%************************************************
\chapter{Background}\label{ch:background}
%************************************************

\input{chapters/2_background_preamble}

\glsresetall{}

\section{Introduction to Constraint Modelling Languages}
\label{sec:back-intro}

A goal shared between all programming languages is to provide a certain level of abstraction to their users.
This reduces the complexity of the programming tasks by hiding unnecessary information.
For example, an assembly language allows its user to abstract from the machine instructions and memory positions that are used by the hardware.
And, early imperative programming languages, like FORTRAN, were the first to offer abstraction from the processor architecture of the targeted system.
Consequently, in current times, writing a computer program requires little knowledge of how the targeted computer system operates.

\textcite{freuder-1997-holygrail} 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.
\Cmls{} operate differently from other computer languages.
The modeller does not describe how to solve a problem, but rather formalizes the requirements of the problem.
It could be said that a \cmodel{} actually describes the answer to the problem.

In a \cmodel{}, instead of specifying the manner in which we find a \gls{sol}, we give a concise description of the problem.
The elements of a \cmodel{} include \prbpars{}, what we already know; \variables{}, what we wish to know; and \constraints{}, the relations that should exist between them.
Through the variation of \prbpars{}, a \cmodel{} describes a full class of problems.
A specific problem is captured by an \instance{}, the combination of a \cmodel{} with a complete \gls{parameter-assignment} (\ie{} a mapping from all \prbpars{} to a value).

The type of problem described by a \cmodel{} is called a \gls{dec-prb}.
The goal of a \gls{dec-prb} is to find a \gls{sol}: a complete \gls{variable-assignment} that satisfies the \constraints{}.
Or, when this is not possible, prove that such an \gls{assignment} cannot exist.
Many \cmls{} also support the modelling of \glspl{opt-prb}, where a \gls{dec-prb} is augmented with an \gls{objective}.
In this case the goal is to find a \gls{sol} that satisfies all \constraints{} while minimizing (or maximizing) the value of the \gls{objective}.

Although a \cmodel{} does not contain any instructions on how to find a suitable \gls{sol}, \instances{} of \cmodels{} can generally be given to a dedicated \solver{}.
To solve these \instances{}, however, they can go through a \gls{rewriting} process to arrive at a \gls{slv-mod}, input accepted by a \solver{}.
The \solver{} then uses a dedicated algorithm that finds a \gls{sol} that fits the requirements of the \instance{}.
This process is visualized in \cref{fig:back-cml-flow}.

\begin{figure}
	\centering
	\includegraphics[width=\columnwidth]{assets/img/back_cml_flow}
	\caption{\label{fig:back-cml-flow}A flow diagram of typical process of a \cml{}.}
\end{figure}

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

	As an 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 enjoys playing with some toys more than others, we try to pick the toys that bring Audrey the most amount of joy, but still fit in the car.
	The following set of equations describes this ``knapsack'' problem as an \gls{opt-prb}:

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

	In these equations \(S\) is a set \variable{}.
	It represents the selection of toys to be packed for the trip.
	The \gls{objective} evaluates the quality of the \gls{sol} through the \variable{} \(z\).
	And, \(z\) is bound to the amount of joy that the selection of toys will bring.
	This is to be maximized.
	The \prbpars{} \(T\) is the set of all available toys.
	The \(joy\) and \(space\) functions are \prbpars{} used to map toys, \( t \in T\), to a numeric value describing the amount of enjoyment and space required respectively.
	Finally, the \prbpar{} \(C\) gives the numeric value of the total space that is left in the car before packing the toys.

	This \cmodel{} gives an abstract mathematical definition of the \gls{opt-prb} that would be easy to adjust to changes in the requirements.
	To solve \instances{} of this problem, however, the \instances{} have to be rewritten into input accepted by a \solver{}.
	\Cmls{} are designed to allow the modeller to express combinatorial problems similar to the above mathematical definition and generate input that can be directly used by dedicated \solvers{}.
\end{example}

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

\minizinc{} is a high-level, \solver{}- and data-independent \cml{} for discrete decision and \glspl{opt-prb} \autocite{nethercote-2007-minizinc}.
Its expressive language and extensive library of \glspl{global} allow users to easily model complex problems.

\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 \prbpars{}.
	\Lref{line:back:knap:toys} declares the enumerated type \mzninline{TOYS} that represents all possible toys, \(T\) in the mathematical model in \cref{ex:back-knapsack}.
	\Lrefrange{line:back:knap:joy}{line:back:knap:space} declare the \glspl{array} \mzninline{toy_joy} and \mzninline{toy_space}, that map toys to integer values.
	These represent the functional mappings \(joy\) and \(space\).
	Finally, \lref{line:back:knap:left} declares the integer \prbpar{} \mzninline{total_space} to represent the remaining car capacity, equivalent to \(C\).

	The model then declares its \variables{}.
	\Lref{line:back:knap:sel} declares the main \variable{} \mzninline{selection}, which represents the selection of toys to be packed.
	This is	\(S\) 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.

	The model then contains a \constraint{}, on \lref{line:back:knap:con}, which ensures we do not exceed the given capacity.
	Finally, it states the goal for the \solver{}: to maximize the value of the \variable{} \mzninline{total_joy}.
\end{example}

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

Note that, although more textual and explicit, the \minizinc{} model definition is very similar to our earlier mathematical definition.

A \minizinc{} model cannot be solved directly.
It first needs to be transformed into a \gls{slv-mod}: a list of \variables{} and \constraints{} that are directly supported as input by the \solver{}.
We call these types of \variables and \constraints{} \gls{native} to the \solver{}.

Given complete \gls{parameter-assignment}, a \minizinc{} model forms a \minizinc{} instance.
The process to transform a \minizinc{} instance into a \gls{slv-mod} is called \gls{rewriting}.
\Glspl{slv-mod} for \minizinc{} \solvers{} are generally represented in \flatzinc{}.
\flatzinc{} is a strict subset of \minizinc{} specifically chosen to represent \glspl{slv-mod}.
It is the primary way in which \minizinc{} communicates with \solvers{}.

\begin{example}
	For example, the \minizinc{} model in \cref{lst:back-mzn-knapsack} and the following \gls{assignment} can form a \minizinc{} \instance{}. 

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

	The modeller can then choose a \solver{}.
	Let us assume we choose a \gls{mip} \solver{}, whose \gls{native} \variables{} are only integer \variables{} and whose \gls{native} \constraints{} are only linear \constraints{}.
	\Gls{rewriting} the \instance{} would result in the \flatzinc{} \gls{slv-mod} in \cref{lst:back-fzn-knapsack}.
	The set type \variable{} \mzninline{selection} is now represented using three integer \variables{}, \mzninline{selection_@\(i\)@}.
	Each represent whether an element is present in the set.
	They take the value one if the element is present, and zero otherwise.
	The sum-\constraints{} have been transformed into integer linear \constraints{}, \mzninline{int_lin_le} and \mzninline{int_lin_eq}.
	The former constrains that the selection \variables{} multiplied by the space required for the represented element and ensures their sum is smaller than the available space.
	The latter calculates the value of the \mzninline{total_joy} \variable{} by adding together selection \variables{} multiplied by the joy value of the represented element.
	In these \constraints{}, the \prbpars{} are merely represented by the values given in the \gls{parameter-assignment}.
	Their names are not present in the \gls{slv-mod}.

	This \gls{slv-mod} is then passed to the targeted \solver{}.
	The \solver{} attempts to determine a complete \gls{variable-assignment} and maximize the \gls{assignment} of the \mzninline{total_joy} \variable{}.
	If such an \gls{assignment} does not exist, then it reports that the \gls{slv-mod} is \gls{unsat}.
\end{example}
\begin{listing}
	\mznfile{assets/listing/back_knapsack.fzn}
	\caption{\label{lst:back-fzn-knapsack} A \flatzinc{} \gls{slv-mod} for a \gls{mip} \solver{}, resulting from \gls{rewriting} \cref{lst:back-mzn-knapsack} with a given complete \gls{parameter-assignment}.}
\end{listing}


\subsection{Model Structure}%
\label{subsec:back-mzn-structure}

The structure of a \minizinc{} model can be described directly in terms of a \cmodel{}:

\begin{itemize}
	\item \variables{} and \prbpars{} are found in the form of variable declarations,
	\item \constraints{} are explicitly defined using their own keyword,
	\item and the \gls{objective} is included as a solving goal.
\end{itemize}

More complex models also include definitions for custom types, user-defined functions, and a custom output format.
These items are not constrained to occur in any particular order.
We briefly discuss the most important model items.
Note that these items already refer to \minizinc{} expressions, which will be discussed in \cref{subsec:back-mzn-expr}.
For a detailed overview of the structure of \minizinc{} models the full syntactic structure of \minizinc{} 2.5.5 can be consulted in \cref{ch:minizinc-grammar}.
\textcite{nethercote-2007-minizinc} offer a detailed discussion of \minizinc{}.
And, much of its history can be learned from the description of its predecessor, \zinc{} \autocite{marriott-2008-zinc}.

\paragraph{Variable Declaration Items} Variables are declared using variable declaration items.
The term ``variable'' has two overlapping, and slightly confusing meanings.
As a programming language, \minizinc uses it to describe a distinct object that contains (currently unknown) information.
As such, a variable in \minizinc{} can be used to represent either a \variable{}, as defined before, or a \gls{parameter} variable.
The latter is used to represent any information that will be known during \gls{rewriting}.
This includes \prbpars{}, but also the result of introspection, or the result of calculations over other \parameters{}.
In the remainder of this thesis we will refer to \parameter{} variables merely as \parameters{}, but will distinguish them from \variables{}.

Variable declarations are stated in the form \mzninline{@\(T\)@: @\(I\)@ = @\(E\)@}, where:
\begin{itemize}
	\item \(T\) is the type instance of the declared value,
	\item \(I\) is a new identifier used to reference the declared value,
	\item and the modeller can functionally define the value using an expression \(E\).
\end{itemize}
The syntax \mzninline{= @\(E\)@} is optional.
It is omitted when a \variable{} has is not functionally defined, or when a \parameter{} is a \prbpar{} and assigned externally.
The identifier used in a top-level variable declaration must be unique.
Two declarations with the same identifier result in an error during the \gls{rewriting} process.

The main types used in \minizinc{} are Boolean, integer, floating point numbers, sets of integers, and (user-defined) enumerated types.
The declaration of \parameters{} and \variables{} are distinguished through the instance of these types.
In the type instance of a \variable{}, the type is preceded by the \mzninline{var} keyword.
In the type instance of a \parameter{}, the type can similarly be marked by the \mzninline{par} keyword, but this is not required. 
\minizinc{} allows collections of these types to be contained in \glspl{array}.
Unlike other languages, \glspl{array} have a user-defined index set, which can start at any value, but they have to be a continuous range.
For example, the following declaration declares an array going from 5 to 10 of new Boolean \variables{}.

\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 constrain a declaration, not just to a certain type, but also to a set of values.
This set of values is generally referred to as the \gls{domain} of a \variable{}.
In \minizinc{} any expression that has a set type can be used as a type expression.
For example, the following two declarations declare two integer \variables{}.
The first takes the values from three to five and the second takes the values one, three, and five.

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

If a declaration includes a functional definition, then the identifier is merely a name given to the expression.
However, if the declaration also includes a type expression, then this places an implicit \constraint{} on the expression.
It forces the result of the expression to take a value according to the type expression.

\paragraph{Constraint Items} \Constraints{} in a \minizinc{} model are specified using the syntax: \mzninline{constraint @\(E\)@}.
A \constraint{} item contains only a single expression \(E\) of Boolean type.
The \gls{rewriting} of the \instance{} translates the expressions in \constraints{} into \constraints{} and potentially additional \variables{} that are \gls{native} to the \solver{}.
It is important that the \gls{native} expressions are \gls{eqsat}.
This means that the \constraints{} in the \solver{} are \gls{satisfied} if-and-only-if the original \constraint{} would have been \gls{satisfied}.

\paragraph{Solving Goal Item} A \minizinc{} model can contain a single solving goal item.
This item signals the \solver{} to perform one of three actions:

\begin{description}
	\item[\mzninline{solve satisfy}] to find an \gls{assignment} to the \variables{} that satisfies the \constraints{},
	\item[\mzninline{solve minimize @\(E\)@}] to find an \gls{assignment} to the \variables{} that satisfies the \constraints{} and minimizes the value of the expression \(E\), or
	\item[\mzninline{solve maximize @\(E\)@}] to similarly maximize the value of the expression \(E\).
\end{description}

\noindent{}The first type of goal indicates that the problem is a \gls{dec-prb}.
The other two types of goals are used when the model describes an \gls{opt-prb}.
If the model does not contain a goal item, then the problem is assumed to be a \gls{dec-prb}.

