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%************************************************
\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 its users.
Reduce the complexity by hiding unnecessary information.
For example, an assembly language allows you 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 different from other computer languages.
The modeller does not describe how to solve a problem, but rather formalises 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 \parameters{}, what we already know; \variables{}, what we wish to know; and \constraints{}, the relationships that should exist between them.
Through the variation of \parameters{}, a \cmodel{} describes a full class of problems.
A specific problem is captured by a \instance{}, the combination of a \cmodel{} with a complete \gls{parameter-assignment} (\ie{} a mapping from all parameters to values).
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}: an complete \gls{variable-assignment} that satisfy the \constraints{}.
Or, when this is not possible, prove that no such \gls{assignment} exists.
Many \cmls{} also support the modelling of \gls{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 minimising (or maximising) 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 might have to 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 of a \cml{} is visualised 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 and pick the toys that bring Audrey the most amount of joy, but still fit in the car.
The following set of equations describe this knapsack problem as an \gls{opt-prb}:
\begin{equation*}
\text{maximise}~z~\text{subject to}~
\begin{cases}
S \subseteq T \\
z = \sum_{i \in S} joy(i) \\
\sum_{i \in S} space(i) \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 maximised.
The \parameter{} \(T\) is the set of all available toys.
The \(joy\) and \(space\) functions are \parameters{} used to map toys, \( t \in T\), to a numeric value describing the amount of enjoyment and space required respectively.
Finally, the \parameter{} \(C\) 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}
% \begin{listing}
% \pyfile{assets/listing/2_dyn_knapsack.py}
% \caption{\label{lst:2-dyn-knapsack} A Python program that solves a 0-1 knapsack problem using dynamic programming}
% \end{listing}
\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 \parameters{}.
\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 arrays \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 \parameter{} \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 maximise the value of the \variable{} \mzninline{total_joy}.
\end{example}
\begin{listing}
\mznfile{assets/listing/back_knapsack.mzn}
\caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack problem}
\end{listing}
One might 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 need to be transformed into a \gls{slv-mod}: a list of \variables{} and \constraints{} \gls{native} to the \solver{}.
Type of \variables{} and \constraints{} are said to be \gls{native} to the \solver{} when they are directly supported as input 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 a 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, given the \minizinc{} model in \cref{lst:back-mzn-knapsack} and the \gls{assignment}
\begin{mzn}
TOYS = {football, tennisball, stuffed_elephant};
toy_joy = [63, 12, 100];
toy_space = [32, 8, 40];
space_left = 44;
\end{mzn}
\noindent{}the following \flatzinc{} \gls{slv-mod} might be the result of \gls{rewriting}:
\begin{mzn}
var 0..1: selection_0;
var 0..1: selection_1;
var 0..1: selection_2;
var 0..175: total_joy:: is_defined_var;
constraint int_lin_le([32,8,40], [selection_0,selection_1,selection_2], 44);
constraint int_lin_eq(
[63,12,100,-1],
[selection_0,selection_1,selection_2,total_joy],
0
);
solve maximize total_joy;
\end{mzn}
This \gls{slv-mod} is then passed to a \solver{}.
The \solver{} attempts to determine a complete \gls{variable-assignment}, \mzninline{selection_@\(i\)@} and \mzninline{total_joy}, that satisfies all constraints and maximises \mzninline{total_joy}.
If there is no such \gls{assignment}, then it reports that the \gls{slv-mod} is \gls{unsat}.
\end{example}
\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 \parameters{} are found in the form of value 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 might 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 the \minizinc{}.
And, much of its history can be learned by the description of its predecessor, \zinc{} \autocite{marriott-2008-zinc}.
\paragraph{Declaration Items} Values in \minizinc{} are declared in the form \mzninline{@\(T\)@: @\(I\)@ = @\(E\)@}, where:
\begin{itemize}
\item \(T\) is the type 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 no functional definition, or when a \parameter{} is assigned externally.
The identifier used in a top-level value definition 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 distinguishes through their type.
The type of a \variable{} is preceded by the \mzninline{var} keyword.
The type of a \parameter{} might similarly be marked by the \mzninline{par} keyword, but this is not required.
These types are used both as normal \parameters{} and as \variables{}.
