Add partial draft for background chapter

assets/bibliography/references.bib

@@ -1,31 +1,36 @@
-
+@article{freuder-1997-holygrail,
-@book{marriott_programming_1998,
+  author          = {Eugene C. Freuder},
-	location = {Cambridge, Mass},
+  title           = {In Pursuit of the Holy Grail},
-	title = {Programming with constraints: an introduction},
+  journal         = {Constraints An Int. J.},
-	isbn = {978-0-262-13341-8},
+  volume          = 2,
-	shorttitle = {Programming with constraints},
+  number          = 1,
-	pagetotal = {467},
+  pages           = {57--61},
-	publisher = {{MIT} Press},
+  year            = 1997,
-	author = {Marriott, Kim and Stuckey, Peter J.},
+  url             = {https://doi.org/10.1023/A:1009749006768},
-	date = {1998},
+  doi             = {10.1023/A:1009749006768},
-	keywords = {Constraint programming (Computer science), Logic programming},
+  timestamp       = {Fri, 13 Mar 2020 10:58:24 +0100},
+  biburl          = {https://dblp.org/rec/journals/constraints/Freuder97.bib},
+  bibsource       = {dblp computer science bibliography, https://dblp.org}
 }
 
-@incollection{hooker_solver-independent_2018,
+@book{marriott-1998-clp,
-	location = {Cham},
+  location        = {Cambridge, Mass},
-	title = {Solver-Independent Large Neighbourhood Search},
+  title           = {Programming with constraints: an introduction},
-	volume = {11008},
+  isbn            = {978-0-262-13341-8},
-	rights = {All rights reserved},
+  shorttitle      = {Programming with constraints},
-	isbn = {978-3-319-98333-2 978-3-319-98334-9},
+  pagetotal       = 467,
-	url = {http://link.springer.com/10.1007/978-3-319-98334-9_6},
+  publisher       = {{MIT} Press},
-	pages = {81--98},
+  author          = {Marriott, Kim and Stuckey, Peter J.},
-	booktitle = {Principles and Practice of Constraint Programming},
+  date            = 1998,
-	publisher = {Springer International Publishing},
+  keywords        = {Constraint programming (Computer science), Logic
-	author = {Dekker, Jip J. and de la Banda, Maria Garcia and Schutt, Andreas and Stuckey, Peter J. and Tack, Guido},
+                  programming},
-	editor = {Hooker, John},
+}
-	urldate = {2020-12-22},
+
-	date = {2018},
+@book{silvano-1990-knapsack,
-	langid = {english},
+  author          = {Martello, Silvano and Toth, Paolo},
-	doi = {10.1007/978-3-319-98334-9_6},
+  title           = {Knapsack Problems: Algorithms and Computer Implementations},
-	note = {Series Title: Lecture Notes in Computer Science},
+  year            = 1990,
+  isbn            = 0471924202,
+  publisher       = {John Wiley \& Sons, Inc.},
+  address         = {USA}
 }

assets/glossary.tex

@@ -8,3 +8,25 @@
 %     make use of the services and resources provided by another particular software
 %     program that implements that API},
 % }
+
+\newglossaryentry{constraint}{
+  name={constraint},
+  description={TODO},
+}
+\newglossaryentry{decision-variable}{
+  name={decision variable},
+  description={TODO},
+}
+\newglossaryentry{minizinc}{
+  name={MiniZinc},
+  description={TODO},
+}
+\newcommand{\minizinc}{\gls{minizinc}}
+\newglossaryentry{solver}{
+  name={solver},
+  description={TODO},
+}
+\newglossaryentry{problem-parameter}{
+  name={problem parameter},
+  description={TODO},
+}

assets/mzn/2_knapsack.mzn

@@ -0,0 +1,15 @@
+% Problem parameters
+enum TOYS = {football, tennisball, stuffed_lama, stuffed_elephant};
+array[TOYS] of int: toy_joy = [63, 12, 50, 100];
+array[TOYS] of int: toy_space = [32, 8, 16, 40];
+int: space_left = 64;
+
+% Decision variables
+var set of TOYS: selection;
+var int: total_joy = sum(toy in selection)(toy_joy[toy]);
+
+% Constraints
+constraint sum(toy in selection)(toy_space[toy]) < space_left;
+
+% Goal
+solve maximize total_joy;

