 | "if" "(" BExp ")"
+                   Block "else" Block         [strict(1)]
+                 | "while" "(" BExp ")" Block
+                 | "int" Ids ";"
+                 | "print" "(" AExps ")" ";"
+                 | "halt" ";"
+                 > "join" AExp ";"            [strict]
+
+  syntax Ids   ::= List{Id,","}               [klabel(exps)]
+  syntax AExps ::= List{AExp,","}             [klabel(exps)]
+  syntax Stmts ::= List{Stmt,""}
+  syntax AExps ::= Ids
+endmodule
+
+
+module IMP
+  imports IMP-SYNTAX
+  imports DOMAINS
+
      +

      Semantics

      +

      We next give the semantics of IMP++. We start by first defining its +configuration.

      +

      Configuration

      +

      The original configuration of IMP has been extended to include +all the various additional cells needed for IMP++. +To facilitate the semantics of threads, more specifically +to naturally give them access to their parent's variables, we prefer a +(rather conventional) split of the program state into an +environment and a store. An environment maps +variable names into locations, while a store maps locations +into values. Stores are also sometimes called states, or +heaps, or memory, in the literature. Like values, locations +can be anything. For simplicity, here we assume they are natural +numbers. Moreover, each thread has its own environment, so it knows +where all the variables that it has access to are located in the store +(that includes its locally declared variables as well as the variables +of its parent thread), and its own unique identifier. The store is +shared by all threads. For simplicity, we assume a sequentially consistent +memory model in IMP++. Note that the thread cell has multiplicity +*, meaning that there could be zero, one, or more instances of that cell +in the configuration at any given time. This multiplicity information +is important for K's configuration abstraction process: it tells +K how to complete rules which, in order to increase the modularity of the +definition, choose to not mention the entire configuration context. +The in and out cells hold the input and the output +buffers as lists of items.

      +
      k
        configuration <T color="yellow">
+                  <threads color="orange">
+                    <thread multiplicity="*" color="blue" type="Map">
+                      <id color="black"> 0 </id>
+                      <k color="green"> $PGM:Stmts </k>
+                      <env color="LightSkyBlue"> .Map </env>
+                    </thread>
+                  </threads>
+//                <br/>
+                  <store color="red"> .Map </store>
+//                  <input color="magenta"> .List </input>
+//                  <output color="Orchid"> .List </output>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="Orchid" stream="stdout"> .List </output>
+                </T>
+// Replace the <input/> and <output/> cells with the next two in order to
+// initialize the input buffer through krun
+//     <input color="magenta"> $IN:List </input>
+//     <output color="Orchid"> .List </output>
+// Replace the <input/> and <output/> cells with the next two to connect the
+// input/output buffers to stdin/stdout through krun
+//     <input color="magenta" stream="stdin"> .List </input>
+//     <output color="Orchid" stream="stdout"> .List </output>
+// Replace the <input/> and <output/> cells with the next two to connect the
+// input/output buffers to stdin/stdout and also allow input through krun
+//     <input color="magenta" stream="stdin"> $IN:List </input>
+//     <output color="Orchid" stream="stdout"> .List </output>
+
      +

      We can also use configuration variables to initialize +the configuration through krun. For example, we may want to +pass a few list items in the in cell when the program makes +use of read(), so that the semantics does not get stuck. +Recall from IMP that configuration variables start with a ParseError: KaTeX parse error: Expected group after '_' at position 63: …, for example, +_̲PGM) and can be initialized with any string by +krun; or course, the string should parse to a term of the +corresponding sort, otherwise errors will be generated. +Moreover, K allows you to connect list cells to the standard input or +the standard output. For example, if you add the attribute +stream="stdin" to the in cell, then krun +will prompt the user to pass input when the in cell is empty +and any semantic rule needs at least one item to be present there in +order to match. Similarly but dually, if you add the attribute +stream="stdout" to the out cell, then any item +placed into this cell by any rule will be promptly sent to the +standard output. This way, krun can be used to obtain +interactive interpreters based directly on the K semantics of the +language. For example:

      +
      shell
      sh$ krun sum-io.imp --output none
+Add numbers up to (<= 0 to quit)? 10
+Sum = 55
+Add numbers up to (<= 0 to quit)? 1000
+Sum = 500500
+Add numbers up to (<= 0 to quit)? 0
+sh$
+
      +

      The option --output none instructs krun to not +display the resulting configuration after the program executes. The +input/output streaming works with or without this option, although +if you don't use the option then a configuration with empty +in and out cells will be displayed after the program +is executed. You can also initialize the configuration using +configuration variables and stream the contents of the cells to +standard input/output at the same time. For example, if you use a +configuration variable in the in cell and pass contents to it +through krun, then that contents will be first consumed and +then the user will be prompted to introduce additional input if the +program's execution encounters more read() constructs.

      +

      The old IMP constructs

      +

      The semantics of the old IMP constructs is almost identical to their +semantics in the original IMP language, except for those constructs +making use of the program state and for those whose syntax has slightly +changed. Indeed, the rules for variable lookup and assignment in IMP +accessed the state cell, but that cell is not available in IMP++ +anymore. Instead, we have to use the combination of environment and store +cells. Thanks to K's implicit configuration abstraction, we do not have +to mention the thread and threads cells: these are +automatically inferred (and added by the K tool at compile time) from the +definition of the configuration above, as there is only one correct +way to complete the configuration context of these rules in order to +match the configuration declared above. In our case here, "correct way" +means that the k and env cells will be considered as +being part of the same thread cell, as opposed to each being part +of a different thread. Configuration abstraction is crucial for modularity, +because it gives us the possibility to write our definitions in a way that +may not require us to revisit existing rules when we change the configuration. +Changes in the configuration are quite frequent in practice, typically +needed in order to accommodate new language features. For example, +imagine that we initially did not have threads in IMP++. There +would be no need for the thread and threads cells in +the configuration then, the cells k and env being simply +placed at the top level in the T cell, together with the +already existing cells. Then the rules below would be exactly the +same. Thus, configuration abstraction allows you to not have to +modify your rules when you make structural changes in your language +configuration.

      +

      Below we list the semantics of the old IMP constructs, referring the +reader to the K semantics of IMP for their meaning. Like we tagged the +addition and the division rules above in the syntax, we also tag the lookup +and the assignment rules below (as members of the groups lookup and +assignment), because we want to refer to them when we generate the +language model (with the kompile tool), basically to allow them to +generate (possibly non-deterministic) transitions. Indeed, these two rules, +unlike the other rules corresponding to old IMP constructs, can yield +non-deterministic behaviors when more threads are executed concurrently. +In terms of rewriting, these two rules can "compete" with each other on +some program configurations, in the sense that they can both match at the +same time and different behaviors may be obtained depending upon which of +them is chosen first.

      +
      k
        syntax KResult ::= Int | Bool
+
      +

      Variable lookup

      +
      k
        rule <k> X:Id => I ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> I ...</store>  [group(lookup)]
+
      +

      Arithmetic constructs

      +
      k
        rule I1 / I2 => I1 /Int I2  when I2 =/=Int 0
+  rule I1 + I2 => I1 +Int I2
+  rule - I => 0 -Int I
+
      +

      Boolean constructs

      +
      k
        rule I1 <= I2 => I1 <=Int I2
+  rule ! T => notBool T
+  rule true && B => B
+  rule false && _ => false
+
      +

      Variable assignment

      +

      Note that the old IMP assignment statement X = I; is now composed of two +constructs: an assignment expression construct X = I, followed by a +semicolon ; turning the expression into a statement. The rationale behind +this syntactic restructuring has been explained in Lesson 7. Here is the +semantics of the two constructs:

      +
      k
        rule _:Int; => .
+  rule <k> X = I:Int => I ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> (_ => I) ...</store>  [group(assignment)]
+
      +

      Sequential composition

      +

      Sequential composition has been defined as a whitespace-separated syntactic +list of statements. Recall that syntactic lists are actually syntactic +sugar for cons-lists. Therefore, the following two rules eventually +sequentialize a syntactic list of statements s1 s2 ... sn.. into the +corresponding computation s1 ~> s2 ~> ... ~> sn.

      +
      k
         rule .Stmts => .
+   rule S:Stmt Ss:Stmts => S ~> Ss
+
      +

      Conditional statement

      +
      k
        rule if (true)  S else _ => S
+  rule if (false) _ else S => S
+
      +

      While loop

      +

      The only thing to notice here is that the empty block has been replaced +with the block holding the explicit empty sequence. That's because in +the semantics all empty lists become explicit corresponding dots +(to avoid parsing ambiguities)

      +
      k
        rule while (B) S => if (B) {S while (B) S} else {.Stmts}
+
      +

      The new IMP++ constructs

      +

      We next discuss the semantics of the new IMP++ constructs.

      +

      Strings

      +

      First, we have to state that strings are also results. +Second, we give the semantics of IMP++ string concatenation (which +uses the already existing addition symbol + from IMP) by +reduction to the built-in string concatenation operation.

      +
      k
        syntax KResult ::= String
+  rule Str1 + Str2 => Str1 +String Str2
+
      +

      Variable increment

      +

      Like variable lookup, this is also meant to be a supercool transition: we +want it to count both in the non-determinism due to strict operations above +it in the computation and in the non-determinism due to thread +interleavings. This rule also relies on K's configuration abstraction. +Without abstraction, you would have to also include the thread and +threads cells.

      +
      k
        rule <k> ++X => I +Int 1 ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> (I => I +Int 1) ...</store>  [group(increment)]
+
      +

      Read

      +

      The read() construct evaluates to the first integer in the +input buffer, which it consumes. Note that this rule is put in the group +increment. This is because we will include it in the set of +potentially non-deterministic transitions when we kompile the definition; +we want to do that because two or more threads can "compete" on +reading the next integer from the input buffer, and different choices +for the next transition can lead to different behaviors.

      +
      k
        rule <k> read() => I ...</k>
+       <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
      +

      Print

      +

      The print statement is strict, so all its arguments are +eventually evaluated (recall that print is variadic). We +append each of its evaluated arguments, in order, to the output buffer, +and structurally discard the residual print statement with an +empty list of arguments. We only want to allow printing integers and +strings, so we define a Printable syntactic category including +only these and define the print statement to only print +Printable elements. Alternatively, we could have had two +similar rules, one for integers and one for strings. Recall that, +currently, K's lists are cons-lists, so we cannot simply rewrite the +head of a list (P) into a list (.). The first rule below is tagged, +because we want to include it in the list of transitions when we kompile; +different threads may compete on the output buffer and we want to capture +all behaviors.

      +
      k
        syntax Printable ::= Int | String
+
+/* currently it is necessary to subsort Printable to AExp,
+  but future K should be able to infer that automatically. */
+  syntax AExp ::= Printable
+
+  context print(HOLE:AExp, _AEs:AExps);
+
+  rule <k> print(P:Printable,AEs => AEs); ...</k>
+       <output>... .List => ListItem(P) </output>  [group(print)]
+  rule print(.AExps); => .
+
      +

      Halt

      +

      The halt statement empties the computation, so the rewriting process +simply terminates as if the program terminated normally. Interestingly, once +we add threads to the language, the halt statement as defined below +will terminate the current thread only. If you want an abrupt termination +statement that halts the entire program, then you need to discard the entire +contents of the threads cell, so the entire computation abruptly +terminates the entire program, no matter how many concurrent threads it has, +because there is nothing else to rewrite.

      +
      k
        rule <k> halt; ~> _ => . </k>
+
      +

      Spawn thread

      +

      A spawned thread is passed its parent's environment at creation time. +The spawn expression in the parent thread is immediately +replaced by the unique identifier of the newly created thread, so the +parent thread can continue its execution. We only consider a sequentially +consistent shared memory model for IMP++, but other memory models can also +be defined in K; see, for example, the definition of KERNELC. Note that +the rule below does not need to be tagged in order to make it a transition +when we kompile, because the creation of the thread itself does not interfere +with the execution of other threads. Also, note that K's configuration +abstraction is at heavy work here, in two different places. First, the +parent thread's k and env cells are wrapped within a +thread cell. Second, the child thread's k, env +and id cells are also wrapped within a thread cell. Why +that way and not putting all these four cells together within the +same thread, or even create an additional threads cell at top +holding a thread cell with the new k, env +and id? Because in the original configuration we declared +the multiplicity of the thread cell to be *, which +effectively tells the K tool that zero, one or more such cells can +co-exist in a configuration at any moment. The other cells have the +default multiplicity one, so they are not allowed to multiply. +Thus, the only way to complete the rule below in a way consistent with +the declared configuration is to wrap the first two cells in a +thread cell, and the latter two cells under the . +also in a thread cell. Once the rule applies, the spawning +thread cell will add a new thread cell next to it, which is consistent +with the declared configuration cell multiplicity. The unique identifier +of the new thread is generated using the fresh side condition.

      +
      k
        rule <k> spawn S => !T:Int +Int 1 ...</k> <env> Rho </env>
+       (.Bag => <thread>... <k> S </k> <env> Rho </env> <id> !T +Int 1 </id> ...</thread>)
+
      +

      Join thread

      +

      A thread who wants to join another thread T has to wait until +the computation of T becomes empty. When that happens, the +join statement is simply dissolved. The terminated thread is not removed, +because we want to allow possible other join statements to also dissolve.

      +
      k
        rule <k> join(T); => . ...</k> <thread>... <k>.</k> <id>T</id> ...</thread>
+
      +

      Blocks

      +

      The body statement of a block is executed normally, making sure +that the environment at the block entry point is saved in the computation, +in order to be recovered after the block body statement. This step is +necessary because blocks can declare new variables having the same +name as variables which already exist in the environment, and our +semantics of variable declarations is to update the environment map in +the declared variable with a fresh location. Thus, variables which +are shadowed lose their original binding, which is why we take a +snapshot of the environment at block entrance and place it after the +block body (see the semantics of environment recovery at the end of +this module). Note that any store updates through variables which are +not declared locally are kept at the end of the block, since the store +is not saved/restored. An alternative to this environment save/restore +approach is to actually maintain a stack of environments and to push a +new layer at block entrance and pop it at block exit. The variable +lookup/assign/increment operations then also need to change, so we do +not prefer that non-modular approach. Compilers solve this problem by +statically renaming all local variables into fresh ones, to completely +eliminate shadowing and thus environment saving/restoring.

      +
      k
        rule <k> {Ss} => Ss ~> Rho ...</k> <env> Rho </env>
+
      +

      Variable declaration

      +

      We allocate a fresh location for each newly declared variable and +initialize it with 0.

      +
      k
        rule <k> int (X,Xs => Xs); ...</k>
+       <env> Rho => Rho[X <- !N:Int] </env>
+       <store>... .Map => !N |-> 0 ...</store>
+  rule int .Ids; => .
+
      +

      Auxiliary operations

      +

      We only have one auxiliary operation in IMP++, the environment +recovery. Its role is to discard the current environment in the +env cell and replace it with the environment that it holds.

      +
      k
        rule <k> Rho => . ...</k> <env> _ => Rho </env>
+
      +

      If you want to avoid useless environment recovery steps and keep the size +of the computation structure smaller, then you can also add the rule

      +
        rule (_:Map => .) ~> _:Map
+
      +

      This rule acts like a ``tail recursion'' optimization, but for blocks. */

      +
      k
      // verification ids
+  syntax Id ::= "n"     [token]
+              | "sum"   [token]
+              | "a"     [token]
+              | "b"     [token]
+              | "c"     [token]
+endmodule
+
      +

      On Kompilation Options

      +

      We are done with the IMP++ semantics. The next step is to kompile the +definition using the kompile tool, this way generating a language +model. Depending upon for what you want to use the generated language model, +you may need to kompile the definition using various options. We here discuss +these options.

      +

      To tell the K tool to exhaustively explore all the behaviors due to the +non-determinism of addition, division, and threads, we have to kompile +with the command:

      +
      shell
      kompile imp.k --transition="addition division lookup assignment increment read print"
+
      +

      As already mentioned, the syntax and rule groups play no theoretical or +foundational role in K. They are only a means to allow kompile to +refer to them in its options, like we did above. By default, kompile's +transition option is empty, because this yields the fastest language model when +executed. Transitions may slow down the execution, but they instrument +the language model to allow for formal analysis of program behaviors, even for +exhaustive analysis.

      +

      Theoretically, the heating/cooling rules in K are fully reversible and +unconstrained by side conditions as we showed in the semantics of IMP. +For example, the theoretical heating/cooling rules corresponding to the +strict attribute of division are the following:

      +
      E₁ / E₂ ⇒ E₁ ⤳ □ / E₂
+E₁ ⤳ □ / E₂ ⇒ E₁ / E₂
+E₁ / E₂ ⇒ E₂ ⤳ E₁ / □
+E₂ ⤳ E₁ / □ ⇒ E₁ / E₂
+
      +

      The other semantic rules apply modulo such structural rules. +For example, using heating rules we can bring a redex (a subterm which +can be reduced with semantic rules) to the front of the computation, +then reduce it, then use cooling rules to reconstruct a term over the +original syntax of the language, then heat again and +non-deterministically pick another redex, and so on and so forth +without losing any opportunities to apply semantic rules. +Nevertheless, these unrestricted heating/cooling rules may create an +immense, often unfeasibly large space of possibilities to analyze. +The --transition option implements an optimization which works +well with other implementation choices made in the current K tool. +Recall from the detailed description of the IMP language semantics that +(theoretical) reversible rules like above are restricted by default +to complementary conditional rules of the form

      +
      E₁ / E₂ ⇒ E₁ ⤳ □ / E₂
+   if E₁ not in KResult
+E₁ ⤳ □ / E₂ ⇒ E₁ / E₂
+   if E₁ in KResult
+E₁ / E₂ ⇒ E₂ ⤳ E₁ / □
+   if E₂ not in KResult
+E₂ ⤳ E₁ / □  ⇒ E₁ / E₂
+   if  E₂ in KResult
+
      +

      Therefore, our tool eagerly heats and lazily cools the computation. +In other words, heating rules apply until a redex gets placed on the +top of the computation, then some semantic rule applies and rewrites +that into a result, then a cooling rule is applied to plug the +obtained result back into its context, then another argument may be +chosen and completely heated, and so on. This leads to efficient +execution, but it may and typically does hide program behaviors. +Using the --transition option allows you to interfere with this +process and to obtain all possible non-deterministic behaviors as if +the theoretical heating/cooling rules were applied. Optimizations +of course happen under the hood, but you need not be aware of them. +Used carefully, this mechanism allows us to efficiently explore more of +the non-deterministic behaviors of a program, even all of them (like here). +For example, with the semantics of IMP++ given above, the krun +command with the --search option detects all five behaviors +of the following IMP++ program (x can be 0, 1, 2, 3, or undefined +due to division-by-zero):

      +
        int x,y;
+  x = 1;
+  y = ++x / (++x / x);
+
      +

      Besides non-determinism due to underspecified argument evaluation +orders, which the current K tool addresses as explained above, there +is another important source of non-determinism in programming languages: +non-determinism due to concurrency/parallelism. For example, when two +or more threads are about to access the same location in the store and at +least one of these accesses is a write (i.e., an instance of the variable +assignment rule), there is a high chance that different choices for +the next transition lead to different program behaviors. While in the +theory of K all the count as computational steps +and hereby as transitions in the transition system associated to the +program, in practice that may yield a tremendous number of step +interleavings to consider. Most of these interleavings are behaviorally +equivalent for most purposes. For example, the fact that a thread computes +a step 8+3 ⇒ 11 is likely irrelevant for the other +threads, so one may not want to consider it as an observable transition in +the space of interleavings. Since the K tool cannot know without help which +transitions need to be explored and which do not, our approach is to +let the user say so explicitly using the transition option of +kompile.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/4_imp++/lesson_8/index.html b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_8/index.html new file mode 100644 index 00000000000..5fa41630eb5 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/4_imp++/lesson_8/index.html @@ -0,0 +1,404 @@ + + + + + + + + + + + + + + +Wrapping up Larger Languages | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Wrapping up Larger Languages

      +

      In this lesson we wrap up IMP++'s semantics and also generate its poster. +While doing so, we also learn how to display larger configurations in order +to make them easier to read and print.

      +

      Note that we rearrange a bit the semantics, to group the semantics of old +IMP's constructs together, and separate it from the new IMP++'s semantics.

      +

      There is a detailed discussion at the end of the document about the +--transition option of kompile, because that is important and we want +the poster to include everything we learned in this part of the tutorial.

      +

      You can go even further and manually edit the generated Latex document. +You typically want to do that when you want to publish your language +definition, or parts of it, and you need to finely tune it to fit the +editing requirements. For example, you may want to insert some negative +spaces, etc.

      +

      Part 4 of the tutorial is now complete. At this moment you should know most +of K framework's features and how to use the K tool. You can now define or +design your own programming languages, and then execute and analyze programs.

      +

      MOVIE (out of date) [06'26"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/index.html b/k-distribution/pl-tutorial/1_k/5_types/index.html new file mode 100644 index 00000000000..d62626560ee --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/index.html @@ -0,0 +1,392 @@ + + + + + + + + + + + + + + +Part 5: Defining Type Systems | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Part 5: Defining Type Systems

      +

      In this part of the tutorial we will show that defining type systems for +languages is essentially no different from defining semantics. The major +difference is that programs and fragments of programs now rewrite to their +types, instead of to concrete values. In terms of K, we will learn how +to use it for a certain particular but important kind of applications.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_1.9/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_1.9/NOTES/index.html new file mode 100644 index 00000000000..fda3200d915 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_1.9/NOTES/index.html @@ -0,0 +1,383 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      This currently does NOT work, because of the rules

      +
      rule _:Int => int                            [anywhere]
+rule _:Bool => bool                          [anywhere]
+
      +

      which now rewrite ANY integer ANYWHERE to "int", including integers +that appear in the internal data-structures/functions of the builtins. +We will need to allow a strategy where "anywhere" means anywhere in one +or more computational cells.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_1/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_1/NOTES/index.html new file mode 100644 index 00000000000..9e10486001d --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_1/NOTES/index.html @@ -0,0 +1,376 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Explain the lack of tenv(...)?

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_1/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_1/index.html new file mode 100644 index 00000000000..276446b289c --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_1/index.html @@ -0,0 +1,492 @@ + + + + + + + + + + + + + + +Imperative, Environment-Based Type Systems | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Imperative, Environment-Based Type Systems

      +

      In this lesson you learn how to define a type system for an imperative +language (the IMP++ language defined in Part 4 of the tutorial), using a style +based on type environments.

      +

      Let us copy the imp.k file from Part 4 of the tutorial, Lesson 7, which holds +the semantics of IMP++, and modify it into a type system. The resulting type +system, when executed, yields a type checker.

      +

      We start by defining the new strictness attributes of the IMP++ syntax. +While doing so, remember that programs and fragments of programs now reduce +to their types. So types will be the new results of our new (type) semantics. +We also clean up the semantics by removing the unnecessary tags, and also +use strict instead of seqstrict wherever possible, because strict gives +implementations more freedom. Interestingly, note that spawn is strict now, +because the code of the child thread should type in the current parent's type +environment. Note that this is not always the case for threads, see for example +SIMPLE in the languages tutorial, but it works here for our simpler IMP++.

      +

      From a typing perspective, the && construct is strict in both its arguments; +its short-circuit (concrete) semantics is irrelevant for its (static) type +system. Similarly, both the conditional and the while loop are strict +constructs when regarded through the typing lenses.

      +

      Finally, the sequential composition is now sequentially strict! Indeed, +statements are now going to reduce to their type, stmt, and it is critical +for sequential composition to type its argument statements left-to-right; +for example, imagine that the second argument is a variable declaration (whose +type semantics will modify the type environment).

      +

      We continue by defining the new results of computations, that is, the actual +types. In this simple imperative language, we only have a few constant types: +int, bool, string, block and stmt.

      +

      We next define the new configuration, which is actually quite simple. Besides +the <k/> cell, all we need is a type environment cell, <tenv/>, which will +hold a map from identifiers to their types. A type environment is therefore +like a state in the abstract domain of type values.

      +

      Let us next modify the semantic rules, turning them into a type system. In +short, the idea is to reduce the basic values to their types, and then have a +rule for each language construct reducing it to its result type whenever its +arguments have the expected types.

      +

      We write the rules in the order given by the syntax declarations, to make +sure we do not forget any construct.

      +

      Integers reduce to their type, int.

      +

      So do the strings.

      +

      Variables are now looked up in the type environment and reduced to their type +there. Since we only declare integer variables in IMP++, their type in tenv +will always be int. Nevertheless, we write the rule generically, so that we +would not have to change it later if we add other type declarations to IMP++. +Note that we reject programs which lookup undeclared variables. Rejection, +in this case, means rewriting getting stuck.

      +

      Variable increment types to int, provided the variable has type int.

      +

      Read types to int, because we only allow integer input.

      +

      Division is only allowed on integers, so it rewrites to int provided that its +arguments rewrite to int. Note, however, that in order to write int / int, +we have to explicitly add int to the syntax of arithmetic expressions. +Otherwise, the K parser rightfully complains, because / was declared on +arithmetic expressions, not on types. One simple and generic way to allow +types to appear anywhere, is to define Type as a syntactic subcategory of all +the other syntactic categories. Let's do it on a by-need basis, though.

      +

      Addition is overloaded, so we add two typing rules for it: one for integers +and another for strings.

      +

      As discussed, spawn types to stmt provided that its argument types to +block.

      +

      The assignment construct was strict(2); its typing policy is that the declared +type of X should be identical to the type of the assigned value. Like for +lookup, we define this rule more generically than needed for IMP++, for any +type, not only for int.

      +

      The typing rules for Boolean expression constructs are in the same spirit. +Note that we need only one rule for &&.

      +

      The typing of blocks is a bit trickier. First, note that we still need to +recover the environment after the block is typed, because we do not want the +block-local variables to be visible in the outer type environment. We recover +the type environment only after the block-enclosed statements type; moreover, +we also opportunistically yield a block type on the computation when we +discard the type environment recovery item. To account for the fact that the +block-enclosed statement can itself be a block (e.g., {{S}}), we would need an +additional rule. Since we do not like repetition, we instead group the types +block and stmt into one syntactic category, BlockOrStmtType, and now we +can have only one rule. We also include BlockOrStmtType in Type, as a +replacement for the two basic types.

      +

      The expression statement types as expected. Recall that we only allow +arithmetic expressions, which type to int, to be used as statements in IMP++.

      +

      The conditional was declared strict in all its arguments. Its typing policy +is that its first argument types to bool and its two branches to block. +If that is the case, then it yields a stmt type.

      +

      For while, its first argument should type to bool and its second to block.

      +

      Variable declarations add new bindings to the type environment. Recall that +we can only declare variables of integer type in IMP++.

      +

      The typing policy of print is that it can only print integer or string values, +and in that case it types to stmt. Like for BlockOrStmtType, to avoid +having two similar rules, one for int and another for string, we prefer to +introduce an additional syntactic category, PrintableType, which includes both +int and string types.

      +

      halt types to stmt; so its subsequent code is also typed.

      +

      join types to stmt, provided that its argument types to int.

      +

      Sequential composition was declared as a whitespace-separated sequentially +strict list. Its typing policy is that all the statements in the list should +type to stmt or block in order for the list to type to stmt. Since +lists are maintained internally as cons-lists, this is probably the simplest +way to do it:

      +
      rule .Stmts => stmt
+rule _:BlockOrStmtType Ss => Ss
+
      +

      Note that the first rule, which types the empty sequence of statements to stmt, +is needed anyway, to type empty blocks {} (together with the block rule).

      +

      kompile imp.k and krun all the programs in Part 4 of the tutorial. They +should all type to stmt.

      +

      In the next lesson we will define a substitution-based type system for LAMBDA.

      +

      Go to Lesson 2, Type Systems: Substitution-Based Higher-Order Type Systems.

      +

      MOVIE (out of date) [10'11"]

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_2/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_2/NOTES/index.html new file mode 100644 index 00000000000..9bbd746df7c --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_2/NOTES/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Maybe we should define simply-typed lambda calculus in some earlier +lecture in tutorial 1, and then reuse its examples here.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_2/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_2/index.html new file mode 100644 index 00000000000..6ffa7a16f44 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_2/index.html @@ -0,0 +1,447 @@ + + + + + + + + + + + + + + +Substitution-Based Higher-Order Type Systems | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Substitution-Based Higher-Order Type Systems

      +

      In this lesson you learn how to define a substitution-based type system for +a higher-order language, namely the LAMBDA language defined in Part 1 of the +tutorial.

      +

      Let us copy the definition of LAMBDA from Part 1 of the tutorial, Lesson 8. +We are going to modify it into a type systems for LAMBDA.

      +

      Before we start, it is important to clarify an important detail, namely that +our type system will yield a type checker when executed, not a type +inferencer. In particular, we are going to change the LAMBDA syntax +to allow us to associate a type to each declared variable. The +constructs which declare variables are lambda, let, letrec and mu. +The syntax of all these will therefore change.

      +

      Since here we are not interested in a LAMBDA semantics anymore, we take the +freedom to eliminate the Val syntactic category, our previous results. +Our new results are going to be the types, because programs will now reduce +to their types.

      +

      As explained, the syntax of the lambda construct needs to change, to also +declare the type of the variable that it binds. We add the new syntactic +category Type, with the following constructs: int, bool, the function +type (which gives it its higher-order status), and parentheses as bracket. +Also, we make types our K results.

      +

      We are now ready to define the typing rules.

      +

      Let us start with the typing rule for lambda abstraction: lambda X : T . E +types to the function type T -> T', where T' is the type obtained by further +typing E[T/X]. This can be elegantly achieved by reducing the lambda +abstraction to T -> E[T/X], provided that we extend the function type construct +to take expressions, not only types, as arguments, and to be strict. +This can be easily achieved by redeclaring it as a strict expression construct +(strictness in the second argument would suffice in this example, but it is +more uniform to define it strict overall).

      +

      The typing rule for application is as simple as it can get: (T1->T2) T1 => T2.

      +

      Let us now give the typing rules of arithmetic and Boolean expression +constructs. First, let us get rid of Val. Second, rewrite each value to its +type, similarly to the type system for IMP++ in the previous lesson. Third, +replace each semantic rule by its typing rule. Fourth, make sure you +do not forget to subsort Type to Exp, so your rules above will parse.

      +

      The typing policy of the conditional statement is that its first argument +should type to bool and its other two arguments should type to the same type +T, which will also be the result type of the conditional. So we make the +conditional construct strict in all its three arguments and we write the +obvious rule: if bool then T:Type else T => T. We want a runtime check that +the latter arguments are actually typed, so we write T:Type.

      +

      There is nothing special about let, except that we have to make sure we +change its syntax to account for the type of the variable that it binds. +This rule is a macro, so the let is desugared statically.

      +

      Similarly, the syntax of letrec and mu needs to change to account for the +type of the variable that they bind. The typing of letrec remains based on +its desugaring to mu; we have to make sure the types are also included now.

      +

      The typing policy of mu is that its body should type to the same type T of +its variable, which is also the type of the entire mu expression. This can +be elegantly achieved by rewriting it to (T -> T) E[T/X]. Recall that +application is strict, so E[T/X] will be eventually reduced to its type. +Then the application types correctly only if that type is also T, and in +that case the result type will also be T.

      +

      kompile and krun some programs. You can, for example, take the LAMBDA +programs from the first tutorial, modify them by adding types to their +variable declarations, and then type check them using krun.

      +

      In the next lesson we will discuss an environment-based type system +for LAMBDA.

      +

      Go to Lesson 3, Type Systems: Environment-Based Higher-Order Type Systems.

      +

      MOVIE (out of date) [6'52"]

      +
      +
      + + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_3/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_3/index.html new file mode 100644 index 00000000000..2a7de6c2530 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_3/index.html @@ -0,0 +1,434 @@ + + + + + + + + + + + + + + +Environment-Based Higher-Order Type Systems | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Environment-Based Higher-Order Type Systems

      +

      In this lesson you learn how to define an environment-based type system for +a higher-order language, namely the LAMBDA language defined in Part 1 of the +tutorial.

      +

      The simplest and fastest way to proceed is to copy the substitution-based +type system of LAMBDA from the previous lesson and modify it into an +environment-based one. A large portion of the substitution-based definition +will remain unchanged. We only have to modify the rules that use +substitution.

      +

      We do not need the substitution anymore, so we can remove the require and +import statements. The syntax of types and expressions stays unchanged, but +we can now remove the binder tag of lambda.

      +

      Like in the type system of IMP++ in Lesson 1, we need a configuration that +contains, besides the <k/> cell, a <tenv/> cell that will hold the type +environment.

      +

      In an environment-based definition, unlike in a substitution-based one, we +need to lookup variables in the environment. So let us start with the +type lookup rule:

      +
      rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</k>
+
      +

      The type environment is populated by the semantic rule of lambda:

      +
      rule <k> lambda X : T . E => (T -> E) ~> Rho ...</k>
+     <tenv> Rho => Rho[X <- T] </tenv>
+
      +

      So X is bound to its type T in the type environment, and then T -> E +is scheduled for processing. Recall that the arrow type construct has been +extended into a strict expression construct, so E will be eventually reduced +to its type. Like in other environment-based definitions, we need to make +sure that we recover the type environment after the computation in the scope +of the declared variable terminates.

      +

      The typing rule of application does not change, so it stays as elegant as it +was in the substitution-based definition:

      +
      rule (T1 -> T2) T1 => T2
+
      +

      So do the rules for arithmetic and Boolean constructs, and those for the +if, and let, and letrec.

      +

      The mu rule needs to change, because it was previously defined using +substitution. We modify it in the same spirit as we modified the lambda +rule: bind X to its type in the environment, schedule its body for typing +in its right context, and then recover the type environment.

      +

      Finally, we give the semantics of environment recovery, making sure +the environment is recovered only after the preceding computation is +reduced to a type:

      +

      rule _:Type ~> (Rho => .) ... _ => Rho

      +

      The changes that we applied to the substitution-based definition were +therefore quite systematic: each substitution invocation was replaced with +an appropriate type environment update/recovery.

      +

      Go to Lesson 4, Type Systems: A Naive Substitution-Based Type Inferencer.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_4/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_4/NOTES/index.html new file mode 100644 index 00000000000..bd2316143df --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_4/NOTES/index.html @@ -0,0 +1,380 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The README needs to be changed to reflect the fact that we now have a builtin +unification procedure. We may even want to merge this lecture with the +next one, and eliminate the approach where we throw equalities on the computation. +This needs some more thinking, though, especialy on how to smoothly glue it +with Lesson 6, where we also use equalities.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_4/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_4/index.html new file mode 100644 index 00000000000..9491f66b59c --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_4/index.html @@ -0,0 +1,577 @@ + + + + + + + + + + + + + + +A Naive Substitution-Based Type Inferencer | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      A Naive Substitution-Based Type Inferencer

      +

      In this lesson you learn how to define a naive substitution-based type +inferencer for a higher-order language, namely the LAMBDA language +defined in Part 1 of the tutorial.

      +

      Unlike in the type checker defined in Lessons 2 and 3, where we had to +associate a type with each declared variable, a type inferencer +attempts to infer the types of all the variables from the way those +variables are used. Let us take a look at this program, say plus.lambda:

      +
      lambda x . lambda y . x + y
+
      +

      Since x and y are used in an integer addition context, we can infer +that they must have the type int and the result of the addition is +also an int, so the type of the entire expression is int -> int -> int. +Similarly, the program if.lambda

      +
      lambda x . lambda y . lambda z .
+  if x then y else z
+
      +

      can only make sense when x has type bool and y and z have the same +type, say t, in which case the type of the entire expression is +bool -> t -> t -> t. Since the type t can be anything, we say that +the type of this expression is polymorphic. That means that the code +above can be used in different contexts, where t can be an int, a +bool, a function type int -> int, and so on.

      +

      In the identity.lambda program

      +
      let f = lambda x . x
+in f 1
+
      +

      f has such a polymorphic type, which is then applied to an integer, +so this program is type-safe and its type is int.

      +

      A typical polymorphic expression is the composition

      +
      lambda f . lambda g . lambda x .
+  g (f x)
+
      +

      which has the type (t1 -> t2) -> (t2 -> t3) -> (t1 -> t3), polymorphic +in 3 types.

      +

      Let us now define our naive type inferencer and then we discuss more +examples. The idea is quite simple: we conceptually do the same +operations like we did within the type checker defined in Lesson 2, +with two important differences:

      +
        +
      1. instead of declaring a type with each declared variable, we assume +a fresh type for that variable; and
        2. +
      2. instead of checking that the types of expressions satisfy the +type properties of the context in which they are used, we impose +those properties as type equality constraints. A general-purpose +unification-based constraint solving mechanism is then used to solve +the generated type constraints.
        3. +
      +

      Let us start with the syntax, which is essentially identical to that +of the type checker in Lesson 2, except that bound variables are not +declared a type anymore. Also, to keep things more compact, we put +all the Exp syntax declarations in one syntax declaration this time.

      + +

      Before we modify the rules, let us first define our machinery for +adding and solving constraints. First, we require and import the +unification procedure. We do not discuss unification here, but if you +are interested you can consult the unification.k files under +k-distribution/include/kframework/builtin, which contains our current generic +definition of unification, which is written also in K. The generic unification +provides a sort, Mgu, for most-general-unifier, an operation +updateMgu(Mgu,T1,T2) which updates Mgu with additional constraints +generated by forcing the terms T1 and T2 to be equal, and an operation +applyMgu(Mgu,T) which applies Mgu to term T. For our use +of unification here, we do not even need to know how Mgu terms are +represented internally.

      +

      We define a K item construct, =, which takes two Type terms and +enforces them to be equal by means of updating the current Mgu. +Once the constraints are added to the Mgu, the equality dissolves +itself. With this semantics of = in mind, we can now go ahead and +modify the rules of the type checker systematically into rules +for a type inferencer. The changes are self-explanatory and +mechanical: for example, the rule

      +
      rule int * int => int
+
      +

      changes into rule

      +
      rule T1:Type  * T2:Type => T1 = int ~> T2 = int ~> int
+
      +

      generating the constraints that the two arguments of multiplication +have the type int, and the result type is int. Recall that each type +equality on the <k/> cell updates the current Mgu appropriately and +then dissolves itself; thus, the above says that after imposing the +constraints T1=int and T2=int, multiplication yields a type int.

      +

      As mentioned above, since types of variables are not declared anymore, +but inferred, we have to generate a fresh type for each variable at its +declaration time, and then generate appropriately constraints for it. +For example, the type semantics of lambda and mu become:

      +
      rule lambda X . E => T -> E[T/X]  when fresh(T:Type)
+rule mu X . E => (T -> T) E[T/X]  when fresh(T:Type)
+
      +

      that is, we add a condition stating that the previously declared type +is now a fresh one. This type will be further constrained by how the +variable X is being used within E.

      +

      Interestingly, the previous typing rule for lambda application is not +powerful enough anymore. Indeed, since types are not given anymore, +it may very well be the case that the inferred type of the first +argument of the application construct is not yet a function type +(remember, for example, the program composition.lambda above). What +we have to do is to enforce it to be a function type, by means of +fresh types and constraints. We can introduce a fresh type for the +result of the application, and then write the expected rule as +follows:

      +
      rule T1:Type T2:Type => T1 = (T2 -> T) ~> T  when fresh(T:Type)
+
      +

      The conditional requires that its first argument is a bool and its +second and third arguments have the same type, which is also the +result type.

      +

      The macros do not change, in particular let is desugared into lambda +application. We will next see that this is a significant restriction, +because it limits the polymorphism of our type system.

      +

      We are done. We have a working type inferencer for LAMBDA.

      +

      Let's kompile it and krun the programs above. They all work as +expected. Let us also try some additional programs, to push it to its +limits.

      +

      First, let us test mu by means of a letrec example:

      +
      letrec f x = 3
+in f
+
      +

      We can also try all the programs that we had in our first tutorial, on +lambda, for example the factorial.imp program:

      +
      letrec f x = if x <= 1 then 1 else (x * (f (x + -1)))
+in (f 10)
+
      +

      Those programs are simple enough that they should all work as +expected with our naive type inferencer here.

      +

      Let us next try to type some tricky programs, which involve more +complex and indirect type constraints.

      +

      tricky-1.lambda:

      +
      lambda f . lambda x . lambda y . (
+  (f x y) + x + (let x = y in x)
+)
+
      +

      tricky-2.lambda:

      +
      lambda x .
+  let f = lambda y . if true then y else x
+  in (lambda x . f 0)
+
      +

      tricky-3.lambda:

      +
      lambda x . let f = lambda y . if true then x 7 else x y
+           in f
+
      +

      tricky-4.lambda:

      +
      lambda x . let f = lambda x . x
+           in let d = (f x) + 1
+              in x
+
      +

      tricky-5.lambda:

      +
      lambda x . let f = lambda y . x y
+           in let z = x 0 in f
+
      +

      It is now time to see the limitations of this naive type inferencer. +Consider the program

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      Our type inferencer fails graciously with a clash in the <mgu/> cell +between int and bool. Indeed, the desugaring macro of let turns it +into a lambda and an application, which further enforce id to have a +type of the form t -> t for some fresh type t. The first use of id +in the condition of if will then constrain t to be bool, while the +other uses in the two branches will enforce t to be int. Thus the +clash in the <mgu/> cell.

      +

      Similarly, the program

      +
      let id = lambda x . x
+in id id
+
      +

      yields a different kind of conflict: if id has type t -> t, in order +to apply id to itself it must be the case that its argument, t, equals +t -> t. These two type terms cannot be unified because there is a +circular dependence on t, so we get a cycle in the <mgu/> cell.

      +

      Both limitations above will be solved when we change the semantics of +let later on, to account for the desired polymorphism.

      +

      Before we conclude this lesson, let us see one more interesting +example, where the lack of let-polymorphism leads not to a type error, +but to a less generic type:

      +
      let f1 = lambda x . x in
+  let f2 = f1 in
+    let f3 = f2 in
+      let f4 = f3 in
+        let f5 = f4 in
+          if (f5 true) then f2 else f3
+
      +

      Our current type inferencer will infer the type bool -> bool for the +program above. Nevertheless, since all functions f1, f2, f3, f4, f5 +are the identity function, which is polymorphic, we would expect the +entire program to type to the same polymorphic identity function type.

      +

      This limitation will be also addressed when we define our +let-polymorphic type inferencer.

      +

      Before that, in the next lesson we will show how easily we can turn +the naive substitution-based type inferencer discussed in this lesson +into a similarly naive, but environment-based type inferencer.

      +

      Go to Lesson 5, Type Systems: A Naive Environment-Based Type Inferencer.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_5/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_5/index.html new file mode 100644 index 00000000000..3fbfd3a02af --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_5/index.html @@ -0,0 +1,433 @@ + + + + + + + + + + + + + + +A Naive Environment-Based Type Inferencer | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      A Naive Environment-Based Type Inferencer

      +

      In this lesson you learn how to define a naive environment-based type +inferencer for a higher-order language. Specifically, we take the +substitution-based type inferencer for LAMBDA defined in Lesson 4 and +turn it into an environment-based one.

      +

      Recall from Lesson 3, where we defined an environment-based type +checker for LAMBDA based on the substitution-based one in Lesson 2, +that the transition from a substitution-based definition to an +environment-based one was quite systematic and mechanical: each +substitution occurrence E[T/X] is replaced by E, but at the same time +the variable X is bound to type T in the type environment. One benefit +of using type environments instead of substitution is that we replace +a linear complexity operation (the substitution) with a constant +complexity one (the variable lookup).

      +

      There is not much left to say which has not been already said in +Lesson 3: we remove the unnecessary binder annotations for the +variable binding operations, then add a <tenv/> cell to the +configuration to hold the type environment, then add a new rule for +variable lookup, and finally apply the transformation of substitutions +E[T/X] into E as explained above.

      +

      The resulting type inferencer should now work exactly the same way as +the substitution-based one, except, of course, that the resulting +configurations will contain a <tenv/> cell now.

      +

      As sanity check, let us consider two more LAMBDA programs that test +the static scoping nature of the inferencer. We do that because +faulty environment-based definitions often have this problem. The +program

      +
      let x = 1
+in let f = lambda a . x
+   in let x = true
+      in f 3
+
      +

      should type to int, not to bool, and so it does. Similarly, the +program

      +
      let y = 0
+in letrec f x = if x <= 0
+                then y
+                else let y = true
+                     in f (x + 1)
+   in f 1
+
      +

      should also type to int, not bool, and so it does, too.

      +

      The type inferencer defined in this lesson has the same limitations, +in terms of polymorphism, as the one in Lesson 4. In the next +lesson we will see how it can be parallelized, and in further lessons +how to make it polymorphic.

      +

      Go to Lesson 6, Type Systems: Parallel Type Checkers/Inferencers.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_6/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_6/NOTES/index.html new file mode 100644 index 00000000000..ac413172d03 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_6/NOTES/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The README needs to be changed to reflect the fact that we now have a builtin +unification procedure.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_6/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_6/index.html new file mode 100644 index 00000000000..454ecd52e37 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_6/index.html @@ -0,0 +1,487 @@ + + + + + + + + + + + + + + +Parallel Type Checkers/Inferencers | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Parallel Type Checkers/Inferencers

      +

      In this lesson you learn how to define parallel type checkers or +inferencers. For the sake of a choice, we will parallelize the one in +the previous lesson, but the ideas are general. We are using the same +idea to define type checkers for other languages in the K tool +distribution, such as SIMPLE and KOOL.

