%************************************************
\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 exist 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 a
\gls{objective} \(z\). In this case the goal is to find a 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, 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.
  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 an 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 similar 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{trs} 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.

\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}

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

  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}.

  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 \gls{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 \gls{variable} \mzninline{total_joy}.
\end{example}

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
\glspl{variable} 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 \gls{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{Model Structure}%
\label{subsec:back-mzn-structure}

As we have seen in \cref{ex:back-mzn-knapsack}, a \minizinc\ model generally
contains value declarations, both for \glspl{variable} and input
\glspl{parameter}, \glspl{constraint}, and a solving goal. More complex models
might also include definitions for custom types, user defined functions, and a
custom output format. In \minizinc\ these items are not constrained to occur in
any particular order. We will briefly discuss the most important model items.
For a detailed overview of the structure of \minizinc\ models you can consult
the full syntactic structure of \minizinc\ 2.5.5 in \cref{ch:minizinc-grammar}.
Nethercote et al.\ and Mariott et al.\ offer a detailed discussion of the
\minizinc\ and \zinc\ language, its predecessor, respectively
\autocite*{nethercote-2007-minizinc,marriott-2008-zinc}.

Values in \minizinc\ are declared in the form \mzninline{@\(T\)@: @\(I\)@ =
@\(E\)@;}. \(T\) is the type of the declared value, \(I\) is a new identifier
used to reference the declared value, and, optionally, the modeller can
functionally define the value using an expression \(E\). The identifier used in
a top-level value definition must be unique. Two declarations with the same
identifier will result in an error during the flattening process.

The main types used in \minizinc\ are Boolean, integer, floating point numbers,
sets of integers, and (user-defined) enumerated types. These types can be used
both as normal \glspl{parameter} and as \glspl{variable}. To better structure
models, \minizinc\ allows collections of these types to be contained in arrays.
Unlike other languages, arrays can have a user defined index set, which can
start at any value, but has to be a continuous range. For example, an array
going from 5 to 10 of new boolean \glspl{variable} might be declared as

\begin{mzn}
  array[5..10] of var bool: bs;
\end{mzn}

The type in a declaration can, however, be more complex when the modeller uses a
type expression. These expressions constrain a declaration, not just to a
certain type, but also to a set of value. This set of values is generally
referred to as the \gls{domain} of a \gls{variable}. In \minizinc\ any
expression that has a set type can be used as a type expression. For example,
the following two declarations

\begin{mzn}
  var 3..5: x;
  var {1,3,5}: y;
\end{mzn}

declare two integer \glspl{variable} that can take the values from three to five and
one, three, and five respectively.

If the declaration includes an expression to functionally define the value, then
the identifier can be used as a name for this expression. If, however, the type
of the declaration is given as a type expression, then this places an implicit
\gls{constraint} on the expression, forcing the result of the expression to take
a value according to the type expression.

\gls{constraint} items, \mzninline{constraint @\(E\)@;} contain the top-level
constraint of the \minizinc\ model. A constraint item contains only a single
expression \(E\) of Boolean type. During the flattening of the model the
expressions in constraints are translated into solver level versions of the same
expression. It is important that the solver-level versions of the expressions
are equisatisfiable, meaning they are only satisfied if-and-only-if the original
expression would have been satisfied.

A \minizinc\ model can contain a single goal item. This item can signal the
solver to do one of three actions: to find an assignment to the \glspl{variable}
that satisfies the constraints, \mzninline{solve satisfy;}, to find an
assignment to the \glspl{variable} that satisfies the constraints and minimises
the value of a \gls{variable}, \mzninline{solve minimize x;}, or similarly
maximises the value of a \gls{variable}, \mzninline{solve maximize x;}.