\paragraph{Function Items} Common structures in \minizinc\ are captured using function declarations.
A function is declared using the syntax \mzninline{function @\(T\)@: @\(I\)@(@\(P\)@) = @\(E\)@}, where:
\begin{itemize}
	\item \(T\) is the type of its result;
	\item \(I\) is its identifier;
	\item \(P\) is a list types and identifiers for its arguments;
	\item and \(E\) is the body of the function, an expression that can use the arguments \(P\).
\end{itemize}

\noindent{}\Gls{rewriting} replaces a call to a function by its body instantiated with the arguments given in the call.
The \minizinc{} language allows users to write the keywords \mzninline{predicate} as a shorthand \mzninline{function var bool} and \mzninline{test} as a shorthand for \mzninline{function bool}.

As an example, we can define a function that squares an integer as follows.

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

\Gls{rewriting} (eventually) transforms all \minizinc{} expressions into function calls.
As such, function declarations are the primary method for \solvers{} to specify how to rewrite a \minizinc{} model into a \gls{slv-mod}.
The collection of function declarations to rewrite for a \solver{} is generally referred to as a \solver{} library.
In this library, functions can be declared without a function body.
This marks them as \gls{native} to the \solver{}.
Calls to these functions are directly placed in the \gls{slv-mod}.
For non-\gls{native} functions, a \solver{} provides a function body that rewrites calls into (or towards) \gls{native} functions, a \gls{decomp}.
\Solver{} implementers are, however, not forced to provide a definition for all functions in \minizinc{}'s extensive library.
Instead, they can rely on a set of standard declarations, known as the standard library, that rewrite functions into a minimal subset of \gls{native} functions, known as the \flatzinc{} built-ins.

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

One of the powers of the \minizinc{} language is the extensive expression language.
It helps modellers create \cmodels{} that are intuitive to read, but are transformed to fit the structure best suited to the chosen \solver{}.
We now briefly discuss the most important types of expression in \minizinc{} and the possible methods employed when \gls{rewriting} them.

\paragraph{Global Constraints} It could be said that the basic building blocks of the \minizinc{} language are \glspl{global}.
These expressions capture common (complex) relations between \variables{}.
\Glspl{global} in the \minizinc{} language are used as function calls.
The following is an example of a \gls{global}.

\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 relation, with the following arguments:
\begin{itemize}
	\item \mzninline{w} are the weights of the items,
	\item \mzninline{p} are the profits for each item,
	\item the \variables{} in \mzninline{x} represent how many of each item are present in the knapsack,
	\item and \mzninline{W} and \mzninline{P}, respectively, represent the weight and profit of the knapsack
\end{itemize}

Note that using this \gls{global} as follows would have simplified the \minizinc{} model in \cref{ex:back-mzn-knapsack}.

\begin{mzn}
	var 0..total_space: space_used;
	constraint knapsack(toy_space, toy_joy, set2bool(selection), space_used, total_joy);
\end{mzn}

\noindent{}This has the additional benefit that the knapsack structure of the problem is then known.
The \constraint{} can be rewritten using a specialized \gls{decomp} provided by the \solver{}, or it can be marked as a \gls{native} \constraint{}.

Although \minizinc{} contains an extensive library of \glspl{global}, many problems contain structures that are not covered by a \gls{global}.
There are many other types of expressions in \minizinc{} that help modellers express complex \constraints{}.

\paragraph{Operators} When we express a mathematical formula, we generally do this through the use of \gls{operator} symbols. 
\minizinc{} includes \glspl{operator} for many mathematical, logic, and set operations.
Consider, for example, the following \constraint{}

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

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

\Gls{operator} symbols in \minizinc{} are a shorthand for \minizinc{} functions.
The \glspl{operator} in the above expression are evaluated using the addition, less-than comparison, and Boolean negation functions respectively.
Although the \gls{operator} syntax for \variables{} and \parameters{} is the same, different (overloaded) versions of these functions are used during \gls{rewriting}.
If the arguments to a function consist of only \parameters{}, then the result of the function can be computed directly.
However, \gls{rewriting} functions with \variable{} as arguments results in a new \variable{} that is constrained to the result of the function.

\paragraph{Conditional Expression} The choice between different expressions is often expressed using a \gls{conditional} expression, also known as an ``if-then-else'' expression.
It can, for example, force that the absolute value of \mzninline{a} is larger than \mzninline{b} using the \constraint{} as follows.

\begin{mzn}
	constraint if a >= 0 then a > b else b < a endif;
\end{mzn}

The result of a \gls{conditional} expression is not limited 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 \gls{conditional} expression.
This can be used to, for example, define an absolute value function an integer as follows.

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

When the condition does not contain any \variables{}, then the \gls{rewriting} process merely has to choose the correct side of the expression.
If, however, the condition does contain a \variable{}, then the result of the expression cannot be defined during \gls{rewriting}.
Instead, an \gls{avar} is created and a \constraint{} is added to ensure that it takes the correct value.
A special \mzninline{if_then_else} \glspl{global} is available to implement this relation.

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

The selection of an element from an \gls{array} is in many ways similar to the choice in a \gls{conditional} expression.
Like a \gls{conditional} expression, the selector \mzninline{i} can be both a \parameter{} or a \variable{}.
If the expression is a \gls{variable}, then the expression is rewritten to an \gls{avar} and an \mzninline{element} \constraint{}.
Otherwise, the \gls{rewriting} replaces the \gls{array} access expression by the chosen element of the \gls{array}.

\paragraph{Comprehensions} \Gls{array} \glspl{comprehension} are expressions which allows modellers to construct \gls{array} objects.
The generation of new \gls{array} structures allows modellers adjust, combine, filter, or order values from within the \minizinc{} model.

\Gls{comprehension} expressions \mzninline{[@\(E\)@ | @\(G\)@ where @\(F\)@]} consist of three parts:

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

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

\begin{mzn}
	[ xi * 3 | xi in x where x mod 2 == 0]
\end{mzn}

During \gls{rewriting}, the instantiation of the expression with current generator values is added to the new \gls{array}.
This means that the type of the \gls{array} primarily depends 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 \variable{} type.
Since we then cannot decide during \gls{rewriting} if an element is present in the \gls{array}, the elements are made of an \gls{optional} type.
This means that the \solver{} still will decide if the element is present in the \gls{array} or if it takes a special ``absent'' value (\mzninline{<>}).

\paragraph{Let Expressions} Together with function definitions, \glspl{let} are the primary scoping mechanism in the \minizinc{} language.
A \gls{let} allows a modeller to provide a list of declarations, that can only be used in the body of the \gls{let}.
Additionally, a \gls{let} can contain \constraints{} to constrain the declared values.
There are three main purposes for \glspl{let}.

\begin{enumerate}

	\item It can name an intermediate expression, for it to be used multiple times or to simplify the expression.
		For example, the following \constraint{} constrains that half of \mzninline{x} is even or takes the value one.

		\begin{mzn}
			constraint let { var int: tmp = x div 2; } in tmp mod 2 == 0 \/ tmp = 1;
		\end{mzn}

	\item It can introduce a scoped \variable{}.
	For example, the following \constraint{} constrains that \mzninline{x} and \mzninline{y} are at most two apart.

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


	\item It can constrain the resulting expression.
		For example, the following function returns an \gls{avar} \mzninline{z} that is constrained to be the multiplication of \mzninline{x} and \mzninline{y} by the relational multiplication \constraint{} \mzninline{pred_int_times}.

		\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}

\end{enumerate}

An important detail in \gls{rewriting} \glspl{let} is that any \variables{} that are introduced may need to be renamed in the resulting \gls{slv-mod}.
Different to \variables{} declared directly in declaration items, the \variables{} declared in \glspl{let} can be evaluated multiple times when used in loops, function definitions (that are called multiple times), and \gls{array} \glspl{comprehension}.
The \gls{rewriting} process must assign any \variables{} in the \gls{let} a new name and use this name in any subsequent definitions and in the resulting expression.

\paragraph{Annotations} Although \minizinc{} does not typically prescribe a way to find the \gls{sol} for an \instance{}, it provides the modeller with a way to give ``hints'' to the \solver{}.
It does this through the use of \glspl{annotation}.
Any item or expression can be annotated.
An annotation is indicated by the \mzninline{::} \gls{operator} followed by an identifier or function call.
The same syntax can be repeated to place multiple \glspl{annotation} on the same item or expression.

A common use of \glspl{annotation} in \minizinc{} is to provide a \gls{search-heuristic}.
Through the use of an \gls{annotation} on the solving goal item, the modeller can express an order in which they think values should be tried for an arrangement of \variables{} in the model.

\subsection{Reification}%
\label{subsec:back-reif}

With the rich expression language in \minizinc{}, \constraints{} can consist of complex expressions that cannot be rewritten into a single \constraint{} in the \gls{slv-mod}.
The sub-expressions of a complex expressions are often rewritten into \glspl{avar}. 
If the sub-expression, and therefore \gls{avar}, is of Boolean type, then it needs to be rewritten into a \gls{reified} \constraint{}.
The \gls{reif} of a \constraint{} \(c\) creates a Boolean \gls{avar} \mzninline{b}, also referred to as its \gls{cvar}, constrained to be the truth-value of this \constraint{}: \(\texttt{b} \leftrightarrow{} c\).

\begin{example}
	Consider the following \minizinc{} model.

	\begin{mzn}
		array[1..10] of var 1..15: x;
		constraint all_different(x);
		solve maximize sum(i in 1..10) (x[i] mod 2 == 0);
	\end{mzn}

	This model maximizes the number of even numbers taken by the elements of the \gls{array} \mzninline{x}.
	In this model the expression \mzninline{x[i] mod 2 == 0} has to be \gls{reified}.
	This means that for each \mzninline{i}, a \gls{cvar} \mzninline{b_i} is added, together with a constraint that makes \mzninline{b_i} true if-and-only-if \mzninline{x[i] mod 2 = 0}.
	We can then add up the values of all these \mzninline{b_i}, as required by the maximization.
	Since the elements have a domain from 1 to 15 and are constrained to take different values, not all elements of \mzninline{x} can take even values.
	Instead, the \solver{} is tasked to maximize the number of reified variables it sets to true.
\end{example}

When an expression occurs in a position where it can be directly enforced, we say it occurs in \rootc{} context.
Contrarily, an expression that occurs in non-\rootc{} is reified during the \gls{rewriting} process.
In \minizinc{}, almost all expressions can be used in both \rootc{} and non-\rootc{} contexts.

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

In this subsection we discuss the handling of \gls{partial} functions in \cmls{} as studied by \textcite{frisch-2009-undefinedness}.

When a function has a well-defined result for all its possible inputs it is said to be total.
Some expressions in \cmls{}, however, are rewritten into functions that do not have well-defined results for all possible inputs.
Part of the semantics of a \cml{} is the choice as to how to treat these \gls{partial} functions.

\begin{example}\label{ex:back-undef}
	Consider, for example, the following ``complex \constraint{}''.

	\begin{mzn}
		constraint i <= 4 -> a[i] * x >= 6;
	\end{mzn}

	\noindent{}It requires that if \mzninline{i} takes a value less or equal to four, then the value in the \texttt{i}\(^{th}\) position of \gls{array} \mzninline{a} multiplied by \mzninline{x} must be at least six.

	Suppose the \gls{array} \texttt{a} has index set \mzninline{1..5}, but \mzninline{i} takes the value seven.
	This means the expression \mzninline{a[i]} is undefined.
	If \minizinc{} did not implement any special handling for \gls{partial} functions, then the whole expression would have to be marked as undefined and a \gls{sol} cannot be found.
	However, intuitively if \mzninline{i = 7} the \constraint{} should be trivially true.
\end{example}

Other examples of \gls{partial} functions in \minizinc{} are:

\begin{itemize}
	\item division (or modulus) when the divisor is zero,

		\begin{mzn}
			x div 0 = @??@
		\end{mzn}

	\item finding the minimum or maximum of an empty set,

		\begin{mzn}
			min({}) = @??@
		\end{mzn}

	\item and computing the square root of a negative value.

		\begin{mzn}
			sqrt(-1) = @??@
		\end{mzn}

\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 are resolved, during \gls{rewriting} or by the \solver{}.