      +

      The idea is in fact quite simple. Instead of one monolithic typing +task, we generate many smaller tasks, which can be processed in +parallel. We use the same approach to define parallel semantics as we +used for threads in IMP++ in Part 4 of the tutorial, that is, we add a +cell holding all the parallel tasks, making sure we declare the cell +holding a task with multiplicity *. For the particular type +inferencer that we chose here, the one in Lesson 5, each task will +hold an expression to type together with a type environment (so it +knows where to lookup its free variables). We have the following +configuration then:

      +
      configuration <tasks color="yellow">
+                <task color="orange" multiplicity="*">
+                  <k color="green"> $PGM:Exp </k>
+                  <tenv color="red"> .Map </tenv>
+                </task>
+              </tasks>
+              <mgu color="blue"> .Mgu </mgu>
+
      +

      Now we have to take each typing rule we had before and change it to +yield parallel typing. For example, our rule for typing +multiplication was the following in Lesson 5:

      +
      rule T1:Type * T2:Type => T1 = int ~> T2 = int ~> int
+
      +

      Since * was strict, its two arguments eventually type, and once that +happens the rule above fires. Unfortunately, the strictness of +multiplication makes the typing of the two expressions sequential in +our previous definition. To avoid typing the two expressions +sequentially and instead generating two parallel tasks, we remove the +strict attribute of multiplication and replace the rule above with the +following:

      +
      rule <k> E1 * E2 => int ...</k> <tenv> Rho </tenv>
+     (. => <task> <k> E1 = int </k> <tenv> Rho </tenv> </task>
+           <task> <k> E2 = int </k> <tenv> Rho </tenv> </task>)
+
      +

      Therefore, we generate two tasks for typing E1 and E2 in the same type +environment as the current task, and let the current task continue by +simply optimistically reducing E1*E2 to its expected result type, int. +If E1 or E2 will not type to int, then either their corresponding +tasks will get stuck or the <mgu/> cell will result into a clash or cycle, +so the program will not type overall in spite of the fact that we +allowed the task containing the multiplication to continue. This is +how we get maximum of parallelism in this case.

      +

      Before we continue, note that the new tasks hold equalities in them, +where one of its arguments is an expression, while previously the +equality construct was declared to take types. What we want now is +for the equality construct to possibly take any expressions, and first +type them and then generate the type constraint like before. This can +be done very easily by just extending the equality construct to +expressions and declaring it strict:

      +
      syntax KItem ::= Exp "=" Exp  [strict]
+
      + +

      Unlike before, where we only passed types to the equality construct, +we now need a runtime check that its arguments are indeed types before +we can generate the updateMgu command:

      +
      rule <k> T:Type = T':Type => . ...</k>
+     <mgu> Theta:Mgu => updateMgu(Theta,T,T') </mgu>
+
      +

      Like before, an equality will therefore update the <mgu/> cell and then +it dissolves itself, letting the <k/> cell in the corresponding task +empty. Such empty tasks are unnecessary, so they can be erased:

      +
      rule <task>... <k> . </k> ...</task> => .
+
      +

      We can now follow the same style as for multiplication to write the +parallel typing rules of the other arithmetic constructs, and even for +the conditional.

      +

      To parallelize the typing of lambda we generate two fresh types, one +for the variable and one for the body, and make sure that we generate +the correct type constraint and environment in the body task:

      +
      rule <k> lambda X . E => Tx -> Te ...</k> <tenv> TEnv </tenv>
+     (. => <task> <k> E = Te </k> <tenv> TEnv[Tx/X] </tenv> </task>)
+  when fresh(Tx:Type) andBool fresh(Te:Type)
+
      +

      Note that the above also allows us to not need to change and then +recover the environment of the current cell.

      +

      For function application we also need to generate two fresh types:

      +
      rule <k> E1 E2 => T ...</k> <tenv> Rho </tenv>
+     (. => <task> <k> E1 = T2 -> T </k> <tenv> Rho </tenv> </task>
+           <task> <k> E2 = T2 </k> <tenv> Rho </tenv> </task>)
+  when fresh(T2:Type) andBool fresh(T:Type)
+
      +

      The only rule left is that of mu X . E. In this case we only need one +fresh type, because X, E and mu X . E have all the same type:

      +
      rule <k> mu X . E => T ...</k>  <tenv> TEnv </tenv>
+     (. => <task> <k> E = T </k> <tenv> TEnv[T/X] </tenv> </task>)
+  when fresh(T:Type)
+
      +

      We do not need the type environment recovery operation, so we delete it.

      +

      We can now kompile and krun all the programs that we typed in Lesson 5. +Everything should work.

      +

      In this lesson we only aimed at parallelizing the type inferencer in +Lesson 5, not to improve its expressiveness; it still has the same +limitations in terms of polymorphism. The next lessons are dedicated +to polymorphic type inferencers.

      +

      Go to Lesson 7, Type Systems: A Naive Substitution-based Polymorphic Type Inferencer.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_7/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_7/NOTES/index.html new file mode 100644 index 00000000000..381543928f3 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_7/NOTES/index.html @@ -0,0 +1,378 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      This approach leads to an incorrect type checker, in that programs which +lead to a runtime error will type. See tricky-5 and/or variations of it.

      +

      Discuss monomorphic vs. polymorphic types and type inferencers.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_7/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_7/index.html new file mode 100644 index 00000000000..c9ca2625c19 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_7/index.html @@ -0,0 +1,492 @@ + + + + + + + + + + + + + + +A Naive Substitution-based Polymorphic Type Inferencer | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      A Naive Substitution-based Polymorphic Type Inferencer

      +

      In this lesson you learn how little it takes to turn a naive monomorphic +type inferencer into a naive polymorphic one, basically only changing +a few characters. In terms of the K framework, you will learn that +you can have complex combinations of substitutions in K, both over +expressions and over types.

      +

      Let us start directly with the change. All we have to do is to take +the LAMBDA type inferencer in Lesson 4 and only change the macro

      +
      rule let X = E in E' => (lambda X . E') E  [macro]
+
      +

      as follows:

      +
      rule let X = E in E' => E'[E/X]  [macro]
+
      +

      In other words, we are inlining the beta-reduction rule of +lambda-calculus within the original rule. In terms of typing, +the above forces the type inferencer to type E in place for each +occurrence of X in E'. Unlike in the first rule, where X had to get +one type only which satisfied the constrains of all X's occurrences in +E', we now never associate any type to X anymore.

      +

      Let us kompile and krun some examples. Everything that worked with +the type inferencer in Lesson 4 should still work here, although the +types of some programs can now be more general. For example, reconsider +the nested-lets.lambda program

      +
      let f1 = lambda x . x in
+  let f2 = f1 in
+    let f3 = f2 in
+      let f4 = f3 in
+        let f5 = f4 in
+          if (f5 true) then f2 else f3
+
      +

      which was previously typed to bool -> bool. With the new rule above, +the sequence of lets is iteratively eliminated and we end up with the +program

      +
      if (lambda x . x) true then (lambda x . x) else (lambda x . x)
+
      +

      which now types (with both type inferencers) to a type of the form +t -> t, for some type variable t, which is more general than the +previous bool -> bool type that the program typed to in Lesson 4.

      +

      We can also now type programs that were not typable before, such as

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      and

      +
      let id = lambda x . x
+in id id
+
      +

      Let us also test it on some trickier programs, also not typable +before, such as

      +
      let f = lambda x . x
+in let g = lambda y . f y
+   in g g
+
      +

      which gives us a type of the form t -> t for some type variable t, +and as

      +
      let f = let g = lambda x . x
+        in let h = lambda x . lambda x . (g g g g)
+           in h
+in f
+
      +

      which types to t1 -> t2 -> t3 -> t3 for some type variables t1, t2, t3.

      +

      Here is another program which was not typable before, which is +trickier than the others above in that a lambda-bound variable appears +free in a let-bound expression:

      +
      lambda x . (
+  let y = lambda z . x
+  in if (y true) then (y 1) else (y (lambda x . x))
+)
+
      +

      The above presents no problem now, because once lambda z . x gets +substituted for y we get a well-typed expression which yields that x +has the type bool, so the entire expression types to bool -> bool.

      +

      The cheap type inferencer that we obtained above therefore works as +expected. However, it has two problems which justify a more advanced +solution. First, substitution is typically considered an elegant +mathematical instrument which is not too practical in implementations, +so an implementation of this type inferencer will likely be based on +type environments anyway. Additionally, we mix two kinds of +substitutions in this definition, one where we substitute types and +another where we substitute expressions, which can only make things +harder to implement efficiently. Second, our naive substitution of E +for X in E' can yield an exponential explosion in size of the original +program. Consider, for example, the following classic example which +is known to generate a type whose size is exponential in the size of +the program (and is thus used as an argument for why let-polymorphic +type inference is exponential in the worst-case):

      +
      let f00 = lambda x . lambda y . x in
+  let f01 = lambda x . f00 (f00 x) in
+    let f02 = lambda x . f01 (f01 x) in
+      let f03 = lambda x . f02 (f02 x) in
+        let f04 = lambda x . f03 (f03 x) in
+          // ... you can add more nested lets here
+          f04
+
      +

      The particular instance of the pattern above generates a type which +has 17 type variables! The desugaring of each let doubles the size of +the program and of its resulting type. While such programs are little +likely to appear in practice, it is often the case that functions can +be quite complex and large while their type can be quite simple in the +end, so we should simply avoid retyping each function each time it is +used.

      +

      This is precisely what we will do next. Before we present the classic +let-polymorphic type inferencer in Lesson 9, which is based on +environments, we first quickly discuss in Lesson 8 an intermediate +step, namely a naive environment-based variant of the inferencer +defined here.

      +

      Go to Lesson 8, Type Systems: A Naive Environment-based Polymorphic Type Inferencer.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_8/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_8/NOTES/index.html new file mode 100644 index 00000000000..0880c604e6b --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_8/NOTES/index.html @@ -0,0 +1,377 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      We also have to redeclare lambda and mu as binders. Program +tricky-2.lambda shows why.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_8/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_8/index.html new file mode 100644 index 00000000000..8b1a6a7e89a --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_8/index.html @@ -0,0 +1,425 @@ + + + + + + + + + + + + + + +A Naive Environment-based Polymorphic Type Inferencer | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      A Naive Environment-based Polymorphic Type Inferencer

      +

      In this short lesson we discuss how to quickly turn a naive +environment-based monomorphic type inferencer into a naive let-polymorphic +one. Like in the previous lesson, we only need to change a few +characters. In terms of the K framework, you will learn how to have +both environments and substitution in the same definition.

      +

      Like in the previous lesson, all we have to do is to take the LAMBDA +type inferencer in Lesson 5 and only change the rule

      +
      rule let X = E in E' => (lambda X . E') E
+
      +

      as follows:

      +
      rule let X = E in E' => E'[E/X]
+
      +

      The reasons why this works have already been explained in the previous +lesson, so we do not repeat them here.

      +

      Since our new let rule uses substitution, we have to require the +substitution module at the top and also import SUBSTITUTION in the +current module, besides the already existing UNIFICATION.

      +

      Everything which worked with the type inferencer in Lesson 7 should +also work now. Let us only try the exponential type example,

      +
      let f00 = lambda x . lambda y . x in
+  let f01 = lambda x . f00 (f00 x) in
+    let f02 = lambda x . f01 (f01 x) in
+      let f03 = lambda x . f02 (f02 x) in
+        let f04 = lambda x . f03 (f03 x) in
+          f04
+
      +

      As expected, this gives us precisely the same type as in Lesson 7.

      +

      So the only difference between this type inferencer and the one in +Lesson 7 is that substitution is only used for LAMBDA-to-LAMBDA +transformations, but not for infusing types within LAMBDA programs. +Thus, the syntax of LAMBDA programs is preserved intact, which some +may prefer. Nevertheless, this type inferencer is still expensive and +wasteful, because the let-bound expression is typed over and over +again in each place where the let-bound variable occurs.

      +

      In the next lesson we will discuss a type inferencer based on the +classic Damas-Hindley-Milner type system, which maximizes the reuse of +typing work by means of parametric types.

      +

      Go to Lesson 9, Type Systems: Let-Polymorphic Type Inferencer (Damas-Hindley-Milner).

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_9.5/NOTES/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_9.5/NOTES/index.html new file mode 100644 index 00000000000..b31c063af38 --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_9.5/NOTES/index.html @@ -0,0 +1,376 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      We have to test this as well; we need a test folder with a config.xml.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/5_types/lesson_9/index.html b/k-distribution/pl-tutorial/1_k/5_types/lesson_9/index.html new file mode 100644 index 00000000000..882c36ef4ed --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/5_types/lesson_9/index.html @@ -0,0 +1,542 @@ + + + + + + + + + + + + + + +Let-Polymorphic Type Inferencer (Damas-Hindley-Milner) | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Let-Polymorphic Type Inferencer (Damas-Hindley-Milner)

      +

      In this lesson we discuss a type inferencer based on what we call today +the Damas-Hindley-Milner type system, which is at the core of many +modern functional programming languages. The first variant of it was +proposed by Hindley in 1969, then, interestingly, Milner rediscovered +it in 1978 in the context of the ML language. Damas formalized it as +a type system in his PhD thesis in 1985. More specifically, our type +inferencer here, like many others as well as many implementations of +it, follows more closely the syntax-driven variant proposed by Clement +in 1987.

      +

      In terms of K, we will see how easily we can turn one definition which +is considered naive (our previous type inferencer in Lesson 8) into a +definition which is considered advanced. All we have to do is to +change one existing rule (the rule of the let binder) and to add a new +one. We will also learn some new predefined features of K, which make +the above possible.

      +

      The main idea is to replace the rule

      +
      rule let X = E in E' => E'[E/X]
+
      +

      which creates potentially many copies of E within E' with a rule +which types E once and then reuses that type in each place where X +occurs free in E'. The simplest K way to type E is to declare the +let construct strict(2). Now we cannot simply bind X to the type +of E, because we would obtain a variant of the naive type inferencer +we already discussed, together with its limitations, in Lesson 5 of this +tutorial. The trick here is to parameterize the type of E in all its +unconstrained fresh types, and then create fresh copies of those +parameters in each free occurrence of X in E'.

      +

      Let us discuss some examples, before we go into the technical details. +Consider the first let-polymorphic example which failed to be typed +with our first naive type-inferencer:

      +
      let id = lambda x . x
+in if (id true) then (id 1) else (id 2)
+
      +

      When typing lambda x . x, we get a type of the form t -> t, for some +fresh type t. Instead of assigning this type to id as we did in the +naive type inferencers, we now first parametrize this type in its +fresh variable t, written

      +
      (forall t) t -> t
+
      +

      and then bind id to this parametric type. The intuition for the +parameter is that it can be instantiated with any other type, so this +parametric type stands, in fact, for infinitely many non-parametric +types. This is similar to what happens in formal logic proof systems, +where rule schemas stand for infinitely many concrete instances of +them. For this reason, parametric types are also called type schemas.

      +

      Now each time id is looked up within the let-body, we create a fresh +copy of the parameter t, which can this way be independently +constrained by each local context. Let's suppose that the three id +lookups yield the types t1 -> t1, t2 -> t2, and respectively t3 -> t3. +Then t1 will be constrained to be bool, and t2 and t3 to be int, +so we can now safely type the program above to int.

      +

      Therefore, a type schema comprises a summary of all the typing work +that has been done for typing the corresponding expression, and an +instantiation of its parameters with fresh copies represents an +elegant way to reuse all that typing work.

      +

      There are some subtleties regarding what fresh types can be made +parameters. Let us consider another example, discussed as part of +Lesson 7 on naive let-polymorphism:

      +
      lambda x . (
+  let y = lambda z . x
+  in if (y true) then (y 1) else (y (lambda x . x))
+)
+
      +

      This program should type to bool -> bool, as explained in Lesson 7. +The lambda construct will bind x to some fresh type tx. Then the +let-bound expression lambda z . x types to tz -> tx for some +additional fresh type tz. The question now is what should the +parameters of this type be when we generate the type schema? If we +naively parameterize in all fresh variables, that is in both tz and +tx obtaining the type schema (forall tz,tx) tz -> tx, then there will +be no way to infer that the type of x, tx, must be a bool! The +inferred type of this expression would then wrongly be tx -> t for +some fresh types tx and t. That's because the parameters are replaced +with fresh copies in each occurrence of y, and thus their relationship +to the original x is completely lost. This tells us that we cannot +parameterize in all fresh types that appear in the type of the +let-bound expression. In particular, we cannot parameterize in those +which some variables are already bound to in the current type +environment (like x is bound to tx in our example above). +In our example, the correct type schema is (forall tz) tz -> tx, +which now allows us to correctly infer that tx is bool.

      +

      Let us now discuss another example, which should fail to type:

      +
      lambda x .
+  let f = lambda y . x y
+  in if (f true) then (f 1) else (f 2)
+
      +

      This should fail to type because lambda y . x y is equivalent to x, +so the conditional imposes the conflicting constraints that x should be +a function whose argument is either a bool or an int. Let us try to +type it using our currently informal procedure. Like in the previous +example, x will be bound to a fresh type tx. Then the let-bound +expression types to ty -> tz with ty and tz fresh types, adding also +the constraint tx = ty -> tz. What should the parameters of this type +be? If we ignore the type constraint and simply make both ty and tz +parameters because no variable is bound to them in the type +environment (indeed, the only variable x in the type environment is +bound to tx), then we can wrongly type this program to tx -> tz +following a reasoning similar to the one in the example above. +In fact, in this example, none of ty and tz can be parameters, because +they are constrained by tx.

      +

      The examples above tell us two things: first, that we have to take the +type constraints into account when deciding the parameters of the +schema; second, that after applying the most-general-unifier solution +given by the type constraints everywhere, the remaining fresh types +appearing anywhere in the type environment are consequently constrained +and cannot be turned into parameters. Since the type environment can in +fact also hold type schemas, which already bind some types, we only need +to ensure that none of the fresh types appearing free anywhere in the +type environment are turned into parameters of type schemas.

      +

      Thanks to generic support offered by the K tool, we can easily achieve +all the above as follows.

      +

      First, add syntax for type schemas:

      +
      syntax TypeSchema ::= "(" "forall" Set ")" Type  [binder]
+
      +

      The definition below will be given in such a way that the Set argument +of a type schema will always be a set of fresh types. We also declare +this construct to be a binder, so that we can make use of the generic +free variable function provided by the K tool.

      +

      We now replace the old rule for let

      +
      rule let X = E in E' => E'[E/X]
+
      +

      with the following rule:

      +
      rule <k> let X = T:Type in E => E ~> tenv(TEnv) ...</k>
+     <mgu> Theta:Mgu </mgu>
+     <tenv> TEnv
+      => TEnv[(forall freeVariables(applyMgu(Theta, T)) -Set
+                      freeVariables(applyMgu(Theta, values TEnv))
+              ) applyMgu(Theta, T) / X]
+     </tenv>
+
      +

      So the type T of E is being parameterized and then bound to X in the +type environment. The current mgu Theta, which comprises all the type +constraints accumulated so far, is applied to both T and the types in +the type environment. The remaining fresh types in T which do not +appear free in the type environment are then turned into type parameters. +The function freeVariables returns, as expected, the free variables of +its argument as a Set; this is why we declared the type schema to be a +binder above.

      +

      Now a LAMBDA variable in the type environment can be bound to either a +type or a type schema. In the first case, the previous rule we had +for variable lookup can be reused, but we have to make sure we check +that T there is of sort Type (adding a sort membership, for example). +In the second case, as explained above, we have to create fresh copies +of the parameters. This can be easily achieved with another +predefined K function, as follows:

      +
      rule <k> X:Id => freshVariables(Tvs,T) ...</k>
+     <tenv>... X |-> (forall Tvs) T ...</tenv>
+
      +

      Indeed, freshVariables takes a set of variables and a term, and returns the +same term but with each of the given variables replaced by a fresh copy.

      +

      The operations freeVariables and freshVariables are useful in many K +definitions, so they are predefined in module substitution.k.

      +

      Our definition of this let-polymorphic type inferencer is now +complete. To test it, kompile it and then krun all the LAMBDA +programs discussed since Lesson 4. They should all work as expected.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/1_k/index.html b/k-distribution/pl-tutorial/1_k/index.html new file mode 100644 index 00000000000..83cc91ee0ae --- /dev/null +++ b/k-distribution/pl-tutorial/1_k/index.html @@ -0,0 +1,397 @@ + + + + + + + + + + + + + + +Learning K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Learning K

      +

      We start by introducing the basic features of K by means of a series +of very simple languages. The objective here is neither to learn those +languages nor to study their underlying paradigm, but simply to learn K.

      +
        +
      • LAMBDA: Lambda calculus defined.
        • +
      • IMP: A simple imperative language.
        • +
      • LAMBDA++: LAMBDA extended with control flow.
        • +
      • IMP++: IMP extended with threads and IO.
        • +
      • TYPES: LAMBDA type system.
        • +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/NOTES/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/NOTES/index.html new file mode 100644 index 00000000000..d79e48957d7 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/NOTES/index.html @@ -0,0 +1,393 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      This is not well tested now, and it was not well tested in v3.6 either. +We should add some rules as transitions, too, and then use search on all +the examples.

      +

      Exercises not revised yet.

      +

      .Bag should be . throughout this definition #1772

      +

      There seems to be a problem with defining auxiliary constructs of sort +KItem when we want to use them as a particular sort in rule. We had to +declare them as construct for that sort instead. May want to explain +this a bit in the Latex discussion (related to #1803):

      +
        +
      • syntax Exp ::= lookup(Int)
        • +
      +
        +
      • syntax KItem ::= lookup(Int)
        • +
      +

      We currently add Vals to KResult, but we should have a better pattern for +List{Sort} and in general for any collections, where we make them hybrid +(they become KResults when their elements become KResults)

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/simple-untyped/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/simple-untyped/index.html new file mode 100644 index 00000000000..9d11a06c160 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/1_untyped/simple-untyped/index.html @@ -0,0 +1,1709 @@ + + + + + + + + + + + + + + +SIMPLE — Untyped | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      SIMPLE — Untyped

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K semantic definition of the untyped SIMPLE language. +SIMPLE is intended to be a pedagogical and research language that captures +the essence of the imperative programming paradigm, extended with several +features often encountered in imperative programming languages. +A program consists of a set of global variable declarations and +function definitions. Like in C, function definitions cannot be +nested and each program must have one function called main, +which is invoked when the program is executed. To make it more +interesting and to highlight some of K's strengths, SIMPLE includes +the following features in addition to the conventional imperative +expression and statement constructs:

      +
        +
      • +

        Multidimensional arrays and array references. An array evaluates +to an array reference, which is a special value holding a location (where +the elements of the array start) together with the size of the array; +the elements of the array can be array references themselves (particularly +when the array is multi-dimensional). Array references are ordinary values, +so they can be assigned to variables and passed/received by functions.

        +
        • +
      • +

        Functions and function values. Functions can have zero or +more parameters and can return abruptly using a return statement. +SIMPLE follows a call-by-value parameter passing style, with static scoping. +Function names evaluate to function abstractions, which hereby become ordinary +values in the language, same like the array references.

        +
        • +
      • +

        Blocks with locals. SIMPLE variables can be declared +anywhere, their scope being from the place where they are declared +until the end of the most nested enclosing block.

        +
        • +
      • +

        Input/Output. The expression read() evaluates to the +next value in the input buffer, and the statement write(e) +evaluates e and outputs its value to the output buffer. The +input and output buffers are lists of values.

        +
        • +
      • +

        Exceptions. SIMPLE has parametric exceptions (the value thrown as +an exception can be caught and bound).

        +
        • +
      • +

        Concurrency via dynamic thread creation/termination and +synchronization. One can spawn a thread to execute any statement. +The spawned thread shares with its parent its environment at creation time. +Threads can be synchronized via a join command which blocks the current thread +until the joined thread completes, via re-entrant locks which can be acquired +and released, as well as through rendezvous commands.

        +
        • +
      +

      Like in many other languages, some of SIMPLE's constructs can be +desugared into a smaller set of basic constructs. We do that at the end +of the syntax module, and then we only give semantics to the core constructs.

      +

      Note: This definition is commented slightly more than others, because it is +intended to be one of the first non-trivial definitions that the new +user of K sees. We recommend the beginner user to first check the +language definitions discussed in the K tutorial.

      +
      k
      module SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      We start by defining the SIMPLE syntax. The language constructs discussed +above have the expected syntax and evaluation strategies. Recall that in K +we annotate the syntax with appropriate strictness attributes, thus giving +each language construct the desired evaluation strategy.

      +

      Identifiers

      +

      Recall from the K tutorial that identifiers are builtin and come under the +syntactic category Id. The special identifier for the function +main belongs to all programs, and plays a special role in the semantics, +so we declare it explicitly. This would not be necessary if the identifiers +were all included automatically in semantic definitions, but that is not +possible because of parsing reasons (e.g., K variables used to match +concrete identifiers would then be ambiguously parsed as identifiers). They +are only included in the parser generated to parse programs (and used by the +kast tool). Consequently, we have to explicitly declare all the +concrete identifiers that play a special role in the semantics, like +main below.

      +
      k
        syntax Id ::= "main" [token]
+
      +

      Declarations

      +

      There are two types of declarations: for variables (including arrays) and +for functions. We are going to allow declarations of the form +var x=10, a[10,10], y=23;, which is why we allow the var +keyword to take a list of expressions. The non-terminals used in the two +productions below are defined shortly.

      +
      k
        syntax Stmt ::= "var" Exps ";"
+                | "function" Id "(" Ids ")" Block
+
      +

      Expressions

      +

      The expression constructs below are standard. Increment (++) takes +an expression rather than a variable because it can also increment an array +element. Recall that the syntax we define in K is what we call the syntax +of the semantics: while powerful enough to define non-trivial syntaxes +(thanks to the underlying SDF technology that we use), we typically refrain +from defining precise syntaxes, that is, ones which accept precisely the +well-formed programs (that would not be possible anyway in general). That job +is deferred to type systems, which can also be defined in K. In other words, +we are not making any effort to guarantee syntactically that only variables +or array elements are passed to the increment construct, we allow any +expression. Nevertheless, we will only give semantics to those, so expressions +of the form ++5, which parse (but which will be rejected by our type +system in the typed version of SIMPLE later), will get stuck when executed. +Arrays can be multidimensional and can hold other arrays, so their +lookup operation takes a list of expressions as argument and applies to an +expression (which can in particular be another array lookup), respectively. +The construct sizeOf gives the size of an array in number of elements +of its first dimension. Note that almost all constructs are strict. The only +constructs which are not strict are the increment (since its first argument +gets updated, so it cannot be evaluated), the input read which takes no +arguments so strictness is irrelevant for it, the logical and and or constructs +which are short-circuited, the thread spawning construct which creates a new +thread executing the argument expression and return its unique identifier to +the creating thread (so it cannot just evaluate its argument in place), and the +assignment which is only strict in its second argument (for the same reason as +the increment).

      +
      k
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
      +

      We also need comma-separated lists of identifiers and of expressions. +Moreover, we want them to be strict, that is, to evaluate to lists of results +whenever requested (e.g., when they appear as strict arguments of +the constructs above).

      +
      k
        syntax Ids  ::= List{Id,","}           [klabel(Exps)]
+  syntax Exps ::= List{Exp,","}          [klabel(Exps), strict]  // automatically hybrid now
+  syntax Exps ::= Ids
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(Exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}    [klabel(Exps)]
+  syntax Ids ::= Bottoms
+
      +

      Statements

      +

      Most of the statement constructs are standard for imperative languages. +We syntactically distinguish between empty and non-empty blocks, because we +chose Stmts not to be a (;-separated) list of +Stmt. Variables can be declared anywhere inside a block, their scope +ending with the block. Expressions are allowed to be used for their side +effects only (followed by a semicolon ;). Functions are allowed +to abruptly return. The exceptions are parametric, i.e., one can throw a value +which is bound to the variable declared by catch. Threads can be +dynamically created and terminated, and can synchronize with join, +acquire, release and rendezvous. Note that the +strictness attributes obey the intended evaluation strategy of the various +constructs. In particular, the if-then-else construct is strict only in its +first argument (the if-then construct will be desugared into if-then-else), +while the loop constructs are not strict in any arguments. The print +statement construct is variadic, that is, it takes an arbitrary number of +arguments.

      +
      k
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+// NOTE: print strict allows non-deterministic evaluation of its arguments
+// Either keep like this but document, or otherwise make Exps seqstrict.
+// Of define and use a different expression list here, which is seqstrict.
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
      +

      The reason we allow Stmts as the first argument of for +instead of Stmt is because we want to allow more than one statement +to be executed when the loop is initialized. Also, as seens shorly, macros +may expand one statement into more statements; for example, an initialized +variable declaration statement var x=0; desugars into two statements, +namely var x; x=0;, so if we use Stmt instead of Stmts +in the production of for above then we risk that the macro expansion +of statement var x=0; happens before the macro expansion of for, +also shown below, in which case the latter would not apply anymore because +of syntactic mismatch.

      +
      k
        syntax Stmt ::= Stmt Stmt                          [right]
+
+// I wish I were able to write the following instead, but confuses the parser.
+//
+// syntax Stmts ::= List{Stmt,""}
+// syntax Top ::= Stmt | "function" Id "(" Ids ")" Block
+// syntax Pgm ::= List{Top,""}
+//
+// With that, I could have also eliminated the empty block
+
      +

      Desugared Syntax

      +

      This part desugars some of SIMPLE's language constructs into core ones. +We only want to give semantics to core constructs, so we get rid of the +derived ones before we start the semantics. All desugaring macros below are +straightforward.

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule for(Start Cond; Step) {} => {Start while (Cond) {Step;}}
+  rule var E1:Exp, E2:Exp, Es:Exps; => var E1; var E2, Es;
+  rule var X:Id = E; => var X; X = E;
+
      +

      For the semantics, we can therefore assume from now on that each +conditional has both branches, that there are only while loops, and +that each variable is declared alone and without any initialization as part of +the declaration.

      +
      k
      endmodule
+
+
+module SIMPLE-UNTYPED
+  imports SIMPLE-UNTYPED-SYNTAX
+  imports DOMAINS
+
      +

      Basic Semantic Infrastructure

      +

      Before one starts adding semantic rules to a K definition, one needs to +define the basic semantic infrastructure consisting of definitions for +values and configuration. As discussed in the definitions +in the K tutorial, the values are needed to know when to stop applying +the heating rules and when to start applying the cooling rules corresponding +to strictness or context declarations. The configuration serves as a backbone +for the process of configuration abstraction which allows users to only +mention the relevant cells in each semantic rule, the rest of the configuration +context being inferred automatically. Although in some cases the configuration +could be automatically inferred from the rules, we believe that it is very +useful for language designers/semanticists to actually think of and design +their configuration explicitly, so the current implementation of K requires +one to define it.

      +

      Values

      +

      We here define the values of the language that the various fragments of +programs evaluate to. First, integers and Booleans are values. As discussed, +arrays evaluate to special array reference values holding (1) a location from +where the array's elements are contiguously allocated in the store, and +(2) the size of the array. Functions evaluate to function values as +λ-abstractions (we do not need to evaluate functions to closures +because each function is executed in the fixed global environment and +function definitions cannot be nested). Like in IMP and other +languages, we finally tell the tool that values are K results.

      +
      k
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+               | lambda(Ids,Stmt)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+                   | Vals  // TODO: should not need this
+
      +

      The inclusion of values in expressions follows the methodology of +syntactic definitions (like, e.g., in SOS): extend the syntax of the language +to encompass all values and additional constructs needed to give semantics. +In addition to that, it allows us to write the semantic rules using the +original syntax of the language, and to parse them with the same (now extended +with additional values) parser. If writing the semantics directly on the K +AST, using the associated labels instead of the syntactic constructs, then one +would not need to include values in expressions.

      +

      Configuration

      +

      The K configuration of SIMPLE consists of a top level cell, T, +holding a threads cell, a global environment map cell genv +mapping the global variables and function names to their locations, a shared +store map cell store mapping each location to some value, a set cell +busy holding the locks which have been acquired but not yet released +by threads, a set cell terminated holding the unique identifiers of +the threads which already terminated (needed for join), input +and output list cells, and a nextLoc cell holding a natural +number indicating the next available location. Unlike in the small languages +in the K tutorial, where we used the fresh predicate to generate fresh +locations, in larger languages, like SIMPLE, we prefer to explicitly manage +memory. The location counter in nextLoc models an actual physical +location in the store; for simplicity, we assume arbitrarily large memory and +no garbage collection. The threads cell contains one thread +cell for each existing thread in the program. Note that the thread cell has +multiplicity *, which means that at any given moment there could be zero, +one or more thread cells. Each thread cell contains a +computation cell k, a control cell holding the various +control structures needed to jump to certain points of interest in the program +execution, a local environment map cell env mapping the thread local +variables to locations in the store, and finally a holds map cell +indicating what locks have been acquired by the thread and not released so far +and how many times (SIMPLE's locks are re-entrant). The control cell +currently contains only two subcells, a function stack fstack which +is a list and an exception stack xstack which is also a list. +One can add more control structures in the control cell, such as a +stack for break/continue of loops, etc., if the language is extended with more +control-changing constructs. Note that all cells except for k are +also initialized, in that they contain a ground term of their corresponding +sort. The k cell is initialized with the program that will be passed +to the K tool, as indicated by the $PGM variable, followed by the +execute task (defined shortly).

      +
      k
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+
+  syntax ControlCell
+  syntax ControlCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Map" color="yellow">
+                      <id color="pink"> -1 </id>
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <genv color="pink"> .Map </genv>
+                  <store color="white"> .Map </store>
+                  <busy color="cyan"> .Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                </T>
+
      +

      Declarations and Initialization

      +

      We start by defining the semantics of declarations (for variables, +arrays and functions).

      +

      Variable Declaration

      +

      The SIMPLE syntax was desugared above so that each variable is +declared alone and its initialization is done as a separate statement. +The semantic rule below matches resulting variable declarations of the +form var X; on top of the k cell +(indeed, note that the k cell is complete, or round, to the +left, and is torn, or ruptured, to the right), allocates a fresh +location L in the store which is initialized with a special value +⊥ (indeed, the unit ., or nothing, is matched anywhere +in the map ‒note the tears at both sides‒ and replaced with the +mapping L ↦ ⊥), and binds X to L in the local +environment shadowing previous declarations of X, if any. +This possible shadowing of X requires us to therefore update the +entire environment map, which is expensive and can significantly slow +down the execution of larger programs. On the other hand, since we know +that L is not already bound in the store, we simply add the binding +L ↦ ⊥ to the store, thus avoiding a potentially complete +traversal of the the store map in order to update it. We prefer the approach +used for updating the store whenever possible, because, in addition to being +faster, it offers more true concurrency than the latter; indeed, according +to the concurrent semantics of K, the store is not frozen while +L ↦ ⊥ is added to it, while the environment is frozen during the +update operation Env[L/X]. The variable declaration command is +also removed from the top of the computation cell and the fresh location +counter is incremented. The undefined symbol ⊥ added in the store +is of sort KItem, instead of Val, on purpose; this way, the +store lookup rules will get stuck when one attempts to lookup an +uninitialized location. All the above happen in one transactional step, +with the rule below. Note also how configuration abstraction allows us to +only mention the needed cells; indeed, as the configuration above states, +the k and env cells are actually located within a +thread cell within the threads cell, but one needs +not mention these: the configuration context of the rule is +automatically transformed to match the declared configuration +structure.

      +
      k
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+
      +

      Array Declaration

      +

      The K semantics of the uni-dimensional array declaration is somehow similar +to the above declaration of ordinary variables. First, note the +context declaration below, which requests the evaluation of the array +dimension. Once evaluated, say to a natural number N, then +N +Int 1 locations are allocated in the store for +an array of size N, the additional location (chosen to be the first +one allocated) holding the array reference value. The array reference +value array(L,N) states that the array has size N and its +elements are located contiguously in the store starting with location +L. The operation L … L' ↦ V, defined at the end of this +file in the auxiliary operation section, initializes each location in +the list L … L' to V. Note that, since the dimensions of +array declarations can be arbitrary expressions, this virtually means +that we can dynamically allocate memory in SIMPLE by means of array +declarations.

      +
      k
        context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L => L +Int 1 +Int N </nextLoc>
+    requires N >=Int 0
+
      +

      SIMPLE allows multi-dimensional arrays. For semantic simplicity, we +desugar them all into uni-dimensional arrays by code transformation. +This way, we only need to give semantics to uni-dimensional arrays. +First, note that the context rule above actually evaluates all the array +dimensions (that's why we defined the expression lists strict!): +Upon evaluating the array dimensions, the code generation rule below +desugars multi-dimensional array declaration to uni-dimensional declarations. +To this aim, we introduce two special unique variable identifiers, +$1 and $2. The first variable, $1, iterates +through and initializes each element of the first dimension with an array +of the remaining dimensions, declared as variable $2:

      +
      k
        syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         for(var $1 = 0; $1 <= N1 - 1; ++$1) {
+           var $2[N2, Vs];
+           X[$1] = $2;
+         }
+       }
+
      +

      Ideally, one would like to perform syntactic desugarings like the one +above before the actual semantics. Unfortunately, that was not possible in +this case because the dimension expressions of the multi-dimensional array need +to be evaluated first. Indeed, the desugaring rule above does not work if the +dimensions of the declared array are arbitrary expressions, because they can +have side effects (e.g., a[++x,++x]) and those side effects would be +propagated each time the expression is evaluated in the desugaring code (note +that both the loop condition and the nested multi-dimensional declaration +would need to evaluate the expressions given as array dimensions).

      +

      Function declaration

      +

      Functions are evaluated to λ-abstractions and stored like any other +values in the store. A binding is added into the environment for the function +name to the location holding its body. Similarly to the C language, SIMPLE +only allows function declarations at the top level of the program. More +precisely, the subsequent semantics of SIMPLE only works well when one +respects this requirement. Indeed, the simplistic context-free parser +generated by the grammar above is more generous than we may want, in that it +allows function declarations anywhere any declaration is allowed, including +inside arbitrary blocks. However, as the rule below shows, we are not +storing the declaration environment with the λ-abstraction value as +closures do. Instead, as seen shortly, we switch to the global environment +whenever functions are invoked, which is consistent with our requirement that +functions should only be declared at the top. Thus, if one declares local +functions, then one may see unexpected behaviors (e.g., when one shadows a +global variable before declaring a local function). The type checker of +SIMPLE, also defined in K (see examples/simple/typed/static), +discards programs which do not respect this requirement.

      +
      k
        rule <k> function F(Xs) S => . ...</k>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> lambda(Xs, S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+
      +

      When we are done with the first pass (pre-processing), the computation +cell k contains only the token execute (see the configuration +declaration above, where the computation item execute was placed +right after the program in the k cell of the initial configuration) +and the cell genv is empty. In this case, we have to call +main() and to initialize the global environment by transferring the +contents of the local environment into it. We prefer to do it this way, as +opposed to processing all the top level declarations directly within the global +environment, because we want to avoid duplication of semantics: the syntax of +the global declarations is identical to that of their corresponding local +declarations, so the semantics of the latter suffices provided that we copy +the local environment into the global one once we are done with the +pre-processing. We want this separate pre-processing step precisely because +we want to create the global environment. All (top-level) functions end up +having their names bound in the global environment and, as seen below, they +are executed in that same global environment; all these mean, in particular, +that the functions "see" each other, allowing for mutual recursion, etc.

      +
      k
        syntax KItem ::= "execute"
+  rule <k> execute => main(.Exps); </k>
+       <env> Env </env>
+       <genv> .Map => Env </genv>
+
      +

      Expressions

      +

      We next define the K semantics of all the expression constructs.

      +

      Variable lookup

      +

      When a variable X is the first computational task, and X is bound to some +location L in the environment, and L is mapped to some value V in the +store, then we rewrite X into V:

      +
      k
        rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
      +

      Note that the rule above excludes reading ⊥, because ⊥ is not +a value and V is checked at runtime to be a value.

      +

      Variable/Array increment

      +

      This is tricky, because we want to allow both ++x and ++a[5]. +Therefore, we need to extract the lvalue of the expression to increment. +To do that, we state that the expression to increment should be wrapped +by the auxiliary lvalue operation and then evaluated. The semantics +of this auxiliary operation is defined at the end of this file. For now, all +we need to know is that it takes an expression and evaluates to a location +value. Location values, also defined at the end of the file, are integers +wrapped with the operation loc, to distinguish them from ordinary +integers.

      +
      k
        context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I => I +Int 1) ...</store>  [group(increment)]
+
      +

      Arithmetic operators

      +

      There is nothing special about the following rules. They rewrite the +language constructs to their library counterparts when their arguments +become values of expected sorts:

      +
      k
        rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 requires I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 requires I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+
      +

      The equality and inequality constructs reduce to syntactic comparison +of the two argument values (which is what the equality on K terms does).

      +
      k
        rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+
      +

      The logical negation is clear, but the logical conjunction and disjunction +are short-circuited:

      +
      k
        rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
      +

      Array lookup

      +

      Untyped SIMPLE does not check array bounds (the dynamically typed version of +it, in examples/simple/typed/dynamic, does check for array out of +bounds). The first rule below desugars the multi-dimensional array access to +uni-dimensional array access; recall that the array access operation was +declared strict, so all sub-expressions involved are already values at this +stage. The second rule rewrites the array access to a lookup operation at a +precise location; we prefer to do it this way to avoid locking the store. +The semantics of the auxiliary lookup operation is straightforward, +and is defined at the end of the file.

      +
      k
      // The [anywhere] feature is underused, because it would only be used
+// at the top of the computation or inside the lvalue wrapper. So it
+// may not be worth, or we may need to come up with a special notation
+// allowing us to enumerate contexts for [anywhere] rules.
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+
      +

      Size of an array

      +

      The size of the array is stored in the array reference value, and the +sizeOf construct was declared strict, so:

      +
      k
        rule sizeOf(array(_,N)) => N
+
      +

      Function call

      +

      Function application was strict in both its arguments, so we can +assume that both the function and its arguments are evaluated to +values (the former expected to be a λ-abstraction). The first +rule below matches a well-formed function application on top of the +computation and performs the following steps atomically: it switches +to the function body followed by return; (for the case in +which the function does not use an explicit return statement); it +pushes the remaining computation, the current environment, and the +current control data onto the function stack (the remaining +computation can thus also be discarded from the computation cell, +because an unavoidable subsequent return statement ‒see +above‒ will always recover it from the stack); it switches the +current environment (which is being pushed on the function stack) to +the global environment, which is where the free variables in the +function body should be looked up; it binds the formal parameters to +fresh locations in the new environment, and stores the actual +arguments to those locations in the store (this latter step is easily +done by reducing the problem to variable declarations, whose semantics +we have already defined; the auxiliary operation mkDecls is +defined at the end of the file). The second rule pops the +computation, the environment and the control data from the function +stack when a return statement is encountered as the next +computational task, passing the returned value to the popped +computation (the popped computation was the context in which the +returning function was called). Note that the pushing/popping of the +control data is crucial. Without it, one may have a function that +contains an exception block with a return statement inside, which +would put the xstack cell in an inconsistent state (since the +exception block modifies it, but that modification should be +irrelevant once the function returns). We add an artificial +nothing value to the language, which is returned by the +nulary return; statements.

      +
      k
        syntax KItem ::=  (Map,K,ControlCellFragment)
+
+  rule <k> lambda(Xs,S)(Vs:Vals) ~> K => mkDecls(Xs,Vs) S return; </k>
+       <control>
+         <fstack> .List => ListItem((Env,K,C)) ...</fstack>
+         C
+       </control>
+       <env> Env => GEnv </env>
+       <genv> GEnv </genv>
+
+  rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem((Env,K,C)) => .List ...</fstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+
      +

      Like for division-by-zero, it is left unspecified what happens +when the nothing value is used in domain calculations. For +example, from the the perspective of the language semantics, +7 +Int nothing can evaluate to anything, or +may not evaluate at all (be undefined). If one wants to make sure that +such artificial values are never misused, then one needs to define a static +checker (also using K, like our the type checker in +examples/simple/typed/static) and reject programs that do. +Note that, unlike the undefined symbol ⊥ which had the sort K +instead of Val, we defined nothing to be a value. That +is because, as explained above, we do not want the program to get +stuck when nothing is returned by a function. Instead, we want the +behavior to be unspecified; in particular, if one is careful to never +use the returned value in domain computation, like it happens when we +call a function for its side effects (e.g., with a statement of the +form f(x);), then the program does not get stuck.

      +

      Read

      +

      The read() expression construct simply evaluates to the next +input value, at the same time discarding the input value from the +in cell.

      +
      k
        rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
      +

      Assignment

      +

      In SIMPLE, like in C, assignments are expression constructs and not statement +constructs. To make it a statement all one needs to do is to follow it by a +semi-colon ; (see the semantics for expression statements below). +Like for the increment, we want to allow assignments not only to variables but +also to array elements, e.g., e1[e2] = e3 where e1 evaluates +to an array reference, e2 to a natural number, and e3 to any +value. Thus, we first compute the lvalue of the left-hand-side expression +that appears in an assignment, and then we do the actual assignment to the +resulting location:

      +
      k
        context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+
      +

      Statements

      +

      We next define the K semantics of statements.

      +

      Blocks

      +

      Empty blocks are simply discarded, as shown in the first rule below. +For non-empty blocks, we schedule the enclosed statement but we have to +make sure the environment is recovered after the enclosed statement executes. +Recall that we allow local variable declarations, whose scope is the block +enclosing them. That is the reason for which we have to recover the +environment after the block. This allows us to have a very simple semantics +for variable declarations, as we did above. One can make the two rules below +computational if one wants them to count as computational steps.

      +
      k
        rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
      +

      The basic definition of environment recovery is straightforward and +given in the section on auxiliary constructs at the end of the file.

