Proof read (#103)

* Change introduction of ethics

* Proof read the introduction

* Fix cross referencing

* Remove introduction for the relational model

* Proof read relational model

* Proof read benchmarking databases

* Proof section on category theory

* Proof read databaes implementation

* Proof read implementation chapter

* Proof read benchmark chapter

* Proof read evaluation

* Proof check the conclusion

* Fix significant figures in mean table

* Fix figure and table placements and size

* Fix listing style

* Add information on shuffling table rows

analysis/src/joinbench/benchmark_group_table_generator.py

@@ -30,8 +30,8 @@ def get_percentage_change_of_indexed_equijoin_for_counts(
             p_percent_change = (i_mean - p_mean) / p_mean * 100
             c_percent_change = (i_mean - c_mean) / c_mean * 100
             percentage_change_of_means[f"{benchmark.get_tuple_count()} tuples"] = [
-                p_percent_change,
-                c_percent_change,
+                f"{p_percent_change:.3g}\%",
+                f"{c_percent_change:.3g}\%",
             ]
 
         return percentage_change_of_means

report/.hunspell

@@ -266,3 +266,18 @@ Bina
 Vishnuram
 Elysia
 indexedTableRelAlgOps
+mystyle
+backgroundcolor
+commentstyle
+keywordstyle
+numberstyle
+stringstyle
+basicstyle
+breakatwhitespace
+captionpos
+keepspaces
+showspaces
+showstringspaces
+showtabs
+tabsize
+mystyle

report/background/categorytheory.tex

@@ -1,5 +1,5 @@
 \section{Category theory}
-Category theory will be the main tool in describing the structure and operations
+Category theory will be the main tool for describing the structure and operations
 of our database system. Category theory is a very powerful tool in mathematics
 and can be seen as an ``abstraction of abstractions''. The category theory that
 will appear in this paper is very limited and does not do justice to the full
@@ -18,7 +18,8 @@ \subsection{Categories}
 \begin{categorydef}
   A \emph{category} \cat{C} is a set\footnote{In more rigorous definitions one must be careful of defining the collections of objects as a set lest Russell's paradox comes into play} of \emph{objects} \objs{C}, such as \obj{a}, \obj{b}, \obj{c}, and \emph{morphisms} (or \emph{arrows}) \morphs{C} between them, such as \morph{f}, \morph{g}. We require that:
   \begin{itemize}
-    \item There are two operations; \emph{domain} which associates with every arrow \morph{f} an object $\obj{a} = \domain{f}$ and \emph{codomain} which associates with every arrow \morph{f} an object $\obj{b} = \codomain{f}$. We can now express this information as \explicitmorph{f}{a}{b}.\footnote{Though we emphasise the distinction between a function and a morphism.}
+    \item There are two operations; \emph{domain}, which associates with every
+        arrow \morph{f} an object $\obj{a} = \domain{f}$ and \emph{codomain}, which associates with every arrow \morph{f} an object $\obj{b} = \codomain{f}$. We can now express this information as \explicitmorph{f}{a}{b}.\footnote{Though we emphasise the distinction between a function and a morphism.}
     \item There is a composition rule between morphisms such that given \explicitmorph{f}{a}{b} and \explicitmorph{g}{b}{c}, there is another arrow \explicitmorph{\morph{g} \circ \morph{f}}{a}{c} in \morphs{C}.
     \item Composition of arrows is associative. That is, for an additional object \obj{d} and arrow \explicitmorph{h}{c}{d} the resulting morphisms $\morph{h} \circ \left(\morph{g} \circ \morph{f}\right)$ and $\left(\morph{h} \circ \morph{g}\right) \circ \morph{f}$ coincide in \morphs{C}.
     \item Every object \obj{a} is assigned an arrow $\id{a}: \obj{a} \to \obj{a}$ in \morphs{C}, called the \emph{identity morphism}.
@@ -53,7 +54,9 @@ \subsection{Functors}
 