To better structure models, \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, an array going from 5 to 10 of new Boolean \variables{} might be declared as
\begin{mzn}
array[5..10] of var bool: bs;
\end{mzn}
The type in a declaration can, however, be more complex when the modeller uses a type expression.
These 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 does include 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 model translated the expressions in \constraints{} into versions of the same expression 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 only \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 do one of three actions:
\begin{itemize}
\item \mzninline{solve satisfy} --- to find an assignment to the \variables{} that satisfies the constraints,
\item \mzninline{solve minimize @\(E\)@} --- to find an assignment to the \variables{} that satisfies the \constraints{} and minimises the value of the expression \(E\),
\item or \mzninline{solve maximize @\(E\)@} --- to similarly maximise the value of the expression \(E\).
\end{itemize}
\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 a \gls{opt-prb}.
If the model does not contain a goal item, then it then the problem is assumed to be a satisfaction problem.
\paragraph{Function Items} Common structures in \minizinc\ are captured using function declarations.
Functions are 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 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, the usage of function declarations is the primary method for \solvers{} to specify how to rewrite a \minizinc{} model into a \gls{slv-mod}.
The collection of functions declarations to rewrite for a \solver{} is generally referred to as a \solver{} library.
In this library, functions might 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 might rely on a set of standard declaration, 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 type 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.
An example of a \gls{global} is
\begin{mzn}
predicate knapsack(
array [int] of int: w,
array [int] of int: p,
array [int] of var int: x,
var int: W,
var int: P,
);
\end{mzn}
This \gls{global} expresses the knapsack relationship, where the argument:
\begin{itemize}
\item \mzninline{w} are the weights of the items,
\item \mzninline{p} are the profit for each item,
\item the \variables{} in \mzninline{x} represent the amount of time the items 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 the usage of this \gls{global} might 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 specialised \gls{decomp} provided by the \solver{} or it might even be marked as \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.
For example the constraint
\begin{mzn}
constraint not (a + b < c);
\end{mzn}
\noindent{}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 is 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 result in a new \variable{} to contain its result.
\paragraph{Conditional Expression} The choice between different expressions are often expressed using a \gls{conditional} expression, also known as an ``if-then-else'' expression.
You could, for example, force that the absolute value of \mzninline{a} is bigger 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 contained to Boolean types.
The condition in the expression, the ``if'', must be of a Boolean type, but as long as the different sides of the \gls{conditional} expression are the same type it is a valid conditional expression.
This can be used to, for example, define an absolute value function for integer \parameter{} 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{ivar} is created and a \constraint{} is added to ensure that it takes the correct value.
A special \mzninline{if_then_else} \glspl{global} are available to implement this relationship.
\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 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 way 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{ivar} 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 are used to compose \gls{array} objects.
The generation of new \gls{array} structures allows modellers adjust, combine, filter, or order values from within the \minizinc{} model.
\Gls{generator} expressions, \mzninline{[@\(E\)@ | @\(G\)@ where @\(F\)@]}, consist of three parts:
\begin{description}
\item[\(G\)] The generator expression which assigns the values of collections to identifiers,
\item[\(F\)] an optional filtering condition, which has to evaluate to \mzninline{true} for the iteration to be included in the array,
\item[\(E\)] and the expression that is evaluated for each iteration when the filtering condition succeeds.
\end{description}
The following example composes 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 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 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 its resulting expression.
Additionally, a \gls{let} might contain \constraints{} to constrain the declared values.
There are three main purposes for \glspl{let}:
\begin{enumerate}
\item To name an intermediate expression, for it to be used multiple times or to simplify the expression.
For example, the \constraint{}
\begin{mzn}
constraint let { var int: tmp = x div 2; } in tmp mod 2 == 0 \/ tmp = 1;
\end{mzn}
\noindent{}constrains that half of \mzninline{x} is even or takes the value one.
\item To introduce a scoped \variable{}. For example, the constraint
\begin{mzn}
constraint let {var -2..2: slack;} in x + slack = y;
\end{mzn}
\noindent{}constrains that \mzninline{x} and \mzninline{y} are at most two apart.
\item To constrain the resulting expression.
For example, the following function
\begin{mzn}
function var int: int_times(var int: x, var int: y) =
let {
var int: z;
constraint pred_int_times(x, y, z);
} in z;
\end{mzn}
\noindent{}returns an \gls{ivar} \mzninline{z} that is constrained to be the multiplication of \mzninline{x} and \mzninline{y} by the relational multiplication constraint \mzninline{pred_int_times}.