      +

      There are two common alternatives to the above semantics of blocks. +One is to keep track of the variables which are declared in the block and only +recover those at the end of the block. This way one does more work for +variable declarations but conceptually less work for environment recovery; we +say conceptually because it is not clear that it is indeed the case that +one does less work when AC matching is involved. The other alternative is to +work with a stack of environments instead of a flat environment, and push the +current environment when entering a block and pop it when exiting it. This +way, one does more work when accessing variables (since one has to search the +variable in the environment stack in a top-down manner), but on the other hand +uses smaller environments and the definition gets closer to an implementation. +Based on experience with dozens of language semantics and other K definitions, +we have found that our approach above is the best trade-off between elegance +and efficiency (especially since rewrite engines have built-in techniques to +lazily copy terms, by need, thus not creating unnecessary copies), +so it is the one that we follow in general.

      +

      Sequential composition

      +

      Sequential composition is desugared into K's builtin sequentialization +operation (recall that, like in C, the semi-colon ; is not a +statement separator in SIMPLE — it is either a statement terminator or a +construct for a statement from an expression). Note that K allows +to define the semantics of SIMPLE in such a way that statements eventually +dissolve from the top of the computation when they are completed; this is in +sharp contrast to (artificially) evaluating them to a special +skip statement value and then getting rid of that special value, as +it is the case in other semantic approaches (where everything must evaluate +to something). This means that once S₁ completes in the rule below, S₂ +becomes automatically the next computation item without any additional +(explicit or implicit) rules.

      +
      k
        rule S1:Stmt S2:Stmt => S1 ~> S2
+
      +

      A subtle aspect of the rule above is that S₁ is declared to have sort +Stmts and not Stmt. That is because desugaring macros can indeed +produce left associative sequential composition of statements. For example, +the code var x=0; x=1; is desugared to +(var x; x=0;) x=1;, so although originally the first term of +the sequential composition had sort Stmt, after desugaring it became +of sort Stmts. Note that the attribute [right] associated +to the sequential compositon production is an attribute of the syntax, and not +of the semantics: e.g., it tells the parser to parse +var x; x=0; x=1; as var x; (x=0; x=1;), but it +does not tell the rewrite engine to rewrite (var x; x=0;) x=1; to +var x; (x=0; x=1;).

      +

      Expression statements

      +

      Expression statements are only used for their side effects, so their result +value is simply discarded. Common examples of expression statements are ones +of the form ++x;, x=e;, e1[e2]=e3;, etc.

      +
      k
        rule _:Val; => .
+
      +

      Conditional

      +

      Since the conditional was declared with the strict(1) attribute, we +can assume that its first argument will eventually be evaluated. The rules +below cover the only two possibilities in which the conditional is allowed to +proceed (otherwise the rewriting process gets stuck).

      +
      k
        rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
      +

      While loop

      +

      The simplest way to give the semantics of the while loop is by unrolling. +Note, however, that its unrolling is only allowed when the while loop reaches +the top of the computation (to avoid non-termination of unrolling). The +simple while loop semantics below works because our while loops in SIMPLE are +indeed very basic. If we allowed break/continue of loops then we would need +a completely different semantics, which would also involve the control cell.

      +
      k
        rule while (E) S => if (E) {S while(E)S}
+
      +

      Print

      +

      The print statement was strict, so all its arguments are now +evaluated (recall that print is variadic). We append each of +its evaluated arguments to the output buffer, and discard the residual +print statement with an empty list of arguments.

      +
      k
        rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+
      +

      Exceptions

      +

      SIMPLE allows parametric exceptions, in that one can throw and catch a +particular value. The statement try S₁ catch(X) S₂ +proceeds with the evaluation of S₁. If S₁ evaluates normally, i.e., +without any exception thrown, then S₂ is discarded and the execution +continues normally. If S₁ throws an exception with a statement of the +form throw E, then E is first evaluated to some value V +(throw was declared to be strict), then V is bound to X, then +S₂ is evaluated in the new environment while the reminder of S₁ is +discarded, then the environment is recovered and the execution continues +normally with the statement following the try S₁ catch(X) S₂ statement. +Exceptions can be nested and the statements in the +catch part (S₂ in our case) can throw exceptions to the +upper level. One should be careful with how one handles the control data +structures here, so that the abrupt changes of control due to exception +throwing and to function returns interact correctly with each other. +For example, we want to allow function calls inside the statement S₁ in +a try S₁ catch(X) S₂ block which can throw an exception +that is not caught by the function but instead is propagated to the +try S₁ catch(X) S₂ block that called the function. +Therefore, we have to make sure that the function stack as well as other +potential control structures are also properly modified when the exception +is thrown to correctly recover the execution context. This can be easily +achieved by pushing/popping the entire current control context onto the +exception stack. The three rules below modularly do precisely the above.

      +
      k
        syntax KItem ::= (Id,Stmt,K,Map,ControlCellFragment)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem((X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem((X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
      +

      The catch statement S₂ needs to be executed in the original environment, +but where the thrown value V is bound to the catch variable X. We here +chose to rely on two previously defined constructs when giving semantics to +the catch part of the statement: (1) the variable declaration with +initialization, for binding X to V; and (2) the block construct for +preventing X from shadowing variables in the original environment upon the +completion of S₂.

      +

      Threads

      +

      SIMPLE's threads can be created and terminated dynamically, and can +synchronize by acquiring and releasing re-entrant locks and by rendezvous. +We discuss the seven rules giving the semantics of these operations below.

      +

      Thread creation

      +

      Threads can be created by any other threads using the spawn S +construct. The spawn expression construct evaluates to the unique identifier +of the newly created thread and, at the same time, a new thread cell is added +into the configuration, initialized with the S statement and sharing the +same environment with the parent thread. Note that the newly created +thread cell is torn. That means that the remaining cells are added +and initialized automatically as described in the definition of SIMPLE's +configuration. This is part of K's configuration abstraction mechanism.

      +
      k
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+             ...</thread>)
+
      +

      Thread termination

      +

      Dually to the above, when a thread terminates its assigned computation (the +contents of its k cell) is empty, so the thread can be dissolved. +However, since no discipline is imposed on how locks are acquired and released, +it can be the case that a terminating thread still holds locks. Those locks +must be released, so other threads attempting to acquire them do not deadlock. +We achieve that by removing all the locks held by the terminating thread in its +holds cell from the set of busy locks in the busy cell +(keys(H) returns the domain of the map H as a set, that is, only +the locks themselves ignoring their multiplicity). As seen below, a lock is +added to the busy cell as soon as it is acquired for the first time +by a thread. The unique identifier of the terminated thread is also collected +into the terminated cell, so the join construct knows which +threads have terminated.

      +
      k
        rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
      +

      Thread joining

      +

      Thread joining is now straightforward: all we need to do is to check whether +the identifier of the thread to be joined is in the terminated cell. +If yes, then the join statement dissolves and the joining thread +continues normally; if not, then the joining thread gets stuck.

      +
      k
        rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
      +

      Acquire lock

      +

      There are two cases to distinguish when a thread attempts to acquire a lock +(in SIMPLE any value can be used as a lock):
      +(1) The thread does not currently have the lock, in which case it has to +take it provided that the lock is not already taken by another thread (see +the side condition of the first rule).
      +(2) The thread already has the lock, in which case it just increments its +counter for the lock (the locks are re-entrant). These two cases are captured +by the two rules below:

      +
      k
        rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    requires (notBool(V in Busy))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N => N +Int 1) ...</holds>
+
      +

      Release lock

      +

      Similarly, there are two corresponding cases to distinguish when a thread +releases a lock:
      +(1) The thread holds the lock more than once, in which case all it needs to do +is to decrement the lock counter.
      +(2) The thread holds the lock only once, in which case it needs to remove it +from its holds cell and also from the the shared busy cell, +so other threads can acquire it if they need to.

      +
      k
        rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N -Int 1) ...</holds>
+    requires N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
      +

      Rendezvous synchronization

      +

      In addition to synchronization through acquire and release of locks, SIMPLE +also provides a construct for rendezvous synchronization. A thread whose next +statement to execute is rendezvous(V) gets stuck until another +thread reaches an identical statement; when that happens, the two threads +drop their rendezvous statements and continue their executions. If three +threads happen to have an identical rendezvous statement as their next +statement, then precisely two of them will synchronize and the other will +remain blocked until another thread reaches a similar rendezvous statement. +The rule below is as simple as it can be. Note, however, that, again, it is +K's mechanism for configuration abstraction that makes it work as desired: +since the only cell which can multiply containing a k cell inside is +the thread cell, the only way to concretize the rule below to the +actual configuration of SIMPLE is to include each k cell in a +thread cell.

      +
      k
        rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+
      +

      Auxiliary declarations and operations

      +

      In this section we define all the auxiliary constructs used in the +above semantics.

      +

      Making declarations

      +

      The mkDecls auxiliary construct turns a list of identifiers +and a list of values in a sequence of corresponding variable +declarations.

      +
      k
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+
      +

      Location lookup

      +

      The operation below is straightforward. Note that we place it in the same +lookup group as the variable lookup rule defined above. This way, +both rules will be considered transitions when we include the lookup +tag in the transition option of kompile.

      +
      k
        syntax Exp ::= lookup(Int)
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
      +

      Environment recovery

      +

      We have already discussed the environment recovery auxiliary operation in the +IMP++ tutorial:

      +
      k
      // TODO: eliminate the env wrapper, like we did in IMP++
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k> <env> _ => Env </env>
+
      +

      While theoretically sufficient, the basic definition for environment +recovery alone is suboptimal. Consider a loop while (E)S, +whose semantics (see above) was given by unrolling. S +is a block. Then the semantics of blocks above, together with the +unrolling semantics of the while loop, will yield a computation +structure in the k cell that increasingly grows, adding a new +environment recovery task right in front of the already existing sequence of +similar environment recovery tasks (this phenomenon is similar to the ``tail +recursion'' problem). Of course, when we have a sequence of environment +recovery tasks, we only need to keep the last one. The elegant rule below +does precisely that, thus avoiding the unnecessary computation explosion +problem:

      +
      k
        rule (setEnv(_) => .) ~> setEnv(_)
+
      +

      In fact, the above follows a common convention in K for recovery +operations of cell contents: the meaning of a computation task of the form +cell(C) that reaches the top of the computation is that the current +contents of cell cell is discarded and gets replaced with C. We +did not add support for these special computation tasks in our current +implementation of K, so we need to define them as above.

      +

      lvalue and loc

      +

      For convenience in giving the semantics of constructs like the increment and +the assignment, that we want to operate the same way on variables and on +array elements, we used an auxiliary lvalue(E) construct which was +expected to evaluate to the lvalue of the expression E. This is only +defined when E has an lvalue, that is, when E is either a variable or +evaluates to an array element. lvalue(E) evaluates to a value of +the form loc(L), where L is the location where the value of E +can be found; for clarity, we use loc to structurally distinguish +natural numbers from location values. In giving semantics to lvalue +there are two cases to consider. (1) If E is a variable, then all we need +to do is to grab its location from the environment. (2) If E is an array +element, then we first evaluate the array and its index in order to identify +the exact location of the element of concern, and then return that location; +the last rule below works because its preceding context declarations ensure +that the array and its index are evaluated, and then the rule for array lookup +(defined above) rewrites the evaluated array access construct to its +corresponding store lookup operation.

      +
      k
      // For parsing reasons, we prefer to allow lvalue to take a K
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+// Local variable
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+// Array element: evaluate the array and its index;
+// then the array lookup rule above applies.
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+// Finally, return the address of the desired object member
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
      +

      Initializing multiple locations

      +

      The following operation initializes a sequence of locations with the same +value:

      +
      k
        syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  requires N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  requires N <=Int M
+
      +

      The semantics of SIMPLE is now complete. Make sure you kompile the +definition with the right options in order to generate the desired model. +No kompile options are needed if you only only want to execute the definition +(and thus get an interpreter), but if you want to search for a different +program behaviors then you need to kompile with the transition option +including rule groups such as lookup, increment, acquire, etc. See the +IMP++ tutorial for what the transition option means how to use it.

      +
      k
      endmodule
+
      +

      Go to Lesson 2, SIMPLE typed static

      +
      +
      + + +
      + +
      +
      + + SIMPLE — Untyped + +
      +
      + + Abstract +
      + + Syntax +
      + + Identifiers +
      + + Declarations +
      + + Expressions +
      + + Statements +
      + + Desugared Syntax +
      + + Basic Semantic Infrastructure +
      + + Values +
      + + Configuration +
      + + Declarations and Initialization +
      + + Variable Declaration +
      + + Array Declaration +
      + + Function declaration +
      + + Expressions +
      + + Variable lookup +
      + + Variable/Array increment +
      + + Arithmetic operators +
      + + Array lookup +
      + + Size of an array +
      + + Function call +
      + + Read +
      + + Assignment +
      + + Statements +
      + + Blocks +
      + + Sequential composition +
      + + Expression statements +
      + + Conditional +
      + + While loop +
      + + Print +
      + + Exceptions +
      + + Threads +
      + + Thread creation +
      + + Thread termination +
      + + Thread joining +
      + + Acquire lock +
      + + Release lock +
      + + Rendezvous synchronization +
      + + Auxiliary declarations and operations +
      + + Making declarations +
      + + Location lookup +
      + + Environment recovery +
      + + lvalue and loc +
      + + Initializing multiple locations +
      +
      +
      + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/NOTES/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/NOTES/index.html new file mode 100644 index 00000000000..cf309f6f3a5 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/NOTES/index.html @@ -0,0 +1,380 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      While we disallow global variables with the same name, and that includes +vector variables, we currently do not check that function names are distinct +from each other and from other global variables. Since we can pass functions +around through their names, this can be problematic. May want to make this +into an exercise in the future.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/exercises/functions-with-throws/tests/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/exercises/functions-with-throws/tests/index.html new file mode 100644 index 00000000000..1ad506bcc0f --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/exercises/functions-with-throws/tests/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The programs in this folder are typed variants of the SIMPLE untyped programs. +These programs will be executed both with the dynamic and with the static +semantics of the typed SIMPLE language. Each of the semantics contains its +own results folder showing the expected results of executing these programs.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/simple-typed-static/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/simple-typed-static/index.html new file mode 100644 index 00000000000..a2b19d2275b --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/1_static/simple-typed-static/index.html @@ -0,0 +1,1150 @@ + + + + + + + + + + + + + + +SIMPLE — Typed — Static | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      SIMPLE — Typed — Static

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K definition of the static semantics of the typed SIMPLE +language, or in other words, a type system for the typed SIMPLE +language in K. We do not re-discuss the various features of the +SIMPLE language here. The reader is referred to the untyped version of +the language for such discussions. We here only focus on the new and +interesting problems raised by the addition of type declarations, and +what it takes to devise a type system/checker for the language.

      +

      When designing a type system for a language, no matter in what +paradigm, we have to decide upon the intended typing policy. Note +that we can have multiple type systems for the same language, one for +each typing policy. For example, should we accept programs which +don't have a main function? Or should we allow functions that do not +return explicitly? Or should we allow functions whose type expects +them to return a value (say an int) to use a plain +return; statement, which returns no value, like in C? +And so on and so forth. Typically, there are two opposite tensions +when designing a type system. On the one hand, you want your type +system to be as permissive as possible, that is, to accept as many +programs that do not get stuck when executed with the untyped +semantics as possible; this will keep the programmers using your +language happy. On the other hand, you want your type system to have +a reasonable performance when implemented; this will keep both the +programmers and the implementers of your language happy. For example, +a type system for rejecting programs that could perform +division-by-zero is not expected to be feasible in general. A simple +guideline when designing typing policies is to imagine how the +semantics of the untyped language may get stuck and try to prevent +those situations from happening.

      +

      Before we give the K type system of SIMPLE formally, we discuss, +informally, the intended typing policy:

      +
        +
      • +

        Each program should contain a main() function. Indeed, +the untyped SIMPLE semantics will get stuck on any program which does +not have a main function.

        +
        • +
      • +

        Each primitive value has its own type, which can be int +bool, or string. There is also a type void +for nonexistent values, for example for the result of a function meant +to return no value (but only be used for its side effects, like a +procedure).

        +
        • +
      • +

        The syntax of untyped SIMPLE is extended to allow type +declarations for all the variables, including array variables. This is +done in a C/Java-style. For example, int x; or +int x=7, y=x+3;, or int[][][] a[10,20]; +(the latter defines a 10 × 20 matrix of arrays of integers). +Recall from untyped SIMPLE that, unlike in C/Java, our multi-dimensional +arrays use comma-separated arguments, although they have the array-of-array +semantics.

        +
        • +
      • +

        Functions are also typed in a C/Java style. However, since in SIMPLE +we allow functions to be passed to and returned by other functions, we also +need function types. We will use the conventional higher-order arrow-notation +for function types, but will separate the argument types with commas. For +example, a function returning an array of bool elements and +taking as argument an array x of two-integer-argument functions +returning an integer, is declared using a syntax of the form +bool[] f(((int,int)->int)[] x) { ... } +and has the type ((int,int)->int)[] -> bool[].

        +
        • +
      • +

        We allow any variable declarations at the top level. Functions +can only be declared at the top level. Each function can only access the +other functions and variables declared at the top level, or its own locally +declared variables. SIMPLE has static scoping.

        +
        • +
      • +

        The various expression and statement constructs take only elements of +the expected types.

        +
        • +
      • +

        Increment and assignment can operate both on variables and on array +elements. For example, if f has type int->int[][] and +function g has the type int->int, then the +increment expression ++f(7)[g(2),g(3)] is valid.

        +
        • +
      • +

        Functions should only return values of their declared result +type. To give the programmers more flexibility, we allow functions to +use return; statements to terminate without returning an +actual value, or to not explicitly use any return statement, +regardless of their declared return type. This flexibility can be +handy when writing programs using certain functions only for their +side effects. Nevertheless, as the dynamic semantics shows, a return +value is automatically generated when an explicit return +statement is not encountered.

        +
        • +
      • +

        For simplicity, we here limit exceptions to only throw and catch +integer values. We let it as an exercise to the reader to extend the +semantics to allow throwing and catching arbitrary-type exceptions. +Like in programming languages like Java, one can go even further and +define a semantics where thrown exceptions are propagated through +try-catch statements until one of the corresponding type is found. +We will do this when we define the KOOL language, not here. +To keep the definition if SIMPLE simple, here we do not attempt to +reject programs which throw uncaught exceptions.

        +
        • +
      +

      Like in untyped SIMPLE, some constructs can be desugared into a +smaller set of basic constructs. In general, it should be clear why a +program does not type by looking at the top of the k cells in +its stuck configuration.

      +
      k
      module SIMPLE-TYPED-STATIC-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      The syntax of typed SIMPLE extends that of untyped SIMPLE with support +for declaring types to variables and functions.

      +
      k
        syntax Id ::= "main" [token]
+
      +

      Types

      +

      Primitive, array and function types, as well as lists (or tuples) of types. +The lists of types are useful for function arguments.

      +
      k
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Type "[" "]"
+                | "(" Type ")"             [bracket]
+                > Types "->" Type
+
+  syntax Types ::= List{Type,","}          [klabel(exps)]
+
      +

      Declarations

      +

      Variable and function declarations have the expected syntax. For variables, +we basically just replaced the var keyword of untyped SIMPLE with a +type. For functions, besides replacing the function keyword with a +type, we also introduce a new syntactic category for typed variables, +Param, and lists over it.

      +
      k
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";"
+                | Type Id "(" Params ")" Block
+
      +

      Expressions

      +

      The syntax of expressions is identical to that in untyped SIMPLE, +except for the logical conjunction and disjunction which have +different strictness attributes, because they now have different +evaluation strategies.

      +
      k
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict, left]
+               | Exp "||" Exp            [strict, left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
      +

      Note that spawn has not been declared strict. This may +seem unexpected, because the child thread shares the same environment +with the parent thread, so from a typing perspective the spawned +statement makes the same sense in a child thread as it makes in the +parent thread. The reason for not declaring it strict is because we +want to disallow programs where the spawned thread calls the +return statement, because those programs would get stuck in +the dynamic semantics. The type semantics of spawn below will reject +such programs.

      +

      We still need lists of expressions, defined below, but note that we do +not need lists of identifiers anymore. They have been replaced by the lists +of parameters.

      +
      k
        syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+
      +

      Statements

      +

      The statements have the same syntax as in untyped SIMPLE, except for +the exceptions, which now type their parameter. Note that, unlike in untyped +SIMPLE, all statement constructs which have arguments and are not desugared +are strict, including the conditional and the while. Indeed, from a +typing perspective, they are all strict: first type their arguments and then +type the actual construct.

      +
      k
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                                  [strict]
+                | "if" "(" Exp ")" Block "else" Block      [avoid, strict]
+                | "if" "(" Exp ")" Block                   [macro]
+                | "while" "(" Exp ")" Block                [strict]
+                | "for" "(" Stmt Exp ";" Exp ")" Block     [macro]
+                | "return" Exp ";"                         [strict]
+                | "return" ";"
+                | "print" "(" Exps ")" ";"                 [strict]
+                | "try" Block "catch" "(" Param ")" Block  [strict(1)]
+                | "throw" Exp ";"                          [strict]
+                | "join" Exp ";"                           [strict]
+                | "acquire" Exp ";"                        [strict]
+                | "release" Exp ";"                        [strict]
+                | "rendezvous" Exp ";"                     [strict]
+
      +

      Note that the sequential composition is now sequentially strict, +because, unlike in the dynamic semantics where statements dissolved, +they now reduce to the stmt type, which is a result.

      +
      k
        syntax Stmt ::= Stmt Stmt                             [seqstrict, right]
+
      +

      Desugaring macros

      +

      We use the same desugaring macros like in untyped SIMPLE, but, of +course, including the types of the involved variables.

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule for(Start Cond; Step) {} => {Start while(Cond){Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+endmodule
+
+
+module SIMPLE-TYPED-STATIC
+  imports SIMPLE-TYPED-STATIC-SYNTAX
+  imports DOMAINS
+
      +

      Static semantics

      +

      Here we define the type system of SIMPLE. Like concrete semantics, +type systems defined in K are also executable. However, K type +systems turn into type checkers instead of interpreters when executed.

      +

      The typing process is done in two (overlapping) phases. In the first +phase the global environment is built, which contains type bindings +for all the globally declared variables and functions. For functions, +the declared types will be ``trusted'' during the first phase and +simply bound to their corresponding function names and placed in the +global type environment. At the same time, type-checking tasks that +the function bodies indeed respect their claimed types are generated. +All these tasks are (concurrently) verified during the second phase. +This way, all the global variable and function declarations are +available in the global type environment and can be used in order to +type-check each function code. This is consistent with the semantics +of untyped SIMPLE, where functions can access all the global variables +and can call any other function declared in the same program. The +two phases may overlap because of the K concurrent semantics. For +example, a function task can be started while the first phase is still +running; moreover, it may even complete before the first phase does, +namely when all the global variables and functions that it needs have +already been processed and made available in the global environment by +the first phase task.

      +

      Extended syntax and results

      +

      The idea is to start with a configuration holding the program to type +in one of its cells, then apply rewrite rules on it mixing types and +language syntax, and eventually obtain a type instead of the original +program. In other words, the program reduces to its type using +the K rules giving the type system of the language. In doing so, +additional typing tasks for function bodies are generated and solved +the same way. If this rewriting process gets stuck, then we say that +the program is not well-typed. Otherwise the program is well-typed +(by definition). We did not need types for statements and for blocks +as part of the typed SIMPLE syntax, because programmers are not allowed +to use such types explicitly. However, we are going to need them in the +type system, because blocks and statements reduce to them.

      +

      We start by allowing types to be used inside expressions and statements in +our language. This way, types can be used together with language syntax in +subsequent K rules without any parsing errors. Like in the type system of +IMP++ in the K tutorial, we prefer to group the block and statement types +under one syntactic sub-category of types, because this allows us to more +compactly state that certain terms can be either blocks or statements. Also, +since programs and fragments of program will reduce to their types, in order +for the strictness and context declarations to be executable we state that +types are results (same like we did in the IMP++ tutorial).

      +
      k
        syntax Exp ::= Type
+  syntax Exps ::= Types
+  syntax BlockOrStmtType ::= "block" | "stmt"
+  syntax Type ::= BlockOrStmtType
+  syntax Block ::= BlockOrStmtType
+  syntax KResult ::= Type
+                   | Types    //TODO: remove this, eventually
+
      +

      Configuration

      +

      The configuration of our type system consists of a tasks cell +holding various typing task cells, and a global type environment. +Each task includes a k cell holding the code to type, a tenv +cell holding the local type environment, and a return cell holding +the return type of the currently checked function. The latter is needed in +order to check whether return statements return values of the expected type. +Initially, the program is placed in a k cell inside a +task cell. Since the cells with multiplicity ? are not +included in the initial configuration, the task cell holding +the original program in its k cell will contain no other +subcells.

      +
      k
        configuration <T color="yellow">
+                  <tasks color="orange">
+                    <task multiplicity="*" color="yellow" type="Set">
+                      <k color="green"> $PGM:Stmt </k>
+                      <tenv multiplicity="?" color="cyan"> .Map </tenv>
+                      <returnType multiplicity="?" color="black"> void </returnType>
+                    </task>
+                  </tasks>
+//                  <br/>
+                  <gtenv color="blue"> .Map </gtenv>
+                </T>
+
      +

      Variable declarations

      +

      Variable declarations type as statements, that is, they reduce to the +type stmt. There are only two cases that need to be +considered: when a simple variable is declared and when an array +variable is declared. The macros at the end of the syntax module +above take care of reducing other variable declarations, including +ones where the declared variables are initialized, to only these two +cases. The first case has two subcases: when the variable declaration +is global (i.e., the task cell contains only the k +cell), in which case it is added to the global type environment +checking at the same time that the variable has not been already +declared; and when the variable declaration is local (i.e., a +tenv cell is available), in which case it is simply added to +the local type environment, possibly shadowing previous homonymous +variables. The third case reduces to the second, incrementally moving +the array dimension into the type until the array becomes a simple +variable.

      +
      k
        rule <task> <k> T:Type X:Id; => stmt ...</k> </task>
+       <gtenv> Rho (.Map => X |-> T) </gtenv>
+    requires notBool(X in keys(Rho))
+  rule <k> T:Type X:Id; => stmt ...</k> <tenv> Rho => Rho[X <- T] </tenv>
+
+  context _:Type _::Exp[HOLE::Exps];
+// The rule below may need to sort E to Exp in the future, if the
+// parser gets stricter; without that information, it may not be able
+// to complete the LHS into T E[int,Ts],.Exps; (and similarly for the RHS)
+  rule T:Type E:Exp[int,Ts:Types]; => T[] E[Ts];
+// I want to write the rule below as _:Type (E:Exp[.Types] => E),
+// but the list completion seems to not work well with that.
+  rule T:Type E:Exp[.Types]; => T E;
+
      +

      Function declarations

      +

      Functions are allowed to be declared only at the top level (the +task cell holds only its k subcell). Each function +declaration reduces to a variable declaration (a binding of its name +to its declared function type), but also adds a task into the +tasks cell. The task consists of a typing of the statement +declaring all the function parameters followed by the function body, +together with the expected return type of the function. The +getTypes and mkDecls functions, defined at the end of +the file in the section on auxiliary operations, extracts the list of +types and makes a sequence of variable declarations from a list of +function parameters, respectively. Note that, although in the dynamic +semantics we include a terminating return statement at the +end of the function body to eliminate from the analysis the case when +the function does not provide an explicit return, we do not need to +include such a similar return statement here. That's because +the return statements type to stmt anyway, and the +entire code of the function body needs to type anyway.

      +
      k
        rule <task> <k> T:Type F:Id(Ps:Params) S => getTypes(Ps)->T F; ...</k> </task>
+       (.Bag => <task>
+               <k> mkDecls(Ps) S </k> <tenv> .Map </tenv> <returnType> T </returnType>
+             </task>)
+
      +

      Checking if main() exists}

      +

      Once the entire program is processed (generating appropriate tasks +to type check its function bodies), we can dissolve the main +task cell (the one holding only a k subcell). Since +we want to enforce that programs include a main function, we also +generate a function task executing main() to ensure that it +types (remove this task creation if you do not want your type system +to reject programs without a main function).

      +
      k
        rule <task> <k> stmt => main(.Exps); </k> (.Bag => <tenv> .Map </tenv>) </task>
+
      +

      Collecting the terminated tasks

      +

      Similarly, once a non-main task (i.e., one which contains a +tenv subcells) is completed using the subsequent rules (i.e., +its k cell holds only the block or stmt +type), we can dissolve its corresponding cell. Note that it is +important to ensure that we only dissolve tasks containing a +tenv cell with the rule below, because the main task should +not dissolve this way! It should do what the above rule says. +In the end, there should be no task cell left in the configuration +when the program correctly type checks.

      +
      k
        rule <task>... <k> _:BlockOrStmtType </k> <tenv> _ </tenv> ...</task> => .Bag
+
      +

      Basic values

      +

      The first three rewrite rules below reduce the primitive values to +their types, as we typically do when we define type systems in K.

      +
      k
        rule _:Int => int
+  rule _:Bool => bool
+  rule _:String => string
+
      +

      Variable lookup

      +

      There are three cases to distinguish for variable lookup: (1) if the +variable is bound in the local type environment, then look its type up +there; (2) if a local environment exists and the variable is not bound +in it, then look its type up in the global environment; (3) finally, +if there is no local environment, meaning that we are executing the +top-level pass, then look the variable's type up in the global +environment, too.

      +
      k
        rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</tenv>
+
+  rule <k> X:Id => T ...</k> <tenv> Rho </tenv> <gtenv>... X |-> T ...</gtenv>
+    requires notBool(X in keys(Rho))
+
+  rule <task> <k> X:Id => T ...</k> </task> <gtenv>... X |-> T ...</gtenv>
+
      +

      Increment

      +

      We want the increment operation to apply to any lvalue, including +array elements, not only to variables. For that reason, we define a +special context extracting the type of the argument of the increment +operation only if that argument is an lvalue. Otherwise the rewriting +process gets stuck. The operation ltype is defined at the +end of this file, in the auxiliary operation section. It essentially +acts as a filter, getting stuck if its argument is not an lvalue and +letting it reduce otherwise. The type of the lvalue is expected to be +an integer in order to be allowed to be incremented, as seen in the +rule ++ int => int below.

      +
      k
        context ++(HOLE => ltype(HOLE))
+  rule ++ int => int
+
      +

      Common expression constructs

      +

      The rules below are straightforward and self-explanatory:

      +
      k
        rule int + int => int
+  rule string + string => string
+  rule int - int => int
+  rule int * int => int
+  rule int / int => int
+  rule int % int => int
+  rule - int => int
+  rule int < int => bool
+  rule int <= int => bool
+  rule int > int => bool
+  rule int >= int => bool
+  rule T:Type == T => bool
+  rule T:Type != T => bool
+  rule bool && bool => bool
+  rule bool || bool => bool
+  rule ! bool => bool
+
      +

      Array access and size

      +

      Array access requires each index to type to an integer, and the +array type to be at least as deep as the number of indexes:

      +
      k
      // NOTE:
+// We used to need parentheses in the RHS, to avoid capturing Ts as an attribute
+// Let's hope that is not a problem anymore.
+
+  rule (T[])[int, Ts:Types] => T[Ts]
+  rule T:Type[.Types] => T
+
      +

      sizeOf only needs to check that its argument is an array:

      +
      k
        rule sizeOf(_T[]) => int
+
      +

      Input/Output

      +

      The read expression construct types to an integer, while print types +to a statement provided that all its arguments type to integers or +strings.

      +
      k
        rule read() => int
+
+  rule print(T:Type, Ts => Ts); requires T ==K int orBool T ==K string
+  rule print(.Types); => stmt
+
      +

      Assignment

      +

      The special context and the rule for assignment below are similar +to those for increment: the LHS of the assignment must be an lvalue +and, in that case, it must have the same type as the RHS, which then +becomes the type of the assignment.

      +
      k
        context (HOLE => ltype(HOLE)) = _
+  rule T:Type = T => T
+
      +

      Function application and return

      +

      Function application requires the type of the function and the +types of the passed values to be compatible. Note that a special case +is needed to handle the no-argument case:

      +
      k
        rule (Ts:Types -> T)(Ts) => T requires Ts =/=K .Types
+  rule (void -> T)(.Types) => T
+
      +

      The returned value must have the same type as the declared +function return type. If an empty return is encountered, than +we should check that we are in a function (and not a thread) +context, that is, a return cell must be available:

      +
      k
        rule <k> return T:Type; => stmt ...</k> <returnType> T </returnType>
+  rule <k> return; => stmt ...</k> <returnType> _ </returnType>
+
      +

      Blocks

      +

      To avoid having to recover type environments after blocks, we prefer +to start a new task for block body, making sure that the new task +is passed the same type environment and return cells. The value +returned by return statements must have the same type as +stated in the return cell. The print variadic +function is allowed to only print integers and strings. The thrown +exceptions can only have integer type.

      +
      k
        rule {} => block
+
+  rule <task> <k> {S} => block ...</k> <tenv> Rho </tenv> R </task>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> R </task>)
+
      +

      Expression statement

      +
      k
        rule _:Type; => stmt
+
      +

      Conditional and while loop

      +
      k
        rule if (bool) block else block => stmt
+  rule while (bool) block => stmt
+
      +

      Exceptions

      +

      We currently force the parameters of exceptions to only be integers. +Moreover, for simplicity, we assume that integer exceptions can be +thrown from anywhere, including from functions which do not define +any try-catch block (with the currently unchecked ‒also for +simplicity‒ expectation that the caller functions would catch those +exceptions).

      +
      k
        rule try block catch(int X:Id) {S} => {int X; S}
+  rule try block catch(int X:Id) {} => {int X;}
+  rule throw int; => stmt
+
      +

      Concurrency

      +

      Nothing special about typing the concurrency constructs, except that +we do not want the spawned thread to return, so we do not include any +return cell in the new task cell for the thread statement. +Same like with the functions above, we do not check for thrown +exceptions which are not caught.

      +
      k
        rule <k> spawn S => int ...</k> <tenv> Rho </tenv>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> </task>)
+  rule join int; => stmt
+  rule acquire _:Type; => stmt
+  rule release _:Type; => stmt
+  rule rendezvous _:Type; => stmt
+
+  rule _:BlockOrStmtType _:BlockOrStmtType => stmt
+
      +

      Auxiliary constructs

      +

      The function mkDecls turns a list of parameters into a +list of variable declarations.

      +
      k
        syntax Stmt ::= mkDecls(Params)  [function]
+  rule mkDecls(T:Type X:Id, Ps:Params) => T X; mkDecls(Ps)
+  rule mkDecls(.Params) => {}
+
      +

      The ltype context allows only expressions which have an +lvalue to evaluate.

      +
      k
        syntax LValue ::= Id
+  rule isLValue(_:Exp[_:Exps]) => true
+  syntax Exp ::= LValue  // K should be able to infer this
+                         // if not added, then it gets stuck with an Id on k cell
+
+// Instead of the second LValue production above you can use a rule:
+//  rule isLValue(_:Exp[_:Exps]) => true
+
+  syntax Exp ::= ltype(Exp)
+//  context ltype(HOLE:LValue)
+// The above context does not work due to some error, so we write instead
+  context ltype(HOLE) requires isLValue(HOLE)
+
      +

      The function getTypes is the same as in SIMPLE typed dynamic.

      +
      k
        syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types   // I would like to not use .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+
+endmodule
+
      +

      Go to Lesson 3, SIMPLE typed dynamic

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/2_dynamic/simple-typed-dynamic/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/2_dynamic/simple-typed-dynamic/index.html new file mode 100644 index 00000000000..680c3715dae --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/2_dynamic/simple-typed-dynamic/index.html @@ -0,0 +1,1142 @@ + + + + + + + + + + + + + + +SIMPLE — Typed — Dynamic | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      SIMPLE — Typed — Dynamic

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K dynamic semantics of the typed SIMPLE language. +It is very similar to the semantics of the untyped SIMPLE, the +difference being that we now dynamically check the typing policy +described in the static semantics of typed SIMPLE. Because of the +dynamic nature of the semantics, we can also perform some additional +checks which were not possible in the static semantics, such as +memory leaks due to accessing an array out of its bounds. We will +highlight the differences between the dynamically typed and the +untyped SIMPLE as we proceed with the semantics. We recommend the +reader to consult the typing policy and the syntax of types discussed +in the static semantics of the typed SIMPLE language.

      +
      k
      module SIMPLE-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      The syntax of typed SIMPLE extends that of untyped SIMPLE with support +for declaring types to variables and functions.

      +

      The syntax below is identical to that of the static semantics of typed +SIMPLE. However, the K strictness attributes are like those of the untyped +SIMPLE, to capture the desired evaluation strategies of the various language +constructs.

      +
      k
        syntax Id ::= "main" [token]
+
      +

      Types

      +
      k
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+  syntax Types ::= List{Type,","}        [klabel(exps)]
+
      +

      Declarations

      +
      k
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";"
+                | Type Id "(" Params ")" Block
+
      +

      Expressions

      +
      k
        syntax Exp ::= Int | Bool | String | Id
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
      +

      Like in the static semantics, there is no need for lists of identifiers +(because we now have lists of parameters).

      +
      k
        syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+
      +

      Statements

      +
      k
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"
+                | "try" Block "catch" "(" Param ")" Block
+            | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+
      +

      The same desugaring macros like in the statically typed SIMPLE.

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule for(Start Cond; Step) {} => {Start while(Cond){Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+endmodule
+
+
+module SIMPLE-TYPED-DYNAMIC
+  imports SIMPLE-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS
+
      +

      Semantics

      +

      Values and results

      +

      These are similar to those of untyped SIMPLE, except that the array +references and the function abstrations now also hold their types. +These types are needed in order to easily compute the type of any +value in the language (see the auxiliary typeOf operation at +the end of this module).

      +
      k
        syntax Val ::= Int | Bool | String
+               | array(Type,Int,Int)
+               | lambda(Type,Params,Stmt)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+                   | Vals  // TODO: should not need this
+
      +

      Configuration

      +

      The configuration is almost identical to that of untyped SIMPLE, +except for a return cell inside the control cell. +This return cell will hold, like in the static semantics of +typed SIMPLE, the expected type of the value returned by the function +being executed. The contents of this cell will be set whenever a +function is invoked and will be checked whenever the evaluation of the +function body encounters an explicit return statement.

      +
      k
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+
+  syntax ControlCell
+  syntax ControlCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" color="yellow" type="Map">
+                      <id color="pink"> 0 </id>
+                      <k color="green"> ($PGM:Stmt ~> execute) </k>
+//                      <br/>
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                        <returnType color="LimeGreen"> void </returnType>
+                       </control>
+//                      <br/>
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                    </thread>
+                  </threads>
+//                  <br/>
+                  <genv color="pink"> .Map </genv>
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set</busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                </T>
+
      +

      Declarations and Initialization

      +

      Variable Declaration

      +

      The undefined construct is now parameterized by a type. +A main difference between untyped SIMPLE and dynamically typed SIMPLE +is that the latter assigns a type to each of its locations and that +type cannot be changed during the execution of the program. We do not +do any memory management in our semantic definitions here, so +locations cannot be reclaimed, garbage collected and/or reused. Each +location corresponds precisely to an allocated variable or array +element, whose type was explicitly or implicitly declared in the +program and does not change. It is therefore safe to type each +location and then never allow that type to change. The typed +undefined values effectively assign both a type and an undefined value +to a location.

      +
      k
        syntax KItem ::= undefined(Type)  [latex(\bot_{#1})]
+
+  rule <k> T:Type X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
      +

      Array Declaration

      +

      The dynamic semantics of typed array declarations is similar to that +in untyped SIMPLE, but we have to make sure that we associate the +right type to the allocated locations.

      +
      k
        rule <k> T:Type X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(T, L +Int 1, N)
+                          (L +Int 1)...(L +Int N) |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    when N >=Int 0
+
+  context _:Type _::Exp[HOLE::Exps];
+
      +

      The desugaring of multi-dimensional arrays into unidimensional +ones is also similar to that in untyped SIMPLE, although we have to +make sure that all the declared variables have the right types. The +auxiliary operation T<Vs>, defined at the end of the file, +adds the length of Vs dimensions to the type T.

      +
      k
      // TODO: Check the desugaring below to be consistent with the one for untyped simple
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule T:Type X:Id[N1:Int, N2:Int, Vs:Vals];
+    => T[]<Vs> X[N1];
+       {
+         T[][]<Vs> $1=X;
+         for(int $2=0; $2 <= N1 - 1; ++$2) {
+           T X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
      +

      Function declaration

      +

      Store all function parameters, as well as the return type, as part +of the lambda abstraction. In the spirit of dynamic typing, we will +make sure that parameters are well typed when the function is invoked.

      +
      k
        rule <k> T:Type F:Id(Ps:Params) S => . ...</k>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> lambda(T, Ps, S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+
      +

      Calling main()

      +

      When done with the first pass, call main().

      +
      k
        syntax KItem ::= "execute"
+  rule <k> execute => main(.Exps); </k>
+       <env> Env </env>
+       <genv> .Map => Env </genv>
+
      +

      Expressions

      +

      Variable lookup

      +
      k
        rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
      +

      Variable/Array increment

      +
      k
        context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
      +

      Arithmetic operators

      +
      k
        rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
      +

      Array lookup

      +

      Check array bounds, as part of the dynamic typing policy.

      +
      k
      // Same comment as for simple untyped regarding [anywhere]
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+// Same comment as for simple untyped regarding [anywhere]
+  rule array(_:Type, L:Int, M:Int)[N:Int] => lookup(L +Int N)
+    when N >=Int 0 andBool N <Int M  [anywhere]
+
      +

      Size of an array

      +
      k
        rule sizeOf(array(_,_,N)) => N
+
      +

      Function call

      +

      Define function call and return together, to see their relationship. +Note that the operation mkDecls now declares properly typed +instantiated variables, and that the semantics of return also +checks that that type of the returned value is expected one.

      +
      k
        syntax KItem ::= (Type,Map,K,ControlCellFragment)
+
+  rule <k> lambda(T,Ps,S)(Vs:Vals) ~> K => mkDecls(Ps,Vs) S return; </k>
+       <control>
+         <fstack> .List => ListItem((T',Env,K,C)) ...</fstack>
+         <returnType> T' => T </returnType>
+         C
+       </control>
+       <env> Env => GEnv </env>
+       <genv> GEnv </genv>
+
+  rule <k> return V:Val; ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem((T',Env,K,C)) => .List ...</fstack>
+         <returnType> T => T' </returnType>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+    when typeOf(V) ==K T   // check the type of the returned value
+
      +

      Like the undefined above, nothing also gets +tagged with a type now. The empty return statement is +completed to return the nothing value tagged as expected.

      +
      k
        syntax Val ::= nothing(Type)
+  rule <k> return; => return nothing(T); ...</k> <returnType> T </returnType>
+
      +

      Read

      +
      k
        rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
      +

      Assignment

      +

      The assignment now checks that the type of the assigned location is +preserved:

      +
      k
        context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (V' => V) ...</store>
+    when typeOf(V) ==K typeOf(V')  [group(assignment)]
+
      +

      Statements

      +

      Blocks

      +
      k
        rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
      +

      Sequential composition

      +
      k
        rule S1:Stmt S2:Stmt => S1 ~> S2
+
      +

      Expression statements

      +
      k
        rule _:Val; => .
+
      +

      Conditional

      +
      k
        rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
      +

      While loop

      +
      k
        rule while (E) S => if (E) {S while(E)S}
+
      +

      Print

      +

      We only allow printing integers and strings:

      +
      k
        rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    when typeOf(V) ==K int orBool typeOf(V) ==K string  [group(print)]
+  rule print(.Vals); => .
+
      +

      Exceptions

      +

      Exception parameters are now typed, but note that the semantics below +works correctly only when the thrown exception has the same type as +the innermost try-catch paramete. To keep things simple, for the time +being we can assume that SIMPLE only throws and catches integer +values, in which case our semantics below works fine:

      +
      k
        syntax KItem ::= (Param,Stmt,K,Map,ControlCellFragment)  // Param instead of Id
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(P) S2 => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem((P, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { T X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem((T:Type X:Id, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
      +

      Threads

      +

      Thread creation

      +
      k
         rule <thread>...
+          <k> spawn S => !T:Int +Int 1 ...</k>
+          <env> Env </env>
+        ...</thread>
+        (.Bag => <thread>...
+                <k> S </k>
+                <env> Env </env>
+                <id> !T +Int 1 </id>
+              ...</thread>)
+
      +

      Thread termination

      +
      k
         rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+        <busy> Busy => Busy -Set keys(H) </busy>
+        <terminated>... .Set => SetItem(T) ...</terminated>
+
      +

      Thread joining

      +
      k
         rule <k> join T:Int; => . ...</k>
+        <terminated>... SetItem(T) ...</terminated>
+
      +

      Acquire lock

      +
      k
         rule <k> acquire V:Val; => . ...</k>
+        <holds>... .Map => V |-> 0 ...</holds>
+        <busy> Busy (.Set => SetItem(V)) </busy>
+     when (notBool(V in Busy:Set))  [group(acquire)]
+
+   rule <k> acquire V; => . ...</k>
+        <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
      +

      Release lock

      +
      k
         rule <k> release V:Val; => . ...</k>
+        <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+      when N >Int 0
+
+   rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+        <busy>... SetItem(V) => .Set ...</busy>
+
      +

      Rendezvous synchronization

      +
      k
         rule <k> rendezvous V:Val; => . ...</k>
+        <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+
      +

      Auxiliary declarations and operations

      +

      Turns a list of parameters and a list of instance values for them +into a list of variable declarations.