 \subsection{Natural transformations}
 \theoremstyle{definition}\newtheorem*{nattransdef}{Natural Transformation}
-We now can explore an intuitive way of relating two functors, called an \emph{natural transformation}. A natural transformation can be seen as a projection of one functor space to another and is in essence a family of arrows that describes this translation.
+We now can explore an intuitive way of relating two functors, called a
+\emph{natural transformation}. A natural transformation can be seen as a
+projection from one functor space to another and is in essence a family of arrows that describes this translation.
 \begin{nattransdef}
 Given two functors $\functor{F}, \functor{G} : \cat{C} \to \cat{D}$ a
 \emph{natural transformation} $\explicitnattrans{\tau}{F}{G}$ associates to each element $\obj{a} \in \objs{C}$ an arrow $\nattransapply{\tau}{a} \in \morphs{D}$ such that $\nattransapply{\tau}{a}: \functorobj{F}{a} \to \functorobj{G}{a}$. Additionally, for $\obj{b} \in \cat{C}$ and \explicitmorph{f}{a}{b} a morphism in \cat{C}, we also require that $\nattransapply{\tau}{b} \circ \functormorph{F}{f} = \functormorph{G}{f} \circ \nattransapply{\tau}{a}$.
@@ -66,7 +69,8 @@ \subsection{Natural transformations}
     Given two natural transformations
     \explicitnattrans{\tau}{E}{F} and
     \explicitnattrans{\nu}{F}{G} where $\functor{E, F, G}: \cat{C} \to
-    \cat{D}$ are functors. The vertical composition \cite{RelationalAlgebraByWayOfAdjunctions}
+    \cat{D}$ are functors. The vertical
+    composition~\cite{RelationalAlgebraByWayOfAdjunctions}
     $\explicitnattrans{\verticalcomposition{\tau}{\nu}}{E}{G}$
     is defined by the composition of the underlying arrows for every
     object $\obj{a} \in \cat{C}$. Explicitly, the components
@@ -77,7 +81,8 @@ \subsection{Natural transformations}
 
 
 \subsection{Adjunctions}
-An adjunction expresses an intersection of arrows of two different functions. For instance Say you have the category \commoncatname{Set} of sets \cite{RelationalAlgebraByWayOfAdjunctions} and two functions $\morph{f}, \morph{g}$ with signatures \explicitmorph{f}{X}{A} and \explicitmorph{g}{X}{B} with $\obj{X}, \obj{A}, \obj{B} \in \commoncatname{Set}$.
+An adjunction expresses an intersection of arrows of two different functions.
+For instance Say you have the category \commoncatname{Set} of sets~\cite{RelationalAlgebraByWayOfAdjunctions} and two functions $\morph{f}, \morph{g}$ with signatures \explicitmorph{f}{X}{A} and \explicitmorph{g}{X}{B} with $\obj{X}, \obj{A}, \obj{B} \in \commoncatname{Set}$.
 in \commoncatname{Set} with a common domain $\domain{f} = \domain{g} = \obj{A}
 \in \commoncatname{Set}$, how might we interpret the application of both
 functions on one element. We could create a new function \explicitmorph{h}{X}{A
@@ -96,7 +101,7 @@ \subsection{Adjunctions}
 become clear. We introduce the diagonal functor $\functor{\Delta}:
 \commoncatname{Set} \to \commoncatname{Set}^2$, s.t. $\functorobj{\Delta}{A} =
 (\obj{A}, \obj{A})$ and $\functormorph{\Delta}{f} = (\morph{f}, \morph{f})$.
-Furthermore, the Cartesian product can also be seen view the lens of a functor $\functor{\times}: \commoncatname{Set}^2 \to \commoncatname{Set}$ that takes the pair of elements $(\obj{A}, \obj{B})$ and maps it to the set $\obj{A} \times \obj{B}$ and a pair of functions $(\morph{i}, \morph{j})$ to a function $k: \left(\domain{i} \times \domain{j}\right) \to \left(\codomain{i} \times \codomain{j}\right)$.
+Furthermore, the Cartesian product can also be seen through the lens of a functor $\functor{\times}: \commoncatname{Set}^2 \to \commoncatname{Set}$ that takes the pair of elements $(\obj{A}, \obj{B})$ and maps it to the set $\obj{A} \times \obj{B}$ and a pair of functions $(\morph{i}, \morph{j})$ to a function $k: \left(\domain{i} \times \domain{j}\right) \to \left(\codomain{i} \times \codomain{j}\right)$.
 Considering the above problem again, we can see very clear links between the two
 functors. We can work our problem in \commoncatname{Set} in the domain of
 $\commoncatname{Set}^2$ by considering $\functorobj{\Delta}{A} = (\obj{A},
@@ -110,7 +115,7 @@ \subsection{Adjunctions}
     Given two functors $\functor{L}: \cat{D} \to \cat{C}$ and $\functor{R}:
     \cat{C} \to \cat{D}$, we define an adjunction $\adunction{L}{R}$
     such that there is a natural isomorphism between the
-    hom-sets as follows \cite{RelationalAlgebraByWayOfAdjunctions}:
+    hom-sets as follows~\cite{RelationalAlgebraByWayOfAdjunctions}:
     \[
         \lfloor - \rfloor: \homset{C}{\functorobj{L}{A}}{B} \cong
         \homset{D}{A}{\functorobj{R}{B}} :\lceil - \rceil
@@ -123,7 +128,8 @@ \subsection{Adjunctions}
     \emph{counit}:
     $\explicitnattrans{\epsilon}{L \circ R}{\mathrm{Id}}$ such that
     $\nattransapply{\epsilon}{B} = \lceil \id{\functorobj{R}{A}} \rceil$. We
-    require that the unit and counit obey the `triangle identities' \cite{RelationalAlgebraByWayOfAdjunctions}:
+    require that the unit and counit obey the `triangle
+    identities'~\cite{RelationalAlgebraByWayOfAdjunctions}:
       $
       \verticalcomposition
           {\nattransapply{\eta}{\functor{R}}}
@@ -148,4 +154,4 @@ \subsection{Adjunctions}
 that
 $\nattransapply{\left(\verticalcomposition{\eta}{\epsilon}\right)}{\obj{A}} =
 \morphcomp{\nattransapply{\eta}{\obj{A}}}{\nattransapply{\epsilon}{\obj{A}}}$
-where the right hand side is simple the composition of arrows.
+where the right-hand side is simple the composition of arrows.

