%************************************************
\chapter{Review of Literature}\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
\gls{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 \cml\ 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.

In a constraint model, instead of specifying the manner in which we can find the
solution, we give a concise description of the problem. We describe what we
already know, the \glspl{parameter}, what we wish to know, the \glspl{variable},
and the relationships that should exists between them, the \glspl{constraint}.

This type of combinatorial problem is typically called a \gls{csp}. Many \cmls\
also support the modelling of \gls{cop}, where a \gls{csp} is augmented with an
\gls{objective} \(z\). In this case the goal is to find an solution that
satisfies all \glspl{constraint} while minimising (or maximising) \(z\).

Although a constraint model does not contain any instructions to find a suitable
solution, dedicated solving programs exist

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.

\begin{listing}
  \pyfile{assets/py/2_dyn_knapsack.py}
  \caption{\label{lst:2-dyn-knapsack} A Python program that solves a 0-1 knapsack
    problem using dynamic programming}
\end{listing}

\begin{example}%
\label{ex:back-knapsack}

  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.

  One way to solve this problem is to try all combinations of toys, but this is
  time intensive and quickly grows with the number of toys considered (which one
  would quickly realise trying to pack a car \(2^{|\text{toys}|}\) different
  ways).

  An educated reader in optimisation problems might recognise that this is a
  variation on the widely known \gls{knapsack}, 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{lst:2-dyn-knapsack}. The use of dynamic
  programming avoid the exponential growth of the problem when increasing the
  number of toys.

  Although expert knowledge can sometimes bring you an efficient solution to a
  known problem, it should be noted that not all problems will easily map to
  well known (and studied) problems. Even when part of a problem finds an
  equivalent in a well studied problem, the overall problem might still contain
  requirements that impair you from using known algorithms to solve the problem.
  For example, if wanted to bring toys with different colours, then the
  algorithm in \cref{lst:2-dyn-knapsack} would have to be drastically changed.
  \Gls{constraint-modelling} can offer a more flexible alternative that requires
  less expert knowledge.

  The following set of equations describe this knapsack problem as a \gls{cop}:

  \begin{equation*}
    \text{maximise}~z~\text{subject to}~
    \begin{cases}
      S \subseteq T \\
      z = \sum_{i \in S} joy(i) \\
      \sum_{i \in S} space(i) < C \\
    \end{cases}
  \end{equation*}

  In these equations \(S\) is set \gls{variable}. It contains the selection of
  toys that will be packed for the trip. \(z\) is the objective \gls{variable}
  that is maximised to find the optimal selections of toys to pack. The
  \gls{parameter} \(T\) is the set of all the toys. The \(joy\) and \(space\)
  functions are \glspl{parameter} used to map toys, \( t \in T\), to a value
  depicting the amount of enjoyment and space required respectively. Finally,
  the \gls{parameter} \(C\) is that depicts the total space that is left in the
  car before packing the toys.

  This constraint model gives a abstract mathematical definition of the
  \gls{cop} that would be easy to adjust to changes in the requirements. To
  solve instances of this problem, however, these instances have to be
  transformed into input accepted by a \gls{solver}. \cmls\ are designed to
  allow the modeller to express combinatorial problems in a similar fashion to
  the above mathematical definition and generate a definition that can be used
  by dedicated solvers.

\end{example}

In the remainder of this chapter we will first, in \cref{sec:back-minizinc}
introduce \minizinc\ as the leading \cml\ used within this thesis.
\cref{sec:back-mzn-interpreter} explains the process that the current \minizinc\
interpreter uses to translate a \minizinc\ model into a solver-level constraint
model. Then, \cref{sec:back-other-languages} introduces alternative \cmls\ and
compares their functionality to \minizinc{}. Finally, \cref{sec:back-term} and
\cref{sec:back-clp} survey the closely related fields of \gls{term-rewriting}
and \gls{clp}.

\section{\glsentrytext{minizinc}}%
\label{sec:back-minizinc}

\minizinc\ is a high-level, solver- and data-independent modelling language for
discrete satisfiability and optimisation problems
\autocite{nethercote-2007-minizinc}. Its expressive language and extensive
library of constraints allow users to easily model complex problems.

Let us introduce the language by modelling the problem from
\cref{ex:back-knapsack}. A \minizinc\ model encoding this problem is shown in
\cref{lst:back-mzn-knapsack}.

\begin{listing}
  \mznfile{assets/mzn/back_knapsack.mzn}
  \caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack
    problem}
\end{listing}

The model starts with the declaration of the \glspl{parameter}.
\Lref{line:back:knap:toys} declares an enumerated type that represents all
possible toys, \(T\) in the mathematical model in the example.
\Lref{line:back:knap:joy,line:back:knap:space} declare arrays mapping from toys
to integer values, these represent the functional mappings \(joy\) and
\(space\). Finally, \lref{line:back:knap:left} declares an integer
\gls{parameter} to represent the car capacity as an equivalent to \(C\).

The model then declares its \glspl{variable}. \Lref{line:back:knap:sel} declares
the main \gls{variable} \mzninline{selection}, which represents the selection of
toys to be packed. \(S\) in our earlier model. We also declare the variable
\mzninline{total_joy}, on \lref{line:back:knap:tj}, which is functionally
defined to be the summation of all the joy for the toy picked in our selection.