\textcite{frisch-2009-undefinedness} define three semantic models to deal with the undefinedness in \cmls:

\begin{description}

	\item[Strict] \Cmls{} employing a \gls{strict-sem} do not allow any undefined behaviour during the evaluation of the \cmodel{}.
		If during the \gls{rewriting} or solving process an expression is found to be undefined, then any expressions in which it is used is also marked as undefined.
		Consequently, this means that the occurrence of a single undefined expression causes the full \instance{} to be undefined.

	\item[Kleene] The \Gls{kleene-sem} treat undefined expressions as expressions for which not enough information is available.
		If an expression contains an undefined sub-expression, then it is only marked as undefined if the value of the sub-expression is required to compute its result.
		Take for example the expression \mzninline{false -> @\(E\)@}.
		When \(E\) is undefined the result of the expression is still be said to be true, since the value of \(E\) does not influence the result of the expression.
		However, if we take the expression \mzninline{true /\ @\(E\)@}, then when \(E\) is undefined the overall expression is also undefined since the value of the expression cannot be determined.

	\item[Relational] The \gls{rel-sem} follow from all expressions in a \cml{} eventually becoming part of a relational \constraint{}.
		So even though a (functional) expression does not have a well-defined result, we can still decide whether its surrounding relation holds.
		For example, the expression \mzninline{x div 0} is undefined, but the relation \mzninline{int_div(x, 0, y)} is said to be false.
		Values for \mzninline{x} and \mzninline{y} such that the relation holds do not exist.
		It can be said that the relational semantic makes the closest relational expression that contains an undefined expression false.

\end{description}

In practice, it is often natural to guard against undefined behaviour using Boolean logic.
\Gls{rel-sem} therefore often feel the most natural for the users of \glspl{cml}.
This is why \minizinc{} uses \gls{rel-sem} during its evaluation.

If we, for example, reconsider the \constraint{} from \cref{ex:back-undef}, we will find that \gls{rel-sem} will correspond to the intuitive expectation.
When \mzninline{i} takes the value seven, the expression \mzninline{a[i]} is undefined.
Its closest Boolean context will take the value false.
In this case, that means the right-hand side of the implication becomes false.
However, since the left-hand side of the implication is also false, the \constraint{} is \gls{satisfied}.

\textcite{frisch-2009-undefinedness} also show that different \solvers{} often employ different semantics.
\minizinc{} therefore implements the \gls{rel-sem} by translating any potentially undefined expression into an expression that is always defined, by introducing appropriate \glspl{avar} and \gls{reified} \constraints{}.
That way, the solving result is independent of the semantics of the chosen \solver{}, and always consistent with the \gls{rel-sem} of the model.

\section{Solving Constraint Models}%
\label{sec:back-solving}

There are many prominent techniques to solve a \constraint{} model, but none of them solve \minizinc{} instances directly.
Instead, a \minizinc{} instance is rewritten into a \gls{slv-mod} containing only \constraints{} and types of \variables{} that are \gls{native} to the \solver{}.

\minizinc{} was initially designed as an input language for \gls{cp} \solvers{}.
These \glspl{solver} often directly support many of the \glspl{global} in a \minizinc{} model.
For \gls{cp} solvers the amount of \gls{rewriting} required varies a lot.
It depends on which \constraints{} are \gls{native} to the targeted \gls{cp} \solver{} and which \constraints{} have to be decomposed.

In some ways \gls{cp} \solvers{} work on a similar level as the \minizinc{} language.
Therefore, some techniques used in \gls{cp} \solvers{} are also used during the \gls{rewriting} process.
The usage of these techniques shrinks the \domains{} of \variables{} and eliminates or simplifies \constraints{}.
\Cref{subsec:back-cp} explains the general method employed by \gls{cp} \solvers{} to solve a \cmodel{}.

Throughout the years \minizinc{} has started targeting \solvers{} using different approaches.
Although these \solvers{} allow new classes of problems to be solved using \minizinc{}, they also bring new challenges to the \gls{rewriting} process.
To understand these \gls{rewriting} challenges, the remainder of this section discusses the other dominant technologies used by \minizinc{} \solvers{} and their \glspl{slv-mod}.

\subsection{Constraint Programming}%
\label{subsec:back-cp}
\glsreset{cp}

When given an \instance{} of a \cmodel{}, one may wonder how to find a \gls{sol}.
The simplest algorithm would be to apply ``brute force'': try every value in the \domains{} of all \variables{}.
This is an inefficient approach.
Consider, for example, the following \constraint{}.

\begin{mzn}
	constraint a + b = 5;
\end{mzn}

It is clear that when the value \mzninline{a} is known, then the value of \mzninline{b} can be deduced.
Therefore, only that one value for \mzninline{b} has to be tried.
\gls{cp} is the idea of solving \glspl{dec-prb} by performing an intelligent search by inferring which values are still feasible for each \variable{} \autocite{rossi-2006-cp}.

A \gls{cp} \solver{} performs a depth first search.
Using a mechanism called \gls{propagation} the \solver{} removes values from \domains{} that are impossible.
\Gls{propagation} works through the use of \glspl{propagator}: algorithms dedicated to a specific \constraint{} that prune \domains{} when they contain values that are proven to be inconsistent.
This mechanism can be very efficient because a \gls{propagator} only has to be run again if the \domain{} of one of its \variables{} has changed.
If a \gls{propagator} can prove that it is always \gls{satisfied}, then it is subsumed: it never has to be run again.

In the best case scenario, when \gls{propagation} eliminates values, all \variables{} are \gls{fixed} to a single value.
In this case we have arrived at a \gls{sol}.
Often, \gls{propagation} alone is not enough to find a \gls{sol}.
When we reach a \gls{fixpoint}, where \gls{propagation} cannot reduce any \domains{}, and a \gls{sol} is not found, the \solver{} then has to make a search decision.
It fixes a \variable{} to a value or adds a new \constraint{}.
This search decision is an assumption made by the \solver{} in the hope of finding a \gls{sol}.
If a \gls{sol} is not found using the search decision, then it needs to try making the opposite decision which requires the exclusion of the chosen value or adding the opposite \constraint{}.

Note that there is an important difference between values excluded by \gls{propagation} and making search decisions.
Values excluded by propagation are guaranteed to not occur in any \gls{sol}, whereas, values excluded by a search heuristic are merely removed locally and can still be part of a \gls{sol}.
A \gls{cp} \solver{} is only able to prove that the \instance{} is \gls{unsat} by trying all possible search decisions.

\Gls{propagation} is not only used when starting the search, but also after making each search decision.
This means that some \gls{propagation} depends on the search decision.
Therefore, if the \solver{} needs to reconsider a search decision, then it must also undo all \domain{} changes that were caused by \gls{propagation} dependent on that search decision.

The most common method in \gls{cp} \solvers{} to keep track of \gls{domain} changes uses a \gls{trail} data structure \autocite{warren-1983-wam}.
Every \domain{} change that is made during \gls{propagation}, after the first search decision, is stored in a list.
Whenever a new search decision is made, the current position of the list is tagged.
If the \solver{} now needs to undo a search decision (\ie\ \gls{backtrack}), it reverses all changes until it reaches the change that is tagged with the search decision.
Because all changes before the tagged point on the \gls{trail} were made before the search decision was made, it is guaranteed that these \domain{} changes do not depend on the search decision.
Furthermore, because \gls{propagation} is performed to a \gls{fixpoint}, \gls{propagation} is never duplicated.

The solving method used by \gls{cp} \solvers{} is very flexible.
A \solver{} can support many types of \variables{}: they can range from Boolean, floating point numbers, and integers, to intervals, sets, and functions.
Similarly, \solvers{} do not all have implementations for the same \glspl{propagator}.
Therefore, a \gls{slv-mod} for one \solver{} looks very different from an \gls{eqsat} \gls{slv-mod} for a different \solver{}.
\Cmls{}, like \minizinc{}, serve as a standardized form of input for these \solvers{}.
They allow modellers to always use \glspl{global} and depending on the \solver{} they are either used directly, or they are automatically rewritten using a \gls{decomp}.

\begin{example}%
\label{ex:back-nqueens}
	As an example of the \gls{propagation} mechanism, let us consider the N-Queens problem.
	Given a chessboard of size \(n \times n\), find a placement for \(n\) queen chess pieces such that the queens cannot attack each other.
	This means we can only place one queen per row, one queen per column, and one queen per diagonal.
	The problem can be modelled in \minizinc{} as follows.

	\begin{mzn}
		int: n;
		set of int: ROW = 1..n;
		set of int: COL = 1..n;
		array [COL] of var ROW: q;

		constraint all_different(q);
		constraint all_different([q[i] + i | i in COL]);
		constraint all_different([q[i] - i | i in COL]);
	\end{mzn}

	The \variables{} in the \gls{array} \mzninline{q} decide for each column in which row the queen is placed.
	This automatically restricts every column to only have a single queen.
	As such, the queen will not be able to attack vertically.
	The \constraints{} model the remaining rules of the problem: two queens cannot be placed in the same row, two queens cannot be placed in the same upward diagonal, and two queens cannot be placed in the same downward diagonal.
	For many \gls{cp} \solvers{} this model is close to a \gls{slv-mod}, since integer \variables{} and an \mzninline{all_different} \gls{propagator} are common.

	When solving the problem, initially we cannot eliminate any values from the \glspl{domain} of the \variables{}.
	The first \gls{propagation} happens when the first queen is placed on the board, the first search decision.

	\Cref{fig:back-nqueens} visualizes the \gls{propagation} after placing a queen on the d3 square of an eight by eight chessboard.
	When the queen it placed on the board in \cref{sfig:back-nqueens-1}, it fixes the value of column 4 (d) to the value 3. This implicitly eliminates any possibility of placing another queen in the column.
	Fixing the \domain{} of the column triggers the \glspl{propagator} of the \mzninline{all_different} \constraints{}.
	As shown in \cref{sfig:back-nqueens-2}, the first \mzninline{all_different} \constraint{} now stops other queens from being placed in the same column.
	It eliminates the value 3 from the domains of the queens in the remaining columns.
	Similarly, the other \mzninline{all_different} \constraints{} remove all values that correspond to positions on the same diagonal as the placed queen, shown in \cref{sfig:back-nqueens-3,sfig:back-nqueens-4}.

	The \gls{propagation} after the first placed queen severely limits the positions where a second queen can be placed.
\end{example}

\begin{figure}
	\centering
	\begin{subfigure}[b]{.48\columnwidth}
		\centering
		\includegraphics{assets/img/back_chess1}
		\caption{\label{sfig:back-nqueens-1} Assign a queen to d3.}
	\end{subfigure}%
	\hspace{0.04\columnwidth}%
	\begin{subfigure}[b]{.48\columnwidth}
		\centering
		\includegraphics{assets/img/back_chess2}
		\caption{\label{sfig:back-nqueens-2} Propagate the rows.}
	\end{subfigure}
	\begin{subfigure}[b]{.48\columnwidth}
		\centering
		\includegraphics{assets/img/back_chess3}
		\caption{\label{sfig:back-nqueens-3} Propagate the upwards diagonal.}
	\end{subfigure}%
	\hspace{0.04\columnwidth}%
	\begin{subfigure}[b]{.48\columnwidth}
		\centering
		\includegraphics{assets/img/back_chess4}
		\caption{\label{sfig:back-nqueens-4} Propagate the downward diagonal.}
	\end{subfigure}
	\caption{\label{fig:back-nqueens} An example of domain propagation when a queen gets assigned in the N-Queens problem.}
\end{figure}

In \gls{cp} solving there is a trade-off between the amount of time spent propagating a \constraint{} and the amount of search that is otherwise required.
The gold standard for a \gls{propagator} is to be \gls{domain-con}. 
A \gls{domain-con} \gls{propagator} leaves only the values in a \domain{} for which there is least one possible \gls{variable-assignment} that satisfies its \constraint{}.
Designing such a \gls{propagator} is, however, not an easy task.
The algorithm can require high computational complexity.
Instead, it is sometimes better to use a propagator with a lower level of consistency.
Although it does not eliminate all possible values of the domain, searching the values that are not eliminated may take less time than achieving domain consistency.