Common structures in \minizinc\ can be captured using function declarations. A
user can declare a function \mzninline{function @\(T\)@: @\(I\)@(@\(P\)@) = E;}.
In the function declaration \(T\) is the type of the result of the function,
\(I\) is the identifier for the function, \(P\) is a list types and identifiers
for the parameters of the functions, and finally \(E\) is the expression that
can use the parameters \(P\) and when flattened will give the result of the
function. The \minizinc\ language offers the keywords \mzninline{predicate} and
\mzninline{test} as a shorthand for \mzninline{function var bool} and
\mzninline{function bool} respectively. For example a function that squares an
integer can be defined as follows.

\begin{mzn}
  function int: square(int: a) = a * a;
\end{mzn}

Function declarations are also the main way in which \gls{solver} libraries are
defined. During flattening all \minizinc\ expressions are (eventually) rewritten
to function calls. A solver can then provide its own implementation for these
functions. It is assumed that the implementation of the functions in the
\gls{solver} libraries will ultimately be rewritten into fully relational call.
When a relational constraint is directly supported by a solver the function
should be declared within an expression body. Any call to such function is
directly placed in the flattened model.

\subsection{MiniZinc Expressions}%
\label{subsec:back-mzn-expr}

One of the powers of the \minizinc\ language is the extensive expression
language that it offers to help modellers create models that are intuitive to
read, but are transformed to fit the structure best suited to the chosen
\gls{solver}. We will now briefly discuss the most important \minizinc\
expressions and the general methods employed when flattening them. A detailed
overview of the full syntactic structure of the \minizinc\ expressions in
\minizinc\ 2.5.5 can be found in \cref{sec:mzn-grammar-expressions}. Nethercote
et al.\ and Mariott et al.\ offer a detailed discussion of the expression
language of \minizinc\ and its predecessor \zinc\ respectively
\autocite*{nethercote-2007-minizinc,marriott-2008-zinc}.

\Glspl{global} are the basic building blocks in the \minizinc\ language. These
expressions capture common (complex) relations between \glspl{variable}.
\Glspl{global} in the \minizinc\ language are used as function calls. An example
of a \gls{global} is
\begin{mzn}
predicate knapsack(
  array [int] of int: w,
  array [int] of int: p,
  array [int] of var int: x,
  var int: W,
  var int: P,
);
\end{mzn}

This \gls{global} expresses the knapsack relationship, where the
\glspl{parameter} \mzninline{w} are the weights of the items, \mzninline{p} are
the profit for each item, the \glspl{variable} in \mzninline{x} represent the
amount of time the items are present in the knapsack, and \mzninline{W} and
\mzninline{P}, respectively, represent the weight and profit of the knapsack.

Note that the usage of this \gls{global} might have simplified the \minizinc\
model in \cref{ex:back-mzn-knapsack}:

\begin{mzn}
  constraint knapsack(toy_space, toy_joy, set2bool(selection), total_joy, space);
\end{mzn}

The usage of this \gls{global} has the additional benefit that the knapsack
structure of the problem is then known to the \gls{solver} which might implement
special handling of the relationship.

Although \minizinc\ contains an extensive library of \glspl{global}, many
problems contain constraints that aren't covered by a \gls{global}. There are
many other expression forms in \minizinc\ that can help modellers express a
constraint.

\Gls{operator} symbols in \minizinc\ are used as a shorthand for \minizinc\
functions that can be used to transform or combine other expressions. For
example the constraint

\begin{mzn}
constraint not (a + b < c);
\end{mzn}

contains the infix \glspl{operator} \mzninline{+} and \mzninline{<}, and the
prefix \gls{operator} \mzninline{not}.

These \glspl{operator} will be evaluated using the addition, less-than
comparison, and Boolean negation functions respectively. Although the
\gls{operator} syntax for \glspl{variable} and \glspl{parameter} is the same,
different (overloaded) versions of these functions will be used during
flattening. For \glspl{parameter} types the result of the function can be
directly computed, but when flattening these functions with \glspl{variable}
types a new \gls{variable} for its result must be introduced and a constraint
enforcing the functional relationship.

The choice between different expressions can often be expressed using a
\gls{conditional} expression, sometimes better known as an ``if-then-else''
expressions. You could, for example, force that the absolute value of
\mzninline{a} is bigger than \mzninline{b} using the constraint
\begin{mzn}
  constraint if b >= 0 then a > b else b < a endif;
\end{mzn}

In \minizinc\ the result of a \gls{conditional} expression is, however, not
contained to Boolean types. The condition in the expression, the ``if'', must be
of a Boolean type, but as long as the different sides of the \gls{conditional}
expression are the same type it is a valid conditional expression. This can be
used to, for example, define an absolute value function for integer
\gls{parameter}:

\begin{mzn}
  function int: abs(int: a) =
    if a >= 0 then a else -a endif;
\end{mzn}

When the condition does not contain any \glspl{variable}, then the flattening of
a \gls{conditional} expression will result in one of the side of the
expressions. If, however, the condition does contain a \gls{variable}, then the
result of the condition cannot be defined during the flattening. Instead, the
expression will introduce a new \gls{variable} for the result of the expression
and a constraint to enforce the functional relationship. In \minizinc\ special
\mzninline{if_then_else} \glspl{global} are available to implement this
relationship.