Finally, the model contains a constraint, on \lref{line:back:knap:con}, to
ensure we do not exceed the given capacity and states the goal for the solver:
to maximise the value of the variable \mzninline{total_joy}.

One might note that, although more textual and explicit, the \minizinc\ model
definition is very similar to our earlier mathematical definition.

Given ground assignments to input \glspl{parameter}, a \minizinc\ model is
translated (via a process called \emph{flattening}) into a set of variables and
primitive constraints.

Given the assignments

\begin{mzn}
TOYS = {football, tennisball, stuffed_elephant};
toy_joy = [63, 12, 100];
toy_space = [32, 8, 40];
space_left = 44;
\end{mzn}

the following model is the result of flattening:

\begin{mzn}
var 0..1: selection_0;
var 0..1: selection_1;
var 0..1: selection_2;
var 0..175: total_joy:: is_defined_var;
constraint int_lin_le([32,8,40],[selection_0,selection_1,selection_2],44);
constraint int_lin_eq([63,12,100,-1],[selection_0,selection_1,selection_2,total_joy],0):: defines_var(total_joy);
solve maximize total_joy;
\end{mzn}

This \emph{flat} problem will be passed to some \gls{solver}, which will attempt
to determine an assignment to each decision variable \mzninline{solection_i} and
\mzninline{total_joy} that satisfies all constraints and maximises
\mzninline{total_joy}, or report that there is no such assignment.


\subsection{Handling Undefined Expressions}%
\label{subsec:back-mzn-partial}

Some expressions in the \cmls\ do not always have a well defined result.
Examples of such expressions in \minizinc\ are:

\begin{itemize}
  \item Division (or modulus) when the divisor is zero: \\ \mzninline{x div 0 =
        @??@}

  \item Array access when the index is outside of the given index set: \\
        \mzninline{array1d(1..3, [1,2,3])[0] = @??@}

  \item Finding the minimum or maximum or an empty set: \\ \mzninline{min({}) =
        @??@}

  \item Computing the square root of a negative value: \\ \mzninline{sqrt(-1) = @??@}

\end{itemize}

The existence of undefined expressions can cause confusion in \cmls. There is
both the question what happens when a undefined expressions is evaluated and at
what point during the process undefined values will be resolved, during
flattening or at solving time.

Frisch and Stuckey define three semantic models to deal with the undefinedness
in \cmls\ \autocite*{frisch-2009-undefinedness}:

\begin{description}

  \item[Strict] \cmls\ employing a ``strict'' undefinedness semantic do not allow any undefined behaviour during the evaluation of the constraint model. If during the flattening or solving process an expression is found to be undefined, then any expressions in which it is used is also marked as undefined. In the end, this means that the occurrence of a single undefined expression will mark the full model as undefined.

  \item[Kleene] The ``Kleene'' semantic treat undefined expressions as expressions for which not enough information is available. This if a expressions contains undefined sub-expression, it will only be marked as undefined if the value of the subexpression is required to compute its result. Take for example the expression \mzninline{false -> E}. Here, when \mzninline{E} is undefined the result of the expression can still be said to be \mzninline{true}, since the value of \mzninline{E} does not influence the result of the expression. However, if we take the expression \mzninline{true /\ E}, then when \mzninline{E} is undefined the overall expression is also undefined since the value of the expression cannot be determined.

  \item[Relational] The ``relational'' semantic follows from the fact that all expressions in \cmls\ will eventually become part of a relational constraint. So even though a (functional) expression in itself might not have a well-defined result, we can still decide whether its surrounding relationship holds. For example, the expression \mzninline{x div 0} is undefined, but the relationship \mzninline{int_div(x,0,y)} can be said to be \mzninline{false}. It can be said that the relational semantic will make the closest relational expression that contains an undefined expression \mzninline{false}.

\end{description}

In practice, it is often natural to guard against undefined behaviour using
Boolean logic. Relational semantics therefore often feel the most natural for
the users of constraint modelling languages. This is why the \minizinc\ uses
relational semantics during its evaluation.

For example, one might might deal with a zero divisor using a disjunction:

\begin{mzn}
  constraint d == 0 \/ a div d < 3;
\end{mzn}

In this case we expect the undefinedness of the division to be contained within
the second part of the disjunction. This corresponds to ``relational''
semantics. \jip{TODO:\@ This also corresponds to Kleene semantics, maybe I should
  use a different example}

Frisch and Stuckey also show that different \glspl{solver} often employ
different semantical models \autocite*{frisch-2009-undefinedness}. It is
therefore important that, during the flattening process, any potentially
undefined expression gets replaced by an equivalent model that is still valid
under a strict semantic. Essentially eliminating the existence of undefined
expressions in the \gls{solver} model.

\section{The Current \glsentrytext{minizinc} Interpreter}%
\label{sec:back-mzn-interpreter}

\section{Other Constraint Modelling Languages}%
\label{sec:back-other-languages}

\section{ACD Term Rewriting}%
\label{sec:back-term}

\section{Constraint Logic Programming}%
\label{sec:back-clp}