This is, for example, the case for integer linear \constraints{} \[ \sum_{i} c_{i} x_{i} = d\] where \(c_{i}\) and \(d\) are integer \parameters{} and \(x_{i}\) are integer \variable{}.
For these \constraints{}, a realistic \gls{domain-con} \gls{propagator} cannot exist because the problem is \gls{np}-hard \autocite{choi-2006-fin-cons}.
A more feasible problem is to find the minimal and maximal values, or \gls{bounds}, for the \variables{} had they been rational numbers.
A \gls{boundsr-con} \gls{propagator} then ensures that the values in the \domain{} of the integer \variables{} are between their rational \gls{bounds}.
Note that this is a relaxation of calculating the integer \gls{bounds}, to create a \gls{boundsz-con} \gls{propagator}, which is still \gls{np}-hard.
We will see the same relaxation in mathematical programming, discussed in the next section.

Thus far, we have only considered finding \glspl{sol} for \glspl{dec-prb}.
\gls{cp} solving can, however, also be used to solve \glspl{opt-prb} using a method called \gls{bnb}.
The \gls{cp} \solver{} follows the same method as previously described.
However, when it finds a \gls{sol}, it does not yet know if this \gls{sol} is an \gls{opt-sol}.
It is merely an incumbent \gls{sol}.
The \solver{} must therefore resume its search, but it is not interested in just any \gls{sol}, only \glspl{sol} for which the \gls{objective} returns a better value.
This is achieved by adding a new \constraint{} that enforces a better objective value than the incumbent \gls{sol}.
If the search process finds another \gls{sol}, then the incumbent \gls{sol} is updated and the search process continues.
If the search process does not find any other \glspl{sol}, then it is proven that a better \gls{sol} than the current incumbent \gls{sol} cannot exist.
It is an \gls{opt-sol}.

\gls{cp} solvers like \gls{chuffed} \autocite{chu-2011-chuffed}, Choco \autocite{prudhomme-2016-choco}, \gls{gecode} \autocite{schulte-2019-gecode}, and OR-Tools \autocite{perron-2021-ortools} have long been one of the leading methods to solve \minizinc\ instances.

\subsection{Mathematical Programming}%
\label{subsec:back-mip}
\glsreset{lp}
\glsreset{mip}

Mathematical programming \solvers{} are the most prominent solving technique employed in \gls{or} \autocite{schrijver-1998-mip}.
At its foundation lies \gls{lp}.
A linear program describes a problem using \constraints{} in the form of linear (in-)equations over continuous \variables{}.
In general, a linear program can be expressed in the following form.

\begin{align*}
	\text{maximize} \hspace{2em}   & \sum_{j=1}^{V} c_{j} x_{j}                        &                   \\
	\text{subject to} \hspace{2em} & l_{i} \leq \sum_{j=0}^{V} a_{ij} x_{j} \leq u_{i} & \forall_{1 \leq{} i \leq{} C} \\
																 & x_{j} \in \mathbb{R}                              & \forall_{1 \leq{} j \leq{} V}
\end{align*}

In this definition \(V\) and \(C\) represent the number of \variables{} and number of \constraints{} respectively.
The vector \(c\) holds the coefficients of the objective function and the matrix \(a\) holds the coefficients for the \constraints{}.
The vectors \(l\) and \(u\) respectively contain the lower and upper \gls{bounds} of the \constraints{}.
Finally, the \variables{} of the linear program are held in the \(x\) vector.

For problems that are in the form of a linear program, there are proven methods to find an \gls{opt-sol}.
One such method, the simplex method, was first conceived by \textcite{dantzig-1955-simplex} after the second world war. 
It finds the optimal solution of a linear program in worst-case exponential time.
It was questioned whether the same problem could be solved in worst-case polynomial time, until \textcite{karmarkar-1984-interior-point} proved this possible through the use of interior point methods.

Methods for solving linear programs provide the foundation for a harder problem.
In \gls{lp} our \variables{} must be continuous.
If we require that one or more take an integer value (\(x_{i} \in \mathbb{Z}\)), then the problem becomes \gls{np} hard.
The problem is referred to as \gls{mip} (or Integer Programming if \textbf{all} \variables{} must take an integer value).

Unlike \gls{lp}, there is not an algorithm that solves a mixed integer program in polynomial time.
We can, however, adapt \gls{lp} solving methods to solve a mixed integer program.
We do this by treating the mixed integer program as a linear program and find an \gls{opt-sol}.
If the \variables{} happen to take integer values in the \gls{sol}, then we have found an \gls{opt-sol} to the mixed integer program.
Otherwise, we pick one of the \variables{} that needs to be integer but whose value in the \gls{sol} of the linear program is not.
For this \variable{} we create two versions of the linear program: a version where this \variable{} is constrained to be less or equal to the value in the \gls{sol} rounded down to the nearest integer value; and a version where it is constrained to be greater or equal to the value in the \gls{sol} rounded up.
Both versions are solved to find the best \gls{sol}.
The process is repeated recursively until an integer \gls{sol} is found.

Much of the power of this solving method comes from \gls{bounds} that are inferred during the process.
The \gls{sol} to the linear program provides an upper bound for the solution in the current step of the solving process.
Similarly, any integer \gls{sol} found in an earlier branch of the search process provides a lower bound.
When the upper bound given by the linear program is lower that the lower bound from an earlier solution, then we know that any integer \gls{sol} following from the linear program is strictly worse than the incumbent.

Over the years \gls{lp} and \gls{mip} \solvers{} have developed immensely.
Modern \solvers{}, such as \gls{cbc} \autocite{forrest-2020-cbc}, \gls{cplex} \autocite{ibm-2020-cplex}, \gls{gurobi} \autocite{gurobi-2021-gurobi}, and \gls{scip} \autocite{gamrath-2020-scip}, are routinely used to solve very large industrial optimization problems.
These \solvers{} are therefore prime targets to solve \minizinc{} \instances{}.

To solve an \instance{} of a \cmodel{}, it can be rewritten into a mixed integer program.
This process is known as \gls{linearization}.
It does, however, come with its challenges.
Many \minizinc{} models contain \constraint{} that are not linear (in-)equations.
The translation of a single such \constraint{} can introduce many linear \constraints{} and \glspl{avar}.
For example, when a \constraint{} reasons about the value that a variable takes, the \gls{linearization} process introduces indicator variables.
An indicator variable \(y_{i}\) is a \gls{avar} that for a \variable{} \(x\) take the value 1 if \(x = i\) and 0 otherwise.
\Constraints{} reasoning about the value of \(x\) are then rewritten as linear \constraints{} using the \variables{} \(y_{i}\).

\begin{example}

	Let us again consider the N-Queens problem from \cref{ex:back-nqueens}.
	The following model shows an integer program of this problem.

	\begin{align}
		\text{given} \hspace{2em}      & N = {1,\ldots,n}                         &                       \notag{}\\
		\text{maximize} \hspace{2em}   & 0                                        &                       \notag{}\\
		\text{subject to} \hspace{2em} & q_{i} \in N                              & \forall_{i \in N}     \notag{}\\
																	 & y_{ij} \in \{0,1\}                       & \forall_{i,j \in N}   \notag{}\\
		\label{line:back-mip-channel}  & x_{i} = \sum_{j \in N} j * y_{ij}        & \forall_{i \in N}     \\
		\label{line:back-mip-row}      & \sum_{i \in N} y_{ij} \leq 1             & \forall_{j \in N}     \\
		\label{line:back-mip-diag1}    & \sum_{i,j \in N: i + j =k} y_{ij} \leq 1 & \forall_{3 \leq{} k \leq{} 2n-1}  \\
		\label{line:back-mip-diag2}    & \sum_{i,j \in N: i - j =k} y_{ij} \leq 1 & \forall_{-n+2 \leq{} k \leq{} n-2}
	\end{align}

	As we can see, this \cmodel{} only uses integer \variables{} and linear \constraints{}.
	Like the \minizinc{} model, \variables{} \(q\) are used to represent where the queen is located in each column.
	To encode the \mzninline{all_different} \constraints{}, the indicator variables \(y\) are inserted to reason about the value of \(q\).
	The \cref{line:back-mip-channel} is used to connect the \(q\) and \(y\) \variables{} and make sure that their values correspond.
	\Cref{line:back-mip-row} ensures that only one queen is placed in the same column.
	Finally, \cref{line:back-mip-diag1,line:back-mip-diag2} constrain all diagonals to contain only one queen.

\end{example}

\subsection{Boolean Satisfiability}%
\label{subsec:back-sat}
\glsreset{sat}
\glsreset{maxsat}

The study of the \gls{sat} problem is one of the oldest in computer science.
The DPLL algorithm that is still the basis for modern \gls{sat} solving stems from the 1960s \autocite{davis-1960-dpll, davis-1962-dpll}, and \gls{sat} was the first problem to be proven to be \gls{np}-complete \autocite{cook-1971-sat}.
The problem asks if there is an \gls{assignment} for the \variables{} of a given Boolean formula, such that the formula is \gls{satisfied}.
This problem is a restriction of the general \gls{dec-prb} where all \variables{} have a Boolean type and all \constraints{} are simple Boolean formulas.

There is a field of research dedicated to solving \gls{sat} problems \autocite{biere-2021-sat}.
In this field a \gls{sat} problem is generally standardized to be in \gls{cnf}.
A \gls{cnf} is formulated in terms of literals.
These are Boolean \variables{} \(x\) or their negations \(\neg x\).
These literals are then used in a conjunction of disjunctive clauses: a Boolean formula in the form \(\forall \exists b_{i}\).
To solve the \gls{sat} problem, the \solver{} has to find an \gls{assignment} for the \variables{} where at least one literal is true in every clause.

Even though the problem is proven to be hard to solve, much progress has been made towards solving even very complex \gls{sat} problems.
Modern day \gls{sat} solvers, like Clasp \autocite{gebser-2012-clasp}, Kissat \autocite{biere-2021-kissat} and MiniSAT \autocite{een-2003-minisat}, solve instances of the problem with thousands of \variables{} and clauses.

Many real world problems modelled using \cmls{} directly correspond to \gls{sat}.
However, even problems that contain \variables{} with types other than Boolean can still be encoded as a \gls{sat} problem.
This process is known as \gls{booleanization}.
Depending on the problem, using a \gls{sat} \solvers{} to solve the encoded problem can still be the most efficient way to solve the problem.

\begin{example}
	Let us once more consider the N-Queens problem presented in \cref{ex:back-nqueens}.
	A possible \gls{sat} encoding for this problem is the following.

	\begin{align}
		\text{given} \hspace{2em}                                    & N = {1,\ldots,n}                         &                                                    \notag{}\\
		\text{find} \hspace{2em}                                     & q_{ij} \in \{\text{true}, \text{false}\} & \forall_{i,j \in N}                                \notag{}\\
		\label{line:back-sat-at-least}\text{subject to} \hspace{2em} & \exists_{j \in N} q_{ij}                 & \forall_{i \in N}                                  \\
		\label{line:back-sat-row}                                    & \neg q_{ij} \lor \neg q_{ik}             & \forall_{i,j \in N} \forall_{j \leq{} k \leq{} n}              \\
		\label{line:back-sat-col}                                    & \neg q_{ij} \lor \neg q_{kj}             & \forall_{i,j \in N}  \forall_{i \leq{} k \leq{} n}             \\
		\label{line:back-sat-diag1}                                  & \neg q_{ij} \lor \neg q_{(i+k)(j+k)}     & \forall_{i,j \in N} \forall_{1 \leq{} k \leq{} \min(n-i, n-j)} \\
		\label{line:back-sat-diag2}                                  & \neg q_{ij} \lor \neg q_{(i+k)(j-k)}     & \forall_{i,j \in N} \forall_{1 \leq{} k \leq{} \min(n-i, j)}
	\end{align}

	The encoding of the problem uses a Boolean \variable{} for every position of the chessboard.
	Each \variable{} represents if a queen is located on this position or not.
	\Cref{line:back-sat-at-least} forces that a queen is placed on every row of the chessboard.
	\Cref{line:back-sat-row,line:back-sat-col} ensure that only one queens is placed in each row and column respectively.
	\Cref{line:back-sat-diag1,line:back-sat-diag2} similarly constrain each diagonal to contain only one queen.
\end{example}

A variation on \gls{sat} is the \gls{maxsat} problem.
In a \gls{sat} problem all clauses need to be \gls{satisfied}, but this is not the case in a \gls{maxsat} problem.
Instead, clauses are given individual weights.
The higher the weight, the more important the clause is for the overall problem.
The goal in the \gls{maxsat} problem is then to find an \gls{assignment} for Boolean \variables{} that maximizes the cumulative weights of the \gls{satisfied} clauses.