      +
      k
        syntax Stmt ::= mkDecls(Params,Vals)  [function]
+  rule mkDecls((T:Type X:Id, Ps:Params), (V:Val, Vs:Vals))
+    => T X=V; mkDecls(Ps,Vs)
+  rule mkDecls(.Params,.Vals) => {}
+
      +

      Location lookup.

      +
      k
        syntax Exp ::= lookup(Int)  // see NOTES.md for why Exp instead of KItem
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
      +

      Environment recovery.

      +
      k
      // TODO: same comment regarding setEnv(...) as for simple untyped
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+
      +

      lvalue and loc

      +
      k
        syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k>  <env>... X |-> L:Int ...</env>
+
+  //context lvalue(_[HOLE])
+  //context lvalue(HOLE[_])
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
      +

      Adds the corresponding depth to an array type

      +
      k
        syntax Type ::= Type "<" Vals ">"  [function]
+  rule T:Type<_,Vs:Vals> => T[]<Vs>
+  rule T:Type<.Vals> => T
+
      +

      Sequences of locations.

      +
      k
        syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  when N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  when N <=Int M
+
+// Type of a value.
+  syntax Type ::= typeOf(K)  [function]
+  rule typeOf(_:Int) => int
+  rule typeOf(_:Bool) => bool
+  rule typeOf(_:String) => string
+  rule typeOf(array(T,_,_)) => (T[])   // () needed! K parses [] as "no tags"
+  rule typeOf(lambda(T,Ps,_)) => getTypes(Ps) -> T
+  rule typeOf(undefined(T)) => T
+  rule typeOf(nothing(T)) => T
+
      +

      List of types of a parameter.

      +
      k
        syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types   // I would like to not use .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+endmodule
+
      +
      +
      + + +
      + +
      +
      + + SIMPLE — Typed — Dynamic + +
      +
      + + Abstract +
      + + Syntax +
      + + Types +
      + + Declarations +
      + + Expressions +
      + + Statements +
      + + Semantics + + +
      +
      + + Declarations and Initialization + + +
      + +
      +
      + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/programs/index.html b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/programs/index.html new file mode 100644 index 00000000000..b2a95c1ad58 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/1_simple/2_typed/programs/index.html @@ -0,0 +1,379 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      The programs in this folder are typed variants of the SIMPLE untyped programs. +These programs will be executed both with the dynamic and with the static +semantics of the typed SIMPLE language. Each of the semantics contains its +own results folder showing the expected results of executing these programs.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/2_kool/1_untyped/kool-untyped/index.html b/k-distribution/pl-tutorial/2_languages/2_kool/1_untyped/kool-untyped/index.html new file mode 100644 index 00000000000..1a601ac7298 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/2_kool/1_untyped/kool-untyped/index.html @@ -0,0 +1,1522 @@ + + + + + + + + + + + + + + +KOOL — Untyped | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      KOOL — Untyped

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K semantic definition of the untyped KOOL language. KOOL +is aimed at being a pedagogical and research language that captures +the essence of the object-oriented programming paradigm. Its untyped +variant discussed here is simpler than the typed one, ignoring several +intricate aspects of types in the presence of objects. A program +consists of a set of class declarations. Each class can extend at +most one other class (KOOL is single-inheritance). A class can +declare a set of fields and a set of methods, all public and called +the class' members. Specifically, KOOL includes the +following features:

      +
        +
      • +

        Class declarations, where a class may or may not explicitly +extend another class. In case a class does not explicitly extend +another class, then it is assumed that it extends the default top-most +and empty (i.e., no members) class called Object. Each class +is required to declare precisely one homonymous method, called its +constructor. Each valid program should contain one class +named Main, whose constructor, Main(), takes no +arguments. The execution of a program consists of creating an object +instance of class Main and invoking the constructor +Main() on it, that is, of executing new Main();.

        +
        • +
      • +

        All features of SIMPLE (see examples/simple/untyped), +i.e., multidimensional arrays, function (here called "method") +abstractions with call-by-value parameter passing style and static +scoping, blocks with locals, input/output, parametric exceptions, and +concurrency via dynamic thread creation/termination and synchronization. +The only change in the syntax of SIMPLE when imported in KOOL is the +function declaration keyword, function, which is changed into +method. The exact same desugaring macros from SIMPLE are +also included in KOOL. We can think of KOOL's classes as embedding +SIMPLE programs (extended with OO constructs, as discussed next).

        +
        • +
      • +

        Object creation using the new C(e1,...,en) +expression construct. An object instance of class C is first +created and then the constructor C(e1,...,en) is implicitly +called on that object. KOOL only allows (and requires) one +constructor per class. The class constructor can be called either +implicitly during a new object creation for the class, or explicitly. +The superclass constructor is not implicitly invoked when a +class constructor is invoked; if you want to invoke the superclass +constructor from a subclass constructor then you have to do it +explicitly.

        +
        • +
      • +

        An expression construct this, which evaluates to the +current object.

        +
        • +
      • +

        An expression construct super, which is used (only) in +combination with member lookup (see next) to refer to a superclass +field or method.

        +
        • +
      • +

        A member lookup expression construct e.x, where e +is an expression (either an expression expected to evaluate to an object +or the super construct) and x is a class member name, +that is, a field or a method name.

        +
        • +
      • +

        Expression constructs e instanceOf C and +(C) e, where e is an expression expected +to evaluate to an object and C a class name. The former +tells whether the class of e is a subclass of C, +that is, whether e can be used as an instance of C, +and the latter changes the class of e to C. These +operations always succeed: the former returns a Boolean value, while +the latter changes the current class of e to C +regardless of whether it is safe to do so or not. The typed version +of KOOL will check the safety of casting by ensuring that the instance +class of the object is a subclass of C. In untyped KOOL we +do not want to perform this check because we want to allow the +programmer maximum of flexibility: if one always accesses only +available members, then the program can execute successfully despite +the potentially unsafe cast.

        +
        • +
      +

      There are some specific aspects of KOOL that need to be discussed.

      +

      First, KOOL is higher-order, allowing function abstractions to be +treated like any other values in the language. For example, if +m is a method of object e then e.m +evaluates to the corresponding function abstraction. The function +abstraction is in fact a closure, because in addition to the method +parameters and body it also encapsulates the object value (i.e., the +environment of the object together with its current class—see below) +that e evaluates to. This way, function abstractions can be +invoked anywhere and have the capability to change the state of their +object. For example, if m is a method of object e +which increments a field c of e when invoked, and if +getm is another method of e which simply returns +m when invoked, then the double application +(e.getm())() has the same effect as e.m(), that is, +increments the counter c of e. Note that the +higher-order nature of KOOL was not originally planned; it came as a +natural consequence of evaluating methods to closures and we decided +to keep it. If you do not like it then do not use it.

      +

      Second, since all the fields and methods are public in KOOL and since +they can be redeclared in subclasses, it is not immediately clear how +to lookup the member x when we write e.x and +e is different from super. We distinguish two cases, +depending on whether e.x occurs in a method invocation +context (i.e., e.x(...)) or in a field context. KOOL has +dynamic method dispatch, so if e.x is invoked as a method +then x will be searched for starting with the instance class of +the object value to which e evaluates. If e.x +occurs in a non-method-invocation context then x will be +treated as a field (although it may hold a method closure due to the +higher-order nature of KOOL) and thus will be searched starting with +the current class of the object value of e (which, because of +this and casting, may be different from its instance class). +In order to achieve the above, each object value will consist of a +pair holding the current class of the object and an environment stack +with one layer for each class in the object's instance class hierarchy.

      +

      Third, although KOOL is dynamic method dispatch, its capabilities +described above are powerful enough to allow us to mimic static +method dispatch. For example, suppose that you want to invoke method +m() statically. Then all you need to do is to declare a +local variable and bind it to m, for example var staticm = m;, and +then call staticm(). This works because +staticm is first bound to the method closure that m +evaluates to, and then looked up as any local variable when invoked. +We only enable the dynamic method dispatch when we have an object +member on an application position, e.g., m().

      +

      In what follows, we limit our comments to the new, KOOL-specific +aspects of the language. We refer the reader to the untyped SIMPLE +language for documentation on the the remaining features, because +those were all borrowed from SIMPLE.

      +
      k
      module KOOL-UNTYPED-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      The syntax of KOOL extends that of SIMPLE with object-oriented +constructs. We removed from the K annotated syntax of SIMPLE two +constructs, namely the one for function declarations (because we want +to call them methods now) and the one for function application +(because application is not strict in the first argument +anymore—needs to initiate dynamic method dispatch). The additional +syntax includes:

      +
        +
      • First, we need a new dedicated identifier, Object, for +the default top-most class.
        • +
      • Second, we rename the function keyword of SIMPLE into method.
        • +
      • Third, we add syntax for class declarations together with a +macro making classes which extend nothing to extend Object.
        • +
      • Fourth, we change the strictness attribute of application +into strict(2).
        • +
      • Finally, we add syntax and corresponding strictness +for the KOOL object-oriented constructs.
        • +
      +
      k
        syntax Id ::= "Object" [token] | "Main" [token]
+
+  syntax Stmt ::= "var" Exps ";"
+                | "method" Id "(" Ids ")" Block  // called "function" in SIMPLE
+                | "class" Id Block               // KOOL
+                | "class" Id "extends" Id Block  // KOOL
+
+  syntax Exp ::= Int | Bool | String | Id
+               | "this"                                 // KOOL
+               | "super"                                // KOOL
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]    // KOOL
+               | "(" Id ")" Exp          [strict(2)]    // KOOL  cast
+               | "new" Id "(" Exps ")"   [strict(2)]    // KOOL
+               | Exp "." Id                             // KOOL
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict(2)]    // was strict in SIMPLE
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Ids  ::= List{Id,","}
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+
+  syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"                          [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "try" Block "catch" "(" Id ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+
      +

      Old desugaring rules, from SIMPLE

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S} => {Start while (Cond) {S Step;}}
+  rule var E1::Exp, E2::Exp, Es::Exps; => var E1; var E2, Es;       [anywhere]
+  rule var X::Id = E; => var X; X = E;                              [anywhere]
+
      +

      New desugaring rule

      +
      k
        rule class C:Id S => class C extends Object S                     // KOOL
+
+endmodule
+
      +

      Semantics

      +

      We first discuss the new configuration of KOOL, which extends that of +SIMPLE. Then we include the semantics of the constructs borrowed from +SIMPLE unchanged; we refrain from discussing those, because they were +already discussed in the K definition of SIMPLE. Then we discuss +changes to SIMPLE's semantics needed for the more general meaning of +the previous SIMPLE constructs (for example for thread spawning, +assignment, etc.). Finally, we discuss in detail the +semantics of the additional KOOL constructs.

      +
      k
      module KOOL-UNTYPED
+  imports KOOL-UNTYPED-SYNTAX
+  imports DOMAINS
+
      +

      Configuration

      +

      KOOL removes one cell and adds two nested cells to the configuration +of SIMPLE. The cell which is removed is the one holding the global +environment, because a KOOL program consists of a set of classes only, +with no global declarations. In fact, since informally speaking each +KOOL class now includes a SIMPLE program, it is safe to say that the +global variables in SIMPLE became class fields in KOOL. Let us now +discuss the new cells that are added to the configuration of SIMPLE.

      +
        +
      • +

        The cell crntObj holds data pertaining to the current +object, that is, the object environment in which the code in cell +k executes: crntClass holds the current class (which +can change as methods of the current object are invoked); +envStack holds the stack of environments as a list, +each layer corresponding to one class in the objects' instance class +hierarchy; location, which is optional, holds the location in +the store where the current object is or has to be located (this is +useful both for method closures and for the semantics of object +creation).

        +
        • +
      • +

        The cell classes holds all the declared classes, each +class being held in its own class cell which contains a name +(className), a parent (extends), and the actual +member declarations (declarations).

        +
        • +
      +
      k
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+  syntax EnvCell
+  syntax ControlCell
+  syntax EnvStackCell
+  syntax CrntObjCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="yellow">
+                      <k color="green"> $PGM:Stmt ~> execute </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                      //<br/> // TODO(KORE): support latex annotations #1799
+                        <crntObj color="Fuchsia">  // KOOL
+                           <crntClass> Object </crntClass>
+                           <envStack> .List </envStack>
+                           <location multiplicity="?"> .K </location>
+                        </crntObj>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                      <id color="pink"> 0 </id>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <classes color="Fuchsia">        // KOOL
+                     <classData multiplicity="*" type="Map" color="Fuchsia">
+                        // the Map has as its key the first child of the cell,
+                        // in this case the className cell.
+                        <className color="Fuchsia"> Main </className>
+                        <baseClass color="Fuchsia"> Object </baseClass>
+                        <declarations color="Fuchsia"> .K </declarations>
+                     </classData>
+                  </classes>
+                </T>
+
      +

      Unchanged Semantics from untyped SIMPLE

      +

      The semantics below is taken over from SIMPLE unchanged. +The semantics of function declaration and invocation, including the +use of the special lambda abstraction value, needs to change +in order to account for the fact that methods are now invoked into +their object's environment. The semantics of function return actually +stays unchanged. Also, the semantics of program initialization is +different: now we have to create an instance of the Main +class which also calls the constructor Main(), while in +SIMPLE we only had to invoke the function Main(). +Finally, the semantics of thread spawning needs to change, too: the +parent thread needs to also share its object environment with the +spawned thread (in addition to its local environment, like in SIMPLE). +This is needed in order to be able to spawn method invokations under +dynamic method dispatch; for example, spawn { run(); } +will need to look up the method run() in the newly created +thread, operation which will most likely fail unless the child thread +sees the object environment of the parent thread. Note that the +spawn statement of KOOL is more permissive than the threads +of Java. In fact, the latter can be implemented in terms of our +spawn—see the program threads.kool for a sketch.

      +

      Below is a subset of the values of SIMPLE, which are also values +of KOOL. We will add other values later in the semantics, such as +object and method closures.

      +
      k
        syntax Val ::= Int | Bool | String
+               | array(Int,Int)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax KResult ::= Vals
+
      +

      The semantics below are taken verbatim from the untyped SIMPLE +definition.

      +
      k
        syntax KItem ::= "undefined"  [latex(\bot)]
+
+  rule <k> var X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
+
+  context var _:Id[HOLE];
+
+  rule <k> var X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(L +Int 1, N)
+                          (L +Int 1) ... (L +Int N) |-> undefined ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    when N >=Int 0
+
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule var X:Id[N1:Int, N2:Int, Vs:Vals];
+    => var X[N1];
+       {
+         var $1=X;
+         for(var $2=0; $2 <= N1 - 1; ++$2) {
+           var X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
+
+  rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+
+  context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
+
+  rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(L,_)[N:Int] => lookup(L +Int N)
+    [anywhere]
+
+
+  rule sizeOf(array(_,N)) => N
+
      +

      The semantics of function application needs to change into dynamic +method dispatch invocation, which is defined shortly. However, +interestingly, the semantics of return stays unchanged.

      +
      k
        rule <k> return(V:Val); ~> _ => V ~> K </k>
+       <control>
+         <fstack> ListItem(fstackFrame(Env,K,XS,<crntObj> CO </crntObj>)) => .List ...</fstack>
+         <xstack> _ => XS </xstack>
+         <crntObj> _ => CO </crntObj>
+       </control>
+       <env> _ => Env </env>
+
+  syntax Val ::= "nothing"
+  rule return; => return nothing;
+
+
+  rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
+
+  context (HOLE => lvalue(HOLE)) = _
+
+  rule <k> loc(L) = V:Val => V ...</k> <store>... L |-> (_ => V) ...</store>
+    [group(assignment)]
+
+
+  rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
+
+  rule S1::Stmt S2::Stmt => S1 ~> S2
+
+  rule _:Val; => .
+
+  rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
+  rule while (E) S => if (E) {S while(E)S}
+
+  rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    [group(print)]
+  rule print(.Vals); => .
+
+
+  syntax KItem ::= xstackFrame(Id,Stmt,K,Map,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Id,Stmt,K,Map,K)
+
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(X) {S2} => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem(xstackFrame(X, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _ => { var X = V; S2 } ~> K </k>
+       <control>
+         <xstack> ListItem(xstackFrame(X, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
      +

      Thread spawning needs a new semantics, because we want the child +thread to also share the object environment with its parent. The new +semantics of thread spawning will be defined shortly. However, +interestingly, the other concurrency constructs keep their semantics +from SIMPLE unchanged.

      +
      k
        // TODO(KORE): ..Bag should be . throughout this definition #1772
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+  /*
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .)
+  */
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
+  rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
+  rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    when (notBool(V in Busy:Set))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
+  rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+    when N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
+  rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+
      +

      Unchanged auxiliary operations from untyped SIMPLE

      +
      k
        syntax Stmt ::= mkDecls(Ids,Vals)  [function]
+  rule mkDecls((X:Id, Xs:Ids), (V:Val, Vs:Vals)) => var X=V; mkDecls(Xs,Vs)
+  rule mkDecls(.Ids,.Vals) => {}
+
+  // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookup(Int)
+  /*
+  syntax KItem ::= lookup(Int)
+  */
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+  // TODO: How can we make sure that the second rule above applies before the first one?
+  //       Probably we'll deal with this using strategies, eventually.
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+
+  rule <k> lvalue(X:Id => loc(L)) ...</k> <env>... X |-> L:Int ...</env>
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
+
+  syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  when N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  when N <=Int M
+
      +

      Changes to the existing untyped SIMPLE semantics

      +

      When we extend a language, sometimes we need to do more than just add +new language constructs and semantics for them. Sometimes we want to +also extend the semantics of existing language constructs, in order to +get more from them.

      +

      Program initialization

      +

      In SIMPLE, once all the global declarations were processed, the +function main() was invoked. In KOOL, the global +declarations are classes, and their specific semantics is given +shortly; essentially, they are pre-processed one by one and added +into the class cell structure in the configuration. +Once all the classes are processed, the computation item +execute, which was placed right after the program in the +initial configuration, is reached. In SIMPLE, the program was +initialized by calling the method main(). In KOOL, the +program is initialized by creating an object instance of class +Main. This will also implicitly call the method +Main() (the Main class constructor). The emptiness +of the env cell below is just a sanity check, to make sure +that the user has not declared anything but classes at the top level +of the program.

      +
      k
        syntax KItem ::= "execute"
+  rule <k> execute => new Main(.Exps); </k> <env> .Map </env>
+
      +

      The semantics of new (defined below) requires the +execution of all the class' declarations (and also of its +superclasses').

      +

      Object and method closures

      +

      Before we can define the semantics of method application (previously +called function application in SIMPLE), we need to add two more values +to the language, namely object and method closures:

      +
      k
        syntax Val ::= objectClosure(Id, List)
+               | methodClosure(Id,Int,Ids,Stmt)
+
      +

      An object value consists of an objectClosure-wrapped bag +containing the current class of the object and the environment stack +of the object. The current class of an object will always be one of +the classes mapped to an environment in the environment stack of the +object. A method closure encapsulates the method's parameters and +code (last two arguments), as well as the object context in which the +method code should execute. This object context includes the current +class of the object (the first argument of methodClosure) and +the object environment stack (located in the object stored at the +location specified as the second argument of methodClosure).

      +

      Method application

      +

      KOOL has a complex mechanism to invoke methods, because it allows both +dynamic method dispatch and methods as first-class-citizen values (the +latter making it a higher-order language). The invocation mechanism +will be defined later. What is sufficient to know for now is that +the two arguments of the application construct eventually reduce to +values, the first being a method closure and the latter a list of +values. The semantics of the method closure application is then as +expected: the local environment and control are stacked, then we +switch to method closure's class and object environment and execute +the method body. The mkDecls construct is the one that came +with the unchanged semantics of SIMPLE above.

      +
      k
        syntax KItem ::= fstackFrame(Map,K,List,K)
+  // TODO(KORE): drop the additional production once parsing issue #1842 is fixed
+                 | (Map,K,K)
+
+  rule <k> methodClosure(Class,OL,Xs,S)(Vs:Vals) ~> K
+           => mkDecls(Xs,Vs) S return; </k>
+       <env> Env => .Map </env>
+       <store>... OL |-> objectClosure(_, EnvStack)...</store>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+          <xstack> XS </xstack>
+          <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj' </crntObj>))
+          ...</fstack>
+          <crntObj> Obj' => <crntClass> Class </crntClass> <envStack> EnvStack </envStack> </crntObj>
+       </control>
+
      +

      Spawn

      +

      We want to extend the semantics of spawn to also share the +current object environment with the child thread, in addition to the +current environment. This extension will allow us to also use method +invocations in the spawned statements, which will be thus looked up as +expected, using dynamic method dispatch. This lookup operation would +fail if the child thread did not have access to its parent's object +environment.

      +
      k
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+         <crntObj> Obj </crntObj>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+               <crntObj> Obj </crntObj>
+             ...</thread>)
+
      +

      Semantics of the new KOOL constructs

      +

      Class declaration

      +

      Initially, the classes forming the program are moved into their +corresponding cells:

      +
      k
        rule <k> class Class1 extends Class2 { S } => . ...</k>
+       <classes>... (.Bag => <classData>
+                            <className> Class1 </className>
+                            <baseClass> Class2 </baseClass>
+                            <declarations> S </declarations>
+                        </classData>)
+       ...</classes>
+
      +

      Method declaration

      +

      Like in SIMPLE, method names are added to the environment and bound +to their code. However, unlike in SIMPLE where each function was +executed in the same environment, namely the program global +environment, a method in KOOL needs to be executed into its object's +environment. Thus, methods evaluate to closures, which encapsulate +their object's context (i.e., the current class and environment stack +of the object) in addition to method's parameters and body. This +approach to bind method names to method closures in the environment +will also allow objects to pass their methods to other objects, to +dynamically change their methods by assigning them other method +closures, and even to allow all these to be done from other objects. +This gives the KOOL programmer a lot of power; one should use this +power wisely, though, because programs can become easily hard to +understand and reason about if one overuses these features.

      +
      k
        rule <k> method F:Id(Xs:Ids) S => . ...</k>
+       <crntClass> Class:Id </crntClass>
+       <location> OL:Int </location>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L |-> methodClosure(Class,OL,Xs,S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+
      +

      New

      +

      The semantics of new consists of two actions: memory +allocation for the new object and execution of the corresponding +constructor. Then the created object is returned as the result of the +new operation; the value returned by the constructor, if any, +is discarded. The current environment and object are stored onto the +stack and recovered after new (according to the semantics of +return borrowed from SIMPLE, when the statement +return this; in the rule below is reached and evaluated), +because the object creation part of new will destroy them. +The rule below also initializes the object creation process by +emptying the local environment and the current object, and allocating +a location in the store where the created object will be eventually +stored (this is what the storeObj task after the object +creation task in the rule below will do—its rule is defined +shortly). The location where the object will be stored is also made +available in the crntObj cell, so that method closures can +refer to it (see rule above).

      +
      k
        syntax KItem ::= "envStackFrame" "(" Id "," Map ")"
+
+  rule <k> new Class:Id(Vs:Vals) ~> K
+           => create(Class) ~> storeObj ~> Class(Vs); return this; </k>
+       <env> Env => .Map </env>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control> <xstack> XS </xstack>
+         <crntObj> Obj
+                   => <crntClass> Object </crntClass>
+                      <envStack> ListItem(envStackFrame(Object, .Map)) </envStack>
+                      <location> L </location>
+         </crntObj>
+         <fstack> .List => ListItem(fstackFrame(Env, K, XS, <crntObj> Obj </crntObj>)) ...</fstack>
+       </control>
+
      +

      The creation of a new object (the memory allocation part only) is +a recursive process, requiring to first create an object for the +superclass. A memory object representation is a layered structure: +for each class on the path from the instance class to the root of the +hierarchy there is a layer including the memory allocated for the +members (both fields and methods) of that class.

      +
      k
        syntax KItem ::= create(Id)
+
+  rule <k> create(Class:Id)
+           => create(Class1) ~> setCrntClass(Class) ~> S ~> addEnvLayer ...</k>
+       <className> Class </className>
+       <baseClass> Class1:Id </baseClass>
+       <declarations> S </declarations>
+
+  rule <k> create(Object) => . ...</k>
+
      +

      The next operation sets the current class of the current object. +This is necessary to be done at each layer, because the current class +of the object is enclosed as part of the method closures (see the +semantics of method declarations above).

      +
      k
        syntax KItem ::= setCrntClass(Id)
+
+  rule <k> setCrntClass(C) => . ...</k>
+       <crntClass> _ => C </crntClass>
+
      +

      The next operation adds a new tagged environment layer to the +current object and gets ready for the next layer by clearing the +environment (note that create expects the environment to be +empty).

      +
      k
        syntax KItem ::= "addEnvLayer"
+
+  rule <k> addEnvLayer => . ...</k>
+       <env> Env => .Map </env>
+       <crntClass> Class:Id </crntClass>
+       <envStack> .List => ListItem(envStackFrame(Class, Env)) ...</envStack>
+
      +

      The following operation stores the created object at the location +reserved by new. Note that the location reserved by +new was temporarily stored in the crntObj cell +precisely for this purpose. Now that the newly created object is +stored at its location and that all method closures are aware of it, +the location is unnecessary and thus we delete it from the +crntObj cell.

      +
      k
        syntax KItem ::= "storeObj"
+
+  rule <k> storeObj => . ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> (<location> L:Int </location> => .Bag) </crntObj>
+       <store>... .Map => L |-> objectClosure(CC, ES) ...</store>
+
      +

      Self reference

      +

      The semantics of this is straightforward: evaluate to the +current object.

      +
      k
        rule <k> this => objectClosure(CC, ES) ...</k>
+       <crntObj> <crntClass> CC </crntClass> <envStack> ES </envStack> </crntObj>
+
      +

      Object member access

      +

      We can access an object member (field or method) either explicitly, +using the construct e.x, or implicitly, using only the member +name x directly. The borrowed semantics of SIMPLE will +already lookup a sole name in the local environment. The first rule +below reduces implicit member access to explicit access when the name +cannot be found in the local environment. There are two cases to +analyze for explicit object member access, depending upon whether the +object is a proper object or it is just a redirection to the parent +class via the construct super. In the first case, we +evaluate the object expression and lookup the member starting with the +current class (static scoping). Note the use of the conditional +evaluation context. In the second case, we just lookup the member +starting with the superclass of the current class. In both cases, +the lookupMember task eventually yields a lookup(L) +task for some appropriate location L, which will be further +solved with the corresponding rule borrowed from SIMPLE. Note that the +current object is not altered by super, so future method +invocations see the entire object, as needed for dynamic method dispatch.

      +
      k
        rule <k> X:Id => this . X ...</k> <env> Env:Map </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id when (HOLE =/=K super)
+
+// TODO: explain how Assoc matching has been replaced with two rules here.
+// Maybe also improve it a bit.
+
+/*  rule objectClosure(<crntClass> Class:Id </crntClass>
+                     <envStack>... envStackFrame(Class,EnvC) EStack </envStack>)
+       . X:Id
+    => lookupMember(envStackFrame(Class,EnvC) EStack, X) */
+
+  rule objectClosure(Class:Id, ListItem(envStackFrame(Class,Env)) EStack)
+       . X:Id
+    => lookupMember(ListItem(envStackFrame(Class,Env)) EStack, X)
+  rule objectClosure(Class:Id, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+       . _X:Id
+    when Class =/=K Class'
+
+/*  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,EnvC) EStack </envStack> */
+  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> super . _X ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+
      +

      Method invocation

      +

      Unlike in SIMPLE, in KOOL application was declared strict only in its +second argument. That is because we want to ensure dynamic method +dispatch when the first argument is a method access. As a +consequence, we need to consider all the cases of interest for the +first argument and to explicitly say what to do in each case. In all +cases except for method access in a proper object (i.e., not +super), we want the same behavior for the first argument as +if it was not in a method invocation position. When it is a member +access (the third rule below), we look it up starting with the +instance class of the corresponding object. This ensures dynamic +dispatch for methods; it actually dynamically dispatches field +accesses, too, which is correct in KOOL, because one can assign method +closures to fields and the field appeared in a method invocation +context. The last context declaration below says that method +applications or array accesses are also allowed as first argument to +applications; that is because methods are allowed to return methods +and arrays are allowed to hold methods in KOOL, since it is +higher-order. If that is the case, then we want to evaluate the +method call or the array access.

      +
      k
        rule <k> (X:Id => V)(_:Exps) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  rule <k> (X:Id => this . X)(_:Exps) ...</k>
+       <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context HOLE._::Id(_) when HOLE =/=K super
+
+  rule (objectClosure(_, EStack) . X
+    => lookupMember(EStack, X:Id))(_:Exps)
+
+/*  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... envStackFrame(Class,_) EStack </envStack> */
+  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> (super . _X)(_:Exps) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack> ListItem(envStackFrame(Class':Id,_)) => .List ...</envStack>
+    when Class =/=K Class'
+
+  // TODO(KORE): fix getKLabel #1801
+  rule (A:Exp(B:Exps))(C:Exps) => A(B) ~> #freezerFunCall(C)
+  rule (A:Exp[B:Exps])(C:Exps) => A[B] ~> #freezerFunCall(C)
+  rule V:Val ~> #freezerFunCall(C:Exps) => V(C)
+  syntax KItem ::= "#freezerFunCall" "(" K ")"
+  /*
+  context HOLE(_:Exps)
+    when getKLabel(HOLE) ==K #klabel(`_(_)`) orBool getKLabel(HOLE) ==K #klabel(`_[_]`)
+  */
+
      +

      Eventually, each of the rules above produces a lookup(L) +task as a replacement for the method. When that happens, we just +lookup the value at location L:

      +
      k
        rule <k> (lookup(L) => V)(_:Exps) ...</k>  <store>... L |-> V:Val ...</store>
+    [group(lookup)]
+
      +

      The value V looked up above is expected to be a method closure, +in which case the semantics of method application given above will +apply. Otherwise, the execution will get stuck.

      +

      Instance Of

      +

      It searches the object environment for a layer corresponding to the +desired class. It returns true iff it can find the class, +otherwise it returns false; it only gets stuck when its first +argument does not evaluate to an object.

      +
      k
        rule objectClosure(_, ListItem(envStackFrame(C,_)) _)
+       instanceOf C => true
+
+  rule objectClosure(_, (ListItem(envStackFrame(C,_)) => .List) _)
+       instanceOf C'  when C =/=K C'
+//TODO: remove the sort cast ::Id of C above, when sort inference bug fixed
+
+  rule objectClosure(_, .List) instanceOf _ => false
+
      +

      Cast

      +

      In untyped KOOL, we prefer to not check the validity of casting. In +other words, any cast is allowed on any object, simply changing the +current class of the object to the desired class. The execution will +get stuck later if one attempts to access a field which is not +available. Moreover, the execution may complete successfully even +in the presence of invalid casts, provided that each accessed member +during the current execution is, or happens to be, available.

      +
      k
        rule (C) objectClosure(_ , EnvStack) => objectClosure(C ,EnvStack)
+
      +

      KOOL-specific auxiliary declarations and operations

      +

      Here we define all the auxiliary constructs used in the above +KOOL-specific semantics (those used in the SIMPLE fragment +have already been defined in a corresponding section above).

      +

      Objects as lvalues

      +

      The current machinery borrowed with the semantics of SIMPLE allows us +to enrich the set of lvalues, this way allowing new means to assign +values to locations. In KOOL, we want object member names to be +lvalues, so that we can assign values to them using the already +existing machinery. The first rule below ensures that the object is +always explicit, the evaluation context enforces the object to be +evaluated, and finally the second rule initiates the lookup for the +member's location based on the current class of the object.

      +
      k
        rule <k> lvalue(X:Id => this . X) ...</k>  <env> Env </env>
+    when notBool(X in keys(Env))
+
+  context lvalue((HOLE . _)::Exp)
+
+/*  rule lvalue(objectClosure(<crntClass> C </crntClass>
+                            <envStack>... envStackFrame(C,EnvC) EStack </envStack>)
+              . X
+              => lookupMember(<envStack> envStackFrame(C,EnvC) EStack </envStack>,
+                              X))  */
+  rule lvalue(objectClosure(Class, ListItem(envStackFrame(Class,Env)) EStack)
+              . X
+              => lookupMember(ListItem(envStackFrame(Class,Env)) EStack,
+                              X))
+  rule lvalue(objectClosure(Class, (ListItem(envStackFrame(Class':Id,_)) => .List) _)
+              . _X)
+    when Class =/=K Class'
+
      +

      Lookup member

      +

      It searches for the given member in the given environment stack, +starting with the most concrete class and going up in the hierarchy.

      +
      k
        // TODO(KORE): clarify sort inferences #1803
+  syntax Exp ::= lookupMember(List, Id)  [function]
+  /*
+  syntax KItem ::= lookupMember(EnvStackCell,Id)  [function]
+  */
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env>... X|->L ...</env>) ...</envStack>, X)
+//    => lookup(L)
+  rule lookupMember(ListItem(envStackFrame(_, X|->L _)) _, X)
+    => lookup(L)
+
+//  rule lookupMember(<envStack> envStackFrame(_, <env> Env </env>) => .List ...</envStack>, X)
+//    when notBool(X in keys(Env))
+  rule lookupMember(ListItem(envStackFrame(_, Env)) Rest, X) =>
+       lookupMember(Rest, X)
+    when notBool(X in keys(Env))
+//TODO: beautify the above
+
+endmodule
+
      +

      Go to Lesson 2, KOOL typed dynamic.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/1_dynamic/kool-typed-dynamic/index.html b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/1_dynamic/kool-typed-dynamic/index.html new file mode 100644 index 00000000000..94e99f0605c --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/1_dynamic/kool-typed-dynamic/index.html @@ -0,0 +1,1385 @@ + + + + + + + + + + + + + + +KOOL — Typed — Dynamic | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      KOOL — Typed — Dynamic

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K dynamic semantics of the typed KOOL language. It is +very similar to the semantics of the untyped KOOL, the difference +being that we now check the typing policy dynamically. Since we have +to now declare the types of variables and methods, we adopt a syntax +for those which is close to Java. Like in the semantics of +untyped KOOL, where we borrowed almost all the semantics of untyped +SIMPLE, we are going to also borrow much of the semantics of +dynamically typed SIMPLE here. We will highlight the differences +between the dynamically typed and the untyped KOOL as we proceed with +the semantics. In general, the type policy of the typed KOOL language +is similar to that of Java. You may find it useful to also read +the discussion in the preamble of the static semantics of typed KOOL +before proceeding.

      +
      k
      module KOOL-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      Like for the untyped KOOL language, the syntax of typed KOOL extends +that of typed SIMPLE with object-oriented constructs. +The syntax below was produced by copying and modifying/extending the +syntax of dynamically typed SIMPLE. In fact, the only change we made +to the existing syntax of dynamically typed SIMPLE was to change the +strictness of the application construct like in untyped KOOL, from +strict to strict(2) (because application is not +strict in the first argument anymore due to dynamic method dispatch). +The KOOL-specific syntactic extensions are identical to those in +untyped KOOL.

      +
      k
        syntax Id ::= "Object" [token] | "Main" [token]
+
      +

      Types

      +
      k
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Id                              // KOOL class
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+  // TODO(KORE): drop klabel once issues #1913 are fixed
+  syntax Types ::= List{Type,","}   [klabel(_,_::Types)]
+  /*
+  syntax Types ::= List{Type,","}
+  */
+
      +

      Declarations

      +
      k
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";" [avoid]
+                | Type Id "(" Params ")" Block    // stays like in typed SIMPLE
+                | "class" Id Block                // KOOL
+                | "class" Id "extends" Id Block   // KOOL
+
      +

      Expressions

      +
      k
        syntax Exp ::= Int | Bool | String | Id
+               | "this"                                 // KOOL
+               | "super"                                // KOOL
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]    // KOOL
+               | "(" Id ")" Exp          [strict(2)]    // KOOL  cast
+               | "new" Id "(" Exps ")"   [strict(2)]    // KOOL
+               | Exp "." Id                             // KOOL
+               > Exp "[" Exps "]"        [strict]
+               > Exp "(" Exps ")"        [strict(2)]    // was strict in SIMPLE
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict(1), left]
+               | Exp "||" Exp            [strict(1), left]
+               > "spawn" Block
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+  syntax Val
+  syntax Vals ::= List{Val,","}          [klabel(exps)]
+
      +

      Statements

      +
      k
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                               [strict]
+                | "if" "(" Exp ")" Block "else" Block   [avoid, strict(1)]
+                | "if" "(" Exp ")" Block                [macro]
+                | "while" "(" Exp ")" Block
+                | "for" "(" Stmt Exp ";" Exp ")" Block  [macro]
+                | "print" "(" Exps ")" ";"              [strict]
+                | "return" Exp ";"                      [strict]
+                | "return" ";"
+                | "try" Block "catch" "(" Param ")" Block
+                | "throw" Exp ";"                       [strict]
+                | "join" Exp ";"                        [strict]
+                | "acquire" Exp ";"                     [strict]
+                | "release" Exp ";"                     [strict]
+                | "rendezvous" Exp ";"                  [strict]
+
+  syntax Stmt ::= Stmt Stmt                          [right]
+
      +

      Desugaring macros

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S::Stmt} => {Start while(Cond){S Step;}}
+  rule T::Type E1::Exp, E2::Exp, Es::Exps; => T E1; T E2, Es;           [anywhere]
+  rule T::Type X::Id = E; => T X; X = E;                                [anywhere]
+
+  rule class C:Id S => class C extends Object S                     // KOOL
+
+endmodule
+
      +

      Semantics

      +

      We first discuss the new configuration, then we include the semantics of +the constructs borrowed from SIMPLE which stay unchanged, then those +whose semantics had to change, and finally the semantics of the +KOOL-specific constructs.

      +
      k
      module KOOL-TYPED-DYNAMIC
+  imports KOOL-TYPED-DYNAMIC-SYNTAX
+  imports DOMAINS
+
      +

      Configuration

      +

      The configuration of dynamically typed KOOL is almost identical to +that of its untyped variant. The only difference is the cell +return, inside the control cell, whose role is to +hold the expected return type of the invoked method. That is because +we want to dynamically check that the value that a method returns has +the expected type.

      +
      k
        // the syntax declarations below are required because the sorts are
+  // referenced directly by a production and, because of the way KIL to KORE
+  // is implemented, the configuration syntax is not available yet
+  // should simply work once KIL is removed completely
+  // check other definitions for this hack as well
+  syntax EnvCell
+  syntax ControlCellFragment
+  syntax EnvStackCell
+  syntax CrntObjCellFragment
+
+  configuration <T color="red">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="yellow">
+                      <k color="green"> ($PGM:Stmt ~> execute) </k>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <control color="cyan">
+                        <fstack color="blue"> .List </fstack>
+                        <xstack color="purple"> .List </xstack>
+                        <returnType color="LimeGreen"> void </returnType>  // KOOL
+                      //<br/> // TODO(KORE): support latex annotations #1799
+                        <crntObj color="Fuchsia">  // KOOL
+                           <crntClass> Object </crntClass>
+                           <envStack> .List </envStack>
+                           <location multiplicity="?"> .K </location>
+                        </crntObj>
+                      </control>
+                    //<br/> // TODO(KORE): support latex annotations #1799
+                      <env color="violet"> .Map </env>
+                      <holds color="black"> .Map </holds>
+                      <id color="pink"> 0 </id>
+                    </thread>
+                  </threads>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <store color="white"> .Map </store>
+                  <busy color="cyan">.Set </busy>
+                  <terminated color="red"> .Set </terminated>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="brown" stream="stdout"> .List </output>
+                  <nextLoc color="gray"> 0 </nextLoc>
+                //<br/> // TODO(KORE): support latex annotations #1799
+                  <classes color="Fuchsia">        // KOOL
+                     <classData multiplicity="*" type="Map" color="Fuchsia">
+                        <className color="Fuchsia"> Main </className>
+                        <baseClass color="Fuchsia"> Object </baseClass>
+                        <declarations color="Fuchsia"> .K </declarations>
+                     </classData>
+                  </classes>
+                </T>
+
      +

      Unchanged semantics from dynamically typed SIMPLE

      +

      The semantics below is taken over from dynamically typed SIMPLE +unchanged. Like for untyped KOOL, the semantics of function/method +declaration and invocation, and of program initialization needs to +change. Moreover, due to subtyping, the semantics of several imported +SIMPLE constructs can be made more general, such as that of the +return statement, that of the assignment, and that of the exceptions. +We removed all these from the imported semantics of SIMPLE below and +gave their modified semantics right after, together with the extended +semantics of thread spawning (which is identical to that of untyped +KOOL).

      +
      k
        syntax Val ::= Int | Bool | String
+               | array(Type,Int,Int)
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax KResult ::= Vals
+
+
+  syntax KItem ::= undefined(Type)  [latex(\bot_{#1})]
+
+  rule <k> T:Type X:Id; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+
+
+  rule <k> T:Type X:Id[N:Int]; => . ...</k>
+       <env> Env => Env[X <- L] </env>
+       <store>... .Map => L |-> array(T, L +Int 1, N)
+                          (L +Int 1)...(L +Int N) |-> undefined(T) ...</store>
+       <nextLoc> L:Int => L +Int 1 +Int N </nextLoc>
+    requires N >=Int 0
+
+  context _:Type _::Exp[HOLE::Exps];
+
+
+  syntax Id ::= "$1" [token] | "$2" [token]
+  rule T:Type X:Id[N1:Int, N2:Int, Vs:Vals];
+    => T[]<Vs> X[N1];
+       {
+         T[][]<Vs> $1=X;
+         for(int $2=0; $2 <= N1 - 1; ++$2) {
+           T X[N2,Vs];
+           $1[$2] = X;
+         }
+       }
+
+
+  rule <k> X:Id => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+
+  context ++(HOLE => lvalue(HOLE))
+  rule <k> ++loc(L) => I +Int 1 ...</k>
+       <store>... L |-> (I:Int => I +Int 1) ...</store>  [group(increment)]
+
+
+  rule I1 + I2 => I1 +Int I2
+  rule Str1 + Str2 => Str1 +String Str2
+  rule I1 - I2 => I1 -Int I2
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 requires I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 requires I2 =/=K 0
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+
+  rule V:Val[N1:Int, N2:Int, Vs:Vals] => V[N1][N2, Vs]
+    [anywhere]
+
+  rule array(_:Type, L:Int, M:Int)[N:Int] => lookup(L +Int N)
+    requires N >=Int 0 andBool N <Int M  [anywhere]
+
+  rule sizeOf(array(_,_,N)) => N
+
+
+  syntax Val ::= nothing(Type)
+  rule <k> return; => return nothing(T); ...</k> <returnType> T </returnType>
+
+
+  rule <k> read() => I ...</k> <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
+
+  context (HOLE => lvalue(HOLE)) = _
+
+
+  rule {} => .
+  rule <k> { S } => S ~> setEnv(Env) ...</k>  <env> Env </env>
+
+
+  rule S1:Stmt S2:Stmt => S1 ~> S2
+
+
+  rule _:Val; => .
+
+
+  rule if ( true) S else _ => S
+  rule if (false) _ else S => S
+
+
+  rule while (E) S => if (E) {S while(E)S}
+
+
+  rule <k> print(V:Val, Es => Es); ...</k> <output>... .List => ListItem(V) </output>
+    requires typeOf(V) ==K int orBool typeOf(V) ==K string  [group(print)]
+  rule print(.Vals); => .
+
+
+  rule (<thread>... <k>.</k> <holds>H</holds> <id>T</id> ...</thread> => .Bag)
+       <busy> Busy => Busy -Set keys(H) </busy>
+       <terminated>... .Set => SetItem(T) ...</terminated>
+
+  rule <k> join T:Int; => . ...</k>
+       <terminated>... SetItem(T) ...</terminated>
+
+  rule <k> acquire V:Val; => . ...</k>
+       <holds>... .Map => V |-> 0 ...</holds>
+       <busy> Busy (.Set => SetItem(V)) </busy>
+    requires (notBool(V in Busy:Set))  [group(acquire)]
+
+  rule <k> acquire V; => . ...</k>
+       <holds>... V:Val |-> (N:Int => N +Int 1) ...</holds>
+
+  rule <k> release V:Val; => . ...</k>
+       <holds>... V |-> (N => N:Int -Int 1) ...</holds>
+    requires N >Int 0
+
+  rule <k> release V; => . ...</k> <holds>... V:Val |-> 0 => .Map ...</holds>
+       <busy>... SetItem(V) => .Set ...</busy>
+
+  rule <k> rendezvous V:Val; => . ...</k>
+       <k> rendezvous V; => . ...</k>  [group(rendezvous)]
+
      +

      Unchanged auxiliary operations from dynamically typed SIMPLE

      +
      k
        syntax Stmt ::= mkDecls(Params,Vals)  [function]
+  rule mkDecls((T:Type X:Id, Ps:Params), (V:Val, Vs:Vals))
+    => T X=V; mkDecls(Ps,Vs)
+  rule mkDecls(.Params,.Vals) => {}
+
+  syntax Exp ::= lookup(Int)
+  rule <k> lookup(L) => V ...</k> <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  syntax KItem ::= setEnv(Map)
+  rule <k> setEnv(Env) => . ...</k>  <env> _ => Env </env>
+  rule (setEnv(_) => .) ~> setEnv(_)
+
+  syntax Exp ::= lvalue(K)
+  syntax Val ::= loc(Int)
+  rule <k> lvalue(X:Id => loc(L)) ...</k>  <env>... X |-> L:Int ...</env>
+
+  context lvalue(_::Exp[HOLE::Exps])
+  context lvalue(HOLE::Exp[_::Exps])
+
+  rule lvalue(lookup(L:Int) => loc(L))
+
+  syntax Type ::= Type "<" Vals ">"  [function]
+  rule T:Type<_,Vs:Vals> => T[]<Vs>
+  rule T:Type<.Vals> => T
+
+  syntax Map ::= Int "..." Int "|->" K
+    [function, latex({#1}\ldots{#2}\mapsto{#3})]
+  rule N...M |-> _ => .Map  requires N >Int M
+  rule N...M |-> K => N |-> K (N +Int 1)...M |-> K  requires N <=Int M
+
+  syntax Type ::= typeOf(K)  [function]
+  rule typeOf(_:Int) => int
+  rule typeOf(_:Bool) => bool
+  rule typeOf(_:String) => string
+  rule typeOf(array(T,_,_)) => (T[])
+  rule typeOf(undefined(T)) => T
+  rule typeOf(nothing(T)) => T
+
+  syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+
      +

      Changes to the existing dynamically typed SIMPLE semantics

      +

      We extend/change the semantics of several SIMPLE constructs in order +to take advantage of the richer KOOL semantic infrastructure and thus +get more from the existing SIMPLE constructs.

