Operational Semantics

1.1 Extended State Representation

In MeTTa, the state of a running program is represented as a complex tuple that encapsulates the entire execution context. This tuple includes:

Expression (E): The current computation or expression being evaluated. It can be a simple literal, a variable, a function call, or a composite expression combining these elements. The evaluation of E is governed by MeTTa's syntax and operational semantics, playing a central role in determining the program's behavior at any given moment.
Agenda: A goal stack containing all pending goals that the program seeks to achieve. Each goal represents an objective or a query that MeTTa aims to resolve. This goal-driven approach is integral to the logic resolution process, guiding the execution flow towards fulfilling these objectives.
Environment (Env): A collection of dynamically scopable variables and their associated values or definitions. It upholds the principle of lexical scoping, allowing functions to access and manipulate variables from their defining scope, thus maintaining state consistency across different parts of the program. Threads use these states to transfer data, such as for job sharing.
Database (DB): An editable repository of facts, rules, types, function definitions, rewrites, and unstarted processes. It is central to MeTTa's operation, being used for type checking, rewriting expressions, defining functions, and managing unstarted processes. The DB's flexible nature allows for runtime modifications, adapting the program's behavior based on new information or changing conditions.
Choice Stack (orStack): A stack of nondeterministic choices for backtracking and retries. It manages decision points and choices made during the program's execution, allowing the program to backtrack and retry different execution paths when necessary. This mechanism is essential for exploring various possibilities and finding solutions in the presence of multiple potential outcomes.
Substitution Environment (θ): A trail stack used in the unification process within MeTTa's logic programming aspects. It tracks variable bindings and substitutions, facilitating pattern matching and logic inference. The trail stack also manages special coroutines associated with variable bindings, which are invoked during unification and cease to exist when the trail is unwound.
Memory Model (Heap): Manages dynamic memory allocations during program execution. It efficiently allocates and deallocates resources, ensuring optimal memory usage. The heap handles the lifecycle of dynamically created entities, ensuring that resources are available when needed and reclaimed when no longer in use.

⟨ E, Agenda, Env, DB, orStack, θ, Heap ⟩

This tuple represents the complete state of a MeTTa program, allowing the interpreter to manage and manipulate the execution flow effectively.

1.2 Detailed Component Analysis

1.2.1 Expression (E)

The Expression (E) is the focal point of computation in MeTTa. It can be:

A simple literal (e.g., numbers, strings).
A variable.
A function call.
A composite expression combining multiple elements.

The evaluation of E determines the next steps in the program's execution and influences the state of other components.

1.2.2 Agenda (Goal Stack)

The Agenda is a stack of pending goals or tasks that the program aims to resolve. It operates on a goal-driven approach, where each goal guides the execution towards achieving specific objectives. As goals are resolved, they are removed from the agenda.

1.2.3 Environment (Env)

The Environment (Env) holds dynamically scopable variables and their associated values or definitions. It allows functions and expressions to access and modify variables within their scope, maintaining consistency across different parts of the program. This is crucial for data sharing between threads or modules.

1.2.4 Database (DB)

The Database (DB) is a dynamic and editable repository containing facts, rules, and definitions. It is central to MeTTa's operation, being used for type checking, rewriting expressions, defining functions, and managing unstarted processes. The DB's flexible nature allows for runtime modifications, adapting the program's behavior based on new information or changing conditions.

1.2.5 Choice Stack (orStack)

The Choice Stack (orStack) manages nondeterministic choices and decision points. It enables the program to:

Backtrack when a computation path fails.
Retry alternative execution paths.
Explore multiple potential outcomes.

This is essential for solving problems with multiple solutions or when the correct path is not known in advance.

1.2.6 Substitution Environment (θ)

The Substitution Environment (θ), also known as the trail stack, is used during unification in logic programming. It:

Tracks variable bindings and substitutions.
Manages special coroutines associated with variable bindings.
Allows variables to be unbound when backtracking occurs.

This ensures that variable states are accurately maintained throughout the computation.

1.2.7 Memory Model (Heap)

The Heap is responsible for dynamic memory management, handling:

Allocation of memory for dynamic data structures.
Deallocation of memory when it is no longer needed.
Management of the lifecycle of dynamically created entities.

Efficient memory management is crucial for performance and resource optimization.