report/background/databaserepresentation.tex

@@ -4,15 +4,20 @@ \section{Evolution of database representation}\label{sec:background:dbrep}
 \fref{chap:database}.
 
 \subsection{Bags}
-\paragraph{Characteristics of a database} We expect our database approximation to not be ordered and admit multiplicities and a finite bag of values is one of the simplest constructions that does so. Like a finite set, a bag contains a collection of unordered values. However, unlike a set, bags can contain duplicate elements.  This multiplicity is key for processing non-idempotent aggregations. For instance, if summing up the ages of a database of people, without admitting multiplicity we would only sum each unique age once.
+\paragraph{Characteristics of a database} We expect our database approximation
+to not be ordered and admit multiplicities and a finite bag of values is one of
+the simplest constructions that does so. Like a finite set, a bag contains a
+collection of unordered values. However, unlike a set, bags can contain
+duplicate elements.  This multiplicity is key for processing non-idempotent
+aggregations. For instance, summing the ages of a database of people, without admitting multiplicity would only sum each unique age once.
 
 \subparagraph{Generalisation} Furthermore, going forward we generalise to bags
 of any types instead of the classical ``bags of records''. This also allows us
 to deal with intermediate tables that contain non-record values which, again,
 may be useful for describing intermediate states of aggergations or projections.
 
 In \fref{tab:BagRelAlgOps} we summarise the implementation of relational algebra operators with bags
-as their bulk type \cite{RelationalAlgebraByWayOfAdjunctions}.
+as their bulk type~\cite{RelationalAlgebraByWayOfAdjunctions}.
 \begin{table}[h]
     \centering
     \begin{tabular}{r|l}
@@ -48,7 +53,7 @@ \subsection{Indexed tables}
 
 We now have the mathematical tools required to define a map. In its finite form a map is widely known in computer science by many other names such as a dictionary, association lists or key-value maps.
 
-Let $\keyset$ be a set and $\valset$ a pointed set. To those already familiar with maps, it may help to think of $\keyset$ as keys and $\valset$ as values.
+Let $\keyset$ be a set and $\valset$ a pointed set. For those already familiar with maps, it may help to think of $\keyset$ as keys and $\valset$ as values.
 \begin{mapdef}
   A map of type $\map{\keyset}{\valset}$ is a total function from K to V.
 \end{mapdef}
@@ -71,11 +76,11 @@ \subsection{Indexed tables}
 \end{split}
 \end{equation*}
 
-The functions above tell us some extremely important information on creating
-empty maps and calculating their unions. As you can see $empty$ returns a
+The functions above tell us some extremely important information about creating
+empty maps and calculating the unions of maps with the same key type. As you can see $empty$ returns a
 function that maps any key to the neutral element $()$. This is to be expected
 as there are no values in an empty map. More interestingly, we see the merge
-of two maps as a function that returns a function that maps a key to a pair of
+of two maps is function that maps a key to a pair of
 values, each of which holds the result of the key lookup in the respective
 table.