\end{enumerate}
An important detail in \gls{rewriting} \glspl{let} is that any \variables{} that are introduced might need to be renamed in the resulting \gls{slv-mod}.
Different \variables{} declared directly in declaration items, the \variables{} declared in \glspl{let} might be flattened 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 does provide the modeller 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 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}.
Different parts of a complex expression are rewritten into different \constraints{}.
However, not all part of the expression are autonomous \constraints{} that can be directly enforced by the solver.
Some parts might have to be \reified{} into a Boolean variable.
The \gls{reif} of a \constraint{} \(c\) creates an \gls{ivar} \mzninline{b} 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 maximises the number of even numbers taken by the elements of the array \mzninline{x}.
In this model the expression \mzninline{x[i] mod 2 == 0} has to be reified.
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 maximise the number of reified variables it sets to \mzninline{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 partial expressions in \cmls{} as studied by \textcite{frisch-2009-undefinedness}.
Some expressions in \cmls{} do not have well-defined results.
Part of the semantics of a \cml{} is the choice as to how to treat these 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{}which requires that if \mzninline{i} takes a value less or equal to four, then the value in the \texttt{i}\(^{th}\) position of array \mzninline{a} multiplied by \mzninline{x} must be at least 6.
Suppose the array \texttt{a} has index set \mzninline{1..5}, but \texttt{i} takes the value \texttt{7}.
The expression \mzninline{a[i]} fails and no \gls{sol} is found.
However, intuitively if \mzninline{i = 7} the constraint should be trivially true.
\end{example}
Other examples of \minizinc{} expressions that result in partial functions 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 or an empty set:
\begin{mzn}
min({}) = @??@
\end{mzn}
\item 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{}.
Frisch and Stuckey 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} treats undefined expressions as expressions for which not enough information is available.
If an expression contains 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 \mzninline{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} follows from all expressions in a \cml{} eventually becoming part of a relational \constraint{}.
So even though a (functional) expression in itself might not have a well-defined result, we can still decide whether its surrounding relationship holds.
For example, the expression \mzninline{x div 0} is undefined, but the relationship \mzninline{int_div(x, 0, y)} is said to be \mzninline{false}.
It can be said that the relational semantic makes the closest relational expression that contains an undefined expression \mzninline{false}.
\end{description}
In practice, it is often natural to guard against undefined behaviour using Boolean logic.
\Glspl{rel-sem} therefore often feel the most natural for the users of constraint modelling languages.
This is why the \minizinc{} uses \glspl{rel-sem} during its evaluation.
For example, one might deal with a zero divisor using a disjunction:
\begin{mzn}
constraint d == 0 \/ a div d < 3;
\end{mzn}
In this case we expect the undefinedness of the division to be contained within the second part of the disjunction.
This corresponds to \glspl{rel-sem}.
Frisch and Stuckey also show that different \solvers{} often employ different semantics.
It is therefore important that the \gls{rewriting} process replaces any potentially undefined expression by an \gls{eqsat} expression that is valid under a \gls{strict-sem}.
This essentially eliminates the existence of undefined expressions in the \gls{slv-mod}.
\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 \constraint{}.
\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 that 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{} \solver{} and their \glspl{slv-mod}.
\subsection{Constraint Programming}%
\label{subsec:back-cp}
\glsreset{cp}
When given an \instance{} of a \cmodel{}, one might wonder how to find a \gls{sol}.
The simplest solution would be to apply ``brute force'': try every value in the \domains{} of all \variables{}.
This is an inefficient approach.
Given, for example, the 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{csp} 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 no longer possible.
\Gls{propagation} works through the use of \glspl{propagator}: algorithms dedicated to a specific \constraint{} that prune \domains{} when its contains 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 \domains{} of one of its \variables{} has changed.
In the best case scenario, \gls{propagation} eliminates all impossible values and all \variables{} have been \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} and we reach a \gls{fixpoint}, where no more \domain{} reductions are performed.
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 no \gls{sol} is 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 might 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 using a \gls{trail}.
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}, it is guaranteed that no duplicate \gls{propagation} is required.
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 access to the same \glspl{propagator}.
Therefore, a \gls{slv-mod} for one \solver{} might look very different from an \gls{eqsat} \gls{slv-mod} for a different \solver{}.