For the selection of an element from an \gls{array}, instead of between
different expressions, the \minizinc\ language uses an \gls{array} access syntax
similar to most other languages. The expression \mzninline{a[i]} selects the
element with index \mzninline{i} from the array \mzninline{a}. Note this is not
necessarily the \(\mzninline{i}^{\text{th}}\) element because \minizinc\ allows
modellers to provide a custom index set.

Like the previous expressions, the selector \mzninline{i} can be both a
\gls{parameter} or a \gls{variable}. If the expression is a \gls{variable}, then
the expression is flattened as being an \mzninline{element} function. Otherwise,
the flattening will replace the \gls{array} access expression by the element
referenced by expression.

\Gls{array} \glspl{comprehension} are expressions can be used to compose
\gls{array} objects. This allows modellers to create \glspl{array} that are not
given directly as input to the model or are a declared collection of \glspl{variable}.

\Gls{generator} expressions, \mzninline{[E | G where F]}, consist of three
parts:

\begin{description}
  \item[\mzninline{G}] The generator expression which assigns the values of
  collections to identifiers,
  \item[\mzninline{F}] an optional filtering condition, which has to evaluate to
    \mzninline{true} for the iteration to be included in the array,
  \item[\mzninline{E}] and the expression that is evaluation for each iteration
    when the filtering condition succeeds.
\end{description}

The following example composes an \gls{array} that contains the doubled even values of
an \gls{array} \mzninline{x}.

\begin{mzn}
  [ xi * 2 | xi in x where x mod 2 == 0]
\end{mzn}

The evaluated expression will be added to the new array. This means that the
type of the array will primarily depend on the type of the expression. However,
in recent versions of \minizinc\ both the collections over which we iterate and
the filtering condition could have a  \gls{variable} type. Since we then cannot
decide during flattening if an element is present in the array, the elements
will be made of a \gls{optional} type. This means that the solver still will
decide if the element is present in the array or if it takes a special
``absent'' value (\mzninline{<>}).

Finally, \glspl{let} are the primary scoping mechanism in the \minizinc\
language, together with function definitions. A \gls{let} allows a modeller to
provide a list of definitions, flattened in order, that can be used in its
resulting definition. There are three main purposes for \glspl{let}:

\begin{enumerate}
  \item To name an intermediate expression, so it can be used multiple times or
  to simplify the expression. For example, the constraint 

  \begin{mzn}
    constraint let { var int: tmp = x div 2; } in tmp mod 2 == 0 \/ tmp = 0;
  \end{mzn}

  constrains that half of \mzninline{x} is even or zero.

  \item To introduce a scoped \gls{variable}. For example, the constraint

  \begin{mzn}
    let {var -2..2: slack;} in x + slack = y;
  \end{mzn}

  constrains that \mzninline{x} and \mzninline{y} are at most two apart.

  \item To constrain the resulting expression. For example, the following function

  \begin{mzn}
    function var int: int_times(var int: x, var int: y) =
      let {
        var int: z;
        constraint pred_int_times(x, y, z);
      } in z;
  \end{mzn}

  returns a new \gls{variable} \mzninline{z} that is constrained to be the
  multiplication of \mzninline{x} and \mzninline{y} by the relational
  multiplication constraint \mzninline{pred_int_times}.
\end{enumerate}

An important detail in flattening \glspl{let} is that any \glspl{variable} that
are introduced might need to be renamed in the resulting solver level model.
Different from top-level definitions, the \glspl{variable} declared in
\glspl{let} can be flattened multiple times when used in loops, function
definitions (that are called multiple times), and \gls{array}
\glspl{comprehension}. In these cases the flattener must assign any
\glspl{variable} in the \gls{let} a new name and use this name in any subsequent
definitions and in the resulting expression.