      +

      Program initialization

      +

      Like in untyped KOOL.

      +
      k
        syntax KItem ::= "execute"
+  rule <k> execute => new Main(.Exps); </k> <env> .Map </env>
+
      +

      Method application

      +

      The only change to untyped KOOL's values is that method closures are +now typed (their first argument holds their type):

      +
      k
       syntax Val ::= objectClosure(Id,List)
+              | methodClosure(Type,Id,Int,Params,Stmt)
+
      +

      The type held by a method clossure will be the entire type of the +method, not only its result type like the lambda-closure of typed +SIMPLE. The reason for this change comes from the the need to +dynamically upcast values when passed to contexts where values of +superclass types are expected; since we want method closures to be +first-class-citizen values in our language, we have to be able to +dynamically upcast them, and in order to do that elegantly it is +convenient to store the entire ``current type'' of the method closure +instead of just its result type. Note that this was unnecessary in +the semantics of the dynamically typed SIMPLE language.

      +

      Method closure application needs to also set a new return type in +the return cell, like in dynamically typed SIMPLE, in order +for the values returned by its body to be checked against the return +type of the method. To do this correctly, we also need to stack the +current status of the return cell and then pop it when the +method returns. We have to do the same with the current object +environment, so we group them together in the stack frame.

      +
      k
        syntax KItem ::= fstackFrame(Map, K, List, Type, K)
+
+  rule <k> methodClosure(_->T,Class,OL,Ps,S)(Vs:Vals) ~> K
+           => mkDecls(Ps,Vs) S return; </k>
+       <env> Env => .Map </env>
+       <store>... OL |-> objectClosure(_, EStack)...</store>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+          <fstack> .List => ListItem(fstackFrame(Env, K, XS, T', <crntObj> Obj' </crntObj>)) ...</fstack>
+          <xstack> XS </xstack>
+          <returnType> T' => T </returnType>
+          <crntObj> Obj' => <crntClass> Class </crntClass> <envStack> EStack </envStack> </crntObj>
+       </control>
+
      +

      At method return, we have to check that the type of the returned +value is a subtype of the expected return type. Moreover, if that is +the case, then we also upcast the returned value to one of the +expected type. The computation item unsafeCast(V,T) changes +the typeof V to T without any additional checks; however, it only +does it when V is an object or a method, otherwise it returns V +unchanged.

      +
      k
        rule <k> return V:Val; ~> _
+           => subtype(typeOf(V), T) ~> true? ~> unsafeCast(V, T) ~> K
+       </k>
+       <control>
+         <fstack> ListItem(fstackFrame(Env, K, XS, RT, <crntObj> CO </crntObj>)) => .List ...</fstack>
+         <xstack> _ => XS </xstack>
+         <returnType> T:Type => RT </returnType>
+         <crntObj> _ => CO </crntObj>
+       </control>
+       <env> _ => Env </env>
+
      +

      Assignment

      +

      Typed KOOL allows to assign subtype instance values to supertype +lvalues. The semantics of assignment below is similar in spirit to +dynamically typed SIMPLE's, but a check is performed that the assigned +value's type is a subtype of the location's type. If that is the +case, then the assigned value is returned as a result and stored, but +it is upcast appropriately first, so the context will continue to see +a value of the expected type of the location. Note that the type of a +location is implicit in the type of its contents and it never changes +during the execution of a program; its type is assigned when the +location is allocated and initialized, and then only type-preserving +values are allowed to be stored in each location.

      +
      k
        rule <k> loc(L) = V:Val
+           => subtype(typeOf(V),typeOf(V')) ~> true?
+              ~> unsafeCast(V, typeOf(V')) ...</k>
+       <store>... L |-> (V' => unsafeCast(V, typeOf(V'))) ...</store>
+    [group(assignment)]
+
      +

      Typed exceptions

      +

      Exceptions are propagated now until a catch that can handle them is +encountered.

      +
      k
        syntax KItem ::= xstackFrame(Param, Stmt, K, Map, K)
+  syntax KItem ::= "popx"
+
+  rule <k> (try S1 catch(P) S2 => S1 ~> popx) ~> K </k>
+       <control>
+         <xstack> .List => ListItem(xstackFrame(P, S2, K, Env, C)) ...</xstack>
+         C
+       </control>
+       <env> Env </env>
+
+  rule <k> popx => . ...</k>
+       <xstack> ListItem(_) => .List ...</xstack>
+
+  rule <k> throw V:Val; ~> _
+        => if (subtype(typeOf(V),T)) { T X = V; S2 } else { throw V; } ~> K
+       </k>
+       <control>
+         <xstack> ListItem(xstackFrame(T:Type X:Id, S2, K, Env, C)) => .List ...</xstack>
+         (_ => C)
+       </control>
+       <env> _ => Env </env>
+
      +

      Spawn

      +

      Like in untyped KOOL.

      +
      k
        rule <thread>...
+         <k> spawn S => !T:Int ...</k>
+         <env> Env </env>
+         <crntObj> Obj </crntObj>
+       ...</thread>
+       (.Bag => <thread>...
+               <k> S </k>
+               <env> Env </env>
+               <id> !T </id>
+               <crntObj> Obj </crntObj>
+             ...</thread>)
+
      +

      Semantics of the new KOOL constructs

      +

      Class declaration

      +

      Like in untyped KOOL.

      +
      k
        rule <k> class Class1 extends Class2 { S } => . ...</k>
+       <classes>... (.Bag => <classData>
+                            <className> Class1 </className>
+                            <baseClass> Class2 </baseClass>
+                            <declarations> S </declarations>
+                        </classData>)
+       ...</classes>
+
      +

      Method declaration

      +

      Methods are now typed and we need to store their types in their +closures, so that their type contract can be checked at invocation +time. The rule below is conceptually similar to that of untyped KOOL; +the only difference is the addition of the types.

      +
      k
        rule <k> T:Type F:Id(Ps:Params) S => . ...</k>
+       <crntClass> C </crntClass>
+       <location> OL </location>
+       <env> Env => Env[F <- L] </env>
+       <store>... .Map => L|->methodClosure(getTypes(Ps)->T,C,OL,Ps,S) ...</store>
+       <nextLoc> L => L +Int 1 </nextLoc>
+
      +

      New

      +

      The semantics of new in dynamically typed KOOL is also +similar to that in untyped KOOL, the main difference being the +management of the return types. Indeed, when a new object is created +we also have to stack the current type in the return cell in +order to be recovered after the creation of the new object. Only the +first rule below needs to be changed; the others are identical to +those in untyped KOOL.

      +
      k
        syntax KItem ::= envStackFrame(Id, Map)
+
+  rule <k> new Class:Id(Vs:Vals) ~> K
+           => create(Class) ~> (storeObj ~> ((Class(Vs)); return this;)) </k>
+       <env> Env => .Map </env>
+       <nextLoc> L:Int => L +Int 1 </nextLoc>
+     //<br/> // TODO(KORE): support latex annotations #1799
+       <control>
+         <xstack> XS </xstack>
+         <crntObj> Obj
+                   => <crntClass> Object </crntClass>
+                      <envStack> ListItem(envStackFrame(Object, .Map)) </envStack>
+                      <location> L </location>
+         </crntObj>
+         <returnType> T => Class </returnType>
+         <fstack> .List => ListItem(fstackFrame(Env, K, XS, T, <crntObj>Obj</crntObj>)) ...</fstack>
+       </control>
+
+  syntax KItem ::= create(Id)
+
+  rule <k> create(Class:Id)
+           => create(Class1) ~> setCrntClass(Class) ~> S ~> addEnvLayer ...</k>
+       <className> Class </className>
+       <baseClass> Class1:Id </baseClass>
+       <declarations> S </declarations>
+
+  rule <k> create(Object) => . ...</k>
+
+  syntax KItem ::= setCrntClass(Id)
+
+  rule <k> setCrntClass(C) => . ...</k>
+       <crntClass> _ => C </crntClass>
+
+  syntax KItem ::= "addEnvLayer"
+
+  rule <k> addEnvLayer => . ...</k>
+       <env> Env => .Map </env>
+       <crntClass> Class:Id </crntClass>
+       <envStack> .List => ListItem(envStackFrame(Class, Env)) ...</envStack>
+
+  syntax KItem ::= "storeObj"
+
+  rule <k> storeObj => . ...</k>
+       <crntObj>
+         <crntClass> Class </crntClass>
+         <envStack> EStack </envStack>
+         (<location> L:Int </location> => .Bag)
+       </crntObj>
+       <store>... .Map => L |-> objectClosure(Class, EStack) ...</store>
+
      +

      Self reference

      +

      Like in untyped KOOL.

      +
      k
        rule <k> this => objectClosure(Class, EStack) ...</k>
+       <crntObj>
+         <crntClass> Class </crntClass>
+         <envStack> EStack </envStack>
+         ...
+       </crntObj>
+
      +

      Object member access

      +

      Like in untyped KOOL.

      +
      k
        rule <k> X:Id => this . X ...</k> <env> Env:Map </env>
+    requires notBool(X in keys(Env))
+
+  context HOLE . _::Id requires (HOLE =/=K super)
+
+/*  rule objectClosure(<crntObj> <crntClass> Class:Id </crntClass>
+                     <envStack>... ListItem((Class,EnvC:EnvCell)) EStack </envStack> </crntObj>)
+       . X:Id
+    => lookupMember(<envStack> ListItem((Class,EnvC)) EStack </envStack>, X) */
+  rule objectClosure(Class:Id,
+                     ListItem(envStackFrame(Class,Env)) EStack)
+       . X:Id
+    => lookupMember(ListItem(envStackFrame(Class,Env)) EStack, X)
+  rule objectClosure(Class:Id,
+                     (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack)
+       . _X:Id
+    requires Class =/=K Class'
+
+/*  rule <k> super . X => lookupMember(<envStack>EStack</envStack>, X) ...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... ListItem((Class,EnvC:EnvCell)) EStack </envStack> */
+  rule <k> super . X => lookupMember(EStack, X) ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> super . _X ...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack </envStack>
+    requires Class =/=K Class'
+
      +

      Method invocation

      +

      The method lookup is the same as in untyped KOOL.

      +
      k
        rule <k> (X:Id => V)(_:Exps) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V:Val ...</store>  [group(lookup)]
+
+  rule <k> (X:Id => this . X)(_:Exps) ...</k>
+       <env> Env </env>
+    requires notBool(X in keys(Env))
+
+  context HOLE._::Id(_) requires HOLE =/=K super
+
+  rule (objectClosure(_, EStack) . X
+    => lookupMember(EStack, X:Id))(_:Exps)
+
+/*  rule <k> (super . X
+            => lookupMember(<envStack>EStack</envStack>,X))(_:Exps)...</k>
+       <crntClass> Class </crntClass>
+       <envStack>... ListItem((Class,_)) EStack </envStack> */
+  rule <k> (super . X
+            => lookupMember(EStack,X))(_:Exps)...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> ListItem(envStackFrame(Class,_)) EStack </envStack>
+  rule <k> (super . _X)(_:Exps)...</k>
+       <crntClass> Class:Id </crntClass>
+       <envStack> (ListItem(envStackFrame(Class':Id,_)) => .List) _EStack </envStack>
+    requires Class =/=K Class'
+
+  // TODO(KORE): fix getKLabel #1801
+  rule (A:Exp(B:Exps))(C:Exps) => A(B) ~> #freezerFunCall(C)
+  rule (A:Exp[B:Exps])(C:Exps) => A[B] ~> #freezerFunCall(C)
+  rule V:Val ~> #freezerFunCall(C:Exps) => V(C)
+  syntax KItem ::= "#freezerFunCall" "(" K ")"
+  /*
+  context HOLE(_:Exps)
+    requires getKLabel HOLE ==KLabel '_`(_`) orBool getKLabel HOLE ==KLabel '_`[_`]
+  */
+
+  rule <k> (lookup(L) => V)(_:Exps) ...</k>  <store>... L |-> V:Val ...</store>
+    [group(lookup)]
+
      +

      Instance of

      +

      Like in untyped KOOL.

      +
      k
        rule objectClosure(_, ListItem(envStackFrame(C,_)) _)
+       instanceOf C => true
+
+  rule objectClosure(_, (ListItem(envStackFrame(C::Id,_)) => .List) _)
+       instanceOf C'  requires C =/=K C'
+
+  rule objectClosure(_, .List) instanceOf _ => false
+
      +

      Cast

      +

      Unlike in untyped KOOL, in typed KOOL we actually check that the object +can indeed be cast to the claimed type.

      +
      k
        rule (C:Id) objectClosure(Irrelevant, EStack)
+    => objectClosure(Irrelevant, EStack) instanceOf C ~> true?
+       ~> objectClosure(C, EStack)
+
      +

      KOOL-specific auxiliary declarations and operations

      +

      Objects as lvalues

      +

      Like in untyped KOOL.

      +
      k
        rule <k> lvalue(X:Id => this . X) ...</k>  <env> Env </env>
+    requires notBool(X in keys(Env))
+
+  context lvalue((HOLE . _)::Exp)
+
+/*  rule lvalue(objectClosure(<crntObj> <crntClass> C </crntClass>
+                            <envStack>... ListItem((C,EnvC:EnvCell)) EStack </envStack> </crntObj>)
+              . X
+              => lookupMember(<envStack> ListItem((C,EnvC)) EStack </envStack>,
+                              X)) */
+  rule lvalue(objectClosure(C:Id,
+                            ListItem(envStackFrame(C,Env)) EStack)
+              . X
+              => lookupMember(ListItem(envStackFrame(C,Env)) EStack,
+                              X))
+  rule lvalue(objectClosure(C,
+                            (ListItem(envStackFrame(C',_)) => .List) _EStack)
+              . _X)
+    requires C =/=K C'
+
      +

      Lookup member

      +

      Like in untyped KOOL.

      +
      k
        syntax Exp ::= lookupMember(List,Id)  [function]
+
+  rule lookupMember(ListItem(envStackFrame(_, X |-> L _)) _, X) => lookup(L)
+
+  // TODO: fix rule below as shown once we support functions with deep rewrites
+  // rule lookupMember(<envStack> ListItem((_, <env> Env </env>)) => .List
+  //                     ...</envStack>, X)
+  //   requires notBool(X in keys(Env))
+  rule lookupMember(ListItem(envStackFrame(_, Env)) L, X)
+    => lookupMember(L, X)
+    requires notBool(X in keys(Env))
+
      +

      typeOf for the additional values}

      +
      k
        rule typeOf(objectClosure(C,_)) => C
+  rule typeOf(methodClosure(T:Type,_,_,_Ps:Params,_)) => T
+
      +

      Subtype checking

      +

      The subclass relation induces a subtyping relation.

      +
      k
        syntax Exp ::= subtype(Types,Types)
+
+  rule subtype(T:Type, T) => true
+
+  rule <k> subtype(C1:Id, C:Id) => subtype(C2, C) ...</k>
+       <className> C1 </className>
+       <baseClass> C2:Id </baseClass>
+    requires C1 =/=K C
+
+  rule subtype(Object,Class:Id) => false
+    requires Class =/=K Object
+
+  rule subtype(Ts1->T2,Ts1'->T2') => subtype(((T2)::Type,Ts1'),((T2')::Type,Ts1))
+
+// Note that the following rule would be wrong!
+//  rule subtype(T[],T'[]) => subtype(T,T')
+
+  rule subtype((T:Type,Ts),(T':Type,Ts')) => subtype(T,T') && subtype(Ts,Ts')
+    requires Ts =/=K .Types
+  rule subtype(.Types,.Types) => true
+
      +

      Unsafe Casting

      +

      Performs unsafe casting. One should only use it in combination with +the subtype relation above.

      +
      k
        syntax Val ::= unsafeCast(Val,Type)  [function]
+
+  rule unsafeCast(objectClosure(_,EStack), C:Id)
+    => objectClosure(C,EStack)
+
+  rule unsafeCast(methodClosure(_T',C,OL,Ps,S), T) => methodClosure(T,C,OL,Ps,S)
+
+  rule unsafeCast(V:Val, T:Type) => V  requires typeOf(V) ==K T
+
      +

      Generic guard

      +

      A generic computational guard: it allows the computation to continue +only if a prefix guard evaluates to true.

      +
      k
        syntax KItem ::= "true?"
+  rule true ~> true? => .
+
+endmodule
+
      +

      Go to Lesson 3, KOOL typed static.

      +
      +
      + + +
      + +
      +
      + + KOOL — Typed — Dynamic + +
      +
      + + Abstract +
      + + Syntax +
      + + Types +
      + + Declarations +
      + + Expressions +
      + + Statements +
      + + Desugaring macros +
      + + Semantics +
      + + Configuration +
      + + Unchanged semantics from dynamically typed SIMPLE +
      + + Unchanged auxiliary operations from dynamically typed SIMPLE +
      + + Changes to the existing dynamically typed SIMPLE semantics +
      + + Program initialization +
      + + Method application +
      + + Assignment +
      + + Typed exceptions +
      + + Spawn +
      + + Semantics of the new KOOL constructs +
      + + Class declaration +
      + + Method declaration +
      + + New +
      + + Self reference +
      + + Object member access +
      + + Method invocation +
      + + Instance of +
      + + Cast +
      + + KOOL-specific auxiliary declarations and operations +
      + + Objects as lvalues +
      + + Lookup member +
      + + typeOf for the additional values} +
      + + Subtype checking +
      + + Unsafe Casting +
      + + Generic guard +
      +
      +
      + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/NOTES/index.html b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/NOTES/index.html new file mode 100644 index 00000000000..edef1815783 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/NOTES/index.html @@ -0,0 +1,380 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Why is the following happening at line 347? It should infer the sort Stmts for S:

      +

      rule {S} => block ... Rho R +(.Bag => S Rho R )

      +

      [Error] Critical: Could not infer a sort for variable 'S' to match every location.

      +

      Similarly at line 517.

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/kool-typed-static/index.html b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/kool-typed-static/index.html new file mode 100644 index 00000000000..5e3b1ec6187 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/2_kool/2_typed/2_static/kool-typed-static/index.html @@ -0,0 +1,1424 @@ + + + + + + + + + + + + + + +KOOL — Typed — Static | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      KOOL — Typed — Static

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K static semantics of the typed KOOL language. +It extends the static semantics of typed SIMPLE with static semantics +for the object-oriented constructs. Also, the static semantics of +some of the existing SIMPLE constructs need to change, in order to +become more generous with regards to the set of accepted programs, +mostly due to subtyping. For example, the assignment construct +x = e required that both the variable x and the +expression e had the same type in SIMPLE. In KOOL, the type +of e can be a subtype of the type of x. +Specifically, we define the following typing policy for KOOL, +everything else not mentioned below borrowing its semantics from +SIMPLE:

      +
        +
      • +

        Each class C yields a homonymous type, which can be +explicitly used in programs to type variables and methods, possibly in +combination with other types.

        +
        • +
      • +

        Since now we have user-defined types, we check that each type +used in a KOOL program is well-formed, that is, it is constructed only +from primitive and class types corresponding to declared classes.

        +
        • +
      • +

        Class members and their types form a class type +environment. Each class will have such a type environment. +Each member in a class is allowed to be declared only once. Since in +KOOL we allow methods to be assigned to fields, we make no distinction +between field and method members; in other words, we reject programs +declaring both a field and a method with the same name.

        +
        • +
      • +

        If an identifier is not found in the local type environment, it +will be searched for in the current class type environment. If not +there, then it will be searched for in its superclass' type +environment. And so on and so forth. If not found until the +Object class is reached, a typing error is reported.

        +
        • +
      • +

        The assignment allows variables to be assigned values of +more concrete types. The result type of the assignment expression +construct will be the (more abstract) type of the assigned variable, +and not the (more concrete) type of the expression, like in Java.

        +
        • +
      • +

        Exceptions are changed (from SIMPLE) to allow throwing and +catching only objects, like in Java. Also, unlike in SIMPLE, we do +not check whether the type of the thrown exception matches the type of +the caught variable, because exceptions can be caught by other +try/catch blocks, even by ones in other methods. To avoid +having to annotate each method with what exceptions it can throw, we +prefer to not check the type safety of exceptions (although this is an +excellent homework!). We only check that the try block +type-checks and that the catch block type-checks after we bind +the caught variable to its claimed type.

        +
        • +
      • +

        Class declarations are not allowed to have any cycles in their +extends relation. Such cycles would lead to non-termination of +new, as it actually does in the dynamic semantics of KOOL +where no such circularity checks are performed.

        +
        • +
      • +

        Methods overriding other methods should be in the right subtyping +relationship with the overridden methods: co-variant in the codomain +and contra-variant in the domain.

        +
        • +
      +
      k
      module KOOL-TYPED-STATIC-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      The syntax of statically typed KOOL is identical to that of +dynamically typed KOOL, they both taking as input the same programs. +What differs is the K strictness attributes. Like in statically +typed SIMPLE, almost all language constructs are strict now, since we +want each to type its arguments almost all the time. Like in the +other two KOOL definitions, we prefer to copy and then modify/extend +the syntax of statically typed SIMPLE.

      +

      Note: This paragraph is old, now we can do things better. We keep +it here only for historical reasons, to see how much we used to suffer :)

      +

      Annoying K-tool technical problem: +Currently, the K tool treats the "non-terminal" productions (i.e., +productions consisting of just one non-terminal), also called +"subsorting" production, differently from the other productions. +Specifically, it does not insert a node in the AST for them. This may +look desirable at first, but it has a big problem: it does not allow +us to treat the subsort differently in different context. For +example, since we want Id to be both a type (a class name) and a +program variable, and since we want expressions to reduce to their +types, we are in an impossible situations in which we do not know how +to treat an identifier in the semantics: as a type, i.e., a result of +computations, or as a program variable, i.e., a non-result. Ideally, +we would like to tag the identifiers at parse-time with their local +interpretation, but that, unfortunately, is not possible with the +current parsing capabilities of the K tool, because it requires to +insert additional information in the AST for the subsort productions. +This will be fixed soon. Until then, unfortunately, we have to do the +job of the parser manually. Instead of subsorting Id directly +to Type, we "wrap" it first, say with a wrapper called +class(...), exactly how the parser should have done. +The major drawback of this is that all the typed KOOL programs +in kool/typed/programs need to also be modified to always +declare class types accordingly. The modified programs can be found +in kool/typed/static/programs. So make sure you execute the +static semantics of KOOL using the modified programs. To avoid seeing +the wrapper in the generated documentation, we associate it an +"invisibility" latex attribute below.

      +
      k
        syntax Id ::= "Object" [token] | "Main" [token]
+
      +

      Types

      +
      k
        syntax Type ::= "void" | "int" | "bool" | "string"
+                | Id                     [klabel("class"), symbol, avoid]  // see next
+                | Type "[" "]"
+                | "(" Type ")"           [bracket]
+                > Types "->" Type
+
+  syntax Types ::= List{Type,","}        [klabel(exps)]
+
      +

      Declarations

      +
      k
        syntax Param ::= Type Id
+  syntax Params ::= List{Param,","}
+
+  syntax Stmt ::= Type Exps ";" [avoid]
+                | Type Id "(" Params ")" Block
+                | "class" Id Block
+                | "class" Id "extends" Id Block
+
      +

      Expressions

      +
      k
        syntax FieldReference ::= Exp "." Id          [strict(1)]
+  syntax ArrayReference ::= Exp "[" Exps "]"    [strict]
+
+  syntax Exp ::= Int | Bool | String | Id
+               | "this"
+               | "super"
+               | "(" Exp ")"             [bracket]
+               | "++" Exp
+               | Exp "instanceOf" Id     [strict(1)]
+               | "(" Id ")" Exp          [strict(2)]
+               | "new" Id "(" Exps ")"   [strict(2)]
+               > Exp "(" Exps ")"        [strict]
+               | "-" Exp                 [strict]
+               | "sizeOf" "(" Exp ")"    [strict]
+               | "read" "(" ")"
+               > left:
+                 Exp "*" Exp             [strict, left]
+               | Exp "/" Exp             [strict, left]
+               | Exp "%" Exp             [strict, left]
+               > left:
+                 Exp "+" Exp             [strict, left]
+               | Exp "-" Exp             [strict, left]
+               > non-assoc:
+                 Exp "<" Exp             [strict, non-assoc]
+               | Exp "<=" Exp            [strict, non-assoc]
+               | Exp ">" Exp             [strict, non-assoc]
+               | Exp ">=" Exp            [strict, non-assoc]
+               | Exp "==" Exp            [strict, non-assoc]
+               | Exp "!=" Exp            [strict, non-assoc]
+               > "!" Exp                 [strict]
+               > left:
+                 Exp "&&" Exp            [strict, left]
+               | Exp "||" Exp            [strict, left]
+               > "spawn" Block  // not strict: to check return and exceptions
+               > Exp "=" Exp             [strict(2), right]
+
+  syntax Exp ::= FieldReference | ArrayReference
+  syntax priority _.__KOOL-TYPED-STATIC-SYNTAX > _[_]_KOOL-TYPED-STATIC-SYNTAX > _(_)_KOOL-TYPED-STATIC-SYNTAX
+
+  syntax Exps ::= List{Exp,","}          [strict, klabel(exps)]
+
      +

      Statements

      +
      k
        syntax Block ::= "{" "}"
+                | "{" Stmt "}"
+
+  syntax Stmt ::= Block
+                | Exp ";"                                 [strict]
+                | "if" "(" Exp ")" Block "else" Block     [avoid, strict]
+                | "if" "(" Exp ")" Block                  [macro]
+                | "while" "(" Exp ")" Block               [strict]
+                | "for" "(" Stmt Exp ";" Exp ")" Block    [macro]
+                | "return" Exp ";"                        [strict]
+                | "return" ";"
+                | "print" "(" Exps ")" ";"                [strict]
+                | "try" Block "catch" "(" Param ")" Block [strict(1)]
+                | "throw" Exp ";"                         [strict]
+                | "join" Exp ";"                          [strict]
+                | "acquire" Exp ";"                       [strict]
+                | "release" Exp ";"                       [strict]
+                | "rendezvous" Exp ";"                    [strict]
+
+  syntax Stmt ::= Stmt Stmt                            [seqstrict, right]
+
      +

      Desugaring macros

      +
      k
        rule if (E) S => if (E) S else {}
+  rule for(Start Cond; Step) {S:Stmt} => {Start while(Cond){S Step;}}
+  rule T:Type E1:Exp, E2:Exp, Es:Exps; => T E1; T E2, Es;               [anywhere]
+  rule T:Type X:Id = E; => T X; X = E;                                  [anywhere]
+
+  rule class C:Id S => class C extends Object S
+
+endmodule
+
      +

      Static semantics

      +

      We first discuss the configuration, then give the static semantics +taken over unchanged from SIMPLE, then discuss the static semantics of +SIMPLE syntactic constructs that needs to change, and in the end we +discuss the static semantics and additional checks specifically +related to the KOOL proper syntax.

      +
      k
      module KOOL-TYPED-STATIC
+  imports KOOL-TYPED-STATIC-SYNTAX
+  imports DOMAINS
+
      +

      Configuration

      +

      The configuration of our type system consists of a tasks +cell with the same meaning like in statically typed SIMPLE, of an +out cell streamed to the standard output that will be used to +display typing error messages, and of a cell classes holding +data about each class in a separate class cell. The +task cells now have two additional optional subcells, namely +ctenvT and inClass. The former holds a temporary +class type environment; its contents will be transferred into the +ctenv cell of the corresponding class as soon as all the +fields and methods in the task are processed. In fact, there will be +three types of tasks in the subsequent semantics, each determined by +the subset of cells that it holds:

      +
        +
      1. +

        Main task, holding only a k cell holding the +original program as a set of classes. The role of this task is to +process each class, generating a class task (see next) for each.

        +
        2. +
      2. +

        Class task, holding k, ctenvT, and +inClass subcells. The role of this task type is to process +a class' contents, generating a class type environment in the +ctenvT cell and a method task (see next) for each method in +the class. To avoid interference with object member lookup rules +below, it is important to add the class type environment to a class +atomically; this is the reason for which we use ctenvT +temporary cells within class tasks (instead of adding each member +incrementally to the class' type environment).

        +
        3. +
      3. +

        Method task, holding k, tenv and +return cells. These tasks are similar to SIMPLE's function +tasks, so we do not discuss them here any further.

        +
        4. +
      +

      Each class cell hods its name (in the className +cell) and the name of the class it extends (in the extends +cell), as well as its type environment (in the ctenv cell) +and the set of all its superclasses (in the extendsAll cell). +The later is useful for example for checking whether there are cycles +in the class extends relation.

      +
      k
        configuration <T multiplicity="?" color="yellow">
+                  <tasks color="orange" multiplicity="?">
+                    <task multiplicity="*" color="yellow" type="Set">
+                      <k color="green"> $PGM:Stmt </k>
+                      <tenv multiplicity="?" color="cyan"> .Map </tenv>
+                      <ctenvT multiplicity="?" color="blue"> .Map </ctenvT>
+                      <returnType multiplicity="?" color="black"> void </returnType>
+                      <inClass multiplicity="?" color="Fuchsia"> .K </inClass>
+                    </task>
+                  </tasks>
+//                  <br/>
+                  <classes color="Fuchsia">
+                    <classData multiplicity="*" type="Map">
+                      <className color="Fuchsia"> Object </className>
+                      <baseClass color="Fuchsia"> .K </baseClass>
+                      <baseClasses color="Fuchsia"> .Set </baseClasses>
+                      <ctenv multiplicity="?" color="blue"> .Map </ctenv>
+                    </classData>
+                  </classes>
+                </T>
+                <output color="brown" stream="stdout"> .List </output>
+
      +

      Unchanged semantics from statically typed SIMPLE

      +

      The syntax and rules below are borrowed unchanged from statically +typed SIMPLE, so we do not discuss them much here.

      +
      k
        syntax Exp ::= Type
+  syntax Exps ::= Types
+  syntax BlockOrStmtType ::= "block" | "stmt"
+  syntax Type ::= BlockOrStmtType
+  syntax Block ::= BlockOrStmtType
+  syntax KResult ::= Type
+                   | Types  // TODO: should not be needed
+
+
+  context _:Type _::Exp[HOLE::Exps];
+
+  rule T:Type E:Exp[int,Ts:Types]; => T[] E[Ts];
+  rule T:Type E:Exp[.Types]; => T E;
+
+
+  rule <task>... <k> _:BlockOrStmtType </k> <tenv> _ </tenv> ...</task> => .Bag
+
+
+  rule _:Int => int
+  rule _:Bool => bool
+  rule _:String => string
+
+
+  rule <k> X:Id => T ...</k> <tenv>... X |-> T ...</tenv>
+
+
+  context ++(HOLE => ltype(HOLE))
+  rule ++ int => int
+  rule int + int => int
+  rule string + string => string
+  rule int - int => int
+  rule int * int => int
+  rule int / int => int
+  rule int % int => int
+  rule - int => int
+  rule int < int => bool
+  rule int <= int => bool
+  rule int > int => bool
+  rule int >= int => bool
+  rule T:Type == T => bool
+  rule T:Type != T => bool
+  rule bool && bool => bool
+  rule bool || bool => bool
+  rule ! bool => bool
+
+
+  rule (T[])[int, Ts:Types] => T[Ts]
+  rule T:Type[.Types] => T
+
+  rule sizeOf(_T[]) => int
+
+
+  rule read() => int
+
+  rule print(T:Type, Ts => Ts); when T ==K int orBool T ==K string
+  rule print(.Types); => stmt
+
+
+  context (HOLE => ltype(HOLE)) = _
+
+
+  rule <k> return; => stmt ...</k> <returnType> _ </returnType>
+
+
+  rule {} => block
+
+  rule <task> <k> {S:Stmt} => block ...</k> <tenv> Rho </tenv> R </task>
+       (.Bag => <task> <k> S </k> <tenv> Rho </tenv> R </task>)
+
+  rule _:Type; => stmt
+  rule if (bool) block else block => stmt
+  rule while (bool) block => stmt
+
+  rule join int; => stmt
+  rule acquire _:Type; => stmt
+  rule release _:Type; => stmt
+  rule rendezvous _:Type; => stmt
+
+  syntax Stmt ::= BlockOrStmtType
+  rule _:BlockOrStmtType _:BlockOrStmtType => stmt
+
      +

      Unchanged auxiliary operations from dynamically typed SIMPLE

      +
      k
        syntax Stmt ::= mkDecls(Params)  [function]
+  rule mkDecls(T:Type X:Id, Ps:Params) => T X; mkDecls(Ps)
+  rule mkDecls(.Params) => {}
+
+  syntax LValue ::= Id
+                  | FieldReference
+                  | ArrayReference
+  syntax Exp ::= LValue
+
+  syntax Exp ::= ltype(Exp)
+// We would like to say:
+//  context ltype(HOLE:LValue)
+// but we currently cannot type the HOLE
+  context ltype(HOLE) when isLValue(HOLE)
+
+// OLD approach:
+//  syntax Exp ::= ltype(Exp)  [function]
+//  rule ltype(X:Id) => X
+//  rule ltype(E:Exp [Es:Exps]) => E[Es]
+
+  syntax Types ::= getTypes(Params)  [function]
+  rule getTypes(T:Type _:Id) => T, .Types
+  rule getTypes(T:Type _:Id, P, Ps) => T, getTypes(P,Ps)
+  rule getTypes(.Params) => void, .Types
+
      +

      Changes to the existing statically typed SIMPLE semantics

      +

      Below we give the new static semantics for language constructs that +come from SIMPLE, but whose SIMPLE static semantics was too +restrictive or too permissive and thus had to change.

      +

      Local variable declaration

      +

      Since we can define new types in KOOL (corresponding to classes), the +variable declaration needs to now check that the claimed types exist. +The operation checkType, defined at the end of this module, +checks whether the argument type is correct (it actually works with +lists of types as well).

      +
      k
        rule <k> T:Type X:Id; => checkType(T) ~> stmt ...</k>
+       <tenv> Rho => Rho[X <- T] </tenv>
+
      +

      Class member declaration

      +

      In class tasks, variable declarations mean class member declarations. +Since we reduce method declarations to variable declarations (see +below), a variable declaration in a class task can mean either a field +or a method declaration. Unlike local variable declarations, which +can shadow previous homonymous local or member declarations, member +declarations are regarded as a set, so we disallow multiple +declarations for the same member (one could improve upon this, like in +Java, by treating members with different types or number of arguments +as different, etc., but we do not do it here). We also issue an error +message if one attempts to redeclare the same class member. The +framed variable declaration in the second rule below should be read +"stuck". In fact, it is nothing but a unary operation called +stuck, which takes a K-term as argument and does nothing +with it; this stuck operation is displayed as a frame in this +PDF document because of its latex attribute (see the ASCII .k file, +at the end of this module).

      +
      k
        rule <k> T:Type X:Id; => checkType(T) ~> stmt ...</k>
+       <ctenvT> Rho (.Map => X |-> T) </ctenvT>
+    when notBool(X in keys(Rho))
+
+  rule <k> T:Type X:Id; => stuck(T X;) ...</k>
+       <ctenvT>... X |-> _ ...</ctenvT>
+       <inClass> C:Id </inClass>
+//       <br/>
+       <output>... .List => ListItem("Member \"" +String Id2String(X)
+                              +String "\" declared twice in class \""
+                              +String Id2String(C) +String "\"!\n") </output>
+
      +

      Method declaration

      +

      A method declaration requires two conceptual checks to be performed: +first, that the method's type is consistent with the type of the +homonymous method that it overrides, if any; and second, that its body +types correctly. At the same time, it should also be added to the +type environment of its class. The first conceptual task is performed +using the checkMethod operation defined below, and the second +by generating a corresponding method task. To add it to the class +type environment, we take advantage of the fact that KOOL is higher +order and reduce the problem to a field declaration problem, which we +have already defined. The role of the ctenvT cell in the +rule below is to structurally ensure that the method declaration takes +place in a class task (we do not want to allow methods to be declared, +for example, inside other methods).

      +
      k
        rule <k> T:Type F:Id(Ps:Params) S
+        => checkMethod(F, getTypes(Ps)->T, C')
+           ~> getTypes(Ps)->T F; ...</k>
+//       <br/>
+       <inClass> C </inClass>
+       <ctenvT> _ </ctenvT> // to ensure we are in a class pass
+       <className> C </className>
+       <baseClass> C' </baseClass>
+//       <br/>
+       (.Bag => <task>
+               <k> mkDecls(Ps) S </k>
+               <inClass> C </inClass>
+               <tenv> .Map </tenv>
+               <returnType> T </returnType>
+             </task>)
+
      +

      Assignment

      +

      A more concrete value is allowed to be assigned to a more abstract +variable. The operation checkSubtype is defined at the end +of the module and it also works with pairs of lists of types.

      +
      k
        rule T:Type = T':Type => checkSubtype(T', T) ~> T
+
      +

      Method invocation and return

      +

      Methods can be applied on values of more concrete types than their +arguments:

      +
      k
        rule (Ts:Types -> T:Type) (Ts':Types) => checkSubtype(Ts',Ts) ~> T
+
      +

      Similarly, we allow values of more concrete types to be returned by +methods:

      +
      k
        rule <k> return T:Type; => checkSubtype(T,T') ~> stmt ...</k>
+       <returnType> T':Type </returnType>
+
      +

      Exceptions

      +

      Exceptions can throw and catch values of any types. Since unlike in Java +KOOL's methods do not declare the exception types that they can throw, +we cannot test the full type safety of exceptions. Instead, we +only check that the try and the catch statements +type correctly.

      +
      k
        rule try block catch(T:Type X:Id) S => {T X; S}
+  rule throw _T:Type ; => stmt
+
      +

      Spawn

      +

      The spawned cell needs to also be passed the parent's class.

      +
      k
      // explain why
+
+  rule <k> spawn S:Block => int ...</k>
+       <tenv> Rho </tenv>
+       <inClass> C </inClass>
+       (.Bag => <task>
+               <k> S </k>
+               <tenv> Rho </tenv>
+               <inClass> C </inClass>
+             </task>)
+
      +

      Semantics of the new KOOL constructs

      +

      Class declaration

      +

      We process each class in the main task, adding the corresponding data +into its class cell and also adding a class task for it. We +also perform some well-formedness checks on the class hierarchy.

      +

      Initiate class processing
      +We create a class cell and a class task for each task. Also, we start +the class task with a check that the class it extends is declared +(this delays the task until that class is processed using another +instance of this rule).

      +
      k
      // There seems to be some error with the configuration concretization,
+// as the rule below does not work when rewriting . to both the task
+// and the class cells; I had to include two separate . rewrites
+
+// TODO: the following fails krun; see #2117
+  rule <task> <k> class C:Id extends C':Id { S:Stmt } => stmt ...</k> </task>
+       (.Bag => <classData>...
+               <className> C </className>
+               <baseClass> C' </baseClass>
+             ...</classData>)
+//       <br/>
+       (.Bag => <task>
+                <k> checkType(`class`(C')) ~> S </k>
+                <inClass> C </inClass>
+                <ctenvT> .Map </ctenvT>
+             </task>)
+
+// You may want to try the thing below, but that failed, too
+/*
+syntax Type ::= "stmtStop"
+
+  rule <tasks>...
+       <task> <k> class C:Id extends C':Id { S:Stmt } => stmtStop ...</k> </task>
+       (.Bag => <task>
+                <k> checkType(`class`(C')) ~> S </k>
+                <inClass> C </inClass>
+                <ctenvT> .Map </ctenvT>
+             </task>)
+       ...</tasks>
+       <classes>...
+       .Bag => <classData>...
+               <className> C </className>
+               <baseClass> C' </baseClass>
+             ...</classData>
+       ...</classes>
+//       <br/>
+*/
+
      +

      Check for unique class names

      +
      k
        rule (<T>...
+          <className> C </className>
+          <className> C </className>
+        ...</T> => .Bag)
+       <output>... .List => ListItem("Class \"" +String Id2String(C)
+                                  +String "\" declared twice!\n") </output>
+
      +

      Check for cycles in class hierarchy
      +We check for cycles in the class hierarchy by transitively closing the +class extends relation using the extendsAll cells, and +checking that a class will never appear in its own extendsAll +cell. The first rule below initiates the transitive closure of the +superclass relation, the second transitively closes it, and the third +checks for cycles.

      +
      k
        rule <baseClass> C </baseClass>
+       <baseClasses> .Set => SetItem(C) </baseClasses>  [priority(25)]
+
+  rule <classData>...
+         <baseClasses> SetItem(C) Cs:Set (.Set => SetItem(C')) </baseClasses>
+       ...</classData>
+       <classData>... <className>C</className> <baseClass>C'</baseClass> ...</classData>
+    when notBool(C' in (SetItem(C) Cs))  [priority(25)]
+
+  rule (<T>...
+          <className> C </className>
+          <baseClasses>... SetItem(C) ...</baseClasses>
+        ...</T> => .Bag)
+       <output>... .List => ListItem("Class \"" +String Id2String(C)
+                                  +String "\" is in a cycle!\n") </output>
+    [group(inheritance-cycle), priority(25)]
+
      +

      New

      +

      To type new we only need to check that the class constructor +can be called with arguments of the given types, so we initiate a call +to the constructor method in the corresponding class. If that +succeeds, meaning that it types to stmt, then we discard the +stmt type and produce instead the corresponding class type of +the new object. The auxiliary discard operation is defined +also at the end of this module.

      +
      k
        rule new C:Id(Ts:Types) => `class`(C) . C (Ts) ~> discard ~> `class`(C)
+
      +

      Self reference

      +

      The typing rule for this is straightforward: reduce to the +current class type.

      +
      k
        rule <k> this => `class`(C) ...</k>
+       <inClass> C:Id </inClass>
+
      +

      Super

      +

      Similarly, super types to the parent class type. +Note that for typing concerns, super can be considered as an object +(recall that this was not the case in the dynamic semantics).

      +
      k
         rule <k> super => `class`(C') ...</k>
+        <inClass> C:Id </inClass>
+        <className> C </className>
+        <baseClass> C':Id </baseClass>
+
      +

      Object member access

      +

      There are several cases to consider here. First, if we are in a class +task, we should lookup the member into the temporary class type +environemnt in cell ctenvT. That is because we want to allow +initialized field declarations in classes, such as int x=10;. +This is desugared to a declaration of x, which is added to +ctenvT during the class task processing, followed by an +assignment of x to 10. In order for the assignment to type +check, we need to know that x has been declared with type +int; this information can only be found in the +ctenvT cell. Second, we should redirect non-local variable +lookups in method tasks to corresponding member accesses (the +local variables are handled by the rule borrowed from SIMPLE). +This is what the second rule below does. Third, we should allow +object member accesses as lvalues, which is done by the third rule +below. These last two rules therefore ensure that each necessary +object member access is explicitly allowed for evaluation. Recall +from the annotated syntax module above that the member access +operation is strict in the object. That means that the object is +expected to evaluate to a class type. The next two rules below define +the actual member lookup operation, moving the search to the +superclass when the member is not found in the current class. Note +that this works because we create the class type environments +atomically; thus, a class either has its complete type environment +available, in which case these rules can safely apply, or its cell +ctenv is not yet available, in which case these rules have to +wait. Finally, the sixth rule below reports an error when the +Object class is reached.

      +
      k
        rule <k> X:Id => T ...</k>
+       <ctenvT>... X |-> T ...</ctenvT>
+
+  rule <k> X:Id => this . X ...</k>
+       <tenv> Rho </tenv>
+    when notBool(X in keys(Rho))
+
+// OLD approach:
+//  rule ltype(E:Exp . X:Id) => E . X
+
+  rule <k> `class`(C:Id) . X:Id => T ...</k>
+       <className> C </className>
+       <ctenv>... X |-> T:Type ...</ctenv>
+
+  rule <k> `class`(C1:Id => C2) . X:Id ...</k>
+       <className> C1 </className>
+       <baseClass> C2:Id </baseClass>
+       <ctenv> Rho </ctenv>
+    when notBool(X in keys(Rho))
+
+  rule <k> `class`(Object) . X:Id => stuck(`class`(Object) . X) ...</k>
+       <inClass> C:Id </inClass>
+//      <br/>
+       <output>... .List => ListItem("Member \"" +String Id2String(X)
+                              +String "\" not declared! (see class \""
+                              +String Id2String(C) +String "\")\n") </output>
+
      +

      Instance of and casting

      +

      As it is hard to check statically whether casting is always safe, +the programmer is simply trusted from a typing perspective. We only +do some basic upcasting and downcasting checks, to reject casts which +will absolutely fail. However, dynamic semantics or implementations +of the language need to insert runtime checks for downcasting to be safe.