Part 2: Expression Evaluation and Resolution Mechanics

2.1 Composite Expression Evaluation

In MeTTa, expressions can be a blend of functional and logical elements, demonstrating the language's hybrid nature. This functionality allows MeTTa to handle both deterministic and nondeterministic computations.

Example:

Combining a mathematical operation with a logic-based query, where the result of one influences the other, leading to a dynamic and interactive computational process.

⟨ (E₁ E₂), Agenda, Env, DB, orStack, θ, Heap ⟩
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Before Evaluation: The state includes expressions E₁ and E₂ to be evaluated together.
After Evaluation: The state transitions to E', with updated components reflecting the outcome of the evaluation. The evaluation process considers the current state of the agenda, environment, database, choice stack, substitution environment, and memory model, yielding new states for each.

2.2 Advanced Unification in Functional Context

Unification within lambda expressions is a hallmark of MeTTa's prowess in integrating functional programming with logical problem-solving. This feature allows variables within lambda expressions to be dynamically matched and bound to values, facilitating pattern matching essential in logic programming.

Use Case:

A lambda expression that filters a list based on a predicate.

⟨ (λ x . E[x]), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────→
⟨ E'[x ↦ v], Agenda', Env', DB', orStack', θ', Heap' ⟩

Explanation: The variable x within E[x] is unified with a value v.
Outcome: The expression E' is evaluated with x bound to v, updating the program state accordingly.

Part 3: Nondeterminism, Backtracking, and Functional Constructs

3.1 Enhanced Nondeterministic Control Flow and Backtracking

MeTTa handles nondeterminism through a mechanism similar to Prolog's backtracking:

Failure Handling: When a computation path results in failure, MeTTa doesn't halt. Instead, it backtracks to the most recent decision point in the orStack.
Choice Exploration: The program retries alternative paths stored in the orStack.

This is akin to a puzzle solver who, upon hitting a dead end, backtracks to the last point where a different choice could be made.

⟨ fail, Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Before Failure: The program encounters a fail state.
After Backtracking: It retrieves the next choice from the orStack and updates the state to E'.

3.2 Advanced Functional Control Structures

MeTTa incorporates functional programming constructs such as let bindings and lambda abstractions:

⟨ (let x = E₁ in E₂), Agenda, Env, DB, orStack, θ, Heap ⟩
────────────────────────────────────────────────────────→
⟨ E₂[x ↦ v], Agenda', Env', DB', orStack', θ', Heap' ⟩

let Binding Purpose: Introduces a local variable x bound to the value of E₁.
Outcome: Evaluates E₂ with x substituted by v.

These constructs promote modularity, code reuse, and maintainability.

Part 4: Recursive Function Definitions and Fixed Points

4.1 Fixed-Point Semantics for Recursion

MeTTa employs fixed-point combinators (like the Y combinator) to enable recursion without explicit self-reference:

Y = λ f. (λ x. f (x x)) (λ x. f (x x))

Explanation: The Y combinator allows a function to be applied to itself, facilitating recursive definitions.
Benefit: Simplifies recursive function creation, aligning with functional programming principles.

4.2 Logical Inference within Functions

MeTTa integrates logical reasoning within functional constructs:

⟨ (λ x . logical_goal(x)), Agenda, Env, DB, orStack, θ, Heap ⟩
────────────────────────────────────────────────────────────→
⟨ resolve(x, logical_goal), Agenda', Env', DB', orStack', θ', Heap' ⟩

Process: The lambda abstraction includes a logical goal. The program resolves this goal, updating its state accordingly.

Part 5: Stratified Negation and Lambda Calculus Integration

5.1 Complex Stratified Negation

MeTTa handles negation carefully to maintain logical consistency:

⟨ (λ x . not(P(x))), Agenda, Env, DB, orStack, θ, Heap ⟩
────────────────────────────────────────────────────────→
⟨ negate(P, x), Agenda', Env', DB', orStack', θ', Heap' ⟩

Explanation: The predicate P(x) is negated within a lambda expression.
Outcome: The program proceeds with the negated logic, ensuring valid reasoning.

5.2 Lambda Calculus Extensions for Logical Operations

MeTTa extends lambda calculus to support advanced logical operations:

⟨ (λ x . ∀ y. P(x, y)), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────────────→
⟨ forall_resolve(P, x, y), Agenda', Env', DB', orStack', θ', Heap' ⟩

Universal Quantification: The lambda expression asserts that P(x, y) holds for all y. The program resolves this, integrating higher-order logic into computation.