\cmls{}, like \minizinc{}, serve as a standardised form of input for these \solvers{}.
They allow modellers to use 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 as an example the N-Queens problem.
Given a chessboard of size \(n \times n\), find a placement for \(n\) queen chess pieces such that no queen can attack another.
Meaning only 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 restrict every column to only have a single queen.
As such, no queen will be able to attack vertically.
The \constraints{} in the model the remaining rules of the problem: no two queens can be placed in the same row, no two queen can be place in the same upward diagonal, and no two queens can be place in the same downward diagonal.
This model can be directly used in most \gls{cp} \solvers{}, since integer \variables{} and an \mzninline{all_different} \gls{propagator} are common.
When solving the problem, initially no values can be eliminated from the \glspl{domain} of the \variables{}.
The first propagation happens when the first queen is placed on the board, the first search decision.
\Cref{fig:back-nqueens} visualises 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 show 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 rows}
\end{subfigure}
\begin{subfigure}[b]{.48\columnwidth}
\centering
\includegraphics{assets/img/back_chess3}
\caption{\label{sfig:back-nqueens-3} Propagate 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 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 spend propagating a \constraint{} and the amount of search that is otherwise required.
The golden standard for a \gls{propagator} is to be \gls{domain-con}.
A \gls{domain-con} \gls{propagator} leaves only values in the \domains{} when there is at least one possible \gls{variable-assignment} that satisfies its \constraint{}.
Designing such a \gls{propagator} is, however, not an easy task.
The algorithm might require high computational complexity.
Instead, it is sometimes be better to use a propagator with a lower level of consistency.
Although it might not eliminate all possible values of the domain, searching the values that are not eliminated might take less time than achieving \gls{domain-con}.
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, no realistic \gls{domain-con} \gls{propagator} exists because the problem is \gls{np}-hard \autocite{choi-2006-fin-cons}.
Instead, \solvers{} generally use a \gls{bounds-con} \gls{propagator}, which guarantee only that the minimum and maximum values in the \glspl{domain} of the \variables{} are used in at least one possible assignment that satisfies the constraint.
Thus far, we have only considered finding \glspl{sol} for \glspl{dec-prb}.
\gls{cp} solving can, however, also be used to solve \gls{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 no longer interested in just any \gls{sol}, only \glspl{sol} that for which the \gls{objective} returns a better value.
This is achieved by adding a new \gls{propagator} 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 there are no better \glspl{sol} than the current incumbent \gls{sol}.
It is an \gls{opt-sol}.
\gls{cp} solvers like Chuffed \autocite{chuffed-2021-chuffed}, Choco \autocite{prudhomme-2016-choco}, \gls{gecode} \autocite{gecode-2021-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}
\gls{lp} is the foundation of maybe the prominent solving techniques for \glspl{opt-prb} \autocite{schrijver-1998-mip}.
A linear program describes a problem using \constraints{} in the form of linear equations over continuous \variables{}.
In general, a linear program can be expressed in the form:
\begin{align*}
\text{maximise} \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_{i} \in \mathbb{R} & \forall_{1 \leq{} i \leq{} V}
\end{align*}
\noindent{}where \(V\) 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 bounds of the constraints.
Finally, the \variables{} of the linear program 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-1982-simplex} during 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.
These methods provide the foundation for a harder problem.
In \gls{lp} our \variables{} must be continuous.
If we require that one or more take a discrete value (\(x_{i} \in \mathbb{Z}\)), then the problem suddenly becomes much harder.
The problem is referred to as \gls{mip} (or Integer Programming if \textbf{all} \variables{} must take a discrete value).
Unlike \gls{lp}, there is no 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 integer \variables{} already have discrete values, then we have found an \gls{opt-sol}.
Otherwise, we pick one of the integer \variables{} that does not yet have a discrete value.
For this \variable{} we create two versions of the linear program: a version where this \variable{} is less or equal to the value in the \gls{sol} rounded down to the nearest discrete value; and a version where it is 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 a discrete \gls{sol} is found.
Much of the power of this solving method comes from 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 discrete \gls{sol} found in an earlier branch of the search process provide a lower bound.
When the upper bound given by the simplex method is lower that the lower bound from an earlier solution, then we know that any discrete \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{cplex-2020-cplex}, \gls{gurobi} \autocite{gurobi-2021-gurobi}, and \gls{scip} \autocite{gamrath-2020-scip}, solve many complex problems.