      +
      k
        rule `class`(_C1:Id) instanceOf _C2:Id => bool
+  rule (C:Id) `class`(C) => `class`(C)
+  rule <k> (C2:Id) `class`(C1:Id) => `class`(C2) ...</k>
+       <className> C1 </className>
+       <baseClasses>...SetItem(C2)...</baseClasses>    // upcast
+  rule <k> (C2:Id) `class`(C1:Id) => `class`(C2) ...</k>
+       <className> C2 </className>
+       <baseClasses>...SetItem(C1)...</baseClasses>    // downcast
+  rule <k> (C2) `class`(C1:Id) => stuck((C2) `class`(C1)) ...</k>
+       <classData>...
+         <className> C1 </className>
+         <baseClasses> S1 </baseClasses>
+       ...</classData>
+       <classData>...
+         <className> C2 </className>
+         <baseClasses> S2 </baseClasses>
+       ...</classData>
+       <output>... .List => ListItem("Classes \"" +String Id2String(C1)
+                              +String "\" and \"" +String Id2String(C2)
+                              +String "\" are incompatible!\n") </output>
+    when notBool(C1 in S2) andBool notBool(C2 in S1)
+
      +

      Cleanup tasks

      +

      Finally, we need to clean up the terminated tasks. Each of the three +types of tasks is handled differently. The main task is replaced by a +method task holding new main();, which will ensure that a +main class with a main() method actually exists +(first rule below). A class task moves its temporary class type +environment into its class' cell, and then it dissolves itself (second +rule). A method task simply dissolves when terminated (third rule); +the presence of the tenv cell in that rule ensures that that +task is a method task. +Finally, when all the tasks are cleaned up, we can also remove the +tasks cell, issuing a corresponding message. Note that +checking for cycles or duplicate methods can still be performed after +the tasks cell has been removed.

      +
      k
      // discard main task when done, issuing a "new main();" command to
+// make sure that the class main and the method main() are declared.
+
+  rule <task> <k> stmt => new Main(.Exps); </k>
+              (.Bag => <tenv> .Map </tenv>
+                    <returnType> void </returnType>
+                    <inClass> Main </inClass>)
+       </task>
+
+// discard class task when done, adding a ctenv in class
+
+  rule (<task>
+          <k> stmt </k>
+          <ctenvT> Rho </ctenvT>
+          <inClass> C:Id </inClass>
+        </task> => .Bag)
+        <className> C </className>
+        (.Bag => <ctenv> Rho </ctenv>)
+
+// discard method task when done
+
+  rule <task>...
+         <k> stmt </k>
+         <tenv> _ </tenv>  // only to ensure that this is a method task
+       ...</task> => .Bag
+
+// cleanup tasks and output a success message when done
+
+  rule (<T>... <tasks> .Bag </tasks> ...</T> => .Bag)
+       <output>... .List => ListItem("Type checked!\n") </output>
+
      +

      KOOL-specific auxiliary declarations and operations

      +

      Subtype checking

      +

      The subclass relation introduces a subtyping relation.

      +
      k
        syntax KItem ::= checkSubtype(Types,Types)
+
+  rule checkSubtype(T:Type, T) => .
+
+  rule <k> checkSubtype(`class`(C:Id), `class`(C':Id)) => . ...</k>
+       <className> C </className>
+       <baseClasses>... SetItem(C') ...</baseClasses>
+
+  rule checkSubtype(Ts1->T2,Ts1'->T2')
+    => checkSubtype(((T2)::Type,Ts1'),((T2')::Type,Ts1))
+
+// note that the following rule would be wrong!
+//  rule checkSubtype(T[],T'[]) => checkSubtype(T,T')
+
+  rule checkSubtype((T:Type,Ts),(T':Type,Ts'))
+    => checkSubtype(T,T') ~> checkSubtype(Ts,Ts')
+    when Ts =/=K .Types
+
+  rule checkSubtype(.Types,.Types) => .
+  rule checkSubtype(.Types,void) => .
+
      +

      Checking well-formedness of types

      +

      Since now any Id can be used as the type of a class, we need to +check that the types used in the program actually exists

      +
      k
        syntax KItem ::= checkType(Types)
+
+  rule checkType(T:Type,Ts:Types) => checkType(T) ~> checkType(Ts)
+    when Ts =/=K .Types
+  rule checkType(.Types) => .
+  rule checkType(int) => .
+  rule checkType(bool) => .
+  rule checkType(string) => .
+  rule checkType(void) => .
+  rule <k> checkType(`class`(C:Id)) => . ...</k> <className> C </className>
+  rule checkType(`class`(Object)) => .
+  rule checkType(Ts:Types -> T:Type) => checkType(T,Ts)
+  rule checkType(T:Type[]) => checkType(T)
+
      +

      Checking correct overiding of methods

      +

      The checkMethod operation below searches to see whether +the current method overrides some other method in some superclass. +If yes, then it issues an additional check that the new method's type +is more concrete than the overridden method's. The types T and T' +below can only be function types. See the definition of +checkSubtype on function types at the end of this module (it +is co-variant in the codomain and contra-variant in the domain).

      +
      k
        syntax KItem ::= checkMethod(Id,Type,Id)
+
+  rule <k> checkMethod(F:Id, T:Type, C:Id) => checkSubtype(T, T') ...</k>
+       <className> C </className>
+       <ctenv>... F |-> T':Type ...</ctenv>
+
+  rule <k> checkMethod(F:Id, _T:Type, (C:Id => C')) ...</k>
+       <className> C </className>
+       <baseClass> C':Id </baseClass>
+       <ctenv> Rho </ctenv>
+    when notBool(F in keys(Rho))
+
+  rule checkMethod(_:Id,_,Object) => .
+
      +

      Generic operations which could be part of the K framework

      +
      k
        syntax KItem ::= stuck(K)  [latex(\framebox{${#1}$})]
+
+  syntax KItem ::= "discard"
+  rule _:KResult ~> discard => .
+
+endmodule
+
      +
      +
      + + +
      + +
      +
      + + KOOL — Typed — Static + +
      +
      + + Abstract +
      + + Syntax +
      + + Types +
      + + Declarations +
      + + Expressions +
      + + Statements +
      + + Desugaring macros +
      + + Static semantics +
      + + Configuration +
      + + Unchanged semantics from statically typed SIMPLE +
      + + Unchanged auxiliary operations from dynamically typed SIMPLE +
      + + Changes to the existing statically typed SIMPLE semantics +
      + + Local variable declaration +
      + + Class member declaration +
      + + Method declaration +
      + + Assignment +
      + + Method invocation and return +
      + + Exceptions +
      + + Spawn +
      + + Semantics of the new KOOL constructs +
      + + Class declaration +
      + + Check for unique class names +
      + + New +
      + + Self reference +
      + + Super +
      + + Object member access +
      + + Instance of and casting +
      + + Cleanup tasks +
      + + KOOL-specific auxiliary declarations and operations +
      + + Subtype checking +
      + + Checking well-formedness of types +
      + + Checking correct overiding of methods +
      + + Generic operations which could be part of the K framework +
      +
      +
      + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/1_environment/fun-untyped/index.html b/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/1_environment/fun-untyped/index.html new file mode 100644 index 00000000000..acc4ca0f33c --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/1_environment/fun-untyped/index.html @@ -0,0 +1,1225 @@ + + + + + + + + + + + + + + +FUN — Untyped — Environment | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      FUN — Untyped — Environment

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K semantic definition of the untyped FUN language. +FUN is a pedagogical and research language that captures the essence +of the functional programming paradigm, extended with several features +often encountered in functional programming languages. +Like many functional languages, FUN is an expression language, that +is, everything, including the main program, is an expression. +Functions can be declared anywhere and are first class values in the +language. +FUN is call-by-value here, but it has been extended (as student +homework assignments) with other parameter-passing styles. +To make it more interesting and to highlight some of K's strengths, +FUN includes the following features:

      +
        +
      • +

        The basic builtin data-types of integers, booleans and strings.

        +
        • +
      • +

        Builtin lists, which can hold any elements, including other lists. +Lists are enclosed in square brackets and their elements are +comma-separated; e.g., [1,2,3].

        +
        • +
      • +

        User-defined data-types, by means of constructor terms. +Constructor names start with a capital letter (while any other +identifier in the language starts with a lowercase letter), and they +can be followed by an arbitrary number of comma-separated arguments +enclosed in parentheses; parentheses are not needed when the +constructor takes no arguments. +For example, Pair(5,7) is a constructor term holding two +numbers, Cons(1,Cons(2,Cons(3,Nil))) is a list-like +constructor term holding 3 elements, and +Tree(Tree(Leaf(1), Leaf(2)), Leaf(3)) is a tree-like +constructor term holding 3 elements. +In the untyped version of the FUN language, no type checking or +inference is performed to ensure that the data constructors are used +correctly. +The execution will simply get stuck when they are misused. +Moreover, since no type checking is performed, the data-types are not +even declared in the untyped version of FUN.

        +
        • +
      • +

        Functions and let/letrec binders can take +multiple space-separated arguments, but these are desugared to +ones that only take one argument, by currying. For example, the +expressions

        +
        fun x y -> x y
+let x y = y in x
+
        +

        are desugared, respectively, into the following expressions:

        +
        fun x -> fun y -> x y
+let x = fun y -> y in x
+
        +
        • +
      • +

        Functions can be defined using pattern matching over the +available data-types. For example, the program

        +
        letrec max = fun [h] -> h
+             |   [h|t] -> let x = max t
+                          in  if h > x then h else x
+in max [1, 3, 5, 2, 4, 0, -1, -5]
+
        +

        defines a function max that calculates the maximum element of +a non-empty list, and the function

        +
        letrec ack = fun Pair(0,n) -> n + 1
+             |   Pair(m,0) -> ack Pair(m - 1, 1)
+             |   Pair(m,n) -> ack Pair(m - 1, ack Pair(m, n - 1))
+in ack Pair(2,3)
+
        +

        calculates the Ackermann function applied to a particular pair of numbers. +Patterns can be nested. Patterns can currently only be used in function +definitions, and not directly in let/letrec binders. +For example, this is not allowed:

        +
        letrec Pai(x,y) = Pair(1,2) in x+y
+
        +

        But this is allowed:

        +
        let f Pair(x,y) = x+y in f Pair(1,2)
+
        +

        because it is first reduced to

        +
        let f = fun Pair(x,y) -> x+y in f Pair(1,2)
+
        +

        by uncurrying of the let binder, and pattern matching is +allowed in function arguments.

        +
        • +
      • +

        We include a callcc construct, for two reasons: first, +several functional languages support this construct; second, some +semantic frameworks have difficulties defining it. Not K.

        +
        • +
      • +

        Finally, we include mutables by means of referencing an +expression, getting the reference of a variable, dereferencing and +assignment. We include these for the same reasons as above: there are +languages which have them, and they are not easy to define in some +semantic frameworks.

        +
        • +
      +

      Like in many other languages, some of FUN's constructs can be +desugared into a smaller set of basic constructs. We do that as usual, +using macros, and then we only give semantics to the core constructs.

      +

      Note:
      +We recommend the reader to first consult the dynamic semantics of the +LAMBDA++ language in the first part of the K Tutorial. +To keep the comments below small and focused, we will not re-explain +functional or K features that have already been explained in there.

      +

      Syntax

      +
      k
      //require "modules/pattern-matching.k"
+
+module FUN-UNTYPED-COMMON
+  imports DOMAINS-SYNTAX
+
      +

      FUN is an expression language. The constructs below fall into +several categories: names, arithmetic constructs, conventional +functional constructs, patterns and pattern matching, data constructs, +lists, references, and call-with-current-continuation (callcc). +The arithmetic constructs are standard; they are present in almost all +our K language definitions. The meaning of FUN's constructs are +discussed in more depth when we define their semantics in the next +module.

      +

      The Syntactic Constructs

      +

      We start with the syntactic definition of FUN names. +We have several categories of names: ones to be used for functions and +variables, others to be used for data constructors, others for types and +others for type variables. We will introduce them as needed, starting +with the former category. We prefer the names of variables and functions +to start with lower case letters. We take the freedom to tacitly introduce +syntactic lists/sequences for each nonterminal for which we need them:

      +
      k
        syntax Name                                      [token]
+  syntax Names ::= List{Name,","}                  [klabel(exps)]
+
      +

      Expression constructs will be defined throughtout the syntax module. +Below are the very basic ones, namely the builtins, the names, and the +parentheses used as brackets for grouping. Lists of expressions are +declared strict, so all expressions in the list get evaluated whenever +the list is on a position which can be evaluated:

      +
      k
        syntax Exp ::= Int | Bool | String | Name
+               | "(" Exp ")"                       [bracket]
+  syntax Exps  ::= List{Exp,","}                   [strict, klabel(exps)]
+  syntax Val
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax Vals ::= List{Val,","}                    [klabel(exps)]
+  syntax Bottom
+  syntax Bottoms ::= List{Bottom,","}              [klabel(exps)]
+
      +

      We next define the syntax of arithmetic constructs, together with +their relative priorities and left-/non-associativities. We also +tag all these rules as members of a new group, "arith", so we can more easily +define global syntax priorities later (at the end of the syntax module).

      +
      k
        syntax Exp ::= left:
+                 Exp "*" Exp                       [strict, group(arith)]
+               | Exp "/" Exp                       [strict, group(arith)]
+               | Exp "%" Exp                       [strict, group(arith)]
+               > left:
+                 Exp "+" Exp                       [strict, left, group(arith)]
+               | Exp "^" Exp                       [strict, left, group(arith)]
+// left attribute should not be necessary; currently a parsing bug
+               | Exp "-" Exp                       [strict, prefer, group(arith)]
+// the "prefer" attribute above is to not parse x-1 as x(-1)
+// Due to some parsing problems, we currently cannot add unary minus:
+               | "-" Exp                           [strict, group(arith)]
+               > non-assoc:
+                 Exp "<" Exp                       [strict, group(arith)]
+               | Exp "<=" Exp                      [strict, group(arith)]
+               | Exp ">" Exp                       [strict, group(arith)]
+               | Exp ">=" Exp                      [strict, group(arith)]
+               | Exp "==" Exp                      [strict, group(arith)]
+               | Exp "!=" Exp                      [strict, group(arith)]
+               > "!" Exp                           [strict, group(arith)]
+               > Exp "&&" Exp                      [strict(1), left, group(arith)]
+               > Exp "||" Exp                      [strict(1), left, group(arith)]
+
      +

      The conditional construct has the expected evaluation strategy, +stating that only the first argument is evaluate:

      +
      k
        syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+
      +

      FUN's builtin lists are formed by enclosing comma-separated +sequences of expressions (i.e., terms of sort Exps) in square +brackets. The list constructor cons adds a new element to the +top of the list, head and tail get the first element +and the tail sublist of a list if they exist, respectively, and get +stuck otherwise, and null?? tests whether a list is empty or +not; syntactically, these are just expression constants. +In function patterns, we are also going to allow patterns following the +usual head/tail notation; for example, the pattern [x_1,...,x_n|t] +binds x_1, ..., x_n to the first elements of the matched list, +and t to the list formed with the remaining elements. We define list +patterns as ordinary expression constructs, although we will make sure that +we do not give them semantics if they appear in any other place then in a +function case pattern.

      +
      k
        syntax Exp ::= "[" Exps "]"                             [strict, klabel(list)]
+               | "head" [macro] | "tail" [macro] | "null?" [macro]
+               | "[" Exps "|" Exp "]"
+  syntax Val ::= "[" Vals "]"                             [klabel(list)]
+  syntax Cons ::= "cons"
+  syntax Val ::= Cons
+  syntax Val ::= Cons Val                                 [klabel(apply)]
+
      +

      Data constructors start with capital letters and they may or may +not have arguments. We need to use the attribute "prefer" to make +sure that, e.g., Cons(a) parses as constructor Cons with +argument a, and not as the expression Cons (because +constructor names are also expressions) regarded as a function applied +to the expression a. Also, note that the constructor is strict +in its second argument, because we want to evaluate its arguments but +not the constuctor name itsef.

      +
      k
        syntax ConstructorName                         [token]
+  syntax Exp ::= ConstructorName
+               | ConstructorName "(" Exps ")"    [prefer, strict(2), klabel(constructor)]
+  syntax Val ::= ConstructorName "(" Vals ")"    [klabel(constructor)]
+
      +

      A function is essentially a |-separated ordered +sequence of cases, each case of the form pattern -> expression, +preceded by the language construct fun. Patterns will be defined +shortly, both for the builtin lists and for user-defined constructors. +Recall that the syntax we define in K is not meant to serve as a +ultimate parser for the defined language, but rather as a convenient +notation for K abstract syntax trees, which we prefer when we write +the semantic rules. It is therefore often the case that we define a +more ``generous'' syntax than we want to allow programs to use. +We do it here, too. Specifically, the syntax of Cases +below allows any expressions to appear as pattern. This syntactic +relaxation permits many wrong programs to be parsed, but that is not a +problem because we are not going to give semantics to wrong combinations, +so those programs will get stuck; moreover, our type inferencer will reject +those programs anyway. Function application is just concatenation of +expressions, without worrying about type correctness. Again, the type +system will reject type-incorrect programs.

      +
      k
        syntax Exp ::= "fun" Cases
+               | Exp Exp                              [strict, left, klabel(apply)]
+// NOTE: We would like eventually to also have Exp "(" Exps ")
+  syntax Case  ::= Exp "->" Exp
+  syntax Cases ::= List{Case, "|"}
+
      +

      The let and letrec binders have the usual syntax +and functional meaning. We allow multiple and-separated bindings. +Like for the function cases above, we allow a more generous syntax for +the left-hand sides of bindings, noting that the semantics will get stuck +on incorrect bindings and that the type system will reject those programs.

      +
      k
        syntax Exp ::= "let" Bindings "in" Exp
+               | "letrec" Bindings "in" Exp                 [prefer]
+// The "prefer" attribute for letrec currently needed due to tool bug,
+// to make sure that "letrec" is not parsed as "let rec".
+  syntax Binding  ::= Exp "=" Exp
+  syntax Bindings ::= List{Binding,"and"}
+
      +

      References are first class values in FUN. The construct ref +takes an expression, evaluates it, and then it stores the resulting value +at a fresh location in the store and returns that reference. Syntactically, +ref is just an expression constant. The construct & +takes a name as argument and evaluates to a reference, namely the store +reference where the variable passed as argument stores its value; this +construct is a bit controversial and is further discussed in the +environment-based semantics of the FUN language, where we desugar +ref to it. The construct @ takes a reference +and evaluates to the value stored there. The construct := takes +two expressions, the first expected to evaluate to a reference; the value +of its second argument will be stored at the location to which the first +points (the old value is thus lost). Finally, since expression evaluation +now has side effects, it makes sense to also add a sequential composition +construct, which is sequentially strict. This evaluates to the value of +its second argument; the value of the first argument is lost (which has +therefore been evaluated only for its side effects.

      +
      k
        syntax Exp ::= "ref"                             [macro]
+               | "&" Name
+               | "@" Exp                                     [strict]
+               | Exp ":=" Exp                                [strict]
+               | Exp ";" Exp                       [strict(1), right]
+
      +

      Call-with-current-continuation, named callcc in FUN, is a +powerful control operator that originated in the Scheme programming +language, but it now exists in many other functional languages. It works +by evaluating its argument, expected to evaluate to a function, and by +passing the current continuation, or evaluation context (or computation, +in K terminology), as a special value to it. When/If this special value +is invoked, the current context is discarded and replaced with the one +held by the special value and the computation continues from there. +It is like taking a snapshot of the execution context at some moment +in time and then, when desired, being able to get back in time to that +point. If you like games, it is like saving the game now (so you can +work on your homework!) and then continuing the game tomorrow or whenever +you wish. To issustrate the strength of callcc, we also +allow exceptions in FUN by means of a conventional try-catch +construct, which will desugar to callcc. We also need to +introduce the special expression contant throw, but we need to +use it as a function argument name in the desugaring macro, so we define +it as a name instead of as an expression constant:

      +
      k
        syntax Exp ::= "try" Exp "catch" "(" Name ")" Exp [macro]
+  syntax Val ::= "callcc"
+  syntax Name ::= "throw" [token]
+
      +

      Finally, FUN also allows polymorphic datatype declarations. These +will be useful when we define the type system later on.

      +
      k
        syntax Exp ::= "datatype" Type "=" TypeCases Exp [macro]
+// NOTE: In a future version of K, we want the datatype declaration
+// to be a construct by itself, but that is not possible currently
+// because K's parser wronly identifies the __ operation allowing
+// a declaration to appear in front of an expression with the function
+// application construct, giving ambiguous parsing errors.
+
      +

      We next need to define the syntax of types and type cases that appear +in datatype declarations.

      +

      Like in many functional languages, type parameters/variables in +user-defined types are quoted identifiers.

      +
      k
        syntax TypeVar                        [token]
+  syntax TypeVars ::= List{TypeVar,","} [klabel(types)]
+
      +

      Types can be basic types, function types, or user-defined +parametric types. In the dynamic semantics we are going to simply ignore +all the type declations, so here the syntax of types below is only useful +for generating the desired parser. To avoid syntactic ambiguities with +the arrow construct for function cases, we use the symbol --> as +a constructor for function types:

      +
      k
        syntax TypeName [token]
+  syntax Type ::= "int" | "bool" | "string"
+                | Type "-->" Type                            [right]
+                | "(" Type ")"                             [bracket]
+                | TypeVar
+                | TypeName             [klabel(TypeName), avoid]
+                | Type TypeName   [klabel(Type-TypeName), symbol, macro]
+                | "(" Types ")" TypeName                    [prefer]
+  syntax Types ::= List{Type,","} [klabel(types)]
+  syntax Types ::= TypeVars
+
+  syntax TypeCase ::= ConstructorName
+                    | ConstructorName "(" Types ")"
+  syntax TypeCases ::= List{TypeCase,"|"}     [klabel(_|TypeCase_)]
+
      +

      Additional Priorities

      +
      k
        syntax priorities @__FUN-UNTYPED-COMMON
+                  > apply
+                  > arith
+                  > _:=__FUN-UNTYPED-COMMON
+                  > let_in__FUN-UNTYPED-COMMON
+                    letrec_in__FUN-UNTYPED-COMMON
+                    if_then_else__FUN-UNTYPED-COMMON
+                  > _;__FUN-UNTYPED-COMMON
+                  > fun__FUN-UNTYPED-COMMON
+                  > datatype_=___FUN-UNTYPED-COMMON
+endmodule
+
+module FUN-UNTYPED-MACROS
+  imports FUN-UNTYPED-COMMON
+
      +

      Desugaring macros

      +

      We desugar the list non-constructor operations to functions matching +over list patterns. In order to do that we need some new variables; for +those, we follow the same convention like in the K tutorial, where we +added them as new identifier constructs starting with the character $, +so we can easily recognize them when we debug or trace the semantics.

      +
      k
        syntax Name ::= "$h" [token] | "$t" [token]
+  rule head => fun [$h|$t] -> $h
+  rule tail => fun [$h|$t] -> $t
+  rule null? => fun [.Exps] -> true | [$h|$t] -> false
+
      +

      Multiple-head list patterns desugar into successive one-head patterns:

      +
      k
        rule [E1,E2,Es:Exps|T] => [E1|[E2,Es|T]]                   [anywhere]
+
      +

      Uncurrying of multiple arguments in functions and binders:

      +
      k
        rule P1 P2 -> E => P1 -> fun P2 -> E                       [anywhere]
+  rule F P = E => F = fun P -> E                             [anywhere]
+
      +

      We desugar the try-catch construct into callcc:

      +
      k
        syntax Name ::= "$k" [token] | "$v" [token]
+  rule try E catch(X) E'
+    => callcc (fun $k -> (fun throw -> E)(fun X -> $k E'))
+
      +

      For uniformity, we reduce all types to their general form:

      +
      k
        rule `Type-TypeName`(T:Type, Tn:TypeName) => (T) Tn
+
      +

      The dynamic semantics ignores all the type declarations:

      +
      k
        rule datatype _T = _TCs E => E
+
+endmodule
+
+
+module FUN-UNTYPED-SYNTAX
+  imports FUN-UNTYPED-COMMON
+  imports BUILTIN-ID-TOKENS
+
+  syntax Name ::= r"[a-z][_a-zA-Z0-9]*"           [token, prec(2)]
+                | #LowerId                        [token]
+  syntax ConstructorName ::= #UpperId             [token]
+  syntax TypeVar  ::= r"['][a-z][_a-zA-Z0-9]*"    [token]
+  syntax TypeName ::= Name                        [token]
+endmodule
+
      +

      Semantics

      +

      The semantics below is environment-based. A substitution-based +definition of FUN is also available, but that drops the & +construct as explained above.

      +
      k
      module FUN-UNTYPED
+  imports FUN-UNTYPED-COMMON
+  imports FUN-UNTYPED-MACROS
+  imports DOMAINS
+  //imports PATTERN-MATCHING
+
      +

      Configuration

      +

      The k, env, and store cells are standard +(see, for example, the definition of LAMBDA++ or IMP++ in the first +part of the K tutorial).

      +
      k
        configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <env color="violet"> .Map </env>
+                  <store color="white"> .Map </store>
+                </T>
+
      +

      Values and results

      +

      We only define integers, Booleans and strings as values here, but will +add more values later.

      +
      k
        syntax Val ::= Int | Bool | String
+  syntax Vals ::= Bottoms
+  syntax KResult ::= Val
+
      +

      Lookup

      +
      k
        rule <k> X:Name => V ...</k>
+       <env>... X |-> L ...</env>
+       <store>... L |-> V ...</store>
+
      +

      Arithmetic expressions

      +
      k
        rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule I1 + I2 => I1 +Int I2
+  rule S1 ^ S2 => S1 +String S2
+  rule I1 - I2 => I1 -Int I2
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
      +

      Conditional

      +
      k
        rule if  true then E else _ => E
+  rule if false then _ else E => E
+
      +

      Lists

      +

      We have already declared the syntactic list of expressions strict, so +we can assume that all the elements that appear in a FUN list are +evaluated. The only thing left to do is to state that a list of +values is a value itself, that is, that the list square-bracket +construct is indeed a constructor, and to give the semantics of +cons. Since cons is a builtin function and is +expected to take two arguments, we have to also state that +cons itself is a value (specifically, a function/closure +value, but we do not need that level of detail here), and also that +cons applied to a value is a value (specifically, it would be +a function/closure value that expects the second, list argument):

      +
      k
        rule cons V:Val [Vs:Vals] => [V,Vs]
+
      +

      Data Constructors

      +

      Constructors take values as arguments and produce other values:

      +
      k
        syntax Val ::= ConstructorName
+
      +

      Functions and Closures

      +

      Like in the environment-based semantics of LAMBDA++ in the first part +of the K tutorial, functions evaluate to closures. A closure includes +the current environment besides the function contents; the environment +will be used at execution time to lookup all the variables that appear +free in the function body (we want static scoping in FUN).

      +
      k
        syntax Val ::= closure(Map,Cases)
+  rule <k> fun Cases => closure(Rho,Cases) ...</k>  <env> Rho </env>
+
      +

      Note: The reader may want to get familiar with +how the pre-defined pattern matching works before proceeding. +The best way to do that is to consult +k/include/modules/pattern-matching.k.

      + +

      We distinguish two cases when the closure is applied. +If the first pattern matches, then we pick the first case: switch to +the closed environment, get the matching map and bind all its +variables, and finally evaluate the function body of the first case, +making sure that the environment is properly recovered afterwards. +If the first pattern does not match, then we drop it and thus move on +to the next one.

      +
      k
        rule (. => getMatching(P, V)) ~> closure(_, P->_ | _) V:Val
+  rule <k> matchResult(M:Map) ~> closure(Rho, _->E | _) _
+           => bindMap(M) ~> E ~> setEnv(Rho') ...</k>
+       <env> Rho' => Rho </env>
+  rule (matchFailure => .) ~> closure(_, (_->_ | Cs:Cases => Cs)) _
+//  rule <k> closure(Rho, P->E | _) V:Val
+//           => bindMap(getMatching(P,V)) ~> E ~> setEnv(Rho') ...</k>
+//       <env> Rho' => Rho </env>  when isMatching(P,V)
+//  rule closure(_, (P->_ | Cs:Cases => Cs)) V:Val  when notBool isMatching(P,V)
+
      +

      Let and Letrec

      +

      To highlight the similarities and differences between let and +letrec, we prefer to give them direct semantics instead of +to desugar them like in LAMBDA. See the formal definitions of +bindTo, bind, and assignTo at the end of +this module. Informally, bindTo(Xs, Es) first +evaluates the expressions Es in Exps in the current +environment (i.e., it is strict in its second argument), then it binds +the variables in Xs in Names to new locations and adds +those bindings to the environment, and finally writes the values +previously obtained after evaluating the expressions Es to those +new locations; bind(Xs) does only the bindings of +Xs to new locations and adds those bindings to the environment; +and assignTo(Xs,Es) evaluates the expressions +Es in the current environment and then it writes the resulting +values to the locations to which the variables Xs are already +bound to in the environment.

      +

      Therefore, let Xs = Es in E first +evaluates Es in the current environment, then adds new +bindings for Xs to fresh locations in the environment, then +writes the values of Es to those locations, and finally +evaluates E in the new environment, making sure that the +environment is properly recovered after the evaluation of E. +On the other hand, letrec does the same things but in a +different order: it first adds new bindings for Xs to fresh +locations in the environment, then it evaluates Es in the new +environment, then it writes the resulting values to their +corresponding locations, and finally it evaluates E and +recovers the environment. The crucial difference is that the +expressions Es now see the locations of the variables Xs +in the environment, so if they are functions, which is typically the +case with letrec, their closures will encapsulate in their +environments the bindings of all the bound variables, including +themselves (thus, we may have a closure value stored at location +L, whose environment contains a binding of the form +F ↦ L; this way, the closure can invoke +itself).

      +
      k
        rule <k> let Bs in E
+        => bindTo(names(Bs),exps(Bs)) ~> E ~> setEnv(Rho) ...</k>
+       <env> Rho </env>
+
+  rule <k> letrec Bs in E
+        => bind(names(Bs))~>assignTo(names(Bs),exps(Bs))~>E~>setEnv(Rho)...</k>
+       <env> Rho </env>
+
      +

      Recall that our syntax allows let and letrec to +take any expression in place of its binding. This allows us to use +the already existing function application construct to bind names to +functions, such as, e.g., let x y = y in .... +The desugaring macro in the syntax module uncurries such declarations, +and then the semantic rules above only work when the remaining +bindings are identifiers, so the semantics will get stuck on programs +that misuse the let and letrec binders.

      +

      References

      +

      The semantics of references is self-explanatory, except maybe for the +desugaring rule of ref, which is further discussed. Note +that &X grabs the location of X from the environment. +Sequential composition, which is needed only to accumulate the +side effects due to assignments, was strict in the first argument. +Once evaluated, its first argument is simply discarded:

      +
      k
        syntax Name ::= "$x" [token]
+  rule ref => fun $x -> & $x
+  rule <k> & X => L ...</k>  <env>... X |-> L ...</env>
+  rule <k> @ L:Int => V:Val ...</k>  <store>... L |-> V ...</store>
+  rule <k> L:Int := V:Val => V ...</k>  <store>... L |-> (_=>V) ...</store>
+  rule _V:Val; E => E
+
      +

      The desugaring rule of ref (first rule above) works +because & takes a variable and returns its location (like in C). +Note that some ``pure'' functional programming researchers strongly dislike +the & construct, but favor ref. We refrain from having +a personal opinion on this issue here, but support & in the +environment-based definition of FUN because it is, technically speaking, +more powerful than ref. From a language design perspective, it +would be equally easy to drop & and instead give a direct +semantics to ref. In fact, this is precisely what we do in the +substitution-based definition of FUN, because there appears to be no way +to give a substitution-based definition to the & construct.

      +

      Callcc

      +

      As we know it from the LAMBDA++ tutorial, call-with-current-continuation +is quite easy to define in K. We first need to define a special +value wrapping an execution context, that is, an environment saying +where the variables should be looked up, and a computation structure +saying what is left to execute (in a substitution-based definition, +this special value would be even simpler, as it would only need to +wrap the computation structure---see, for example, the +substitution-based semantics of LAMBDA++ in the the first part of the +K tutorial, or the substitution-based definition of FUN). Then +callcc creates such a value containing the current +environment and the current remaining computation, and passes it to +its argument function. When/If invoked, the special value replaces +the current execution context with its own and continues the execution +normally.

      +
      k
        syntax Val ::= cc(Map,K)
+  rule <k> (callcc V:Val => V cc(Rho,K)) ~> K </k>  <env> Rho </env>
+  rule <k> cc(Rho,K) V:Val ~> _ => V ~> K </k>  <env> _ => Rho </env>
+
      +

      Auxiliary operations

      +

      Environment recovery

      +

      The environment recovery operation is the same as for the LAMBDA++ +language in the K tutorial and many other languages provided with the +K distribution. The first ``anywhere'' rule below shows an elegant +way to achieve the benefits of tail recursion in K.

      +
      k
        syntax KItem ::= setEnv(Map)  // TODO: get rid of env
+  //rule (setEnv(_) => .) ~> setEnv(_)  [anywhere]
+  rule <k> _:Val ~> (setEnv(Rho) => .) ...</k> <env> _ => Rho </env>
+
      +

      bindTo, bind and assignTo

      +

      The meaning of these operations has already been explained when we +discussed the let and letrec language constructs +above.

      +
      k
        syntax KItem ::= bindTo(Names,Exps)         [strict(2)]
+                 | bindMap(Map)
+                 | bind(Names)
+
+  rule (. => getMatchingAux(Xs,Vs)) ~> bindTo(Xs:Names,Vs:Vals)
+  rule matchResult(M:Map) ~> bindTo(_:Names, _:Vals) => bindMap(M)
+
+  rule bindMap(.Map) => .
+  rule <k> bindMap((X:Name |-> V:Val => .Map) _:Map) ...</k>
+       <env> Rho => Rho[X <- !L:Int] </env>
+       <store>... .Map => !L |-> V ...</store>
+
+  rule bind(.Names) => .
+  rule <k> bind(X:Name,Xs => Xs) ...</k>
+       <env> Rho => Rho[X <- !_L:Int] </env>
+
+  syntax KItem ::= assignTo(Names,Exps)  [strict(2)]
+
+  rule <k> assignTo(.Names,.Vals) => . ...</k>
+  rule <k> assignTo((X:Name,Xs => Xs),(V:Val,Vs:Vals => Vs)) ...</k>
+       <env>... X |-> L ...</env>
+       <store>... .Map => L |-> V ...</store>
+
      +

      Getters

      +

      The following auxiliary operations extract the list of identifiers +and of expressions in a binding, respectively.

      +
      k
        syntax Names ::= names(Bindings)  [function]
+  rule names(.Bindings) => .Names
+  rule names(X:Name=_ and Bs) => (X,names(Bs))::Names
+
+  syntax Exps ::= exps(Bindings)  [function]
+  rule exps(.Bindings) => .Exps
+  rule exps(_:Name=E and Bs) => E,exps(Bs)
+
+  /* Extra kore stuff */
+  syntax KResult ::= Vals
+  syntax Exps ::= Names
+  syntax Names ::= Bottoms
+
+  /* Matching */
+  syntax MatchResult ::= getMatching(Exp, Val)                      [function]
+                       | getMatchingAux(Exps, Vals)                 [function]
+                       | mergeMatching(MatchResult, MatchResult)    [function]
+                       | matchResult(Map)
+                       | "matchFailure"
+
+  rule getMatching(C:ConstructorName(Es:Exps), C(Vs:Vals)) => getMatchingAux(Es, Vs)
+  rule getMatching([Es:Exps], [Vs:Vals])                   => getMatchingAux(Es, Vs)
+  rule getMatching(C:ConstructorName, C) => matchResult(.Map)
+  rule getMatching(B:Bool, B)            => matchResult(.Map)
+  rule getMatching(I:Int, I)             => matchResult(.Map)
+  rule getMatching(S:String, S)          => matchResult(.Map)
+  rule getMatching(N:Name, V:Val) => matchResult(N |-> V)
+  rule getMatching(_, _) => matchFailure        [owise]
+
+  rule getMatchingAux((E:Exp, Es:Exps), (V:Val, Vs:Vals)) => mergeMatching(getMatching(E, V), getMatchingAux(Es, Vs))
+  rule getMatchingAux(.Exps, .Vals)                       => matchResult(.Map)
+  rule getMatchingAux(_, _) => matchFailure     [owise]
+
+  rule mergeMatching(matchResult(M1:Map), matchResult(M2:Map)) => matchResult(M1 M2)
+    requires intersectSet(keys(M1), keys(M2)) ==K .Set
+  //rule mergeMatching(_, _) => matchFailure      [owsie]
+  rule mergeMatching(matchResult(_:Map), matchFailure) => matchFailure
+  rule mergeMatching(matchFailure, matchResult(_:Map)) => matchFailure
+  rule mergeMatching(matchFailure, matchFailure)       => matchFailure
+
      +

      Besides the generic decomposition rules for patterns and values, +we also want to allow [head|tail] matching for lists, so we add +the following custom pattern decomposition rule:

      +
      k
        rule getMatching([H:Exp | T:Exp], [V:Val, Vs:Vals])
+    => getMatchingAux((H, T), (V, [Vs]))
+endmodule
+
      +

      Go to Lesson 2, FUN untyped, Substitution-Based.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/2_substitution/fun-untyped/index.html b/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/2_substitution/fun-untyped/index.html new file mode 100644 index 00000000000..5e236de1ea1 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/3_fun/1_untyped/2_substitution/fun-untyped/index.html @@ -0,0 +1,720 @@ + + + + + + + + + + + + + + +FUN — Untyped — Substitution | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      FUN — Untyped — Substitution

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the substitution-based definition of FUN. For additional +explanations regarding the semantics of the various FUN constructs, +the reader should consult the emvironment-based definition of FUN.

      +

      Syntax

      +
      k
      require "substitution.md"
+//require "modules/pattern-matching.k"
+
+module FUN-UNTYPED-COMMON
+  imports DOMAINS-SYNTAX
+
      +

      The Syntactic Constructs

      +
      k
        syntax Name
+  syntax Names ::= List{Name,","}
+
+  syntax Exp ::= Int | Bool | String | Name
+               | "(" Exp ")"                       [bracket]
+  syntax Exps  ::= List{Exp,","}                   [strict]
+  syntax Val
+  syntax Vals ::= List{Val,","}
+
+  syntax Exp ::= left:
+                 Exp "*" Exp                       [strict, arith]
+               | Exp "/" Exp                       [strict, arith]
+               | Exp "%" Exp                       [strict, arith]
+               > left:
+                 Exp "+" Exp                       [strict, left, arith]
+               | Exp "^" Exp                       [strict, left, arith]
+               | Exp "-" Exp                       [strict, prefer, arith]
+               | "-" Exp                           [strict, arith]
+               > non-assoc:
+                 Exp "<" Exp                       [strict, arith]
+               | Exp "<=" Exp                      [strict, arith]
+               | Exp ">" Exp                       [strict, arith]
+               | Exp ">=" Exp                      [strict, arith]
+               | Exp "==" Exp                      [strict, arith]
+               | Exp "!=" Exp                      [strict, arith]
+               > "!" Exp                           [strict, arith]
+               > Exp "&&" Exp                      [strict(1), left, arith]
+               > Exp "||" Exp                      [strict(1), left, arith]
+
+  syntax Exp ::= "if" Exp "then" Exp "else" Exp    [strict(1)]
+
+  syntax Exp ::= "[" Exps "]"                      [strict]
+               | "cons" |  "head" | "tail" | "null?"
+               | "[" Exps "|" Exp "]"
+  syntax Val ::= "[" Vals "]"
+
+  syntax ConstructorName
+  syntax Exp ::= ConstructorName
+               | ConstructorName "(" Exps ")"      [prefer, strict(2)]
+  syntax Val ::= ConstructorName "(" Vals ")"
+
+  syntax Exp ::= "fun" Cases
+               | Exp Exp                           [strict, left]
+  syntax Case  ::= Exp "->" Exp                    [binder]
+// NOTE: The binder attribute above is the only difference between this
+// module and the syntax module of environment-based FUN.  We need
+// to fix a bug in order to import modules and override the attributes
+// of operations.
+  syntax Cases ::= List{Case, "|"}
+
+  syntax Exp ::= "let" Bindings "in" Exp
+               | "letrec" Bindings "in" Exp                 [prefer]
+  syntax Binding  ::= Exp "=" Exp
+  syntax Bindings ::= List{Binding,"and"}
+
+  syntax Exp ::= "ref"
+               | "&" Name
+               | "@" Exp                           [strict]
+               | Exp ":=" Exp                      [strict]
+               | Exp ";" Exp                       [strict(1), right]
+
+  syntax Exp ::= "callcc"
+               | "try" Exp "catch" "(" Name ")" Exp
+  syntax Name ::= "throw" [token]
+
+  syntax Exp ::= "datatype" Type "=" TypeCases Exp
+
+  syntax TypeVar
+  syntax TypeVars ::= List{TypeVar,","}
+
+  syntax TypeName
+  syntax Type ::= "int" | "bool" | "string"
+                | Type "-->" Type                            [right]
+                | "(" Type ")"                             [bracket]
+                | TypeVar
+                | TypeName             [klabel(TypeName), avoid]
+                | Type TypeName   [klabel(Type-TypeName), onlyLabel]
+                | "(" Types ")" TypeName                    [prefer]
+  syntax Types ::= List{Type,","}
+  syntax Types ::= TypeVars
+
+  syntax TypeCase ::= ConstructorName
+                    | ConstructorName "(" Types ")"
+  syntax TypeCases ::= List{TypeCase,"|"}     [klabel(_|TypeCase_)]
+
      +

      Additional Priorities

      +
      k
        syntax priorities @__FUN-UNTYPED-COMMON
+                  > ___FUN-UNTYPED-COMMON
+                  > arith
+                  > _:=__FUN-UNTYPED-COMMON
+                  > let_in__FUN-UNTYPED-COMMON
+                    letrec_in__FUN-UNTYPED-COMMON
+                    if_then_else__FUN-UNTYPED-COMMON
+                  > _;__FUN-UNTYPED-COMMON
+                  > fun__FUN-UNTYPED-COMMON
+                  > datatype_=___FUN-UNTYPED-COMMON
+endmodule
+
+module FUN-UNTYPED-MACROS
+  imports FUN-UNTYPED-COMMON
+
      +

      Desugaring macros

      +
      k
        rule P1 P2 -> E => P1 -> fun P2 -> E                       [macro-rec]
+  rule F P = E => F = fun P -> E                             [macro-rec]
+
+  rule [E1,E2,Es:Exps|T] => [E1|[E2,Es|T]]                   [macro-rec]
+
+//  rule 'TypeName(Tn:TypeName) => (.TypeVars) Tn              [macro]
+  rule `Type-TypeName`(T:Type, Tn:TypeName) => (T) Tn          [macro]
+
+  syntax Name ::= "$h" | "$t"
+  rule head => fun [$h|$t] -> $h                             [macro]
+  rule tail => fun [$h|$t] -> $t                             [macro]
+  rule null? => fun [.Exps] -> true | [$h|$t] -> false       [macro]
+
+  syntax Name ::= "$k" | "$v"
+  rule try E catch(X) E'
+    => callcc (fun $k -> (fun throw -> E)(fun X -> $k E'))   [macro]
+
+  rule datatype _T = _TCs E => E                               [macro]
+
      +

      mu needed for letrec, but we put it here so we can also write +programs with mu in them, which is particularly useful for testing.

      +
      k
        syntax Exp ::= "mu" Case
+
+endmodule
+
+
+module FUN-UNTYPED-SYNTAX
+  imports FUN-UNTYPED-COMMON
+  imports BUILTIN-ID-TOKENS
+
+  syntax Name ::= r"[a-z][_a-zA-Z0-9]*"            [token, prec(2)]
+                | #LowerId                         [token]
+  syntax ConstructorName ::= #UpperId              [token]
+  syntax TypeVar  ::= r"['][a-z][_a-zA-Z0-9]*"     [token]
+  syntax TypeName ::= Name                         [token]
+endmodule
+
      +

      Semantics

      +
      k
      module FUN-UNTYPED
+  imports FUN-UNTYPED-COMMON
+  imports FUN-UNTYPED-MACROS
+  imports DOMAINS
+  imports SUBSTITUTION
+  //imports PATTERN-MATCHING
+
+  configuration <T color="yellow">
+                  <k color="green"> $PGM:Exp </k>
+                  <store color="white"> .Map </store>
+                </T>
+
      +

      Both Name and functions are values now:

      +
      k
        syntax Val ::= Int | Bool | String | Name
+  syntax Exp ::= Val
+  syntax Exps ::= Vals
+  syntax KResult ::= Val
+  syntax Exps ::= Names
+  syntax Vals ::= Names
+
+  rule I1 * I2 => I1 *Int I2
+  rule I1 / I2 => I1 /Int I2 when I2 =/=K 0
+  rule I1 % I2 => I1 %Int I2 when I2 =/=K 0
+  rule I1 + I2 => I1 +Int I2
+  rule S1 ^ S2 => S1 +String S2
+  rule I1 - I2 => I1 -Int I2
+  rule - I => 0 -Int I
+  rule I1 < I2 => I1 <Int I2
+  rule I1 <= I2 => I1 <=Int I2
+  rule I1 > I2 => I1 >Int I2
+  rule I1 >= I2 => I1 >=Int I2
+  rule V1:Val == V2:Val => V1 ==K V2
+  rule V1:Val != V2:Val => V1 =/=K V2
+  rule ! T => notBool(T)
+  rule true  && E => E
+  rule false && _ => false
+  rule true  || _ => true
+  rule false || E => E
+
+  rule if  true then E else _ => E
+  rule if false then _ else E => E
+
+  rule isVal(cons) => true
+  rule isVal(cons _V:Val) => true
+  rule cons V:Val [Vs:Vals] => [V,Vs]
+
+  syntax Val ::= ConstructorName
+
+  rule isVal(fun _) => true
+  syntax KVar ::= Name
+  syntax Name ::= freshName(Int)    [freshGenerator, function]
+  rule freshName(I:Int) => {#parseToken("Name", "#" +String Int2String(I))}:>Name
+
+  rule (. => getMatching(P, V)) ~> (fun P->_ | _) V:Val
+  rule matchResult(M:Map) ~> (fun _->E | _) _ => E[M]
+  rule (matchFailure => .) ~> (fun (_->_ | Cs:Cases => Cs)) _
+//  rule (fun P->E | _) V:Val => E[getMatching(P,V)]  when isMatching(P,V)
+//  rule (fun (P->_ | Cs:Cases => Cs)) V:Val  when notBool isMatching(P,V)
+
      +

      We can reduce multiple bindings to one list binding, and then +apply the usual desugaring of let into function application. +It is important that the rule below is a macro, so let is eliminated +immediately, otherwise it may interfere in ugly ways with substitution.