The \gls{maxsat} problem is analogous to an \gls{opt-prb}.
Like an \gls{opt-prb}, the aim of \gls{maxsat} is to find the \gls{opt-sol} to the problem.
The difference is that the weights are given to the \constraints{} instead of the \variables{} or a function over them.
It is, again, possible to rewrite a \cmodel{} with an \gls{objective} as a \gls{maxsat} problem.
A naive approach to encode an integer objective is, for example, to encode each possible result of the \gls{objective} as a Boolean \variable{}.
This Boolean \variable{} then forms a singleton clause with the result value as its weight.

For many problems the use of \gls{maxsat} \solvers{}, such as \gls{openwbo} \autocite{martins-2014-openwbo} and RC2 \autocite{ignatiev-2019-rc2}, offers a very successful method to find an \gls{opt-sol} to a problem.

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

Although \minizinc{} is the \cml{} that is the primary focus of this thesis, there are many other \cmls{}.
Each \cml{} has its own focus and purpose and comes with its own strengths and weaknesses.
Most of the techniques that are discussed in this thesis may be adapted to these languages.

We now discuss some prominent \cmls{} and compare them to \minizinc{} to indicate to the reader where our techniques will need to be adjusted to fit other languages.

A notable difference between all these languages and \minizinc{} is that only \minizinc{} allows modellers to extend the language using their own (user-defined) functions.
In other \cmls{} the modeller is restricted to the expressions and functions provided by the language.

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

One of the most used \cmls\ is \gls{ampl} \autocite{fourer-2003-ampl}.
As the name suggest, \gls{ampl} was designed to allow modellers to express problems through the use of mathematical equations.
It is therefore also described as an ``algebraic modelling language''.
Specifically \gls{ampl} was designed to model linear programs.
These days \gls{ampl} has been extended to allow more advanced \solver{} usage.
Depending on the \gls{solver} targeted by \gls{ampl}, the language gives the modeller access to additional functionality.
Different types of \solvers{} can also have access to different types of \constraints{}, such as quadratic and non-linear \constraints{}.
\gls{ampl} has even been extended to allow certain \glspl{global} when using a \gls{cp} \solver{} \autocite{fourer-2002-amplcp}.

\begin{example}

	Let us consider modelling in \gls{ampl} using the well-known \gls{tsp}.
	In the \gls{tsp}, we are given a list of cities and the distances between all cities.
	The goal of the problem is to find a shortest path that visits all the cities exactly once and returns to its origin.
	A possible \cmodel{} for this problem in \gls{ampl} is shown in \cref{lst:back-ampl-tsp}.

	\begin{listing}
		\amplfile[l]{assets/listing/back_tsp.mod}
		\caption{\label{lst:back-ampl-tsp} An \gls{ampl} model describing the \gls{tsp}.}
	\end{listing}

	\Lrefrange{line:back:ampl:parstart}{line:back:ampl:parend} declare the \parameters{} of the \cmodel{}.
	The \gls{ampl} model requires a set of names of cities, \texttt{Cities}, as input.
	From these cities it declares a set \texttt{Paths} that contains all possible paths between the different cities.
	Note how its definition uses a \gls{comprehension} to eliminate symmetric paths.
	For each possible path the model also requires its cost.

	On \lref{line:back:ampl:var} we find the declaration for the \variables{} of the model.
	For each possible path it decides whether it is used or not.
	The \constraint{} on \lref{line:back:ampl:con1} ensures that for each city, one possible path to the city and one possible path from the city is taken.

	Crucially, this does not yet enforce that the taken variables represent a single path.
	It is still possible for so-called subtours to exist, multiple unconnected paths that together span all the cities.
	\Lrefrange{line:back:ampl:compstart}{line:back:ampl:compend} introduce the classic Miller--Tucker--Zemlin method \autocite{miller-1960-subtour} to the model to eliminate these subtours.
	For each city, this method introduces a variable that represent the order of the cities in the path.
	The \constraint{} forces that if a path is taken from \texttt{i} to \texttt{j}, then \texttt{Order[i] < Order[j]}.
	As such, each \texttt{Order} \variable{} must take a unique value and all cities are on the same path.
	To remove symmetries in the model we set the \texttt{Order} \variable{} of the first city in the set to be one.

	Finally, \lref{line:back:ampl:goal} sets the goal of the \gls{ampl} model: to minimize the cost of all the paths taken.

	This model shows that the \gls{ampl} syntax has many features similar to \minizinc{}.
	Like \minizinc{}, \gls{ampl} has an extensive expression language, which includes \gls{generator} expressions and a large collection of \glspl{operator}.
	The building blocks of the model are also similar: \parameter{} declarations, \variable{} declarations, \constraints{}, and a solving goal.

	\begin{listing}
		\mznfile{assets/listing/back_tsp.mzn}
		\caption{\label{lst:back-mzn-tsp} An \minizinc{} model describing the \gls{tsp}.}
	\end{listing}

	A \minizinc{} model for the same problem is shown in \cref{lst:back-mzn-tsp}.
	Even though \gls{ampl} has a similar syntax to \minizinc{}, the models could not be more different.
	The main reason for this difference is the level at which these models are written.

	The \gls{ampl} model is written to target a \gls{mip} solver.
	In the \gls{ampl} language this means that the modeller is required to use the language functionality that is compatible with the targeted \solver{}; in this case, all expression have to be linear (in-)equations.

	In \minizinc{} the modeller is not constrained in the same way.
	It can use the viewpoint of choosing, from each city, to which city to go next.
	A \mzninline{circuit} \gls{global} is then used to enforce that these decisions form a single path over all cities.
	When targeting a \gls{mip} \solver{}, the \gls{decomp} of the \mzninline{circuit} \constraint{} will use a similar method to Miller--Tucker--Zemlin.

	In \minizinc{}, the modeller is always encouraged to create high-level \cmodels{}.
	\minizinc{} then rewrites these models into compatible \glspl{slv-mod}.
\end{example}

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

\gls{opl} is a \cml{} that aims to combine the strengths of mathematical programming languages like \gls{ampl} with the strengths of \gls{cp} \autocite{van-hentenryck-1999-opl}.
The syntax of \gls{opl} is very similar to the \minizinc{} syntax.

Where \gls{opl} really shines is when modelling scheduling problems.
Resources and activities are separate objects in \gls{opl}.
This allows users to express resource scheduling \constraints{} in an incremental and more natural fashion.
When solving a scheduling problem, \gls{opl} makes use of specialized interval \variables{}, which represent when a task is scheduled.

\begin{example}

	Let us consider modelling in \gls{opl} using the well-known ``job shop'' problem.
	The job shop problem is similar to the open shop problem discussed in the introduction.
	Like the open shop problem, the goal is to schedule jobs with multiple tasks.
	Each task must be performed by an assigned machine.
	A machine can only perform one task at the same time.
	And, only one task of the same job can be scheduled at the same time.
	However, in the job shop problem, the tasks within a job also have a specified order.
	Abstracting from the \parameter{} declarations, the possible formulation of the \variable{} declarations and \constraints{} for the job shop problem in \gls{opl} is shown in \cref{lst:back-opl-jsp}.

	\begin{listing}
		\plainfile[l]{assets/listing/back_jsp.mod}
		\caption{\label{lst:back-opl-jsp} An \gls{opl} model describing the job shop problem, abstracting from \parameter{} declarations.}
	\end{listing}

	\Lref{line:back:opl:task} declares the \texttt{task} \variables{}, the main \variables{} in model.
	These \variables{} have the type \texttt{Activity}, a special type used to declare any scheduling event with a start and end time.
	\Variables{} of this type are influenced by the \texttt{ScheduleHorizon} \parameter{} defined on \lref{line:back:opl:horizon} of the model.
	This \parameter{} restricts the time span in which all activities in the \cmodel{} must be scheduled.
	On \lref{line:back:opl:con2}, we enforce the order between the different tasks for the same job.
	The \constraint{} uses the \texttt{precedes} operator to enforce that one activity takes places before another.

	Another activity is declared on \lref{line:back:opl:makespan}.
	The \texttt{makespan} \variable{} represents the time spanned by all tasks.
	This is enforced by the \constraint{} on \lref{line:back:opl:con1}.
	\Lref{line:back:opl:goal} sets the minimization of \texttt{makespan} to be the goal of the model.

	Resources are important notions in \gls{opl}.
	A resource can be any requirement of an activity.
	On \lref{line:back:opl:resources} a \texttt{UnaryResource} is declared for every machine. 
	A \texttt{UnaryResource} can be used by at most one activity at a time.
	The \constraint{} on \lref{line:back:opl:con3} ensures that at most one task activity can use same machine at the same time.
	The \constraint{} uses the \texttt{requires} operator to bind an activity to a resource.

	\begin{listing}
		\mznfile{assets/listing/back_jsp.mzn}
		\caption{\label{lst:back-mzn-jsp} An \minizinc{} model describing the job shop problem, abstracting from \parameter{} declarations.}
	\end{listing}

	A fragment of a \minizinc{} model, modelling the same parts of the job shop problem, is shown in \cref{lst:back-mzn-jsp}.
	Notably, \minizinc{} does not have explicit activity \variables{}.
	It instead uses integer \variables{} that represent the start times of the tasks and the end time for the \mzninline{makespan} activity that spans all tasks.
	This means that much of the implicit behaviour of the \texttt{Activity} \variables{} has to be defined explicitly.
	Where in the \gls{opl} model we could just state a global scheduling horizon, in \minizinc{} it has to be explicitly included in the \domains{} of the time \variables{}.
	And, instead of a \texttt{precedes} operator, we have to explicitly enforce the precedence of tasks using linear \constraints{}.

	\Gls{opl} model also has the advantage of its resource syntax.
	It first states the resource objects and then merely has to use the \texttt{requires} keyword to force tasks on the same machine to be mutually exclusive.
	In \minizinc{} the same requirement is implemented through the use of \mzninline{disjunctive} \constraints{}.
	Although this has the same effect, all mutually exclusive jobs have to be combined in a single statement in the model.
	This makes it harder in \minizinc{} to write the correct \constraint{} and its meaning is less clear.
\end{example}

The \gls{opl} also contains a specialized search syntax that is used to instruct its solvers \autocite{van-hentenryck-2000-opl-search}.
This syntax gives modellers full programmatic control over how the solver explores the search space depending on the current state of the variables.
This gives the modeller more control over the search in comparison to the \gls{search-heuristic} \glspl{annotation} in \minizinc{}, which only allow modellers to select predefined \glspl{search-heuristic} already implemented in the solver.
Take, for example, the following \gls{opl} search definition:

\begin{plain}
	search {
		try x < y | y >= x endtry;
	}
\end{plain}

This search strategy ensures that we first try to find a \gls{sol} where the \variable{} \mzninline{x} takes a value smaller than \mzninline{y}.
If it does not find a \gls{sol}, then it tries finding a \gls{sol} where the opposite is true.
This search specification, like many others imaginable, cannot be enforced using \minizinc{}'s \gls{search-heuristic} \glspl{annotation}.

To support \gls{opl}'s dedicated search language, the language is tightly integrated with its supported \solvers{}.
Its search syntax requires that the \gls{opl} process directly interacts with the \solver{}'s internal search mechanism and that the \solver{} reasons about search on the same level as the \gls{opl} model.
It is therefore not always possible to connect other \solvers{} to \gls{opl}.

While this advanced search language is an interesting construct, it has been removed from the current version of \gls{opl} \autocite{ibm-2017-opl}.
Instead, \gls{opl} now offers the use of ``search phases'', which function similarly to \minizinc{}'s \gls{search-heuristic} \glspl{annotation}.

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

\gls{essence} \autocite{frisch-2007-essence} is another high-level \cml{}.
It is notable for its adoption of high-level \variable{} types.
In addition to all variable types that are supported by \minizinc{}, \gls{essence} also contains:

\begin{itemize}
	\item Finite sets of non-integer types,
	\item finite multi-sets of any type,
	\item finite (\gls{partial}) functions,
	\item and (regular) partitions of finite types.
\end{itemize}

Since sets, multi-sets, and functions can be defined on any other type, these types can be arbitrarily nested, the modeller could, for example, define a \variable{} that is a set of sets of integers.
Partitions are defined for finite types: Booleans, enumerated types, or a restricted set of integers.

\begin{example}

	Let us consider modelling in \gls{essence} using the well-known ``social golfer'' problem.
	In the social golfer problem, a community of golfers plans games of golf for a set number of weeks.
	Every week all the golfers are split into groups of equal size and each group plays a game of golf.
	The goal of the problem is to find a way to split the groups differently every week, such that two golfers will not meet each other twice.
	Two golfers can only be part of the same group once.
	A \cmodel{} for the social golfer problem in \gls{essence} can be seen in \cref{lst:back-essence-sgp}.

	\begin{listing}
		\plainfile[l]{assets/listing/back_sgp.essence}
		\caption{\label{lst:back-essence-sgp} An \gls{essence} model describing the social golfer problem}
	\end{listing}

	The \gls{essence} model starts with the preamble declaring the version of the language that is used.
	All the \parameters{} of the \cmodel{} are declared on \lref{line:back:essence:pars}: \texttt{weeks} is the number of weeks to be considered, \texttt{groups} is the number of groups of golfers, and \texttt{size} is the size of each group.
	The input for the \parameters{} is checked to ensure that they take a value that is one or higher.
	\Lref{line:back:essence:ntype} then uses the \parameters{} to declare a new type to represent the golfers.