\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 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 of what happens when an undefined expression 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 an
    expression contains undefined sub-expression, it will only be marked as
    undefined if the value of the sub-expression 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 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 semantics \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}

For version 2.5.5 of the \minizinc\ bundle, the \texttt{minizinc} executable is
officially provided tool to solve \minizinc\ instances. A modeller provides the
\texttt{minizinc} executable with a \minizinc\ model, the ground data required
to instantiate the model, and a \gls{solver} definition. Primarily the
\gls{solver} definition defines the \minizinc\ library used to flatten the
\minizinc\ instance and the way in which the \gls{solver} is to be executed. The
process of the \texttt{minizinc} executable can be categorised into the
following stages:

\begin{description}
  \item[Parsing] \texttt{minizinc} parses the input data, the \minizinc\ model,
  and the \gls{solver} library.
  \item[Type checking] \texttt{minizinc} ensures the type consistency of the
  \minizinc\ model, making sure that the types of all expressions match their
  declarations and is allowed in the locations where they are used. \\ In the
  process of type checking the model, all identifiers and calls are connected to
  the declaration that they refer to.
  \item[Flattening] \jip{TODO: This should have something}
  \item[Optimisation] Given the generated \flatzinc{} model, \texttt{minizinc}
  will try optimise this model to try and reduce the number of
  \glspl{constraint} and size of the \glspl{domain} of \glspl{variable} in the
  \flatzinc\ model.
  \item[Solving] The optimised \flatzinc\ model is given to the \gls{solver}.
  Any solutions found by the \gls{solver} are communicated back to the user.
\end{description}

\jip{TODO: Description of the flattening process}

\jip{TODO: Description of the techniques used during the optimisation phase}

\subsection{Propagation}
\label{sub:back-propagation}

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

\subsection{AMPL}%
\label{sub:back-ampl}

\subsection{OPL}%
\label{sub:back-}

\subsection{Essence}%
\label{sub:back-essence}

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

At the heart of the flattening process lies a \glsaccesslong{trs}. A \gls{trs}
\cite{baader-1998-term-rewriting} describes a computational model the full
process can be describe as the application of rules \(l \rightarrow r\), that
replace a \gls{term} \(l\) with another \gls{term} \(r\). A \gls{term} is an
expression with nested sub-expressions consisting of \emph{function} and
\emph{constant} symbols. An example of a term is \(F(0 + 1,F(1,0))\), where
\(F\) and \(+\) are function symbols and \(0\) and \(1\) are constant symbols.
In a term rewriting rule, a term can also contain a \emph{term variable} which
captures a term sub-expression. For example, the following \gls{trs} consists of
some (well-known) rules to handle logical and:

\begin{align*}
(r_{1}):& 0 \land x \rightarrow 0 \\
(r_{2}):& 1 \land x \rightarrow x \\
(r_{3}):& x \land y \rightarrow y \land x
\end{align*}

From these rules it follows that

\[ 1 \land 1 \land 0 \rightarrow^{r_{1}} 1 \land 0 \rightarrow^{r_{3}} 0 \land 1 \rightarrow^{r_{2}} 0 \]

Notice that there can be a choice between different rules. A \gls{trs} can be
non-deterministic. In the example we could also have applied the \(r_{1}\) twice
and arrived at the same result. Two important properties of \gls{trs} are,
therefore, \gls{termination} and \gls{confluence}. A \gls{trs} is said to be
terminating if, no-matter what order the term rewriting rules are applied, you
always arrive at a \gls{normal-form} (\ie, a term where no more rules apply). A
\gls{trs} is confluent if, no-matter what order the term rewriting rules are
applied, you always arrive at the same \gls{normal-form} (if you arrive at a
\gls{normal-form}).

It is trivial to see that our previous example is non-terminating, since you can
repeat rule \(r_{3}\) an infinite amount of times. The system, however, is
confluent as, if it arrives at the same \gls{normal-form}: if the term contains
any \(0\), then the result will be \(0\); otherwise, the result will be \(1\).

\subsection{Constraint Handling Rules}%
\label{sub:back-chr}



\subsection{ACD Term Rewriting}%
\label{subsec:back-acd}

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