      +
      k
        rule let Bs in E => ((fun [names(Bs)] -> E) [exps(Bs)])    [macro]
+
      +

      We only give the semantics of one-binding letrec. +Multipe bindings are left as an exercise.

      +
      k
        // changed because of parsing error
+  //rule mu X:Name -> E => E[(mu X -> E) / X]
+  rule mu X:Name -> E => E[X |-> (mu X -> E)]
+  rule letrec F:Name = E in E' => let F = (mu F -> E) in E'  [macro]
+
      +

      We cannot have & anymore, but we can give direct +semantics to ref. We also have to declare ref to +be a value, so that we will never heat on it.

      +
      k
      //  rule <k> & X => L ...</k>  <env>... X |-> L </env>
+  rule isVal(ref) => true
+  rule <k> ref V:Val => !L:Int ...</k> <store>... .Map => !L |-> V ...</store>
+  rule <k> @ L:Int => V:Val ...</k>  <store>... L |-> V ...</store>
+  rule <k> L:Int := V:Val => V ...</k>  <store>... L |-> (_=>V) ...</store>
+  rule _V:Val; E => E
+
+  syntax Val ::= cc(K)
+  rule isVal(callcc) => true
+  rule <k> (callcc V:Val => V cc(K)) ~> K </k>
+  rule <k> cc(K) V:Val ~> _ => V ~> K </k>
+
      +

      Auxiliary getters

      +
      k
        syntax Names ::= names(Bindings)  [function]
+  rule names(.Bindings) => .Names
+  rule names(X:Name=_ and Bs) => X,names(Bs)
+
+  syntax Exps ::= exps(Bindings)  [function]
+  rule exps(.Bindings) => .Exps
+  rule exps(_:Name=E and Bs) => E,exps(Bs)
+
+  /* Extra kore stuff */
+  syntax KResult ::= Vals
+  syntax Exps ::= Names
+
+  /* Matching */
+  syntax MatchResult ::= getMatching(Exp, Val)                      [function]
+                       | getMatchingAux(Exps, Vals)                 [function]
+                       | mergeMatching(MatchResult, MatchResult)    [function]
+                       | matchResult(Map)
+                       | "matchFailure"
+
+  rule getMatching(C:ConstructorName(Es:Exps), C(Vs:Vals)) => getMatchingAux(Es, Vs)
+  rule getMatching([Es:Exps], [Vs:Vals])                   => getMatchingAux(Es, Vs)
+  rule getMatching(C:ConstructorName, C) => matchResult(.Map)
+  rule getMatching(B:Bool, B)            => matchResult(.Map)
+  rule getMatching(I:Int, I)             => matchResult(.Map)
+  rule getMatching(S:String, S)          => matchResult(.Map)
+  rule getMatching(N:Name, V:Val) => matchResult(N |-> V)
+  rule getMatching(_, _) => matchFailure        [owise]
+
+  rule getMatchingAux((E:Exp, Es:Exps), (V:Val, Vs:Vals)) => mergeMatching(getMatching(E, V), getMatchingAux(Es, Vs))
+  rule getMatchingAux(.Exps, .Vals)                       => matchResult(.Map)
+  rule getMatchingAux(_, _) => matchFailure     [owise]
+
+  rule mergeMatching(matchResult(M1:Map), matchResult(M2:Map)) => matchResult(M1 M2)
+    requires intersectSet(keys(M1), keys(M2)) ==K .Set
+  //rule mergeMatching(_, _) => matchFailure      [owsie]
+  rule mergeMatching(matchResult(_:Map), matchFailure) => matchFailure
+  rule mergeMatching(matchFailure, matchResult(_:Map)) => matchFailure
+  rule mergeMatching(matchFailure, matchFailure)       => matchFailure
+
      +

      Besides the generic decomposition rules for patterns and values, +we also want to allow [head|tail] matching for lists, so we add +the following custom pattern decomposition rule:

      +
      k
        rule getMatching([H:Exp | T:Exp], [V:Val, Vs:Vals])
+    => getMatchingAux((H, T), (V, [Vs]))
+endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/4_logik/basic/logik/index.html b/k-distribution/pl-tutorial/2_languages/4_logik/basic/logik/index.html new file mode 100644 index 00000000000..e75067806ba --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/4_logik/basic/logik/index.html @@ -0,0 +1,710 @@ + + + + + + + + + + + + + + +LOGIK | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      LOGIK

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Author: Traian Florin Șerbănuță (traian.serbanuta@unibuc.ro)
      +Organization: University of Bucharest

      +

      Abstract

      +

      This is the K semantic definition of LOGIK, a trivial language +capturing the essence of the logic programming paradigm. In this +definition, we explicitly focus on simplicity and mathematical +clarity, not on advanced logic programming features or performance. +Those are covered in the LOGIK++ extension under examples/logik++.

      +

      Specifically, a LOGIK program consists of a sequence of Horn clauses +of the form

      +
      P :- P1, P2, ..., Pn .
+
      +

      followed by a query of the form

      +
      ?- Q1, Q2, ..., Qm .
+
      +

      where P, P1, P2, ..., Pn, Q1, Q2, +..., Qm are literals. The +symbol :- is read "if". A literal has the form +p(T1,T2,...,Tk), where p is a predicate symbol +and where T1,T2,...,Tk are terms. Terms are built as +usual, with operation symbols and variables. A common +convention in logic programming languages, also adopted here, is that +variables are capitalized and operation symbols are not. Operations +with zero arguments are called constants and are written without +parentheses, that is, c instead of c(). Horn +clauses without conditions, called facts, are written +without :-, that is, P. instead of P :- ..

      +

      For example, the LOGIK program below gives a few facts about a +parent predicate, then several clauses defining some useful +predicates including an ancestor predicate, and finally a +query asking for those who both have ancestors and are ancestors +themselves in the parent relation:

      +
      parent(david,john).
+parent(jim,david).
+parent(steve,jim).
+parent(nathan,steve).
+
+grandparent(A,B):-
+  parent(A,X),
+  parent(X,B).
+
+ancestor(A,B):-
+  parent(A,X),
+  parents(X,B).
+
+parents(X,X).
+parents(A,B):-
+  ancestor(A,B).
+
+both(X) :- ancestor(A,X), ancestor(X,B).
+
+?- both(X).
+
      +

      Above, we only have constant operation symbols, so these and variables +are the only terms that can be used in predicates. As expected, the +LOGIK program above will give us three solutions for X: +david, steve, and jim. If we inline the +both(X) predicate in the query, that is, if we replace the +query with ?- ancestor(A,X), ancestor(X,B). then we get +10 solutions, one for for each triple A, X, and +B satisfying both predicates ancestor(A,X) and +ancestor(X,B).

      +

      As another example, the program below defines an append +predicate followed by a simple goal:

      +
      append(nil,L,L).
+append(cons(H,T),L,cons(H,Z)) :- append(T,L,Z).
+
+?- append(cons(a,nil), cons(b,nil), V).
+
      +

      Besides the predicate symbol append, the program above also +includes a constant symbol nil and a binary operation symbol +cons. Additionally, the query also includes two more +constants, a and b. The capitalized identifiers are +all variables. As expected, the LOGIK program above yields only one +solution, namely V = cons(a,cons(b,nil)). On the other hand, +if we change the query to:

      +
      ?- append(L1, cons(a,L2), cons(a,cons(b,cons(a,nil)))).
+
      +

      then LOGIK yields two solutions: one where L1 is +cons(a,cons(b,nil)) and L2 is nil, +and another where L1 is nil and L2 is +cons(a,cons(b,nil)).

      +

      The programs above all generated ground solutions, that is, +solutions where the query variables are mapped to ground terms (i.e., +terms without variables). Let us now consider the following query:

      +
      ?- append(cons(a,nil), Y, Z).
+
      +

      There are obviously infinitely many ground solutions for the query +above, e.g.,
      +Y = nil and Z = cons(a,nil),
      +Y = cons(a,nil) and Z = cons(a,cons(a,nil)),
      +Y = cons(b,nil) and Z = cons(a,cons(b,nil)),
      +Y = cons(c,cons(b,nil)) and Z = cons(a,cons(c,cons(b,nil))),
      +etc. However, all the ground solutions for the query above can be +elegantly characterized by the property that Z is bound to a list +starting with a and followed by the list that Y is +bound to. This property can in fact be described as a symbolic solution +to the query: Z = cons(a,Y) or, equivalently, +Y = Symb and Z = cons(a,Symb). It is possible to +define a ``more general than'' relation on such symbolic solutions, +in the sense that the more particular solution can be obtained as a +specialization/substitution of the more general one, and then it can +be shown that the above is the most general solution to the +stated query. Logic programming languages, including our LOGIK, +attempt to always compute such most general solutions.

      +

      Logic programming languages are highly non-deterministic, in that +several Horn clauses may be used at the same time, each possibly +resulting in a different solution. Implementations of logic +programming languages consist of complex, optimized search and +indexing algorithms, which we are not concerned with here. Instead, +we here take advantage of K's builtin support for search. +Specifically, to find all the solutions of a LOGIK program, we have to +use krun with the option --search. However, note +that some programs have infinitely many solutions which cannot relate +to each other by the "more general" relation. For example, the query

      +
      ?- append(L1, cons(a,L2), L3) .
+
      +

      To address such cases and terminate, logic programming languages allow +the user to choose how many solutions to be computed and displayed. +In LOGIK, we can use the --bound option of krun for +this purpose.

      +

      Finally, note that some queries have no solution. In some cases that +is easy to detect by exhaustive analysis, such as for the following +query:

      +
      ?- append(cons(a,L1), L2, cons(b,L3)).
+
      +

      Logic programming languages, including LOGIK, terminate in such cases +and report a no solution answer. However, there are cases where +exhaustive analysis is not sufficient, such as for the query:

      +
      ?- append(cons(a,L), nil, L).
+
      +

      In such cases, logic programming languages do not terminate. While +one may devise techniques to detect non-termination in some cases, +one cannot do it in general (same like for all Turing-complete +languages).

      +
      k
      require "unification.k"
+
+module LOGIK-COMMON
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      The syntax of LOGIK is straightforward: a program is a sequence of +Horn clauses followed by a query:

      +
      k
        syntax Literal
+  syntax Term ::= Literal | Literal "(" Terms ")"
+  syntax Terms ::= List{Term,","}
+  syntax Clause ::= Term ":-" Terms "." | Term "."
+  syntax Query ::= "?-" Terms "."
+  syntax Pgm ::= Query | Clause Pgm
+endmodule
+
+module LOGIK-SYNTAX
+  imports LOGIK-COMMON
+  imports BUILTIN-ID-TOKENS
+
      +

      Variables and literals are defined as tokens following the conventions +used in Prolog (variables start with _ or capital letter, while literals +start with lower case letters):

      +
      k
        syntax #KVariable ::= r"[A-Z_][A-Za-z0-9_]*"   [token, prec(2)]
+                      | #UpperId                 [token]
+  syntax Term ::= #KVariable [klabel(#SemanticCastToTerm)]
+  syntax Literal ::= r"[a-z][a-zA-Z0-9_]*"                 [token]
+                   | #LowerId                              [token]
+endmodule
+
+module LOGIK
+  imports LOGIK-COMMON
+  imports DOMAINS
+  imports UNIFICATION
+
      +

      Unification is at the core of logic programming. Here we are +going to use the predefined unification procedure (the same one we +used in the type inferencers in Tutorial 5).

      +

      Configuration

      +

      The configuration stores each clause in its own cell for easy access, +and the most general unifier in a cell named mgu, same like +the type inferencers. The k cell holds the query and the +fresh cell holds a fresh clause instance to be attempted on +the next query item. To more easily read the solutions, we add a +second top-level cell, solution. Both top cells are +optional. Indeed, we start with the main top cell and, when a +solution is found, we move it into the solution cell and +discard the main cell.

      +
      k
        configuration <T color="yellow" multiplicity="?">
+                  <k color="green"> $PGM:Pgm </k>
+                  <fresh color="orange"> .K </fresh>
+                  <clauses color="red">
+                    <clause color="pink" multiplicity="*"> .K </clause>
+                  </clauses>
+                  <mgu> .K </mgu>
+                </T>
+                <solution multiplicity="?"> .K </solution>
+
      +

      Pre- and post-processing

      +

      Before we launch the semantics, we first scan the given program and +place each clause in its own cell, and then place the query in the +k cell and initialize the mgu with the variables from the query.

      +

      Note that we put a fresh instance of the clause to avoid interference with +the query variables. By a "fresh instance" of a clause we mean one whose +variables are renamed with fresh names; we need that in order to avoid +undesired unification conflicts due to particular names chosen for +variables in the original program, as well as conflicts due to +subsequent uses of the same clause. It is safe to rename the +variables in a clause, because clauses are universally quantified in +their variables. This process of creating a fresh instance of a +clause is similar to how we created fresh instances of type schemas in +the higher-order type inferencer discussed in Tutorial 5. Indeed, we +can safely regard clauses as "clause schemas" comprising infinitely +many instances, one for each context.

      +
      k
        rule <k> C:Clause Pgm => Pgm </k>
+       (.Bag => <clause> #renameVariables(C) </clause>)
+
+  rule <k> ?- Ls:Terms. => Ls ...</k>
+       <mgu> _ => #variablesMap(#variables(Ls)) </mgu>
+
      +

      We also sequentialize the goals for easier processing:

      +
      k
        rule L:Term, Ls:Terms => L ~> Ls
+  rule .Terms => .
+
      +

      When all the goals are solved, indicated by the empty k +cell, the calculated most general unifier (mgu) is in the mgu +cell. In that case, to ease reading of the final solution we move the +mgu in the solution cell and delete the rest of the +configuration:

      +
      k
      rule <T>... <k> . </k> <mgu> Theta </mgu> ...</T>
+  => <solution> Theta </solution>
+
      +

      Since we are not interested in seeing the failed attempts to solve +the query, we collapse all the error configurations into an empty +configuration (recall that both top-level cells in the configuration +were declared optional). This way, if we see an empty configuration +when we search for all solutions, we know that some attempts failed +(but we do not know which ones).

      +
      k
      // this would be nice, but we need feedback from the external unifier
+// for this.
+//  rule <T>... <mgu> _:MguError </mgu> ...</T> => .
+
      +

      Semantics

      +

      Once all the infrastructure is in place, the actual semantics of LOGIK +is quite simple. All we have to do is to pick some (fresh instance of +a) clause, then unify its conclusion with the first query literal, and +then replace that literal with condition of the clause. The intuition +here is the following: to satisfy the first literal in the query, we +need to find some instance of some clause that matches it, and then to +similarly show that we can satisfy the conditions of that clause. +Mathematically, this is an instance of the proof principle called +resolution: if p ∨ q and ¬ p ∨ r hold, then so does +q ∨ r. We let it as an exercise to the reader to see how the two +relate (hint: assume the negation of the goal together with all the +clauses, and then derive false).

      +

      The following two rules are tightly connected and they together +perform the following core task: pick a fresh instance of a clause +which unifies with the first goal item, then add its conditions as new +goals.

      +

      Pick a clause and generate a fresh instance of it when the +fresh cell is empty:

      +
      k
        rule <fresh> . => #renameVariables(C) </fresh> <clause> C </clause>
+       <k> T:Term ...</k>
+  requires #unifiable(T,head(C))
+
+  syntax Term ::= head(Clause) [function]
+  rule head(L.) => L
+  rule head(L:-_.) => L
+
      +

      If the goal is unifiable with the fresh clause's head, replace the goal +with the clause body, and empty the fresh cell (so that +another clause can be chosen using the rule above):

      +
      k
        rule <k> L:Term => . ...</k>
+       <fresh>  L:Term . => . </fresh>
+
+  rule <k> L:Term :KItem => Ls ...</k>
+       <fresh>  L:Term :- Ls:Terms. => . </fresh>
+
      +

      Note that there is no problem if a clause is chosen whose +conclusion literal does not unify with the first goal literal. +The search +option of krun will systematically try all clauses, so no +solution is missed. Of course, the above is not the most efficient +way to implement a logic programming language, but recall that our +objective here was to present a simple and mathematically clean +solution. We encourage the interested reader to consult the LOGIK++ +language definition for a more efficient definition of a richer logic +programming language.

      +
      k
      endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/2_languages/index.html b/k-distribution/pl-tutorial/2_languages/index.html new file mode 100644 index 00000000000..7394b709b63 --- /dev/null +++ b/k-distribution/pl-tutorial/2_languages/index.html @@ -0,0 +1,396 @@ + + + + + + + + + + + + + + +K Languages | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K Languages

      +

      Here we present several "real-world" language examples. These languages +demonstrate many of the features you would expect to find in a full-fledged +programming language.

      +
        +
      • SIMPLE: Imperative programming language with threads.
        • +
      • KOOL: SIMPLE extended with object-oriented features.
        • +
      • FUN: A functional language with algebraic data-types and pattern-matching.
        • +
      • LOGIK: A logical programming language based on clause unification.
        • +
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/NOTES/index.html b/k-distribution/pl-tutorial/NOTES/index.html new file mode 100644 index 00000000000..47380b73360 --- /dev/null +++ b/k-distribution/pl-tutorial/NOTES/index.html @@ -0,0 +1,455 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Dear reader: these NOTES.md documents are mostly for myself (Grigore), to +keep track of changes and updates that need to be made as things evolve +in the K framework. You can safely ignore them. However, if you are curious +how things will change and decide to read them, I would be grateful if you +let me know whenever you find inconsistencies or things that I forgot to +mention here. Or even better, feel free to make pull requests with suggested +changes or updates.

      +

      Global changes that need to be made:

      +
        +
      • Replace when in rules with requires
        • +
      +

      Things to revise each time the structure of the tutorial changes:

      +
        +
      • 1_k\2_imp\lesson_1\README.md refers to Tutorial 1
        • +
      • 1_k\2_imp\lesson_4\README.md refers to Lesson 1
        • +
      • 1_k\2_imp\lesson_4\README.md refers to Tutorial 1, Lesson 2.5
        • +
      • 1_k\3_lambda++\lesson_1\README.md refers to Lesson 8, Tutorial 1
        • +
      • `1_k\3_lambda++\lesson_1\exercises refers to Lesson 1, tests/config.xml
        • +
      • 1_k\3_lambda++\lesson_2\README.md Lesson 1, Tutorial 1; Tutorial 6; Part 1, 2
        • +
      • 1_k\3_lambda++\lesson_3\README.md refers to Lesson 7, Tutorial 1
        • +
      • 1_k\3_lambda++\lesson_4\README.md refers to Lesson 1
        • +
      • 1_k\3_lambda++\lesson_5\README.md refers to Lesson 4, Lesson 8 and Lesson 7 of Tutorial 1
        • +
      • 1_k\3_lambda++\lesson_6\README.md refers to Parts 3 and 4 of the tutorial
        • +
      • 1_k\4_imp++\lesson_1\README.md refers to Lesson 4, Tutorial 2; also Tutorial 3 (at the end)
        • +
      • 1_k\4_imp++\lesson_2\README.md refers to Tutorial 3; Tutorial 2
        • +
      • 1_k\4_imp++\lesson_3\README.md refers to Lesson 1, Lesson 6
        • +
      • 1_k\4_imp++\lesson_4\README.md refers to Tutorial 6
        • +
      • 1_k\4_imp++\lesson_5\README.md refers to Lesson 4; Tutorial 3
        • +
      • 1_k\4_imp++\lesson_6\README.md refers to Lesson 3
        • +
      • 1_k\4_imp++\lesson_7\README.md refers to Lesson 1, 6
        • +
      • 1_k\5_types\lesson_1\README.md refers to Part 4; SIMPLE
        • +
      • 1_k\5_types\lesson_2\NOTES/README.md refer to Tutorial 1
        • +
      • 1_k\5_types\lesson_3\README.md refers to Part 1; Lesson 1,2
        • +
      • 1_k\5_types\lesson_4\README.md refers to Part 1, and to Lessons 2 and 3
        • +
      • 1_k\5_types\lesson_5\README.md refers to Lessons 4, 3, 2
        • +
      • 1_k\5_types\lesson_6\README.md refers to Lesson 5; SIMPLE, KOOL, IMP++
        • +
      • 1_k\5_types\lesson_7\README.md refers to Lesson 4, 8, 9
        • +
      • 1_k\5_types\lesson_8\README.md refers to Lessons 5, 7
        • +
      • 1_k\5_types\lesson_9\README.md refers to Lessons 8, 5, 7, 4
        • +
      • +
      +

      1_k\4_imp++\lesson_2\README.md states "generates a term of the form +symNat(n) of sort Nat", but the representation of symbolic numbers may +have changed

      +

      Describe/use/explain/justify the terminology "the <k/> cell" as opposed to "the k cell".

      +

      Would it be a good idea to make the README files self contained, that is, +to include the entire lang.k code in them, spread over the entire README, as things +are discussed? In case we decide not to, make sure that the code snippets mentioned +in the READMEs are in perfect correspondence to the code in the actual .k definitions. +Maybe add a tag before each code snippet saying what file and what lines in that +file comes from, then we can use a script to check them to be identical.

      +

      1_k\5_types\lesson_4\README.md refers to polymorphism, but some may say that is not precisely +polymorphism, because the types are not universally quantified. Explain that better.

      +

      Modify the entire tutorial to use . or, if needed, .::Map, etc., +instead of .Map, etc.. Check for each instance specifically, because +the surrounding text may also need to be modified.

      +

      We sometimes use "Kompile", or "kompile", as a verb instead of "Compile", +or "compile", to indicate that we mean compilation with K. Similarly for +"Krun", or "krun", instead of "Run" or "run".

      +

      Add citations to:

      +
        +
      • chemical abstract machine
        • +
      • logics, where the distinction between side condition and premise is explained
        • +
      • reduction semantics with evaluation contexts
        • +
      +

      Replace I1 +Int I2, notBool B, etc., with (I1 + I2)@INT, (not B)@Bool, +etc., when we have module qualification in place and working.

      +

      Explain isSort(T) for all sorts Sort, in one place, when it is first used. +Explain also that T:Sort yields a side condition isSort(T).

      +

      Currently all the K collections are "untyped", that is, over the sort K. +In the future we want to have parametric collections. Make sure the tutorial +is systematically changed when this happens.

      +

      See issue #2023 and modify pl-tutorial/1_k/2_imp/lesson_4 accordingly, if needed.

      +

      All definitions, and corresponding READMEs, should be changed to take advantage +of modules and module operations. Ideally, we'd like to have no code repetition +in any examples, except for demonstration purposes.

      +

      In the PL semantics book, define print(AExps) to have the same semantics as in +IMP++: evaluates and prints each of its arguments in order (as opposed to +first evaluate all of them and then print them---for example, if the second +argument performs a division by zero, I still want to print the first argument.)

      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/pl-tutorial/index.html b/k-distribution/pl-tutorial/index.html new file mode 100644 index 00000000000..c5a2fbadfba --- /dev/null +++ b/k-distribution/pl-tutorial/index.html @@ -0,0 +1,578 @@ + + + + + + + + + + + + + + +K PL Tutorial | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K PL Tutorial

      +

      Here you will learn how to use the K tool to define languages by means of a series of screencast movies. It is recommended to do these in the indicated order, because K features already discussed in a previous language definition will likely not be rediscussed in latter definitions. The screencasts follow quite closely the structure of the files under the tutorial folder in the K tool distribution. If you'd rather follow the instructions there and do the tutorial exercises yourself, then go back to https://kframework.org and download the K tool, if you have not done it already. Or, you can first watch the screencasts below and then do the exercises, or do them in parallel.

      +

      K Overview

      +

      Make sure you watch the K overview video before you do the K tutorial:

      + +

      Learning K

      +

      [34'46"] Part 1: Defining LAMBDA

      +

      Here you will learn how to define a very simple functional language in K and the basics of how to use the K tool. The language is a call-by-value variant of lambda calculus with builtins and mu, and its definition is based on substitution.

      + +

      [37'07"] Part 2: Defining IMP

      +

      Here you will learn how to define a very simple, prototypical textbook C-like imperative language, called IMP, and several new features of the K tool.

      + +

      [33'10"] Part 3: Defining LAMBDA++

      +

      Here you will learn how to define constructs which abruptly change the execution control, as well as how to define functional languages using environments and closures. LAMBDA++ extends the LAMBDA language above with a callcc construct.

      + +

      [46'46"] Part 4: Defining IMP++

      +

      Here you will learn how to refine configurations, how to generate fresh elements, how to tag syntactic constructs and rules, how to exhaustively search the space of non-deterministic or concurrent program executions, etc. IMP++ extends the IMP language above with increment, blocks and locals, dynamic threads, input/output, and abrupt termination.

      + +

      [17'03"] Part 5: Defining Type Systems

      +

      Here you will learn how to define various kinds of type systems following various approaches or styles using K.

      + +

      [??'??"] Part 6: Miscellaneous Other K Features

      +

      Here you will learn a few other K features, and better understand how features that you have already seen work.

      +
        +
      • [??'??"] ...
        • +
      +

      Learning Language Design and Semantics using K

      +

      [??'??"] Part 7: SIMPLE: Designing Imperative Programming Languages

      +

      Here you will learn how to design imperative programming languages using K. SIMPLE is an imperative language with functions, threads, pointers, exceptions, multi-dimensional arrays, etc. We first define an untyped version of SIMPLE, then a typed version. For the typed version, we define both a static and a dynamic semantics.

      + +

      [??'??"] Part 8: KOOL: Designing Object-Oriented Programming Languages

      +

      Here woul will learn how to design object-oriented programming languages using K. KOOL is an object-oriented language that extends SIMPLE with classes and objects. We first define an untyped version of KOOL, then a typed version, with both a dynamic and a static semantics.

      + +

      [??'??"] Part 9: FUN: Designing Functional Programming Languages

      +

      H +ere woul will learn how to design functional programming languages using K. FUN is a higher-order functional language with general let, letrec, pattern matching, references, lists, callcc, etc. We first define an untyped version of FUN, then a let-polymorphic type inferencer.

      + +

      [??'??"] Part 10: LOGIK: Designing Logic Programming Languages

      +

      Here you will learn how to design a logic programming language using K.

      + +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/checks/markdownErrorLocation/index.html b/k-distribution/tests/regression-new/checks/markdownErrorLocation/index.html new file mode 100644 index 00000000000..08d0ebeb113 --- /dev/null +++ b/k-distribution/tests/regression-new/checks/markdownErrorLocation/index.html @@ -0,0 +1,394 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      1

      +
      k
      // keep indentation
+module MARKDOWNERRORLOCATION-SYNTAX
+endmodule
+
      +

      7

      +
      .a .b
      9
+
      +

      11

      +
      .k .x
      module MARKDOWNERRORLOCATION
+  imports INT
+
      +
      {
      }
+
      +
      k
       rule 21 // pandoc would think this is line 20, column 7
+
      +
      .y .k
      endmodule // pandoc would miss this last unfinished block
+
+
      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/imp++-llvm/imp/index.html b/k-distribution/tests/regression-new/imp++-llvm/imp/index.html new file mode 100644 index 00000000000..08b4f9c7e71 --- /dev/null +++ b/k-distribution/tests/regression-new/imp++-llvm/imp/index.html @@ -0,0 +1,961 @@ + + + + + + + + + + + + + + +IMP++ | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      IMP++

      +

      Author: Grigore Roșu (grosu@illinois.edu)
      +Organization: University of Illinois at Urbana-Champaign

      +

      Abstract

      +

      This is the K semantic definition of the IMP++ language. +IMP++ extends the IMP language with the features listed below. We +strongly recommend you to first familiarize yourself with the IMP +language and its K definition in Tutorial 2 before proceeding.

      +

      Strings and concatenation of strings. Strings are useful +for the print statement, which is discussed below. For +string concatenation, we use the same + construct that we use +for addition (so we overload it).

      +

      Variable increment. We only add a pre-increment construct: +++x increments variable x and evaluates to the +incremented value. Variable increment makes the evaluation of +expressions have side effects, and thus makes the evaluation strategies +of the various language constructs have an influence on the set +of possible program behaviors.

      +

      Input and output. IMP++ adds a read() expression +construct which reads an integer number and evaluates to it, and +a variadic (i.e., it has an arbitrary number of arguments) statement +construct print(e1,e2,...,en) which evaluates its arguments +and then outputs their values. Note that the K tool allows to +connect the input and output cells to the standard input and output +buffers, this way compiling the language definition into an +interactive interpreter.

      +

      Abrupt termination. The halt statement simply halts +the program. The K tool shows the resulting configuration, as if the +program terminated normally. We therefore assume that an external +observer does not care whether the program terminates normally or +abruptly, same like with exit statements in conventional +programming languages like C.

      +

      Dynamic threads. The expression construct spawn s +starts a new concurrent thread that executes statement s, +which is expected to be a block, and evaluates immediately to a fresh +thread identifier that is also assigned to the newly created thread. +The new thread is given at creation time the environment of its +parent, so it can access all its parent's variables. This allows for +the parent thread, and the child thread to communicate; it also allows +for races and "unexpected" behaviors, so be careful. +For thread synchronization, IMP++ provides a thread join statement +construct join t;, where t evaluates to a thread +identifier, which stalls the current thread until thread t +completes its computation. For simplicity, we here assume a +sequentially consistent shared memory model. To experiment with other +memory models, see the definition of KERNELC.

      +

      Blocks and local variables. IMP++ allows blocks enclosed by +curly brackets. Also, IMP's global variable declaration construct is +generalized to be used anywhere as a statement, not only at the +beginning of the program. As expected, the scope of the declared +variables is from their declaration point till the end of the most +nested enclosing block.

      +

      What You Will Learn Here

      +
        +
      • How to define a less trivial language in K, as explained above.
        • +
      • How to use the superheat and supercool +options of the K tool kompile to exhaustively explore the +non-determinism due to underspecified evaluation strategies.
        • +
      • How to use the transition option of the K tool to +exhaustively explore the non-determinism due to concurrency.
        • +
      • How to connect certain cells in the configuration to the +standard input and standard output, and thus turn the krun +tool into an interactive interpreter for the defined language.
        • +
      • How to exhaustively search for the non-deterministic behaviors +of a program using the search option of krun.
        • +
      +
      k
      module IMP-SYNTAX
+  imports DOMAINS-SYNTAX
+
      +

      Syntax

      +

      IMP++ adds several syntactic constructs to IMP. Also, since the +variable declaration construct is generalized to be used anywhere a +statement can be used, not only at the beginning of the program, we +need to remove the previous global variable declaration of IMP and +instead add a variable declaration statement construct

      +

      We do not re-discuss the constructs which are taken over from IMP, +except when their syntax has been subtly modified (such as, for +example, the syntax of the previous "statement" assignment which +is now obtained by composing the new assignment expression, and the +new expression statement constructs); go the last lesson of +Tutorial 2 if you are interested in IMP's constructs. For execution +purposes, we tag the addition and division operations with the +addition and division tags. These attributes have +no theoretical significance, in that they do not affect the semantics +of the language in any way. They only have practical relevance, +specific to our implementation of the K tool. Specifically, we can +tell the K tool (using its superheat and supercool +options) that we want to exhaustively explore all the non-deterministic +behaviors (due to strictness) of these language constructs. For performance +reasons, by default the K tool chooses an arbitrary but fixed order to +evaluate the arguments of the strict language constructs, thus possibly +losing behaviors due to missed interleavings. This aspect was irrelevant in +IMP, because its expressions had no side effects, but it becomes relevant +in IMP++.

      +

      The syntax of the IMP++ constructs is self-explanatory. Note that assignment +is now an expression construct. Also, print is variadic, taking a +list of expressions as argument. It is also strict, which means that the +entire list of expressions, that is, each expression in the list, will be +evaluated. Note also that we have now defined sequential composition +of statements as a whitespace-separated list of statements, aliased with +the nonterminal Stmts, and block as such a (possibly empty) sequence +of statements surrounded by curly brackets.

      +
      k
        syntax AExp  ::= Int | String | Id
+                 | "++" Id
+                 | "read" "(" ")"
+                 | "-" AExp                   [strict]
+                 | "(" AExp ")"               [bracket]
+                 > AExp "/" AExp              [left, strict, group(division)]
+                 > AExp "+" AExp              [left, strict]
+                 > "spawn" Block
+                 > Id "=" AExp                [strict(2)]
+  syntax BExp  ::= Bool
+                 | AExp "<=" AExp             [seqstrict, latex({#1}\leq{#2})]
+                 | "!" BExp                   [strict]
+                 | "(" BExp ")"               [bracket]
+                 > BExp "&&" BExp             [left, strict(1)]
+  syntax Block ::= "{" Stmts "}"
+  syntax Stmt  ::= Block
+                 | AExp ";"                   [strict]
+                 | "if" "(" BExp ")"
+                   Block "else" Block         [strict(1)]
+                 | "while" "(" BExp ")" Block
+                 | "int" Ids ";"
+                 | "print" "(" AExps ")" ";"
+                 | "halt" ";"
+                 > "join" AExp ";"            [strict]
+
+  syntax Ids   ::= List{Id,","}
+  syntax AExps ::= List{AExp,","}
+  syntax Stmts ::= List{Stmt,""}
+  syntax AExps ::= Ids
+endmodule
+
+
+module IMP
+  imports IMP-SYNTAX
+  imports DOMAINS
+
      +

      Semantics

      +

      We next give the semantics of IMP++. We start by first defining its +configuration.

      +

      Configuration

      +

      The original configuration of IMP has been extended to include +all the various additional cells needed for IMP++. +To facilitate the semantics of threads, more specifically +to naturally give them access to their parent's variables, we prefer a +(rather conventional) split of the program state into an +environment and a store. An environment maps +variable names into locations, while a store maps locations +into values. Stores are also sometimes called states, or +heaps, or memory, in the literature. Like values, locations +can be anything. For simplicity, here we assume they are natural +numbers. Moreover, each thread has its own environment, so it knows +where all the variables that it has access to are located in the store +(that includes its locally declared variables as well as the variables +of its parent thread), and its own unique identifier. The store is +shared by all threads. For simplicity, we assume a sequentially consistent +memory model in IMP++. Note that the thread cell has multiplicity +*, meaning that there could be zero, one, or more instances of that cell +in the configuration at any given time. This multiplicity information +is important for K's configuration abstraction process: it tells +K how to complete rules which, in order to increase the modularity of the +definition, choose to not mention the entire configuration context. +The in and out cells hold the input and the output +buffers as lists of items.

      +
      k
        configuration <T color="yellow">
+                  <threads color="orange">
+                    <thread multiplicity="*" type="Set" color="blue">
+                      <k color="green"> $PGM:Stmts </k>
+                      <env color="LightSkyBlue"> .Map </env>
+                      <id color="black"> 0 </id>
+                    </thread>
+                  </threads>
+//                <br/>
+                  <store color="red"> .Map </store>
+//                  <input color="magenta"> .List </input>
+//                  <output color="Orchid"> .List </output>
+                  <input color="magenta" stream="stdin"> .List </input>
+                  <output color="Orchid" stream="stdout"> .List </output>
+                </T>
+// Replace the <input/> and <output/> cells with the next two in order to
+// initialize the input buffer through krun
+//     <input color="magenta"> $IN:List </input>
+//     <output color="Orchid"> .List </output>
+// Replace the <input/> and <output/> cells with the next two to connect the
+// input/output buffers to stdin/stdout through krun
+//     <input color="magenta" stream="stdin"> .List </input>
+//     <output color="Orchid" stream="stdout"> .List </output>
+// Replace the <input/> and <output/> cells with the next two to connect the
+// input/output buffers to stdin/stdout and also allow input through krun
+//     <input color="magenta" stream="stdin"> $IN:List </input>
+//     <output color="Orchid" stream="stdout"> .List </output>
+
      +

      We can also use configuration variables to initialize +the configuration through krun. For example, we may want to +pass a few list items in the in cell when the program makes +use of read(), so that the semantics does not get stuck. +Recall from IMP that configuration variables start with a ParseError: KaTeX parse error: Expected group after '_' at position 63: …, for example, +_̲PGM) and can be initialized with any string by +krun; or course, the string should parse to a term of the +corresponding sort, otherwise errors will be generated. +Moreover, K allows you to connect list cells to the standard input or +the standard output. For example, if you add the attribute +stream="stdin" to the in cell, then krun +will prompt the user to pass input when the in cell is empty +and any semantic rule needs at least one item to be present there in +order to match. Similarly but dually, if you add the attribute +stream="stdout" to the out cell, then any item +placed into this cell by any rule will be promptly sent to the +standard output. This way, krun can be used to obtain +interactive interpreters based directly on the K semantics of the +language. For example:

      +
      shell
      sh$ krun sum-io.imp --output none
+Add numbers up to (<= 0 to quit)? 10
+Sum = 55
+Add numbers up to (<= 0 to quit)? 1000
+Sum = 500500
+Add numbers up to (<= 0 to quit)? 0
+sh$
+
      +

      The option --output none instructs krun to not +display the resulting configuration after the program executes. The +input/output streaming works with or without this option, although +if you don't use the option then a configuration with empty +in and out cells will be displayed after the program +is executed. You can also initialize the configuration using +configuration variables and stream the contents of the cells to +standard input/output at the same time. For example, if you use a +configuration variable in the in cell and pass contents to it +through krun, then that contents will be first consumed and +then the user will be prompted to introduce additional input if the +program's execution encounters more read() constructs.

      +

      The old IMP constructs

      +

      The semantics of the old IMP constructs is almost identical to their +semantics in the original IMP language, except for those constructs +making use of the program state and for those whose syntax has slightly +changed. Indeed, the rules for variable lookup and assignment in IMP +accessed the state cell, but that cell is not available in IMP++ +anymore. Instead, we have to use the combination of environment and store +cells. Thanks to K's implicit configuration abstraction, we do not have +to mention the thread and threads cells: these are +automatically inferred (and added by the K tool at compile time) from the +definition of the configuration above, as there is only one correct +way to complete the configuration context of these rules in order to +match the configuration declared above. In our case here, "correct way" +means that the k and env cells will be considered as +being part of the same thread cell, as opposed to each being part +of a different thread. Configuration abstraction is crucial for modularity, +because it gives us the possibility to write our definitions in a way that +may not require us to revisit existing rules when we change the configuration. +Changes in the configuration are quite frequent in practice, typically +needed in order to accommodate new language features. For example, +imagine that we initially did not have threads in IMP++. There +would be no need for the thread and threads cells in +the configuration then, the cells k and env being simply +placed at the top level in the T cell, together with the +already existing cells. Then the rules below would be exactly the +same. Thus, configuration abstraction allows you to not have to +modify your rules when you make structural changes in your language +configuration.

      +

      Below we list the semantics of the old IMP constructs, referring the +reader to the K semantics of IMP for their meaning. Like we tagged the +addition and the division rules above in the syntax, we also tag the lookup +and the assignment rules below (with tags lookup and +assignment), because we want to refer to them when we generate the +language model (with the kompile tool), basically to allow them to +generate (possibly non-deterministic) transitions. Indeed, these two rules, +unlike the other rules corresponding to old IMP constructs, can yield +non-deterministic behaviors when more threads are executed concurrently. +In terms of rewriting, these two rules can "compete" with each other on +some program configurations, in the sense that they can both match at the +same time and different behaviors may be obtained depending upon which of +them is chosen first.

      +
      k
        syntax KResult ::= Int | Bool
+
      +

      Variable lookup

      +
      k
        rule <k> X:Id => I ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> I ...</store>  [group(lookup)]
+
      +

      Arithmetic constructs

      +
      k
        rule I1 / I2 => I1 /Int I2  when I2 =/=Int 0
+  rule I1 + I2 => I1 +Int I2
+  rule - I => 0 -Int I
+
      +

      Boolean constructs

      +
      k
        rule I1 <= I2 => I1 <=Int I2
+  rule ! T => notBool T
+  rule true && B => B
+  rule false && _ => false
+
      +

      Variable assignment

      +

      Note that the old IMP assignment statement X = I; is now composed of two +constructs: an assignment expression construct X = I, followed by a +semicolon ; turning the expression into a statement. The rationale behind +this syntactic restructuring has been explained in Lesson 7. Here is the +semantics of the two constructs:

      +
      k
        rule _:Int; => .
+  rule <k> X = I:Int => I ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> (_ => I) ...</store>  [group(assignment)]
+
      +

      Sequential composition

      +

      Sequential composition has been defined as a whitespace-separated syntactic +list of statements. Recall that syntactic lists are actually syntactic +sugar for cons-lists. Therefore, the following two rules eventually +sequentialize a syntactic list of statements s1 s2 ... sn.. into the +corresponding computation s1 ~> s2 ~> ... ~> sn.

      +
      k
         rule .Stmts => .
+   rule S:Stmt Ss:Stmts => S ~> Ss
+
      +

      Conditional statement

      +
      k
        rule if (true)  S else _ => S
+  rule if (false) _ else S => S
+
      +

      While loop

      +

      The only thing to notice here is that the empty block has been replaced +with the block holding the explicit empty sequence. That's because in +the semantics all empty lists become explicit corresponding dots +(to avoid parsing ambiguities)

      +
      k
        rule while (B) S => if (B) {S while (B) S} else {.Stmts}
+
      +

      The new IMP++ constructs

      +

      We next discuss the semantics of the new IMP++ constructs.

      +

      Strings

      +

      First, we have to state that strings are also results. +Second, we give the semantics of IMP++ string concatenation (which +uses the already existing addition symbol + from IMP) by +reduction to the built-in string concatenation operation.

      +
      k
        syntax KResult ::= String
+  rule Str1 + Str2 => Str1 +String Str2
+
      +

      Variable increment

      +

      Like variable lookup, this is also meant to be a supercool transition: we +want it to count both in the non-determinism due to strict operations above +it in the computation and in the non-determinism due to thread +interleavings. This rule also relies on K's configuration abstraction. +Without abstraction, you would have to also include the thread and +threads cells.

      +
      k
        rule <k> ++X => I +Int 1 ...</k>
+       <env>... X |-> N ...</env>
+       <store>... N |-> (I => I +Int 1) ...</store>  [group(increment)]
+
      +

      Read

      +

      The read() construct evaluates to the first integer in the +input buffer, which it consumes. Note that this rule is tagged +increment. This is because we will include it in the set of +potentially non-deterministic transitions when we kompile the definition; +we want to do that because two or more threads can "compete" on +reading the next integer from the input buffer, and different choices +for the next transition can lead to different behaviors.

      +
      k
        rule <k> read() => I ...</k>
+       <input> ListItem(I:Int) => .List ...</input>  [group(read)]
+
      +

      Print

      +

      The print statement is strict, so all its arguments are +eventually evaluated (recall that print is variadic). We +append each of its evaluated arguments, in order, to the output buffer, +and structurally discard the residual print statement with an +empty list of arguments. We only want to allow printing integers and +strings, so we define a Printable syntactic category including +only these and define the print statement to only print +Printable elements. Alternatively, we could have had two +similar rules, one for integers and one for strings. Recall that, +currently, K's lists are cons-lists, so we cannot simply rewrite the +head of a list (P) into a list (.). The first rule below is tagged, +because we want to include it in the list of transitions when we kompile; +different threads may compete on the output buffer and we want to capture +all behaviors.

      +
      k
        syntax Printable ::= Int | String
+
+/* currently it is necessary to subsort Printable to AExp,
+  but future K should be able to infer that automatically. */
+  syntax AExp ::= Printable
+
+  context print(HOLE:AExp, _AEs:AExps);
+
+  rule <k> print(P:Printable,AEs => AEs); ...</k>
+       <output>... .List => ListItem(P) </output>  [group(print)]
+  rule print(.AExps); => .
+
      +

      Halt

      +

      The halt statement empties the computation, so the rewriting process +simply terminates as if the program terminated normally. Interestingly, once +we add threads to the language, the halt statement as defined below +will terminate the current thread only. If you want an abrupt termination +statement that halts the entire program, then you need to discard the entire +contents of the threads cell, so the entire computation abruptly +terminates the entire program, no matter how many concurrent threads it has, +because there is nothing else to rewrite.