assets/mzn/2_knapsack.mzntex

@@ -0,0 +1,17 @@
+\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1,codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8}]
+\PY{c}{\PYZpc{} Problem parameters}
+\PY{k+kt}{enum}\PY{l+s}{ }\PY{n+nv}{TOYS}\PY{l+s}{ }\PY{o}{=}\PY{l+s}{ }\PY{p}{\PYZob{}}\PY{n+nv}{football}\PY{p}{,}\PY{l+s}{ }\PY{n+nv}{tennisball}\PY{p}{,}\PY{l+s}{ }\PY{n+nv}{stuffed\PYZus{}lama}\PY{p}{,}\PY{l+s}{ }\PY{n+nv}{stuffed\PYZus{}elephant}\PY{g+gr}{\PYZcb{}}\PY{p}{;}
+\PY{k+kt}{array}\PY{p}{[}\PY{n+nv}{TOYS}\PY{g+gr}{]}\PY{l+s}{ }\PY{k+kt}{of}\PY{l+s}{ }\PY{k+kt}{int}\PY{p}{:}\PY{l+s}{ }\PY{n+nv}{toy\PYZus{}joy}\PY{l+s}{ }\PY{o}{=}\PY{l+s}{ }\PY{p}{[}\PY{l+m}{63}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{12}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{50}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{100}\PY{g+gr}{]}\PY{p}{;}
+\PY{k+kt}{array}\PY{p}{[}\PY{n+nv}{TOYS}\PY{g+gr}{]}\PY{l+s}{ }\PY{k+kt}{of}\PY{l+s}{ }\PY{k+kt}{int}\PY{p}{:}\PY{l+s}{ }\PY{n+nv}{toy\PYZus{}space}\PY{l+s}{ }\PY{o}{=}\PY{l+s}{ }\PY{p}{[}\PY{l+m}{32}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{8}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{16}\PY{p}{,}\PY{l+s}{ }\PY{l+m}{40}\PY{g+gr}{]}\PY{p}{;}
+\PY{k+kt}{int}\PY{p}{:}\PY{l+s}{ }\PY{n+nv}{space\PYZus{}left}\PY{l+s}{ }\PY{o}{=}\PY{l+s}{ }\PY{l+m}{64}\PY{p}{;}
+
+\PY{c}{\PYZpc{} Decision variables}
+\PY{k+kt}{var}\PY{l+s}{ }\PY{k+kt}{set}\PY{l+s}{ }\PY{k+kt}{of}\PY{l+s}{ }\PY{n+nv}{TOYS}\PY{p}{:}\PY{l+s}{ }\PY{n+nv}{selection}\PY{p}{;}
+\PY{k+kt}{var}\PY{l+s}{ }\PY{k+kt}{int}\PY{p}{:}\PY{l+s}{ }\PY{n+nv}{total\PYZus{}joy}\PY{l+s}{ }\PY{o}{=}\PY{l+s}{ }\PY{n+nb}{sum}\PY{p}{(}\PY{n+nv}{toy}\PY{l+s}{ }\PY{o}{in}\PY{l+s}{ }\PY{n+nv}{selection}\PY{g+gr}{)}\PY{p}{(}\PY{n+nv}{toy\PYZus{}joy}\PY{p}{[}\PY{n+nv}{toy}\PY{g+gr}{]}\PY{g+gr}{)}\PY{p}{;}
+
+\PY{c}{\PYZpc{} Constraints}
+\PY{k}{constraint}\PY{l+s}{ }\PY{n+nb}{sum}\PY{p}{(}\PY{n+nv}{toy}\PY{l+s}{ }\PY{o}{in}\PY{l+s}{ }\PY{n+nv}{selection}\PY{g+gr}{)}\PY{p}{(}\PY{n+nv}{toy\PYZus{}space}\PY{p}{[}\PY{n+nv}{toy}\PY{g+gr}{]}\PY{g+gr}{)}\PY{l+s}{ }\PY{o}{\PYZlt{}}\PY{l+s}{ }\PY{n+nv}{space\PYZus{}left}\PY{p}{;}
+
+\PY{c}{\PYZpc{} Goal}
+\PY{k}{solve}\PY{l+s}{ }\PY{k}{maximize}\PY{l+s}{ }\PY{n+nv}{total\PYZus{}joy}\PY{p}{;}
+\end{Verbatim}

assets/packages.tex

@@ -44,12 +44,34 @@ Scale=1.4
 \usepackage[
 style=apa,
 ]{biblatex}
+\usepackage{cleveref}
 