	Most of the problem is modelled on \lref{line:back:essence:var}.
	It declares a \variable{} that is a set of partitions of the golfers.
	The choice in \variable already contains some implied \constraints{}.
	Because the \variable{} reasons about partitions of the golfers, a correct \gls{assignment} is already guaranteed to correctly split the golfers into groups.
	The usage of a set of size \texttt{weeks} means that we directly reason about that number of partitions that have to be unique.

	The type of the \variable{} does, however, not guarantee that two golfers will not meet each other twice.
	Therefore, a \constraint{} that enforces this is found on \lref{line:back:essence:con}.
	Notably, the \texttt{together} function tests whether two elements are in the same part of a partition.

	A \minizinc{} model for the same problem can be found in \cref{lst:back-mzn-sgp}.
	It starts similarly to the \gls{essence} model, with the declaration of the \parameters{} and a type for the golfers.
	The differences start from the declaration of the \variables{}.
	The \minizinc{} model is unable to use a set of partitions and instead uses an \gls{array} of sets.
	Each set represents a single group in a single week.
	Note that, through the \gls{essence} type system, the first two \constraints{} in the \minizinc{} model are implied in the \gls{essence} model.
	This is an example where the use of high-level types helps the modeller create more concise models.
	Apart from syntax and the \variable{} viewpoint, the \constraint{} that enforces that golfers only occur in the same group once is identical.

\end{example}

\begin{listing}
	\mznfile{assets/listing/back_sgp.mzn}
	\caption{\label{lst:back-mzn-sgp} A \minizinc{} model describing the social golfers problem}
\end{listing}

\Gls{essence} allows the use of many high-level \variable{} types and \constraints{} on these \variables{}.
Since these types of \variables{} are often not \gls{native} to the \solver{}, an extensive \gls{rewriting} process is required to arrive at a \gls{slv-mod}.
Different from the other two languages presented in this section, the implementation of the \gls{essence} toolchain is available open source and has been the subject of published research.
The \gls{rewriting} process of \gls{essence} is split into two steps.
First, an \gls{essence} model is transformed into \gls{ess-prime}.
Then, an \gls{ess-prime} model forms an \instance{}, and is subsequently rewritten into a \gls{slv-mod}.

Compared to \gls{essence}, the \gls{ess-prime} language does not contain the same high-level \variables{}.
As such, the main task of Conjure, the \compiler{} from \gls{essence} to \gls{ess-prime}, is to decide on a representation of these \variables{} in terms of integer and Boolean \variables{} \autocite{akgun-2014-essence}.
However, there are often multiple ways to represent a high-level \variable{}, or how to enforce its implicit \constraints{}.
Although the modeller is able to decide on representation, Conjure has been extended to automatically select among the models it produces \autocite{akgun-2013-auto-ess}.

\paragraph{Essence'}

Once a \cmodel{} is turned into \gls{ess-prime} it is at a very similar level to a \minizinc{} model.
This can be illustrated using the N-Queens problem introduced in \cref{ex:back-nqueens}.
The same problem is modelled in \gls{ess-prime} is shown in \cref{lst:back-ep-nqueens}.
Apart from the syntax used, both languages use the exact same concepts to model the problem.

\begin{listing}
	\plainfile{assets/listing/back_nqueens.eprime}
	\caption{\label{lst:back-ep-nqueens} A \gls{ess-prime} model describing the N-Queens problem.}
\end{listing}

An \instance{} of a \gls{ess-prime} model can be rewritten by Saville Row into a \gls{slv-mod} for a variety of \solvers{}, including \gls{cp}, \gls{sat}, and \gls{maxsat} \solvers{}.
Savile Row \autocite{nightingale-2017-ess-prime}, and its predecessor Tailor \autocite{rendl-2010-thesis}, have often stood at the forefront of \gls{rewriting} \cmodels{}.
As such, at present many simplification techniques used during \gls{rewriting} are shared between \gls{ess-prime} and \minizinc{}.
At its core, however, the \gls{rewriting} of \gls{ess-prime} works very differently from \minizinc{}.
\Gls{ess-prime} is rewritten using a more traditional \compiler{}.
For all concepts in the language the \compiler{} intrinsically knows how to rewrite it for its target \solver{}.

Recently, \textcite{gokberk-2020-ess-incr} have also presented Savile Row as the basis of a \gls{meta-optimization} toolchain.
The authors extend Savile Row to bridge the gap between the incremental assumption interface of \gls{sat} \solvers{} and the modelling language and show how this helps to efficiently solve pattern mining and optimization problems.
Consequently, this research reiterates the use of \gls{meta-optimization} algorithms in \cmls{} and the need for incremental bridging between the modelling language and the \solver{}. 

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

At the heart of the \gls{rewriting} process that transforms a \minizinc{} \instance{} into a \gls{slv-mod}, lies a \gls{trs}.
A \gls{trs} describes a computational model.
The full process can be described as the application of rules \(l \rightarrow r_{1}, \ldots, r_{n}\), that replace a \gls{term} \(l\) with \(n \geq{} 1\) \glspl{term} \(r_{1}, \ldots, r_{n}\) \autocite{baader-1998-term-rewriting}.
A \gls{term} is an expression with nested sub-expressions consisting of function and 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 term variable, which captures a term sub-expression.

\begin{example}

	Consider the following \gls{trs}, which consists of some (well-known) rules to rewrite logical conjunctions.

	\begin{align*}
		(r_{1}):\hspace{5pt} & 0 \land x \rightarrow 0         \\
		(r_{2}):\hspace{5pt} & 1 \land x \rightarrow x         \\
		(r_{3}):\hspace{5pt} & x \land y \rightarrow y \land x
	\end{align*}

	From these rules it follows that

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

	Notice that there is a choice between different rules.
	In general, a \gls{trs} can be non-deterministic.
	We could also have applied the \(r_{2}\) twice and arrived at the same result.
\end{example}

Two properties of a \gls{trs} that are often studied are \gls{termination} and \gls{confluence}.
A \gls{trs} is said to be terminating if, no-matter what order the term rewriting rules are applied, it always arrives at a \gls{normal-form} (\ie{} a set of \glspl{term} for which none of the rules apply).
A \gls{trs} is confluent if, no-matter what order the term rewriting rules are applied, it always arrives at the same \gls{normal-form} (if it arrives at a \gls{normal-form}).

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

These properties could also be studied in the translation process of a \minizinc{} instance into \flatzinc{}.
The \gls{confluence} of the \gls{rewriting} process would ensure that the same \gls{slv-mod} is produced independently of the order in which the \minizinc{} \instance{} is processed.
Although this is a desirable quality, it is not guaranteed since it conflicts with important simplifications, discussed in \cref{sec:back-mzn-interpreter}, used to improve the quality of \gls{slv-mod}.

Many of the techniques used by \solvers{} targeted by \minizinc{} are proven to be complete.
This means that they are guaranteed to (eventually) find a (optimal) \gls{sol} for the \instance{} or prove that there is none.
For this quality to hold for the overall \minizinc{} solving process, it has to be guaranteed that the \minizinc{} \gls{rewriting} process terminates (so the solving process is able to start).
While this property is interesting, it cannot be guaranteed for the \gls{rewriting} process in general.
Since \minizinc{} is a Turing complete language, it is possible to create a \minizinc{} model for which the \gls{rewriting} process will infinitely recurse.

In the remainder of this section we discuss two types of \glspl{trs} that are closely related to \cmls{} and the \gls{rewriting} process: \gls{clp} and \gls{chr}.

\subsection{Constraint Logic Programming}%
\label{subsec:back-clp}
\glsreset{clp}

\gls{clp} \autocite{jaffar-1987-clp} is a predecessor of \cmls{} like \minizinc{}.
This subsection provides a brief introduction into the workings of a \gls{clp} system. 
For a more comprehensive overview on modelling, rewriting, and solving using a \gls{clp} system, we recommend reading ``Programming with constraints: an introduction'' by \textcite{marriott-1998-clp}.

A constraint logic program describes the process in which a \cmodel{} is eventually rewritten into a \gls{slv-mod} and solved by a \solver{}.
Like in \minizinc{}, the user defines \constraint{} predicates to use in the definition of the \cmodel{}.
Different from \minizinc{}, \constraint{} predicates in \gls{clp} can be rewritten in multiple ways.
The goal of a constraint logic program is to rewrite all \constraints{} in such a way that all \gls{native} \glspl{constraint} are \gls{satisfied}.

Variables{} are another notable difference between \cmls{} and \gls{clp}.
In \gls{clp}, like in a conventional \gls{trs}, a variable is merely a name.
The symbol can be replaced or equated with a constant symbol, but, different from \cmls{}, this is not a requirement.
A variable can remain a name in the solution of a constraint logic program.
This means that the solution of a constraint logic program can be a relation between different variables.
In cases where an instantiated solution is required, a special \mzninline{labeling} predicate is used to force a variable to take a constant value.
Similarly, there is a \mzninline{minimize} predicate that is used to find the optimal value for a variable.

The evaluation of a constraint logic program rewrites the list of \constraints{}, called the goal, in the order given by the programmer.
The \gls{rewriting} of the \constraint{} predicates is tried in the order in which the different \gls{rewriting} rules for the \constraint{} predicates are defined.
The process is completed when all \constraints{} are rewritten and the produced \gls{native} \constraints{} are not found to be inconsistent.
If all the possible ways of \gls{rewriting} the program are tried, but all of them prove to be inconsistent, then the program itself is said to be \gls{unsat}.
Even when a correct \gls{rewriting} is found, it is possible to continue the process.
This can discover all possible correct ways to rewrite the program.

To implement this mechanism there is a tight integration between the \solver{}, referred to as the constraint store, and the evaluator of the constraint logic program.
In addition to just adding \constraints{}, the program also inspects the status of the constraint store and retracts \constraints{} from the constraint store.
This allows the program to detect when the constraint store has become inconsistent and then \gls{backtrack} the constraint store to the last decision (\ie\ restore the constraint store to its state before the last decision was made) and try the next rewriting rule.

The strength of the constraint store affects the correctness of a constraint logic program.
Some \solvers{} are incomplete; they are unable to tell if some of their \constraints{} are \gls{satisfied} or not.
This, for example, happens with \solvers{} that work with integer \glspl{domain}.
In these cases the programmer must use the \mzninline{labeling} \constraint{} to force constant values for the variables.
Once the variables have been assigned constant values, the \solver{} is always able to decide if the \constraints{} are \gls{satisfied}.

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

When \constraints{} are seen as terms in a \gls{trs}, then it is not just possible to define rules to rewrite \constraints{} to \gls{native} \constraints{}.
It is also possible to define rules to simplify, propagate, and derive new \constraints{} within the \solver{}.
\gls{chr} \autocite{fruhwirth-1998-chr} follow from this idea.
\gls{chr} are a language extension (targeted at \gls{clp}) to allow for the definition of user-defined \constraints{} within a \solver{}.
\Constraints{} defined using \gls{chr} are rewritten into simpler \constraints{} until they are solved.

Different from \gls{clp}, \gls{chr} allows term rewriting rules that are multi-headed.
This means that, for some rules, multiple terms must match, to apply the rule.

\begin{example}

	Consider the following user-defined \constraint{} for logical implication using \gls{chr}.

	\begin{plain}
		reflexivity   @ X -> Y <=> X = Y | true
		anti-symmetry @ X -> Y, Y -> X <=> X = Y
		transitivity  @ X -> Y, Y -> Z ==> X -> Z
	\end{plain}

	These definitions specify how \texttt{->} simplifies and propagates as a \constraint{}.
	The rules follow the mathematical concepts of reflexivity, anti-symmetry, and transitivity.

	\begin{itemize}
		\item The first rule states that if \texttt{X = Y}, then \texttt{X -> Y} is logically true.
			This rule removes the term \texttt{X -> Y}.
			Since the \constraint{} is said to be logically true, nothing gets added.
			\texttt{X = Y} functions as a guard.
			This \solver{} \gls{native} \constraint{} is required for the rewriting rule to apply.