      +
      k
        rule <k> halt; ~> _ => . </k>
+
      +

      Spawn thread

      +

      A spawned thread is passed its parent's environment at creation time. +The spawn expression in the parent thread is immediately +replaced by the unique identifier of the newly created thread, so the +parent thread can continue its execution. We only consider a sequentially +consistent shared memory model for IMP++, but other memory models can also +be defined in K; see, for example, the definition of KERNELC. Note that +the rule below does not need to be tagged in order to make it a transition +when we kompile, because the creation of the thread itself does not interfere +with the execution of other threads. Also, note that K's configuration +abstraction is at heavy work here, in two different places. First, the +parent thread's k and env cells are wrapped within a +thread cell. Second, the child thread's k, env +and id cells are also wrapped within a thread cell. Why +that way and not putting all these four cells together within the +same thread, or even create an additional threads cell at top +holding a thread cell with the new k, env +and id? Because in the original configuration we declared +the multiplicity of the thread cell to be *, which +effectively tells the K tool that zero, one or more such cells can +co-exist in a configuration at any moment. The other cells have the +default multiplicity one, so they are not allowed to multiply. +Thus, the only way to complete the rule below in a way consistent with +the declared configuration is to wrap the first two cells in a +thread cell, and the latter two cells under the . +also in a thread cell. Once the rule applies, the spawning +thread cell will add a new thread cell next to it, which is consistent +with the declared configuration cell multiplicity. The unique identifier +of the new thread is generated using the fresh side condition.

      +
      k
        rule <k> spawn S => !T:Int ...</k> <env> Rho </env>
+       (.Bag => <thread>... <k> S </k> <env> Rho </env> <id> !T </id> ...</thread>)
+
      +

      Join thread

      +

      A thread who wants to join another thread T has to wait until +the computation of T becomes empty. When that happens, the +join statement is simply dissolved. The terminated thread is not removed, +because we want to allow possible other join statements to also dissolve.

      +
      k
        rule <k> join(T); => . ...</k> <thread>... <k>.</k> <id>T</id> ...</thread>
+
      +

      Blocks

      +

      The body statement of a block is executed normally, making sure +that the environment at the block entry point is saved in the computation, +in order to be recovered after the block body statement. This step is +necessary because blocks can declare new variables having the same +name as variables which already exist in the environment, and our +semantics of variable declarations is to update the environment map in +the declared variable with a fresh location. Thus, variables which +are shadowed lose their original binding, which is why we take a +snapshot of the environment at block entrance and place it after the +block body (see the semantics of environment recovery at the end of +this module). Note that any store updates through variables which are +not declared locally are kept at the end of the block, since the store +is not saved/restored. An alternative to this environment save/restore +approach is to actually maintain a stack of environments and to push a +new layer at block entrance and pop it at block exit. The variable +lookup/assign/increment operations then also need to change, so we do +not prefer that non-modular approach. Compilers solve this problem by +statically renaming all local variables into fresh ones, to completely +eliminate shadowing and thus environment saving/restoring. The rule +below can be structural, because what it effectively does is to take a +snapshot of the current environment; this operation is arguably not a +computational step.

      +
      k
        rule <k> {Ss} => Ss ~> Rho ...</k> <env> Rho </env>
+
      +

      Variable declaration

      +

      We allocate a fresh location for each newly declared variable and +initialize it with 0.

      +
      k
        rule <k> int (X,Xs => Xs); ...</k>
+       <env> Rho => Rho[X <- !N:Int] </env>
+       <store>... .Map => !N |-> 0 ...</store>
+  rule int .Ids; => .
+
      +

      Auxiliary operations

      +

      We only have one auxiliary operation in IMP++, the environment +recovery. Its role is to discard the current environment in the +env cell and replace it with the environment that it holds. +This rule is structural: we do not want them to count as computational +steps in the transition system of a program.

      +
      k
        rule <k> Rho => . ...</k> <env> _ => Rho </env>
+
      +

      If you want to avoid useless environment recovery steps and keep the size +of the computation structure smaller, then you can also add the rule

      +
        rule (_:Map => .) ~> _:Map
+
      +

      This rule acts like a ``tail recursion'' optimization, but for blocks. */

      +
      k
      // verification ids
+  syntax Id ::= "n"     [token]
+              | "sum"   [token]
+              | "a"     [token]
+              | "b"     [token]
+              | "c"     [token]
+endmodule
+
      +

      On Kompilation Options

      +

      We are done with the IMP++ semantics. The next step is to kompile the +definition using the kompile tool, this way generating a language +model. Depending upon for what you want to use the generated language model, +you may need to kompile the definition using various options. We here discuss +these options.

      +

      To tell the K tool to exhaustively explore all the behaviors due to the +non-determinism of addition, division, and threads, we have to kompile +with the command:

      +
      shell
      kompile imp.k --transition="addition division lookup assignment increment read print"
+
      +

      As already mentioned, the syntax and rule tags play no theoretical or +foundational role in K. They are only a means to allow kompile to +refer to them in its options, like we did above. By default, kompile's +transition option is empty, because this yields the fastest language model when +executed. Transitions may slow down the execution, but they instrument +the language model to allow for formal analysis of program behaviors, even for +exhaustive analysis.

      +

      Theoretically, the heating/cooling rules in K are fully reversible and +unconstrained by side conditions as we showed in the semantics of IMP. +For example, the theoretical heating/cooling rules corresponding to the +strict attribute of division are the following:

      +
      E₁ / E₂ ⇒ E₁ ⤳ □ / E₂
+E₁ ⤳ □ / E₂ ⇒ E₁ / E₂
+E₁ / E₂ ⇒ E₂ ⤳ E₁ / □
+E₂ ⤳ E₁ / □ ⇒ E₁ / E₂
+
      +

      The other semantic rules apply modulo such structural rules. +For example, using heating rules we can bring a redex (a subterm which +can be reduced with semantic rules) to the front of the computation, +then reduce it, then use cooling rules to reconstruct a term over the +original syntax of the language, then heat again and +non-deterministically pick another redex, and so on and so forth +without losing any opportunities to apply semantic rules. +Nevertheless, these unrestricted heating/cooling rules may create an +immense, often unfeasibly large space of possibilities to analyze. +The --transition option implements an optimization which works +well with other implementation choices made in the current K tool. +Recall from the detailed description of the IMP language semantics that +(theoretical) reversible rules like above are restricted by default +to complementary conditional rules of the form

      +
      E₁ / E₂ ⇒ E₁ ⤳ □ / E₂
+   if E₁ not in KResult
+E₁ ⤳ □ / E₂ ⇒ E₁ / E₂
+   if E₁ in KResult
+E₁ / E₂ ⇒ E₂ ⤳ E₁ / □
+   if E₂ not in KResult
+E₂ ⤳ E₁ / □  ⇒ E₁ / E₂
+   if  E₂ in KResult
+
      +

      Therefore, our tool eagerly heats and lazily cools the computation. +In other words, heating rules apply until a redex gets placed on the +top of the computation, then some semantic rule applies and rewrites +that into a result, then a cooling rule is applied to plug the +obtained result back into its context, then another argument may be +chosen and completely heated, and so on. This leads to efficient +execution, but it may and typically does hide program behaviors. +Using the --transition option allows you to interfere with this +process and to obtain all possible non-deterministic behaviors as if +the theoretical heating/cooling rules were applied. Optimizations +of course happen under the hood, but you need not be aware of them. +Used carefully, this mechanism allows us to efficiently explore more of +the non-deterministic behaviors of a program, even all of them (like here). +For example, with the semantics of IMP++ given above, the krun +command with the --search option detects all five behaviors +of the following IMP++ program (x can be 0, 1, 2, 3, or undefined +due to division-by-zero):

      +
        int x,y;
+  x = 1;
+  y = ++x / (++x / x);
+
      +

      Besides non-determinism due to underspecified argument evaluation +orders, which the current K tool addresses as explained above, there +is another important source of non-determinism in programming languages: +non-determinism due to concurrency/parallelism. For example, when two +or more threads are about to access the same location in the store and at +least one of these accesses is a write (i.e., an instance of the variable +assignment rule), there is a high chance that different choices for +the next transition lead to different program behaviors. While in the +theory of K all the rules count as computational steps +and hereby as transitions in the transition system associated to the +program, in practice that may yield a tremendous number of step +interleavings to consider. Most of these interleavings are behaviorally +equivalent for most purposes. For example, the fact that a thread computes +a step 8+3 ⇒ 11 is likely irrelevant for the other +threads, so one may not want to consider it as an observable transition in +the space of interleavings. Since the K tool cannot know without help which +transitions need to be explored and which do not, our approach is to +let the user say so explicitly using the transition option of +kompile.

      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/kprove-markdown/set-balance-spec/index.html b/k-distribution/tests/regression-new/kprove-markdown/set-balance-spec/index.html new file mode 100644 index 00000000000..1e1ed4c9ba0 --- /dev/null +++ b/k-distribution/tests/regression-new/kprove-markdown/set-balance-spec/index.html @@ -0,0 +1,433 @@ + + + + + + + + + + + + + + +Balances Module Specifications | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Balances Module Specifications

      +
      keep
      module SET-BALANCE-SPEC
+    imports VERIFICATION
+
      +
      k
      ignore thie code block!
+
      +

      total_balance tests

      +
      keep
          claim <k> totalBalance(AID) => 50 </k>
+          <account>
+            <accountID> AID </accountID>
+            <freeBalance> 30 </freeBalance>
+            <reservedBalance> 20 </reservedBalance>
+            ...
+          </account>
+
      +

      No Zero-Balance Accounts Exist

      +

      This property shows that set_balance will not result in a zero-balance attack. +TODO: Generalize to any EntryAction. +TODO: Assertions about log events.

      +
      discard
          rule <k> set_balance(Root, WHO, FREE_BALANCE', RESERVED_BALANCE') => . ... </k>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE +Int ( FREE_BALANCE' -Int FREE_BALANCE ) +Int ( RESERVED_BALANCE' -Int RESERVED_BALANCE ) </totalIssuance>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> FREE_BALANCE => FREE_BALANCE' </freeBalance>
+           <reservedBalance> RESERVED_BALANCE => RESERVED_BALANCE' </reservedBalance>
+           ...
+         </account>
+      requires #inWidth(96, TOTAL_ISSUANCE +Int (FREE_BALANCE' -Int FREE_BALANCE))
+       andBool #inWidth(96, TOTAL_ISSUANCE +Int (FREE_BALANCE' -Int FREE_BALANCE) +Int (RESERVED_BALANCE' -Int RESERVED_BALANCE))
+       andBool EXISTENTIAL_DEPOSIT <=Int FREE_BALANCE'
+       andBool EXISTENTIAL_DEPOSIT <=Int RESERVED_BALANCE'
+
      +
      keep
          claim <k> set_balance_reserved ( WHO , RESERVED_BALANCE' ) => . ... </k>
+          <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+          <totalIssuance> TOTAL_ISSUANCE +Int ( FREE_BALANCE' -Int FREE_BALANCE ) => TOTAL_ISSUANCE +Int ( FREE_BALANCE' -Int FREE_BALANCE ) +Int ( RESERVED_BALANCE' -Int RESERVED_BALANCE ) </totalIssuance>
+          <account>
+            <accountID> WHO </accountID>
+            <freeBalance> FREE_BALANCE' </freeBalance>
+            <reservedBalance> RESERVED_BALANCE => RESERVED_BALANCE' </reservedBalance>
+            ...
+          </account>
+      requires #inWidth(96, TOTAL_ISSUANCE +Int (FREE_BALANCE' -Int FREE_BALANCE) +Int (RESERVED_BALANCE' -Int RESERVED_BALANCE))
+       andBool EXISTENTIAL_DEPOSIT <=Int RESERVED_BALANCE'
+
      +
      keep
      endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/kprove-markdown/set-balance/index.html b/k-distribution/tests/regression-new/kprove-markdown/set-balance/index.html new file mode 100644 index 00000000000..e72cda8e096 --- /dev/null +++ b/k-distribution/tests/regression-new/kprove-markdown/set-balance/index.html @@ -0,0 +1,1227 @@ + + + + + + + + + + + + + + +set_balance spec | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      set_balance spec

      +

      State Model

      +
      k
      
+module SET-BALANCE
+    imports INT
+    imports DOMAINS
+    imports COLLECTIONS
+
+    configuration
+      <set-balance>
+        <k> $ACTION:Action </k>
+        <now> 0 </now>
+        <events> .List </events>
+        <return-value> .Result </return-value>
+        <call-stack> .List </call-stack>
+        <existentialDeposit> 0 </existentialDeposit>
+        <creationFee> 0 </creationFee>
+        <transferFee> 0 </transferFee>
+        <totalIssuance> 0 </totalIssuance>
+        <accounts>
+          <account multiplicity="*" type="Map">
+            <accountID> .AccountId:AccountId </accountID>
+            <freeBalance> 0 </freeBalance>
+            <reservedBalance> 0 </reservedBalance>
+            <vestingBalance> 0 </vestingBalance>
+            <startingBlock> 0 </startingBlock>
+            <perBlock> 0 </perBlock>
+            <nonce> .Nonce </nonce>
+            <locks> .Set </locks>
+          </account>
+        </accounts>
+      </set-balance>
+
      +

      Data

      +
        +
      • An AccountId is an Int.
        • +
      • An Origin is an AccountId, Root, or None.
        • +
      • A Nonce is an optional Int.
        • +
      • An Event records some happenning.
        • +
      +
      k
          syntax AccountId ::= ".AccountId" | Int
+ // ---------------------------------------
+
+    syntax Origin ::= AccountId | ".Root" | ".None"
+ // -----------------------------------------------
+
+    syntax Nonce ::= ".Nonce" | Int
+ // -------------------------------
+
+    syntax Event ::= DustEvent ( Int )
+ // ----------------------------------
+
      +

      Some predicates which help specifying behavior:

      +
        +
      • #inWidth: Specify that a given number is in some bitwidth.
        • +
      +
      k
          syntax Bool ::= #inWidth(Int, Int) [function, total]
+ // ---------------------------------------------------------
+    rule #inWidth(N, M) => 0 <=Int M andBool M <Int (2 ^Int N)
+
      +

      Results

      +

      A Result is the return value of an execution step.

      +
        +
      • AccountKilled indicates that the free balance goes below the existential threshold.
        • +
      • Updated indicates that an account was updated successfully.
        • +
      +
      k
          syntax Result ::= ".Result" | "AccountKilled" | "Updated"
+ // ---------------------------------------------------------
+
      +

      Public Getters

      +

      total_balance

      +

      Retrieves the total balance of an account. This includes both the free and +reserved balances.

      +
      k
          syntax Int ::= "total_balance" "(" AccountId ")" [function, total]
+ // -----------------------------------------------------------------------
+    rule total_balance(WHO) => free_balance(WHO) +Int reserved_balance(WHO)
+
      +

      free_balance

      +

      Gets the free balance of an account.

      +

      Other than when this module is executing, this will never be strictly between +EXISTENTIAL_DEPOSIT and zero.

      +
      k
          syntax Int ::= "free_balance" "(" AccountId ")" [function, total]
+ // ----------------------------------------------------------------------
+    rule    free_balance(_)   => 0 [owise]
+    rule [[ free_balance(WHO) => FREE_BALANCE ]]
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> FREE_BALANCE </freeBalance>
+           ...
+         </account>
+
      +

      reserved_balance

      +

      Gets the reserved balance of an account.

      +

      Other than when this module is executing, this will never be strictly between +EXISTENTIAL_DEPOSIT and zero.

      +
      k
          syntax Int ::= "reserved_balance" "(" AccountId ")" [function, total]
+ // --------------------------------------------------------------------------
+    rule    reserved_balance(_)   => 0 [owise]
+    rule [[ reserved_balance(WHO) => FREE_BALANCE ]]
+         <account>
+           <accountID> WHO </accountID>
+           <reservedBalance> FREE_BALANCE </reservedBalance>
+           ...
+         </account>
+
      +

      can_slash

      +

      Determines if an account’s free balance is over the value provided. This is +often used to determine if an account has enough balance to cover a potential +slash, hence the name.

      +
      k
          syntax Bool ::= "can_slash" "(" AccountId "," Int ")" [function, total]
+ // ----------------------------------------------------------------------------
+    rule    can_slash(_, _)        => false
+    rule [[ can_slash(WHO, AMOUNT) => FREE_BALANCE >=Int AMOUNT ]]
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> FREE_BALANCE </freeBalance>
+           ...
+         </account>
+
      +

      total_issuance

      +

      Retrieves the total outstanding amount of currency outstanding. This will +always be equal to the sum of all free and reserved balances in all active +accounts, except when the balances module is executing.

      +
      k
          syntax Int ::= "total_issuance" [function, total]
+ // ------------------------------------------------------
+    rule [[ total_issuance => TOTAL_ISSUANCE ]]
+         <totalIssuance> TOTAL_ISSUANCE </totalIssuance>
+
      +

      issue

      +

      Issues currency, creating an imbalance.

      +

      This is not specified, since these semantics do not include the concept of an +imbalance. Without the concept of destructors and move semantics, it would be +almost impossible to use correctly.

      +

      burn

      +

      Burns currency.

      +

      This is not part of the semantics for the same reason burn is not.

      +

      Actions and Results

      +

      An Action is an execution step (or the result of an execution step). +An EntryAction is an Action that can be invoked externally. +A Result is considered an Action, as is an EntryAction.

      +
      k
          syntax Action ::= Result | EntryAction
+ // --------------------------------------
+
      +

      account_exists

      +
      k
          syntax Bool ::= "account_exists" "(" AccountId ")" [function, total]
+ // -------------------------------------------------------------------------
+    rule    account_exists(_)   => false [owise]
+    rule [[ account_exists(WHO) => true ]]
+         <account> <accountID> WHO </accountID> ... </account>
+
      +

      create_account

      +
      k
          syntax Action ::= "create_account" "(" AccountId ")"
+ // ----------------------------------------------------
+    rule <k> create_account(WHO) => . ... </k>
+         <accounts> ( .Bag => <account> <accountID> WHO </accountID> ... </account> ) ... </accounts>
+
      +

      set_free_balance

      +
        +
      • Updates an accounts balance if the new balance is above the existential threshold.
        • +
      • Kills the account if the balance goes below the existential threshold and the reserved balance is non-zero.
        • +
      • Reaps the account if the balance goes below the existential threshold and the reserved balance is zero.
        • +
      +
      k
          syntax Action ::= "set_free_balance" "(" AccountId "," Int ")"
+ // --------------------------------------------------------------
+    rule <k> (. => create_account(WHO)) ~> set_free_balance(WHO, _) ... </k>
+      requires notBool account_exists(WHO)
+
+    rule [free-account-updated]:
+         <k> set_free_balance(WHO, BALANCE) => . ... </k>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> _ => BALANCE </freeBalance>
+           ...
+         </account>
+      requires EXISTENTIAL_DEPOSIT <=Int BALANCE
+
+    rule [free-account-killed]:
+         <k> set_free_balance(WHO, BALANCE) => . ... </k>
+         <events> ... (.List => ListItem(DustEvent(FREE_BALANCE))) </events>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int BALANCE </totalIssuance>
+         <account>
+           <accountID> WHO </accountID>
+           <nonce> _ => .Nonce </nonce>
+           <freeBalance> FREE_BALANCE => 0 </freeBalance>
+           <reservedBalance> RESERVED_BALANCE </reservedBalance>
+           ...
+         </account>
+      requires BALANCE <Int EXISTENTIAL_DEPOSIT
+       andBool 0 <Int RESERVED_BALANCE
+
+    rule [free-account-reaped]:
+         <k> set_free_balance(WHO, BALANCE) => . ... </k>
+         <events> ... (.List => ListItem(DustEvent(FREE_BALANCE))) </events>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int BALANCE </totalIssuance>
+         <accounts>
+           ( <account>
+               <accountID> WHO </accountID>
+               <freeBalance> FREE_BALANCE </freeBalance>
+               <reservedBalance> 0 </reservedBalance>
+               ...
+             </account>
+          => .Bag
+           )
+           ...
+         </accounts>
+      requires BALANCE <Int EXISTENTIAL_DEPOSIT
+
      +

      set_reserved_balance

      +
        +
      • Updates an accounts balance if the new balance is above the existential threshold.
        • +
      • Kills the account if the balance goes below the existential threshold and the free balance is non-zero.
        • +
      • Reaps the account if the balance goes below the existential threshold and the free balance is zero.
        • +
      +
      k
          syntax Action ::= "set_reserved_balance" "(" AccountId "," Int ")"
+ // ------------------------------------------------------------------
+    rule <k> (. => create_account(WHO)) ~> set_reserved_balance(WHO, _) ... </k>
+      requires notBool account_exists(WHO)
+
+    rule [reserved-account-updated]:
+         <k> set_reserved_balance(WHO, BALANCE) => . ... </k>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <account>
+           <accountID> WHO </accountID>
+           <reservedBalance> _ => BALANCE </reservedBalance>
+           ...
+         </account>
+      requires EXISTENTIAL_DEPOSIT <=Int BALANCE
+
+    rule [reserved-account-killed]:
+         <k> set_reserved_balance(WHO, BALANCE) => . ... </k>
+         <events> ... (.List => ListItem(DustEvent(RESERVED_BALANCE))) </events>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int BALANCE </totalIssuance>
+         <account>
+           <accountID> WHO </accountID>
+           <nonce> _ => .Nonce </nonce>
+           <freeBalance> FREE_BALANCE  </freeBalance>
+           <reservedBalance> RESERVED_BALANCE => 0 </reservedBalance>
+           ...
+         </account>
+      requires BALANCE <Int EXISTENTIAL_DEPOSIT
+       andBool 0 <Int FREE_BALANCE
+
+    rule [reserved-account-reaped]:
+         <k> set_reserved_balance(WHO, BALANCE) => . ... </k>
+         <events> ... (.List => ListItem(DustEvent(RESERVED_BALANCE))) </events>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int BALANCE </totalIssuance>
+         <accounts>
+           ( <account>
+               <accountID> WHO </accountID>
+               <freeBalance> 0 </freeBalance>
+               <reservedBalance> RESERVED_BALANCE </reservedBalance>
+               ...
+             </account>
+          => .Bag
+           )
+           ...
+         </accounts>
+      requires BALANCE <Int EXISTENTIAL_DEPOSIT
+
      +

      set_balance

      +
        +
      • Sets the new free balance
        • +
      • Creates suitible imbalances (both positive and negative).
        • +
      • Calls set_free_balance with the new free balance.
        • +
      • Calls set_reserved_balance with the new reserved balance.
        • +
      +
      k
          syntax EntryAction ::= "set_balance" "(" AccountId "," AccountId "," Int "," Int ")"
+ // ------------------------------------------------------------------------------------
+    rule [balance-set]:
+        <k> set_balance(_, WHO, FREE_BALANCE, RESERVED_BALANCE)
+         => set_balance_free(WHO, FREE_BALANCE)
+         ~> set_balance_reserved(WHO, RESERVED_BALANCE)
+        ...
+        </k>
+
      +

      Helpers for calling set_free_balance and set_reserved_balance.

      +
        +
      • Sets the new free balance
        • +
      • Emits an imbalance event
        • +
      • Helper function for set_balance
        • +
      +
      k
          syntax Action ::= "set_balance_free"     "(" AccountId "," Int ")"
+    syntax Action ::= "set_balance_reserved" "(" AccountId "," Int ")"
+ // ------------------------------------------------------------------
+    rule [balance-set-free]:
+         <k> set_balance_free(WHO, FREE_BALANCE') => set_free_balance(WHO, FREE_BALANCE') ... </k>
+         <totalIssuance> ISSUANCE => ISSUANCE +Int (FREE_BALANCE' -Int free_balance(WHO)) </totalIssuance>
+      requires #inWidth(96, ISSUANCE +Int (FREE_BALANCE' -Int free_balance(WHO)))
+
+    rule [balance-set-reserved]:
+         <k> set_balance_reserved(WHO, RESERVED_BALANCE') => set_reserved_balance(WHO, RESERVED_BALANCE') ... </k>
+         <totalIssuance> ISSUANCE => ISSUANCE +Int (RESERVED_BALANCE' -Int reserved_balance(WHO)) </totalIssuance>
+      requires #inWidth(96, ISSUANCE +Int (RESERVED_BALANCE' -Int reserved_balance(WHO)))
+
      +

      transfer_raw

      +

      Transfer some liquid free balance to another account.

      +

      transfer will set the FreeBalance of the sender and receiver. +It will decrease the total issuance of the system by the TransferFee. +If the sender's account is below the existential deposit as a result +of the transfer, the account will be reaped.

      +

      The dispatch origin for this call must be Signed by the transactor.

      +
      k
          syntax ExistenceRequirement ::= "AllowDeath"
+                                  | "KeepAlive"
+
+    syntax EntryAction ::= transfer(Origin, AccountId, Int)
+                         | "transfer_keep_alive" "(" Origin "," AccountId "," Int ")"
+ // ---------------------------------------------------------------------------------
+
+    syntax Action ::= rawTransfer(AccountId, AccountId, Int, ExistenceRequirement)
+ // ------------------------------------------------------------------------------
+    rule [transfer-to-raw]:
+         <k> transfer(ORIGIN:AccountId, DESTINATION, AMOUNT)
+          => rawTransfer(ORIGIN, DESTINATION, AMOUNT, AllowDeath)
+         ...
+         </k>
+
+    rule [transfer-keep-alive]:
+         <k> transfer_keep_alive(ORIGIN:AccountId, DESTINATION, AMOUNT)
+          => rawTransfer(ORIGIN, DESTINATION, AMOUNT, KeepAlive)
+         ...
+         </k>
+
+    rule <k> (. => create_account(DESTINATION)) ~> rawTransfer(ORIGIN, DESTINATION, _, _) ... </k>
+      requires         account_exists(ORIGIN)
+       andBool notBool account_exists(DESTINATION)
+
+    rule [transfer-self]:
+         <k> rawTransfer(ORIGIN:AccountId, ORIGIN, _, _) => . ... </k>
+      requires account_exists(ORIGIN)
+
+    rule [transfer-existing-account]:
+         <k> rawTransfer(ORIGIN, DESTINATION, AMOUNT, EXISTENCE_REQUIREMENT)
+          => set_free_balance(ORIGIN, SOURCE_BALANCE -Int AMOUNT -Int FEE)
+          ~> set_free_balance(DESTINATION, DESTINATION_BALANCE +Int AMOUNT)
+         ...
+         </k>
+         <totalIssuance> ISSUANCE => ISSUANCE -Int FEE </totalIssuance>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <transferFee> FEE </transferFee>
+         <accounts>
+           <account>
+             <accountID> ORIGIN </accountID>
+             <freeBalance> SOURCE_BALANCE </freeBalance>
+             ...
+           </account>
+           <account>
+             <accountID> DESTINATION </accountID>
+             <freeBalance> DESTINATION_BALANCE </freeBalance>
+             ...
+           </account>
+         </accounts>
+      requires ORIGIN =/=K DESTINATION
+       andBool DESTINATION_BALANCE >Int 0
+       andBool SOURCE_BALANCE >=Int (AMOUNT +Int FEE)
+       andBool ensure_can_withdraw(ORIGIN, Transfer, SOURCE_BALANCE -Int AMOUNT -Int FEE)
+       andBool (EXISTENCE_REQUIREMENT ==K AllowDeath orBool SOURCE_BALANCE -Int AMOUNT -Int FEE >Int EXISTENTIAL_DEPOSIT)
+
+    rule [transfer-create-account]:
+         <k> rawTransfer(ORIGIN:AccountId, DESTINATION, AMOUNT, EXISTENCE_REQUIREMENT)
+          => set_free_balance(ORIGIN, SOURCE_BALANCE -Int AMOUNT -Int CREATION_FEE)
+          ~> set_free_balance(DESTINATION, AMOUNT)
+         ...
+         </k>
+         <totalIssuance> ISSUANCE => ISSUANCE -Int CREATION_FEE </totalIssuance>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+         <creationFee> CREATION_FEE </creationFee>
+         <accounts>
+           <account>
+             <accountID> ORIGIN </accountID>
+             <freeBalance> SOURCE_BALANCE </freeBalance>
+             ...
+           </account>
+           <account>
+             <accountID> DESTINATION </accountID>
+             <freeBalance> 0 </freeBalance>
+             <reservedBalance> 0 </reservedBalance>
+             ...
+           </account>
+           ...
+         </accounts>
+      requires ORIGIN =/=K DESTINATION
+       andBool SOURCE_BALANCE >=Int (AMOUNT +Int CREATION_FEE)
+       andBool EXISTENTIAL_DEPOSIT <=Int AMOUNT
+       andBool ensure_can_withdraw(ORIGIN, Transfer, SOURCE_BALANCE -Int AMOUNT -Int CREATION_FEE)
+       andBool (EXISTENCE_REQUIREMENT ==K AllowDeath orBool SOURCE_BALANCE -Int AMOUNT -Int CREATION_FEE >=Int EXISTENTIAL_DEPOSIT)
+
      +

      force_transfer

      +

      Force a transfer from any account to any other account. This can only be done by root.

      +
      k
          syntax EntryAction ::= "force_transfer" "(" Origin "," AccountId "," AccountId "," Int ")"
+ // ------------------------------------------------------------------------------------------
+    rule [force-transfer]:
+         <k> force_transfer(.Root, SOURCE, DESTINATION, AMOUNT) => transfer(SOURCE, DESTINATION, AMOUNT) ... </k>
+
      +

      withdraw

      +

      Withdraw funds from an account.

      +
      k
          syntax EntryAction ::= withdraw(AccountId, Int, WithdrawReason, ExistenceRequirement)
+ // -------------------------------------------------------------------------------------
+    rule [withdraw]: // K really needs where clauses
+         <k> withdraw(WHO, AMOUNT, REASON, EXISTENCE_REQUIREMENT)
+          => withdrawInner(WHO, AMOUNT, AMOUNT -Int free_balance(WHO), REASON, EXISTENCE_REQUIREMENT)
+         ...
+         </k>
+
+    syntax Action ::= withdrawInner(AccountId, Int, Int, WithdrawReason, ExistenceRequirement)
+ // ------------------------------------------------------------------------------------------
+    rule [withdrawInner]:
+         <k> withdrawInner(WHO, AMOUNT, NEW_BALANCE, REASON, EXISTENCE_REQUIREMENT)
+          => set_free_balance(WHO, NEW_BALANCE)
+         ...
+         </k>
+         <totalIssuance> ISSUANCE => ISSUANCE -Int AMOUNT </totalIssuance>
+         <existentialDeposit> EXISTENTIAL_DEPOSIT </existentialDeposit>
+      requires NEW_BALANCE >=Int 0
+       andBool ensure_can_withdraw(WHO, REASON, NEW_BALANCE)
+       andBool (EXISTENCE_REQUIREMENT ==K AllowDeath orBool NEW_BALANCE >=Int EXISTENTIAL_DEPOSIT)
+
      +

      Call Frames

      +

      Function call and return.

      +
      k
          syntax CallFrame ::= frame(continuation: K)
+    syntax Action ::= call   ( Action )
+                    | return ( Result )
+ // -----------------------------------
+    rule [call]:
+         <k> call(Action) ~> CONT => Action </k>
+         <call-stack> .List => ListItem(frame(CONT)) ... </call-stack>
+
+    rule [return]:
+         <k> return(R) ~> _ => CONT </k>
+         <return-value> _ => R </return-value>
+         <call-stack> ListItem(frame(CONT)) => .List ... </call-stack>
+
+    rule [return-unit]:
+         <k> . => CONT </k>
+         <return-value> _ => .Result </return-value>
+         <call-stack> ListItem(frame(CONT)) => .List ... </call-stack>
+
      +

      Ensure that a given amount can be withdrawn from an account.

      +

      FIXME: we do not account for multiple withdrawl reasons, due to K’s +lacking polymorphism.

      +
      k
          syntax WithdrawReason ::= "TransactionPayment"
+                            | "Transfer"
+                            | "Reserve"
+                            | "Fee"
+                            | "Tip"
+ // -------------------------------
+
+
+    syntax Bool ::= "ensure_can_withdraw" "(" AccountId "," WithdrawReason "," Int ")" [function, total]
+ // ---------------------------------------------------------------------------------------------------------
+    rule ensure_can_withdraw(_, _, _) => true [owise]
+
+    rule [[ ensure_can_withdraw(WHO, Transfer #Or Reserve, BALANCE) => false ]]
+         <account>
+           <accountID> WHO </accountID>
+           <vestingBalance> VESTING_BALANCE </vestingBalance>
+           ...
+         </account>
+      requires VESTING_BALANCE <Int BALANCE
+
+    rule [[ ensure_can_withdraw(WHO, REASON, BALANCE) => false ]]
+         <now> NOW </now>
+         <account>
+           <accountID> WHO </accountID>
+           <locks> ACCOUNT_LOCKS </locks>
+           ...
+         </account>
+      requires activeLocks(ACCOUNT_LOCKS, NOW, REASON, BALANCE)
+
+    syntax LockID ::= "Election"
+                    | "Staking"
+                    | "Democracy"
+                    | "Phragmen"
+ // ----------------------------
+
+    syntax AccountLock ::= lock ( id: LockID, until: Int, amount: Int, reasons: Set )
+ // ---------------------------------------------------------------------------------
+
+    syntax Bool ::= activeLock (AccountLock, Int, WithdrawReason, Int      ) [function]
+                  | activeLocks(Set,         Int, WithdrawReason, Int      ) [function]
+                  | activeLocks(List,        Int, WithdrawReason, Int, Bool) [function, klabel(activeLocksAux)]
+ // -----------------------------------------------------------------------------------------------------------
+    rule activeLock(AL, NOW, REASON, BALANCE) => NOW <Int until(AL) andBool BALANCE <Int amount(AL) andBool REASON in reasons(AL)
+
+    rule activeLocks(ALS, NOW, REASON, BALANCE) => activeLocks(Set2List(ALS), NOW, REASON, BALANCE, false)
+
+    rule activeLocks(.List, _, _, _, RESULT) => RESULT
+    rule activeLocks((ListItem(AL) => .List) _, NOW, REASON, BALANCE, RESULT => RESULT orBool activeLock(AL, NOW, REASON, BALANCE))
+
      +

      Slashing and repatriation of reserved balances

      +

      The first of these is also used by slash.

      +
        +
      • slash_reserved
        • +
      • repatriate_reserved
        • +
      +
      k
          syntax Action ::= "slash_reserved" "(" AccountId "," Int ")"
+ // ------------------------------------------------------------
+    rule [slash-reserved]:
+         <k> slash_reserved(ACCOUNT, AMOUNT)
+          => set_reserved_balance(ACCOUNT, maxInt(0, RESERVED_BALANCE -Int AMOUNT))
+         ...
+         </k>
+         <accounts>
+           <account>
+             <accountID> ACCOUNT </accountID>
+             <reservedBalance> RESERVED_BALANCE </reservedBalance>
+             ...
+           </account>
+         </accounts>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int minInt(RESERVED_BALANCE, AMOUNT) </totalIssuance>
+
+    syntax Action ::= "repatriate_reserved" "(" AccountId "," AccountId "," Int ")"
+ // -------------------------------------------------------------------------------
+    rule [repatriate-reserved]:
+         <k> repatriate_reserved(SLASHED, BENEFICIARY, AMOUNT)
+          => set_free_balance(BENEFICIARY, BENEFICIARY_FREE_BALANCE +Int minInt(SLASHED_RESERVED_BALANCE, AMOUNT))
+          ~> set_reserved_balance(SLASHED, SLASHED_RESERVED_BALANCE -Int minInt(SLASHED_RESERVED_BALANCE, AMOUNT))
+         ...
+         </k>
+         <accounts>
+           <account>
+             <accountID> SLASHED </accountID>
+             <reservedBalance> SLASHED_RESERVED_BALANCE </reservedBalance>
+             ...
+           </account>
+           <account>
+             <accountID> BENEFICIARY </accountID>
+             <reservedBalance> BENEFICIARY_RESERVED_BALANCE </reservedBalance>
+             <freeBalance> BENEFICIARY_FREE_BALANCE </freeBalance>
+             ...
+           </account>
+         </accounts>
+      requires BENEFICIARY_FREE_BALANCE +Int BENEFICIARY_RESERVED_BALANCE >Int 0
+       andBool SLASHED =/=K BENEFICIARY
+
+    rule [repatriate-reserved-same-account]:
+         <k> repatriate_reserved(SLASHED, SLASHED, AMOUNT) => unreserve(SLASHED, AMOUNT) ... </k>
+
      +

      Slashing

      +

      Used to punish a node for violating the protocol.

      +
      k
          syntax EntryAction ::= slash ( AccountId , Int )
+ // ------------------------------------------------
+    rule [slash]:
+         <k> slash(ACCOUNT, AMOUNT) => set_free_balance(ACCOUNT, FREE_BALANCE -Int AMOUNT) ... </k>
+         <accounts>
+           <account>
+             <accountID> ACCOUNT </accountID>
+             <freeBalance> FREE_BALANCE </freeBalance>
+             ...
+           </account>
+         </accounts>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int AMOUNT </totalIssuance>
+      requires FREE_BALANCE >=Int AMOUNT
+
+    rule [slash-empty-free]:
+         <k> slash(ACCOUNT, AMOUNT)
+          => set_free_balance(ACCOUNT, 0)
+          ~> slash_reserved(ACCOUNT, AMOUNT -Int FREE_BALANCE)
+         ...
+         </k>
+         <accounts>
+           <account>
+             <accountID> ACCOUNT </accountID>
+             <freeBalance> FREE_BALANCE </freeBalance>
+             ...
+           </account>
+         </accounts>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE -Int FREE_BALANCE </totalIssuance>
+      requires FREE_BALANCE <Int AMOUNT
+
      +

      Reservation and unreservation of balances

      +

      Used to move balance from free to reserved and visa versa.

      +
      k
          syntax Action ::= reserve ( AccountId , Int )
+ // ---------------------------------------------
+    rule [reserve]:
+         <k> reserve(ACCOUNT, AMOUNT)
+          => set_reserved_balance(ACCOUNT, FREE_BALANCE +Int AMOUNT)
+          ~> set_free_balance(ACCOUNT, FREE_BALANCE -Int AMOUNT)
+         ...
+         </k>
+         <accounts>
+           <account>
+             <accountID> ACCOUNT </accountID>
+             <freeBalance> FREE_BALANCE </freeBalance>
+             <reservedBalance> _ </reservedBalance>
+             ...
+           </account>
+         </accounts>
+      requires FREE_BALANCE >=Int AMOUNT
+       andBool ensure_can_withdraw(ACCOUNT, Reserve, FREE_BALANCE -Int AMOUNT)
+
+    syntax Action ::= unreserve ( AccountId , Int )
+ // -----------------------------------------------
+    rule [unreserve]:
+         <k> unreserve(ACCOUNT, AMOUNT)
+          => set_free_balance(ACCOUNT, FREE_BALANCE +Int minInt(AMOUNT, RESERVED_BALANCE))
+          ~> set_reserved_balance(ACCOUNT, FREE_BALANCE -Int minInt(AMOUNT, RESERVED_BALANCE))
+         ...
+         </k>
+         <accounts>
+           <account>
+             <accountID> ACCOUNT </accountID>
+             <freeBalance> FREE_BALANCE </freeBalance>
+             <reservedBalance> RESERVED_BALANCE </reservedBalance>
+             ...
+           </account>
+         </accounts>
+
      +

      Vesting

      +
        +
      • locked_at ― amount currently locked
        • +
      • vesting_balance ― get the balance that cannot currently be withdrawn.
        • +
      +
      k
          syntax Int ::= "locked_at" "(" AccountId ")" [function, total]
+ // -------------------------------------------------------------------
+    rule [[ locked_at(WHO) => maxInt(0, VESTING_BALANCE -Int (PER_BLOCK *Int maxInt(0, NOW -Int STARTING_BLOCK))) ]]
+         <now> NOW </now>
+         <account>
+           <accountID> WHO </accountID>
+           <vestingBalance> VESTING_BALANCE </vestingBalance>
+           <startingBlock> STARTING_BLOCK </startingBlock>
+           <perBlock> PER_BLOCK </perBlock>
+           ...
+         </account>
+
+    syntax Int ::= "vesting_balance" "(" AccountId ")" [function, total]
+ // -------------------------------------------------------------------------
+    rule [[ vesting_balance(WHO) => minInt(FREE_BALANCE, locked_at(WHO)) ]]
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> FREE_BALANCE </freeBalance>
+           ...
+        </account>
+
      +

      Deposits

      +

      Deposit into an existing account.

      +
      k
          syntax EntryAction ::= "deposit_into_existing" "(" AccountId "," Int ")"
+ // ------------------------------------------------------------------------
+    rule [deposit-into-existing]:
+         <k> deposit_into_existing(WHO, AMOUNT) => . ... </k>
+         <totalIssuance> TOTAL_ISSUANCE => TOTAL_ISSUANCE +Int AMOUNT </totalIssuance>
+         <account>
+           <accountID> WHO </accountID>
+           <freeBalance> FREE_BALANCE => FREE_BALANCE +Int AMOUNT </freeBalance>
+           ...
+         </account>
+      requires FREE_BALANCE >Int 0
+
      +

      End of module

      +
      k
      endmodule
+
      +
      +
      + + +
      + + + +
      +
      + +
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      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/markdownSelectors/a-spec/index.html b/k-distribution/tests/regression-new/markdownSelectors/a-spec/index.html new file mode 100644 index 00000000000..26881fb4c7a --- /dev/null +++ b/k-distribution/tests/regression-new/markdownSelectors/a-spec/index.html @@ -0,0 +1,381 @@ + + + + + + + + + + + + + + +K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      + 
      k
      require "test.md"
+
+module A-SPEC
+
+endmodule
+
      +
      +
      + + +
      + +
      +
      + +
      +
      + + + + + + + + + + + + + diff --git a/k-distribution/tests/regression-new/markdownSelectors/test/index.html b/k-distribution/tests/regression-new/markdownSelectors/test/index.html new file mode 100644 index 00000000000..824fa86e382 --- /dev/null +++ b/k-distribution/tests/regression-new/markdownSelectors/test/index.html @@ -0,0 +1,407 @@ + + + + + + + + + + + + + + +Test | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      Test

      +
      k
      module TEST-SYNTAX
+  imports INT
+endmodule
+module TEST
+  imports INT
+
      +
      .k
        configuration <k> $PGM:K </k> <r> 0 </r>
+
      +
      .discard
        rule <r> 0 => 1 </r>
+
      +
      .keep
        rule <k> 0 => 1 </k>
+
      +
      .k .keep
        rule <k> 1 => 2 </k>
+
      +
      .k .discard .numberLines
        rule <k> 2 => 3 </k>
+
      +
      .keep .discard
        rule <r> 0 => 1 </r>
+
      +
      k
      endmodule
+
      +
      +
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      + + + + K + + Semantic Framework + + + Download +
      + + + +
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      The K Framework Demo

      + +
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      + + + + + + + + + + + + diff --git a/overview/index.html b/overview/index.html new file mode 100644 index 00000000000..e7d15ffa33f --- /dev/null +++ b/overview/index.html @@ -0,0 +1,356 @@ + + + + + + + + + + + + + + +K overview | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
      +
      +
      + + +
      + +
      +

      K overview

      + +

      Go to Youtube mirror, if the above does not work.

      +

      Go back to https://kframework.org for further links, the K tool and contact information.

      +
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      + + + + + + + + + + + + diff --git a/projects/index.html b/projects/index.html new file mode 100644 index 00000000000..e558c81ff83 --- /dev/null +++ b/projects/index.html @@ -0,0 +1,447 @@ + + + + + + + + + + + + + + +Projects using K | Runtime Verification Inc + + + + + + + + + + + + + + + + + + + +
      + + + + K + + Semantic Framework + + + Download +
      + + + +
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      + +
      +

      Projects using K

      +

      A list of projects using the K framework. If you are working on something interesting, and you want to share it with the community, +let us know on our socials, and we will feature you on this list.

      +
      + +
      +
        +
      • +

        KAVM (Feb 2022 - Present)

        +

        The Algorand Virtual Machine and TEAL Semantics in K
        +KAVM leverages the K Framework to empower Algorand smart contracts' developers +with property-based testing and formal verification.

        +
        • +
      • +

        KPlutus (2016 - Present)

        +

        The K Semantics of Plutus-Core

        +
        • +
      • +

        Dedukti (Mar 2021 - Present)

        +

        This project aims to translate real K semantics into Dedukti.

        +
        • +
      • +

        KWasm (Aug 2015 - Present)

        +

        KWasm is the K semantics of WebAssembly. +WebAssembly is a low-level (but simple and streamlined) assembly language that was originally developed to provide a fast execution engine for browser-based tools. +More recently, it has been used in several blockchain smart-contract platforms as the underlying language for executing financial agreements. +KWasm has been used for measuring coverage of test-suites over Wasm code and verifying programs which are compiled to Wasm.

        +
        • +
      • +

        KEVM (Sep 2017 - Present)

        +

        KEVM is the K semantics of the Ethereum Virtual Machine. +It passes all the Ethereum Test Suite, and is used for verifying EVM programs.

        +
        • +
      • +

        IELE (Oct 2016 - Present)

        +

        IELE is the underlying VM integrated into the Cardano blockchain. +IELE is a register-based VM (inspired by LLVM), which attempts to avoid many of the missteps in design present in EVM.

        +
        • +
      • +

        K-Michelson (Oct 2019 - Present)

        +

        K-Michelson is the K semantics of Michelson blockchain programming language, which powers the Tezos blockchain. +KMichelson provides additional testing tools for developers, including a unit-testing framework which is extendable to symbolic property testing.

        +
        • +
      • +

        C (Jul 2010 - Present)

        +

        The K semantics of the C programming language specifies the translation, linking, and execution semantics of the C language according to the official C standard. +It has been used to build tools like RV-Match, which detects undefined behaviors in users programs by running their test-suites through the C semantics.

        +
        • +
      +

      Archived

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      +
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