These \solvers{} are therefore prime targets to solve \minizinc{} \instances{}.
To solve a \instance{} of a \cmodel{}, it can be encoded as a mixed integer program.
This process is known as \gls{linearisation}.
It does, however, come with its challenges.
Most \constraints{} in a \minizinc{} model are not linear equations.
The translation of a single \constraint{} might introduce many linear \constraints{} and even new \variables{}.
For example, when a \constraint{} reasons about the value that a variable takes, the \gls{linearisation} process introduces indicator variables.
The indicator variables \(y_{i}\) are \glspl{ivar} 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{maximise} \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}
The encoding of this \cmodel{} uses only integers.
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 might be one of the oldest in computer science.
The DPLL algorithm that is used to this day stems from the 60s \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.
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 standardised 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, a lot of progress has been made towards solving even the biggest the most 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.
Depending on the problem, using a \gls{sat} \solvers{} to solve the encoded problem might 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 place 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 assignment for Boolean \variables{} that maximises the cumulative weights of the \gls{satisfied} clauses.
The \gls{maxsat} problem is analogous to a \gls{opt-prb}.
Like a \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 encode 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{} is then forms a singleton clause with the result value as its weight.
For many problems the use of \gls{maxsat} \solvers{} offers a very successful method to find the \gls{opt-sol} to a problem.
\section{Other 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 strength and weaknesses.
Most of the techniques that are discusses in this thesis might be adapted to these languages.
We now discuss some other prominent \cmls{} and compare them to \minizinc{} to indicate to the reader where techniques might have 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.
For \solvers{} that have a \gls{mip} solving method, the modellers is required to use \variables{} of numeric type.
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 the usage of certain \glspl{global} when using a \gls{cp} \solver{} \autocite{fourer-2002-amplcp}.
\begin{example}
Let us consider modelling in \gls{ampl} by considering 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{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 contain all possible paths between the different cities.
Note how its definition uses a \gls{comprehension} to eliminate symmetric paths.
For each possible paths 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 the it is used, yes or no.
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 yet enforce that the taken variables represent a single path.
It is still possible for so-called subtours to exist, multiple unconnected path that together span all the cities.
A naive way to eliminate these subtours is by forcing that any subset of the set of cities has at least an incoming and an outgoing path.
\Lrefrange{line:back:ampl:compstart}{line:back:ampl:compend} of the model compute all possible subsets of cities.
These are then used by the \constraint{} on \lref{line:back:ampl:con2} to ensure they have the required paths going in and out.
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 vast collection of \glspl{operator}.
The building block 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 the \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 you are required to use the language functionality that is compatible with the targeted \solver{}; in this case, all expression have to be linear 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.
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 has a focus 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 the \gls{opl} really shines is when modelling scheduling problems.
Resources and activities are separate objects in the \gls{opl}.
This allows users express resource scheduling \constraints{} in an incremental and more natural fashion.
When solving a scheduling problem, the \gls{opl} makes use of specialised interval \variables{}, which represent when a task is scheduled.
\begin{example}
Let us consider modelling in \gls{opl}, by considering the well-known \gls{jsp}.
The \gls{jsp} is similar to the open shop problem discussed in the introduction.
Like the open shop problem, we are tasked with scheduling 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 \gls{jsp}, 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 \gls{jsp} in \gls{opl} is shown in \cref{lst:back-opl-jsp}.
\begin{listing}
\plainfile{assets/listing/back_jsp.mod}
\caption{\label{lst:back-opl-jsp} An \gls{opl} model describing the \gls{jsp}, abstracting from \parameter{} declarations.}
\end{listing}
\Lref{line:back:opl:task} declares the \texttt{task} \variables{}, the main \variable{} of in model.
These \variables{} have the type \texttt{Activity}, a special type used to declare any scheduling event with a begin and end time.
\Variables{} of this type are influence 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 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} set the minimisation 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 \gls{jsp}, abstracting from \parameter{} declarations.}
\end{listing}
A fragment of a \minizinc{} model modelling the same parts of \gls{jsp} 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 time at which all tasks are finished.
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{}.
The \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 might make it harder in \minizinc{} to write the correct \constraint{} and its meaning might be less clear.