		\item The second rule implements the anti-symmetry of logical implications; the two implications, \texttt{X -> Y} and \texttt{Y -> X}, are replaced by a \solver{} \gls{native}, \texttt{X = Y}.

		\item Finally, the transitivity rule introduces a derived \constraint{}.
			When it finds the \constraints{} \texttt{X -> Y} and \texttt{Y -> Z}, then it adds another \constraint{} \texttt{X -> Z}.
			Different from the other rules, the matched \constraints{} are not removed.
	\end{itemize}

	Note that the use of multi-headed rewriting rules is essential to define these rules.

\end{example}

The rules in a \gls{chr} system are categorized into three different categories: simplification, propagation, and simpagation.
The first two rules in the previous example are simplification rules: they replace some \constraint{} atoms by others.
The final rule in the example was a propagation rule: based on the existence of certain \constraints{}, new \constraints{} are introduced.
Simpagation rules are a combination of both types of rules in the form:

\[ H_{1}, \ldots H_{l} \backslash H_{l+1}, \ldots, H_{n} \texttt{<=>} G_{1}, \ldots{}, G_{m} | B_{1}, \ldots, B_{o} \]

It is possible to rewrite using a simpagation rule when there are terms matching \(H_{1}, \ldots, H_{n}\) and there are \solver{} \gls{native} \constraints{} \(G_{1}, \ldots{}, G_{m}\).
When the simpagation rule is applied, the terms \(H_{l+1}, \ldots, H_{n}\) are replaced by the terms \(B_{1}, \ldots, B_{o}\).
The terms \(H_{1}, \ldots H_{l}\) are kept in the system.
Since simpagation rules incorporate both the elements of simplification and propagation rules, it is possible to formulate all rules as simpagation rules.

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

Traditionally a \compiler{} is split into three sequential parts: the ``frontend'', the ``middle-end'', and the ``backend''.
It is the job of the frontend to parse the user input, report on any errors or inconsistencies in the input, and transform it into an internal representation.
The middle-end performs the main translation, independent of the compilation target.
It converts the internal representation at the level of the compiler frontend to another internal representation as close as possible to the level required by the compilation targets.
The final transformations into the format required by the compilation target are performed by the backend.
When a \compiler{} is separated into these three steps, then adding support for a new language or compilation target only requires the addition of a frontend or backend respectively.

The \minizinc{} compilation process can be split into the same three parts, as shown in \cref{fig:back-mzn-comp}.
In the frontend, a \minizinc{} model is combined with its data into an \instance{}.
The instance is parsed into an \gls{ast}.
The process then analyses the \gls{ast} to discover the types of all expressions used in the \instance{}.
If an inconsistency is discovered, then an error is reported to the modeller.
Finally, the frontend also preprocesses the \gls{ast}.
This process is used to transform expressions into a common form for the middle-end, removing the ``syntactic sugar''.
For instance, this replaces enumerated types by normal integers.

\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{assets/img/back_compilation_structure}
	\caption{\label{fig:back-mzn-comp} The compilation structure of the \minizinc \compiler{}.}
\end{figure}

The middle-end contains the most important two processes: \gls{rewriting} and optimization.
During the \gls{rewriting} process the \minizinc{} model is rewritten into a \gls{slv-mod}.
It could be noted that the \gls{rewriting} step depends on the compilation target to define its \gls{native} \constraints{}.
Even though the information required for this step is target dependent, we consider it part of the middle-end as the mechanism is the same for all compilation targets.
A full description of this process will follow in \cref{subsec:back-rewriting}.
Once a \gls{slv-mod} is constructed, the \minizinc{} \compiler{} tries to optimize this model: shrink \domains{} of \variables{}, remove \constraints{} that are proven to hold, and remove \variables{} that have become unused.
These optimization techniques are discussed in \cref{subsec:back-fzn-optimization}.

The backend converts the internal \gls{slv-mod} into a format to be used by the targeted \solver{}.
Given the formatted artefact, a \solver{} process, controlled by the backend, is then started.
The \solver{} process produces \glspl{sol} for the \gls{slv-mod}.
Before these are given to the modeller, the backend reformulates these \glspl{sol} to become \glspl{sol} of the modeller's \instance{}.

In this section we discuss the \gls{rewriting} and optimization process as employed by the 2.5.5 version of \minizinc{} \autocite{minizinc-2021-minizinc}.

\subsection{Rewriting}%
\label{subsec:back-rewriting}

The goal of the \gls{rewriting} process is to arrive at a flat \gls{slv-mod}: an \gls{eqsat} \instance{} that only contains \constraints{} that consist of a singular calls, all arguments to calls are \parameter{} literals or \variable{} identifiers, and using only \constraints{} and \variable{} types that are \gls{native} to the target \solver{}.

To arrive at a flat \gls{slv-mod}, the \gls{rewriting} process traverses the declarations, \constraints{}, and the solver goal and rewrites any expression contained in these items.
An expression is rewritten into other \minizinc{} expressions according to the \gls{decomp} given in the target \solver{}'s library.
Enforced by \minizinc{}'s type system, at most one rule applies for any given \constraint{}.
The \gls{rewriting} of expressions is performed bottom-up, we rewrite any sub-expression before its parent expression.
For instance, in a call each argument is rewritten before the call itself is rewritten.

An exception to this bottom-up approach is the \gls{rewriting} of \glspl{comprehension} \mzninline{[@\(E\)@ | @\(i\)@ in @\(S\)@ where @\(F\)@]}.
\Gls{rewriting} \(E\) requires instantiating the identifier \(i\) with the values from the set \(S\), and evaluating the condition \(W\).
The \compiler{} therefore iterates through all values in \(S\), binds the values to the specified identifier(s), and rewrites the condition \(F\).
If \(F\) is true, it rewrites the expression \(E\) and collects the result.
Once the \gls{generator} is exhausted, the compiler rewrites its surrounding expression using the collected values.

The \gls{decomp} system in \minizinc{} is defined in terms of function declarations.
Any call, whose declaration has a function body, is eventually replaced by an instantiation of this function body using the arguments to the call.
Calls are, however, not the only type of expression that are decomposed during the \gls{rewriting} process.
Other expressions, like \gls{operator} expressions, variable \gls{array} access, and \gls{conditional} expressions, may also have to be decomposed for the target \solver{}.
During the \gls{rewriting} process, these expressions are rewritten into equivalent call expressions that start the decomposition process.

A notable effect of the \gls{rewriting} is that sub-expressions are replaced by literals or \glspl{avar}.
If the expression contains only \parameters{}, then the \gls{rewriting} of the expression is merely a calculation to find the value of the literal to be put in its place.
If, however, the expression contains a \variable{}, then this calculation cannot be performed during the \gls{rewriting} process.
Instead, an \gls{avar} must be created to represent the value of the sub-expression, and it must be constrained to take the value corresponding to the expression.
The creation of this \gls{avar} and defining \constraints{} happens in one of two ways:

\begin{itemize}

	\item For Boolean expressions in a non-\rootc{} context, the \gls{avar} is inserted by the \gls{rewriting} process itself.
		To constrain this \gls{avar}, the \compiler{} then adds the \gls{reif} of the \constraint{}.
		This \constraint{} contains a variation of the call that would have been generated for the expression in \rootc{} context.
		The name of the function is appended with \mzninline{_reif} and an extra Boolean \variable{} argument is added to the call.
		The definition of this \constraint{} should implement the \gls{reif} of the original expression: setting the additional argument to true if the \constraint{} is \gls{satisfied}, and false otherwise.
		For example, consider the following \minizinc{} \constraint{}.

		\begin{mzn}
			constraint b \/ this_call(x, y);
		\end{mzn}

		\noindent{}During \gls{rewriting} it will be turned into the following \cmodel{} fragment.

		\begin{mzn}
			var bool: i1;
			constraint this_call_reif(x, y, i1);
			constraint b \/ i1
		\end{mzn}

		\noindent{}Rewriting then continues with the \mzninline{this_call_reif} function (if its declaration has a body), as well as the disjunction operator.

	\item For non-Boolean expressions, the \gls{avar} and defining \constraints{} are introduced in the definition of the function itself.
		For example, the \mzninline{max} function in the standard library, which calculates the maximum of two values, is defined as follows.

		\begin{mzn}
			function var int: max(var int: x, var int: y) =
			let {
				var max(lb(x),lb(y))..max(ub(x),ub(y)): m;
				constraint int_max(x,y,m);
			} in m;
		\end{mzn}

		Using a \gls{let} the function body explicitly creates an \gls{avar}, constrains it to take to correct value, and then returns it.
\end{itemize}

These are the basic steps that are followed to rewrite \minizinc{} instances.
This is, however, not the complete process.
Following these steps alone would result in poor quality \glspl{slv-mod}.
A \gls{slv-mod} containing extra \variables{} and \constraints{} that do not add any information to the solving process can exponentially slow it down.
Therefore, the \minizinc{} \gls{rewriting} process is extended using many techniques to help improve the quality of the \gls{slv-mod}.
In the remainder of this chapter, we discuss the most important techniques used to improve the \gls{rewriting} process.

\subsection{Common Sub-expression Elimination}%
\label{subsec:back-cse}
\glsreset{cse}

Since \minizinc{} is, at its core, a pure functional programming language, the evaluation of a \minizinc{} expression does not have any side effects.
As a consequence, evaluating the same expression twice will always reach an equivalent result.

It is therefore possible to reuse the same result for equivalent expressions.
This simplification is called \gls{cse}.
It is a well understood technique that originates from compiler optimization \autocite{cocke-1970-cse}.
\Gls{cse} has also proven to be very effective in discrete optimization \autocite{marinov-2005-sat-optimisations, araya-2008-cse-numcsp}, including during the evaluation of \cmls{} \autocite{rendl-2009-enhanced-tailoring}.

\begin{example}
\label{ex:back-cse}
	For instance, in the following \constraint{} the same expression, \mzninline{abs(x)}, occurs twice.

	\begin{mzn}
		constraint (abs(x)*2 >= 20) \/ (abs(x)+5 >= 15);
	\end{mzn}

	However, we do not need to create two separate \variables{} (and defining \constraints{}) to represent the absolute value of \mzninline{x}.
	The same \variable{} can be used to represent the \mzninline{abs(x)} in both sides of the disjunction.
\end{example}

Seeing that the same expression occurs multiple times is not always easy.
Some expressions only become syntactically equal when instantiated, as in the following example.

\begin{example}
	Consider the fragment:

	\begin{mzn}
		function var float: pythagoras(var float: a, var float: b) =
			let {
				var float: x = pow(a, 2);
				var float: y = pow(b, 2);
			} in sqrt(x + y);
		constraint pythagoras(i, i) >= 5;
	\end{mzn}

	Although the expressions \mzninline{pow(a, 2)} and \mzninline{pow(b, 2)} are not syntactically equal, the function call \mzninline{pythagoras(i,i)} using the same \variable{} for \mzninline{a} and \mzninline{b} makes them equivalent.
\end{example}

To ensure that two identical instantiations of a function are only evaluated once, the \minizinc{} \compiler{} uses memoization.
After the \gls{rewriting} of an expression, the instantiated expression and its result are stored in a lookup table: the \gls{cse} table.
Then before any consequent expression is rewritten the \gls{cse} table is consulted.
If an equivalent expression is found, then the accompanying result is used; otherwise, the evaluation proceeds as normal.

In our example, the evaluation of \mzninline{pythagoras(i, i)} would proceed as normal to evaluate \mzninline{x = pow(i, 2)}.
However, the expression defining \mzninline{y}, \mzninline{pow(i, 2)}, is then found in the \gls{cse} table and replaced by the earlier stored result: \mzninline{y = x}.

\gls{cse} also has an important interaction with the occurrence of \glspl{reif}.
\Glspl{reif} are often defined in the library in terms of complicated \glspl{decomp} into \gls{native} \constraints{}, or require more complex algorithms in the target \solver{}.
In either case, it can be very beneficial for the efficiency of the solving process if we detect that a \gls{reified} \constraint{} is in fact not required.

If a \constraint{} is present in the \rootc{} context, it means that it must hold globally.
If the same \constraint{} is used in a non-\rootc{} context, \gls{cse} can then replace them with the constant true, avoiding the need for \gls{reif} (or in fact any evaluation).