 % Glossary / Acronyms
-\usepackage[acronym]{glossaries}
+\usepackage[acronym,toc]{glossaries}
+\defglsentryfmt[main]{\ifglsused{\glslabel}{\glsgenentryfmt}{\textit{\glsgenentryfmt}}}
 \makeglossaries{}
 
 % Code formatting
 \usepackage{fancyvrb}
 \usepackage{color}
 \input{assets/pygments_header.tex}
+\DeclareNewTOC[
+type=program,
+float,
+name=Program,
+counterwithin=chapter,
+atbegin={%
+\scriptsize
+}
+]{program}
+\crefname{program}{program}{programs}
+\DeclareNewTOC[
+type=model,
+float,
+name=Model,
+counterwithin=chapter,
+atbegin={%
+\scriptsize
+}
+]{model}
+\crefname{model}{model}{models}

assets/py/2_dyn_knapsack.py

@@ -0,0 +1,28 @@
+toys_joy = [63, 12, 50, 100]
+toys_space = [32, 8, 16, 40]
+space_left = 64
+
+num_toys = len(toys_joy)
+# Initialise an empty table
+table = [
+    [0 for x in range(space_left + 1)]
+    for x in range(num_toys + 1)
+]
+for i in range(num_toys + 1):
+    for j in range(space_left + 1):
+        # If we are out of space or toys we cannot choose a toy
+        if i == 0 or j == 0:
+            table[i][j] = 0
+        # If there is space for the toy, then compare the joy of
+        # picking this toy over picking the next toy with more
+        # space left
+        elif toys_space[i - 1] <= j:
+            table[i][j] = max(
+                toys_joy[i - 1] + table[i - 1][j - wt[i - 1]],
+                table[i - 1][j],
+            )
+        # Otherwise, consider the next toy
+        else:
+            table[i][j] = table[i - 1][j]
+
+optimal_joy = table[num_toys][space_left]

assets/py/2_dyn_knapsack.pytex

@@ -0,0 +1,30 @@
+\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1,codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8}]
+\PY{n}{toys\PYZus{}joy} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{63}\PY{p}{,} \PY{l+m+mi}{12}\PY{p}{,} \PY{l+m+mi}{50}\PY{p}{,} \PY{l+m+mi}{100}\PY{p}{]}
+\PY{n}{toys\PYZus{}space} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{32}\PY{p}{,} \PY{l+m+mi}{8}\PY{p}{,} \PY{l+m+mi}{16}\PY{p}{,} \PY{l+m+mi}{40}\PY{p}{]}
+\PY{n}{space\PYZus{}left} \PY{o}{=} \PY{l+m+mi}{64}
+
+\PY{n}{num\PYZus{}toys} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{toys\PYZus{}joy}\PY{p}{)}
+\PY{c+c1}{\PYZsh{} Initialise an empty table}
+\PY{n}{table} \PY{o}{=} \PY{p}{[}
+    \PY{p}{[}\PY{l+m+mi}{0} \PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{space\PYZus{}left} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
+    \PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{num\PYZus{}toys} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}
+\PY{p}{]}
+\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{num\PYZus{}toys} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
+    \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{space\PYZus{}left} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
+        \PY{c+c1}{\PYZsh{} If we are out of space or toys we cannot choose a toy}
+        \PY{k}{if} \PY{n}{i} \PY{o}{==} \PY{l+m+mi}{0} \PY{o+ow}{or} \PY{n}{j} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
+            \PY{n}{table}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{n}{j}\PY{p}{]} \PY{o}{=} \PY{l+m+mi}{0}
+        \PY{c+c1}{\PYZsh{} If there is space for the toy, then compare the joy of}
+        \PY{c+c1}{\PYZsh{} picking this toy over picking the next toy with more}
+        \PY{c+c1}{\PYZsh{} space left}
+        \PY{k}{elif} \PY{n}{toys\PYZus{}space}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n}{j}\PY{p}{:}
+            \PY{n}{table}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{n}{j}\PY{p}{]} \PY{o}{=} \PY{n+nb}{max}\PY{p}{(}
+                \PY{n}{toys\PYZus{}joy}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{table}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{j} \PY{o}{\PYZhy{}} \PY{n}{wt}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{,}
+                \PY{n}{table}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{j}\PY{p}{]}\PY{p}{,}
+            \PY{p}{)}
+        \PY{c+c1}{\PYZsh{} Otherwise, consider the next toy}
+        \PY{k}{else}\PY{p}{:}
+            \PY{n}{table}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{n}{j}\PY{p}{]} \PY{o}{=} \PY{n}{table}\PY{p}{[}\PY{n}{i} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{j}\PY{p}{]}
+
+\PY{n}{optimal\PYZus{}joy} \PY{o}{=} \PY{n}{table}\PY{p}{[}\PY{n}{num\PYZus{}toys}\PY{p}{]}\PY{p}{[}\PY{n}{space\PYZus{}left}\PY{p}{]}
+\end{Verbatim}