\end{example}
The \gls{opl} also contains a specialised search syntax that is used to instruct its solvers \autocite{van-hentenryck-2000-opl-search}.
This syntax allows the modellers full programmatic control over how the solver explores the search space depending on the current state of the variables.
This offers to 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 and find a solution where the \variable{} \mzninline{x} takes a value smaller than \mzninline{y}, if it does not find a solution, then it tries finding a solution where the opposite is true.
This search specification, like many others imaginable, cannot be enforced using \minizinc\ \gls{search-heuristic} \glspl{annotation}.
To support \gls{opl}'s dedicated search language, the language is tightly integrated with its dedicated \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 possible to connect other \solvers{} to \gls{opl}.
\subsection{Essence}%
\label{sub:back-essence}
\gls{essence} \autocite{frisch-2007-essence} is another high-level \cml{}.
It is cherished for its adoption of high-level \variable{} types.
In addition to all variable types that are contained in \minizinc{}, \gls{essence} also contains:
\begin{itemize}
\item Finite sets of non-integer types,
\item finite multi-sets of any type,
\item finite (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 arbitrary nested and the modeller could, for example, define a \variable{} that is a sets 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} by considering the well-known \gls{sgp}.
The \gls{sgp} considers a community of golfers that 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 goals of the problem is find a way to split the groups different every week, such that no two golfers will meet each other twice.
Two golfers can only be part of the same group once.
A \cmodel{} for the \gls{sgp} in \gls{essence} can be seen in \cref{lst:back-essence-sgp}.
\begin{listing}
\plainfile{assets/listing/back_sgp.essence}
\caption{\label{lst:back-essence-sgp} An \gls{essence} model describing the \gls{sgp}}
\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 line \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 declares a new type to represent the golfers.
Most of the problem is modelled in the \variable{} declared 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 of the implied constraints.
Because the \variable{} reasons about partitions of the golfers, a correct 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 the no 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{} the same problem could be modelled can be found in \cref{lst:back-mzn-sgp}.
The \minizinc{} model start similar to the essence model, in 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 a 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{} are implied in the \gls{essence} model.
This is an example where the use of high-level types helps give 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.
\begin{listing}
\mznfile{assets/listing/back_sgp.mzn}
\caption{\label{lst:back-mzn-sgp} An \minizinc{} model describing the \gls{sgp}}
\end{listing}
\end{example}
\Gls{essence} allows the use of many high-level \variable{} and \constraints{} on these \variables{}.
Since these types of \variables{} are often not \gls{native} to the \solver{}.
\Gls{rewriting} will translate these types into \gls{native} \variables{} and \constraints{}.
However, along the way there can be many decisions about the representation used for the \solver{}.
There are often multiple ways to represent a high-level \variable{}, or how to enforce its implicit \constraints{}.
Unlike \minizinc{}, \Gls{essence} does not allow modellers or \solvers{} to influence this process directly.
The process is internal to the \gls{essence} compiler.
\section{Term Rewriting}%
\label{sec:back-term}
\glsreset{trs}
At the heart of the \gls{rewriting} process, to transform 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 one or more \glspl{term} \(r_{1}, \ldots, r_{n}\) \autocite{baader-1998-term-rewriting}.
A \gls{term} is an expression with nested sub-expressions consisting of \emph{function} and \emph{constant} symbols.
An example of a term is \(F(0 + 1,F(1,0))\), where \(F\) and \(+\) are function symbols and \(0\) and \(1\) are constant symbols.
In a term rewriting rule, a term can also contain a \emph{term variable} which captures a term sub-expression.
\begin{example}
Consider the following \gls{trs} consists of some (well-known) rules to rewrite logical and operations.
\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, you always arrive at a \gls{normal-form} (\ie, a set of \glspl{term} for which no more rules apply).
A \gls{trs} is confluent if, no-matter what order the term rewriting rules are applied, you always arrive at the same \gls{normal-form} (if you arrive at a \gls{normal-form}).
It is trivial to see that our previous example is non-terminating, since you can repeat rule \(r_{3}\) an infinite amount of times.
The system, however, is confluent as, if it arrives at the same \gls{normal-form}: if the term contains any \(0\), then the result is \(0\); otherwise, the result is \(1\).
These properties are also interesting when in the translation process of a \minizinc{} instance into \flatzinc{}.