We harness \gls{cse} to store the evaluation context when a \constraint{} is added, and detect when the same \constraint{} is used in both contexts.
Whenever a lookup in the \gls{cse} table is successful, action is taken depending on both the current and stored evaluation context.
If the stored expression was in \rootc{} context, then the constant true is used, independent of the current context.
Otherwise, if the stored expression was in non-\rootc{} context and the current context is non-\rootc{}, then the stored result variable is used.
Finally, if the stored expression was in non-\rootc{} context and the current context is \rootc{} context, then the previous result is replaced by the constant true and the evaluation proceeds as normal with the \rootc{} context \constraint{}.

\begin{example}
	Consider the following \minizinc{} fragment.

	\begin{mzn}
		constraint b0 <-> q(x);
		constraint b1 <-> t(x,y);
		function var bool: p(var int: x, var int: y) = q(x) /\ r(y);
		constraint b1 <-> p(x,y);
	\end{mzn}

	If we process the \constraints{} in order, we create a \gls{reified} call to \mzninline{q(x)} when \gls{rewriting} the first \constraint{}.
	Suppose when we rewrite the second \constraint{}, we discover \mzninline{t(x,y)} is true, fixing \mzninline{b1}.
	When we then process \mzninline{q(x)} in instantiation of the call \mzninline{p(x,y)}, it is in the \rootc{} context.
	So \gls{cse} needs to set \mzninline{b0} to true.
\end{example}

\subsection{Constraint Propagation}%
\label{subsec:back-adjusting-dom}

Sometimes a \constraint{} can be detected to be true based on its semantics, and the known \glspl{domain} of \variables{}.
For example, consider the \constraint{} \mzninline{3*x + 7*y > 8}, and assume that both \mzninline{x} and \mzninline{y} cannot be smaller than one.
In this case, we can determine that the \constraint{} is always \gls{satisfied}, and remove it from the model without changing satisfiability.
This is a simple form of \gls{propagation}, which, as discussed in \cref{subsec:back-cp}, also tightens the \glspl{domain} of \variables{} in the presence of a \constraint{}.

The principles of \gls{propagation} can also be applied during the \gls{rewriting} of a \minizinc{} model.
It is generally a good idea to detect cases where we can directly change the \gls{domain} of a \variable{}.
Sometimes this means that the \constraint{} does not need to be added at all and that constricting the \gls{domain} is enough.
Tight domains also allow us to avoid the creation of \glspl{reif} when the truth-value of a reified \constraint{} can be determined from the \domains{}.
Finally, it can also be helpful for \solvers{} as they may need to decide on a representation of \variables{} based on their initial \domain{}.

\begin{example}%
\label{ex:back-adj-dom}
	Consider the following \minizinc{} model:

	\begin{mzn}
		var 1..10: a;
		var 1..5: b;

		constraint a < b;
		constraint (a > 5) -> (a + b > 12);
	\end{mzn}

	Given the \domain{} specified in the model, the second \constraint{} is rewritten using \gls{reified} \constraints{}.
	One for each side of the implication.

	If we however consider the first \constraint{}, then we deduce that \mzninline{a} must always take a value that is 4 or lower.
	When the compiler adjusts the domain of \mzninline{a} while \gls{rewriting} the first \constraint{}, then the second \constraint{} can be simplified.
	The expression \mzninline{a > 5} is known to be false, which means that the \constraint{} is simplified to true.
\end{example}

During \gls{rewriting}, the \minizinc{} compiler actively removes values from the \domain{} when it encounters \constraints{} that trivially reduce it.
For example, it detects \constraints{} with a single comparison expression between a \variable{} and a \parameter{} (\eg\ \mzninline{x != 5}), \constraints{} with a single comparison between two \variables{} (\eg\ \mzninline{x >= y}), \constraints{} that directly change the domain (\eg\ \mzninline{x in 3..5}).
It even performs more complex \gls{propagation} for some known \constraints{}.
For example, it will reduce the \domains{} for \mzninline{int_times} and \mzninline{int_div}, and we will see in the next subsection how \gls{aggregation} will help simplify certain \constraints{}.

However, \gls{propagation} is only performed locally, when the \constraint{} is recognized.
During \gls{rewriting}, the \gls{propagation} of one \constraint{}, will not trigger the \gls{propagation} of another.

Crucially, the \minizinc{} compiler also handles equality \constraints{}.
During the \gls{rewriting} process we are in a unique position to perform effective equality \gls{propagation}.
If the \compiler{} finds that two \variables{} \mzninline{x} and \mzninline{y} are equal, then only a single \variable{} is required in the \gls{slv-mod} to represent them both.
Whenever any equality \constraint{} is found, it is removed and one of the \variables{} is replaced by the other.

This is often beneficial for the \solver{}, since it reduced the number of \variables{} and some \solver{} are not able to perform this replacement, forcing the propagation of an equality \constraint{}.
Moreover, replacing one variable for another can improve the effectiveness of \gls{cse}.

\begin{example}
	Consider the following \minizinc{} fragment.

	\begin{mzn}
		var 1..5: a;
		var 1..5: b;

		constraint a = b;
		constraint this(a);
		constraint this(b);
	\end{mzn}

	The equality \constraint{}, replaces the \variable{} \mzninline{b} with the \variable{} \mzninline{a}.
	Now the two calls to \mzninline{this} suddenly become equivalent, and the second will be found in the \gls{cse} table.
\end{example}

Note that if the equality \constraint{} in the example would have be found after both calls, then both calls would have already been rewritten.
The \minizinc{} compiler would be unable to revisit the rewriting of second calls after the equality is found.
It is therefore important that equalities are found early in the \gls{rewriting} process.

\subsection{Constraint Aggregation}%
\label{subsec:back-aggregation}

Complex \minizinc{} expressions sometimes result in the creation of many \glspl{avar} to represent intermediate results.
This is in particular true for linear and Boolean equations that are generally written using \minizinc{} operators.

\begin{example}%
\label{ex:back-agg}
	For example the evaluation of the linear \constraint{} \mzninline{x + 2*y <= z} could result in the following \flatzinc{}:

	\begin{nzn}
		var int: x;
		var int: y;
		var int: z;
		var int: i1;
		var int: i2;
		constraint int_times(y, 2, i1);
		constraint int_plus(x, i1, i2);
		constraint int_le(i2, z);
	\end{nzn}

	This \flatzinc{} model is correct, but, at least for pure \gls{cp} solvers, the existence of the \gls{avar} is likely to have a negative impact on the \solver{}'s performance.
	These \solvers{} would likely perform better had they directly received the equivalent linear \constraint{}:

	\begin{mzn}
		constraint int_lin_le([1,2,-1], [x,y,z], 0)
	\end{mzn}

	This \constraint{} directly represents the initial \constraint{} in the \cmodel{} without the use of \gls{avar}.
\end{example}

This can be resolved using \gls{aggregation}.
When we aggregate \constraints{} we collect multiple \minizinc{} expressions that would each have been rewritten separately, and combine them into a singular structure that eliminates the need for \gls{avar}.
For example, arithmetic expressions are combined into linear \constraints{}, Boolean logic expressions are combined into clauses, and counting \constraints{} are combined into global cardinality \constraints{}.

The \minizinc{} \compiler{} aggregates expressions whenever possible.
When the \minizinc{} \compiler{} reaches an expression that could potentially be part of an aggregated \constraint{}, the \compiler{} does not rewrite the expression.
The \compiler{} instead performs a search of its sub-expressions to collect all other expressions to form an aggregated \constraint{}.
The \gls{rewriting} process continues by \gls{rewriting} this aggregated \constraint{}.

\subsection{Delayed Rewriting}%
\label{subsec:back-delayed-rew}

Adding \gls{propagation} during \gls{rewriting} means that the system becomes non-confluent, and some orders of execution may produce ``better'', \ie{} more compact or more efficient, \flatzinc{}.

\begin{example}
	The following example is similar to code found in the \minizinc{} libraries of \gls{mip} \solvers{}.

	\begin{mzn}
		function var int: lq_zero_if_b(var int: x, var bool: b) =
			x <= ub(x)*(1-b);
	\end{mzn}

	This predicate expresses the \constraint{} \mzninline{b -> x<=0}, using a well-known technique called the ``Big M method''.
	The expression \mzninline{ub(x)} returns a valid upper bound for \mzninline{x}, \ie{} a fixed value known to be greater than or equal to \mzninline{x}.
	This could be the initial upper bound \mzninline{x} was declared with, or the current value adjusted by the \minizinc{} \compiler{}.
	If \mzninline{b} takes the value 0, the expression \mzninline{ub(x)*(1-b)} is equal to \mzninline{ub(x)}, and the \constraint{} \mzninline{x <= ub(x)} holds trivially.
	If \mzninline{b} takes the value 1, \mzninline{ub(x)*(1-b)} is equal to 0, enforcing the \constraint{} \mzninline{x <= 0}.
\end{example}

For \gls{mip} solvers, it is quite important to enforce tight \gls{bounds} in order to improve efficiency and sometimes even numerical stability.
It would therefore be useful to rewrite the \mzninline{lq_zero_if_b} predicate only after the \domains{} of the involved \variables{} have been reduced as much as possible, in order to take advantage of the tightest possible \gls{bounds}.
On the other hand, evaluating a predicate may also impose new \gls{bounds} on \variables{}, so it is not always clear which order of evaluation is best.

The same problem occurs with \glspl{reif} that are produced during \gls{rewriting}.
Other \constraints{} could fix the \domain{} of the reified \variable{} and make the \gls{reif} unnecessary.
Instead, the \constraint{} (or its negation) can be rewritten in \rootc{} context.
This could avoid the use of a big \gls{decomp} or an expensive \gls{propagator}.

To tackle this problem, the \minizinc{} \compiler{} employs \gls{del-rew}.
When a linear \constraint{} is aggregated or a relational \gls{reif} \constraint{} is introduced it is not immediately rewritten.
Instead, these \constraints{} are appended to the end of the current \gls{ast}.
All other \constraints{}, that are not yet rewritten and could change the relevant \domains{}, are rewritten first.

Note that this heuristic does not guarantee that \variables{} have their tightest possible \gls{domain}.
One delayed \constraint{} can still influence the \domains{} of \variables{} used by other delayed \constraints{}.

Delaying the rewriting of \constraints{} also interferes with \gls{aggregation}.
Since \gls{aggregation} is eagerly performed only when a \constraint{} is first encountered, it cannot aggregate any \constraints{} that follow from delayed values.
For example, when aggregating Boolean logic expressions, we can come across an expression that needs to be \gls{reified}.
A Boolean \gls{avar} is created and the reified \constraint{} is delayed.
Although the Boolean \gls{avar} can be used in the aggregated \constraints{}, we did not consider the body of the \gls{reif}.
If the body of the \gls{reif} was defined in terms of Boolean logic, then it would have been aggregated as well.
However, since the rewriting of the body was delayed and the \compiler{} does not revisit \gls{aggregation}, this does not happen.

This problem can be solved through the use of \minizinc{}'s multi-pass compilation \autocite{leo-2015-multipass}.
It can rewrite (and propagate) an \instance{} multiple times, remembering information about the earlier iteration(s).
As such, information normally discovered later in the \gls{rewriting} process, such as the final \domains{} of \variables{}, whether two \variables{} are equal, and whether an expression must be \gls{reified}, can be used from the start. 

\subsection{FlatZinc Optimization}%
\label{subsec:back-fzn-optimization}

After the \compiler{} has finished \gls{rewriting} the \minizinc{} instance, it enters the optimization phase.
The primary role of this phase is to further perform \gls{propagation} until \gls{fixpoint}.
However, this phase operates at the level of the \gls{slv-mod}, where all \constraints{} are now \gls{native} \constraints{} of the target \solver{}.
This means that, depending on the target \solver{}, the \compiler{} will not be able to understand the meaning of all \constraints{}.
It only understands the meaning of the standard \flatzinc{} \constraints{}, but not any of the \solver{} specific \gls{native} \constraints{}.
For the standard \flatzinc{} \constraints{}, it employs \gls{propagation} methods, as discussed in \cref{subsec:back-cp}, are used to eliminate values from \domains{} and simplify \constraints{}.

In the current implementation, the \minizinc{} \compiler{} propagates mostly Boolean \constraints{} in this phase.
It tries to reduce the number of Boolean \variables{} and tries to reduce the number of literals in clauses and conjunctions.