Part 6: Type System and Higher-Order Functions

6.1 Robust Type System Integration

MeTTa's type system supports both static and dynamic typing:

⟨ type_infer(E), Agenda, Env, DB, orStack, θ, Heap ⟩
────────────────────────────────────────────────────→
⟨ T, Agenda', Env', DB', orStack', θ', Heap' ⟩

Type Inference: The program infers the type T of expression E, ensuring type safety and correctness.

6.2 Higher-Order Functions and Logic Integration

Higher-order functions are first-class citizens in MeTTa:

⟨ (λ f . λ x . f(P(x))), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Explanation: A higher-order function takes a function f and an argument x, applying f to the result of a predicate P(x).

This integration allows for the creation of complex, reusable logic constructs.

Part 7: Concurrency and Parallelism in a Nondeterministic Setting

7.1 Concurrency in MeTTa

MeTTa exploits parallelism to enhance performance:

⟨ concurrent_and(E₁, E₂), Agenda, Env, DB, orStack, θ, Heap ⟩

⟨ concurrent_or(E₁, E₂), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────────────→
⟨ E'₁ ∥ E'₂, Agenda', Env', DB', orStack', θ', Heap' ⟩

AND Parallelism: Both E₁ and E₂ must succeed.
OR Parallelism: Either E₁ or E₂ can succeed.
Outcome: Expressions are evaluated in parallel, leveraging multi-core architectures.

7.2 Enhanced AND/OR Parallelism

MeTTa handles complex logical operations with combined parallelism:

⟨ and_or_parallel(E₁, E₂), Agenda, Env, DB, orStack, θ, Heap ⟩
────────────────────────────────────────────────────────────→
⟨ (E'₁ ∧ E'₂) ∨ (E'₁ ∨ E'₂), Agenda', Env', DB', orStack', θ', Heap' ⟩

Explanation: Executes E₁ and E₂ in parallel, considering both their conjunction and disjunction.

Part 8: Memory Management and Advanced Computational Models

8.1 Advanced Memory Management Techniques

The Heap in MeTTa manages dynamic memory efficiently:

Allocates memory for dynamic data structures as needed.
Deallocates memory when it goes out of scope or is no longer needed.
Optimizes resource usage to prevent memory leaks and enhance performance.

8.2 Support for Advanced Computational Models

MeTTa supports lazy evaluation:

⟨ freeze(E), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────→
⟨ frozen(E), Agenda', Env', DB', orStack', θ', Heap' ⟩

Lazy Evaluation: Computations are deferred until their results are required, reducing unnecessary processing.

Part 9: Interactivity and Real-World Interfacing

9.1 Interactive and Reactive Programming Support

MeTTa is suitable for building interactive applications:

⟨ event_handle(E, event), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Event Handling: Responds to user inputs or external events in real-time, ensuring applications remain responsive.

9.2 External System Interfacing and Integration

MeTTa can interface with external systems and languages:

⟨ py_call(E), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Integration: Executes external code (e.g., Python functions), enhancing functionality by leveraging external libraries.

Part 10: MeTTa-Programming and Reflection

10.1 MeTTa-Programming Capabilities

MeTTa allows programs to manipulate their own code:

⟨ add/remove-atom(E), Agenda, Env, DB, orStack, θ, Heap ⟩
──────────────────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

MeTTa-Programming: Generates or modifies code at runtime, enabling dynamic behavior and adaptability.

10.2 Reflection and Introspection Mechanisms

MeTTa supports reflection:

⟨ match(E), Agenda, Env, DB, orStack, θ, Heap ⟩
─────────────────────────────────────────────→
⟨ E', Agenda', Env', DB', orStack', θ', Heap' ⟩

Reflection: Examines and modifies the program's own structure and behavior, useful for debugging and adaptive algorithms.

Part 11: Optimization and Efficiency in Execution

11.1 Performance Optimization Techniques

Optimization is a key aspect of MeTTa's design:

Code Transformation: Rewriting code for better performance.
Resource Management: Efficiently handling memory and computational resources.
Algorithmic Enhancements: Implementing efficient algorithms to reduce complexity.

These techniques ensure that MeTTa programs run efficiently, making the language suitable for both small scripts and large-scale applications.