The \gls{confluence} of the \gls{rewriting} process ensures that the same \gls{slv-mod} is produced independently of the order in which the \minizinc{} \instance{} is processed.
This is a desirable quality as it makes the behaviour of the \gls{rewriting} process more predictable.
Many of the techniques used by \solvers{} targeted by \minizinc{} are proven to be complete.
Meaning 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).
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 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 \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 might 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 relationship between different variables.
In cases where an instantiated solution is required, a special \mzninline{labeling} \constraint{} is used to force a variable to take a constant value.
Similarly, there is a \mzninline{minimize} \constraint{} 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 no inconsistency is detected between the \gls{native} \constraints{} that are produced.
If all the possible ways of 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 rewriting is found, it is possible to continue the process.
This way you find all possible correct ways to rewrite the program.
To implement this mechanism there is a tight integration between the \solver{}, referred to as the \emph{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{} 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 influence the correctness of a constraint logic program.
Some \solvers{} are incomplete; it is unable to tell if some of its \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 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 rules 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, no \constraints{} are 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 categorised 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 \emph{frontend}, the \emph{middle-end}, and the \emph{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 to the level required by the compilation targets.
The final transformation to the format required by the compilation target are performed by the backend.
When a \compiler{} is separated into these few steps, then adding support for new language or compilation target only require the addition of a frontend or backend respectively.
The \minizinc{} compilation process categorised in the same three categories, as shown in \cref{fig:back-mzn-comp}.
In the frontend, a \minizinc{} model is combined with its data into an \instance{}.
The instance 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, \eg\ remove the ``syntactic sugar''.
For instance, replacing the usage of 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: the flattening and the optimisation.
During the flattening process the \minizinc{} model is rewritten into a \gls{slv-mod}.
It could be noted that the 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 optimise this model: shrink \domains{} of \variables{}, remove \constraints{} that are proven to hold, and remove \variables{} that have become unused.
These optimisation techniques are discussed in \cref{subsec:back-fzn-optimisation}.
The backend converts the internal \gls{slv-mod} into a format that to be used by the targeted \solver{}.
Given the formatted artefact, a solver process, controlled by the backend, is then be started.
Whenever the solver process produces a \gls{sol}, the backend reconstructs the \gls{sol} to the specification of the original \minizinc{} model.
In this section we discuss the \gls{rewriting} and optimisation 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 call instruction, 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 transverses the declarations, \constraints{}, and the solver goal and rewrite any expression contained in these items.
During the \gls{rewriting} of an expression, the expression 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 \gls{generator} expressions.
Expression containing \glspl{generator}, such as array \glspl{comprehension} and loops, have to be instantiated before their sub-expression is rewritten.
The compiler exhaustively binds the values of the \gls{generator} to the specified identifiers.
For each iteration the compiler rewrites the sub-expression and collects its 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 functions 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 expression, like \gls{operator} expressions, variable array access, and if-then-else expressions, might 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-expression are replaced by literals or identifiers.
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, a new \variable{} 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 new \variable{} and defining \constraints{} happens in one of two ways:
\begin{itemize}
\item For Boolean expressions in a non-\rootc{} context, the new \variable{} is inserted by the \gls{rewriting} process itself.
To constrain this \variable{}, 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 \mzninline{true} if the constraint is \gls{satisfied}, and \mzninline{false} otherwise.
For example, the constraint in \minizinc{}
\begin{mzn}
constraint b \/ this_call(x, y);
\end{mzn}
\noindent{} is during \gls{rewriting} turned into:
\begin{mzn}
var bool: i1;
constraint this_call_reif(x, y, i1);
constraint b \/ i1
\end{mzn}
\item For other expressions, the \variable{} 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} it explicitly creates a new \variable{}, constrains this \variable{} to take to correct value, and returns the newly created \variable{}.
\end{itemize}
These are the basic steps that are followed to rewrite \minizinc{} instance.
This is, however, not the complete process.
The quality of the resulting \gls{slv-mod} is of the utmost importance.
A \gls{slv-mod} containing extra \variables{} and \constraints{} that do not add any information to the solving process might 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 \crefrange{subsec:back-cse}{subsec:back-delayed-rew}, we discuss the most important techniques used to improve the \gls{rewriting} process.
\subsection{Common Sub-expression Elimination}%
\label{subsec:back-cse}
\glsreset{cse}
The evaluation of a \minizinc{} expression cannot have any side effects.