chapters/2_background.tex

@@ -1,3 +1,69 @@
 %************************************************
 \chapter{Modelling with Constraints}\label{ch:background}
 %************************************************
+
+A goal shared between all programming languages is to provide a certain level of
+abstraction: an assembly language allows you to abstract from the binary
+instructions and memory positions; Low-level imperial languages, like FORTRAN,
+were the first to allow you to abstract from the processor architecture of the
+target machine; and nowadays writing a program requires little knowledge of the
+actual workings of the hardware. Freuder states that the ``Holy Grail'' of
+programming languages would be where the user merely states the problem, and the
+computer solves it and that constraint modelling is one of the biggest steps
+towards this goal to this day \autocite*{freuder-1997-holygrail}. Different
+from imperative (and even other declarative) languages, in a constraint
+modelling language the modeller does not describe how to solve the problem, but
+rather provides the problem requirements. You could say that a constraint model
+actually describes the solution to the problem.
+
+For example, let us consider the following scenario: Packing for a weekend trip,
+I have to decide which toys to bring for my dog, Audrey. We only have a small
+amount of space left in the car, so we cannot bring all the toys. Since Audrey
+gets enjoys playing with some toys more than others, we can now try and pick the
+toys that bring Audrey the most amount of joy, but still fit in the car.
+
+\begin{program}[ht]
+  \input{assets/py/2_dyn_knapsack.pytex}
+  \caption{\label{prog:dyn-knapsack} A Python program that solves a 0-1 knapsack
+    problem using dynamic programming}
+\end{program}
+
+A well educated reader in optimisation problems might immediately recognise that
+this is a variation on the widely known \textit{knapsack problem}, more
+specifically a \textit{0-1 knapsack problem}
+\autocite[13--67]{silvano-1990-knapsack}. A commonly used solution to this
+problem is based on dynamic programming. An implementation of this approach is
+shown in \cref{prog:dyn-knapsack}. In a naive recursive approach we would try
+all different combinations of toys to find the combination that will give the
+most joy, but using a dynamic programming approach this exponential behaviour
+(on the number of toys) can be avoided.
+
+\begin{model}[ht]
+  \input{assets/mzn/2_knapsack.mzntex}
+  \caption{\label{model:knapsack} A \minizinc\ model describing a 0-1 knapsack
+    problem}
+\end{model}
+
+A constraint model offers a different view of the problem. Instead of specifying
+the manner in which we can find the solution, we give a concise description of
+the problem in terms of what we already know, the \glspl{problem-parameter},
+what we wish to know, the \glspl{decision-variable}, and the relationships that
+should exists between them, the \glspl{constraint}. \Cref{model:knapsack} shows
+a \minizinc\ model of the knapsack problem, where the different elements of the
+constraint model are separated. Although a constraint model does not contain any
+instructions to find a suitable solutions, these models can generally be given
+to a dedicated solving program, or \gls{solver} for short, that can find a
+solution that fits the requirements of the model.
+
+
+\section{Constraint Modelling Basics}
+\label{sec:2-constraint-modelling-basics}
+
+\section{Solving Techniques}
+\label{sec:2-solving-techniques}
+
+\section{A Comparison of Constraint Modelling Languages}
+\label{sec:2-different-languages}
+
+\section{What Makes a ``Good'' Model?}
+\label{sec:2-model-quality}