Having evaluated one expression cannot influence the result of evaluating another expression.
Similarly, the evaluation of the same expression 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 optimisation \autocite{cocke-1970-cse}.
\Gls{cse} has also proven to be very effective in discrete optimisation \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 \constraint{}
\begin{mzn}
constraint (abs(x)*2 >= 20) \/ (abs(x)+5 >= 15);
\end{mzn}
\noindent{}the expression \mzninline{abs(x)} occurs twice.
There is however no 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 the same instantiation of a function are only evaluated once, the \minizinc{} \compiler{} uses memorisation.
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 flattened 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 \gls{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 solving process if we detect that a 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, it is then replaced with the constant \mzninline{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 \mzninline{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 \mzninline{true} and the evaluation proceeds as normal with the \rootc{} context \constraint{}.
\begin{example}
Consider the fragment:
\begin{mzn}
function var bool: p(var int: x, var int: y) = q(x) /\ r(y);
constraint b0 <-> q(x);
constraint b1 <-> t(x,y);
constraint b1 <-> p(x,y);
\end{mzn}
If we process the \constraints{} in order we create a reified call to \mzninline{q(x)} for the original call.
Suppose processing the second \constraint{} we discover \mzninline{t(x,y)} is \mzninline{true}, fixing \mzninline{b1}.
When we then process \mzninline{q(x)} in instantiation of the call \mzninline{p(x,y)}, we find it is the \rootc{} context.
So \gls{cse} needs to set \mzninline{b0} to \mzninline{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 might mean that the \constraints{} 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 might 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 a reified \constraints{} 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 \mzninline{false}, which means that the \constraint{} is simplified to \mzninline{true}.
\end{example}
During \gls{rewriting}, the \minizinc{} compiler actively removes values from the \domain{} when it encounters \constraints{} that trivially reduces it.
For example, \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, however, does not perform more complex \gls{propagation}, like the \gls{propagation} of \glspl{global}.
\subsection{Constraint Aggregation}%
\label{subsec:back-aggregation}
Complex \minizinc{} expressions sometimes result in the creation of many \glspl{ivar} 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 \glspl{ivar} 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 \glspl{ivar}.
Since many \solvers{} support linear \constraints{}, it is often an additional burden to have \glspl{ivar} that have to be given a value in the solution.
\end{example}
This can be resolved using the \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 \glspl{ivar}.
For example, the arithmetic definitions are combined into linear \constraints{}, Boolean logic 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-expression 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 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{} has been reduced as much as possible, in order to take advantage of the tightest possible bounds.
On the other hand, evaluating a predicate may also \emph{impose} new 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) might be flattened 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 might also interfere with the \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, if when aggregating Boolean clauses comes across an expression that needs to be reified, then a new Boolean \variable{} is created and the reified \constraint{} is delayed.
The problem is, however, that if the definition of this \gls{reif} turn out to be in terms of Boolean clauses as well, then this definition could have been aggregated as well.
Because the \compiler{} does not revisit \gls{aggregation}, this does not happen.
\subsection{FlatZinc Optimisation}%
\label{subsec:back-fzn-optimisation}
After the \compiler{} is done \gls{rewriting} the \minizinc{} instance, it enters the optimisation phase.
This phase occurs at the level at which the targeted \solver{} operates.
Depending on this \solver{}, the \minizinc{} \compiler{} might still understand the meaning of certain \constraints{}.
In these cases, \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 main focus of the \compiler{} is to propagate Boolean \constraints{}.
The \compiler{} tries to reduce the number of Boolean \variables{} and tries to reduce the number of literals in clauses and conjunctions.
The additional \gls{propagation} might fix the result of the \gls{reif} of a \constraint{}.
If this \constraint{} is not yet rewritten, then the \solver{} knows to use a direct \constraint{} instead of a reified version.
Even more important than the Boolean \constraints{}, are equality \constraints{}.
During the \gls{rewriting} process we are in a unique position to perform effective equality propagation.
Since they both have to take the same value, only a single \variable{} is required in the \gls{slv-mod} to represent them both.
Whenever any (recognisable) equality \constraint{} is found during the optimisation phase, it is removed and one of the \variables{} is replaced by the other.
Once initialised, it is generally not possible for \solvers{} to perform this replacement internally.
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