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chapters/2_background.tex
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@ -2,8 +2,8 @@
\chapter{Background}\label{ch:background} \chapter{Background}\label{ch:background}
%************************************************ %************************************************
A goal shared between all programming languages is to provide a certain level of A goal shared between all programming languages is to provide a certain level
abstraction: an assembly language allows you to abstract from the binary of abstraction: an assembly language allows you to abstract from the binary
instructions and memory positions; Low-level imperial languages, like FORTRAN, instructions and memory positions; Low-level imperial languages, like FORTRAN,
were the first to allow you to abstract from the processor architecture of the were the first to allow you to abstract from the processor architecture of the
target machine; and nowadays writing a program requires little knowledge of the target machine; and nowadays writing a program requires little knowledge of the
@ -11,21 +11,22 @@ actual workings of the hardware on which the program is executed.
Freuder states that the ``Holy Grail'' of programming languages would be where Freuder states that the ``Holy Grail'' of programming languages would be where
the user merely states the problem, and the computer solves it and that the user merely states the problem, and the computer solves it and that
\gls{constraint-modelling} is one of the biggest steps towards this goal to this \gls{constraint-modelling} is one of the biggest steps towards this goal to
day \autocite*{freuder-1997-holygrail}. Different from imperative (and even this day \autocite*{freuder-1997-holygrail}. Different from imperative (and
other declarative) languages, in a \cml{} the modeller does not describe how to even other declarative) languages, in a \cml{} the modeller does not describe
solve the problem, but rather provides the problem requirements. You could say how to solve the problem, but rather provides the problem requirements. You
that a constraint model actually describes the solution to the problem. could say that a constraint model actually describes the solution to the
problem.
In a constraint model, instead of specifying the manner in which we can find the In a constraint model, instead of specifying the manner in which we can find
solution, we give a concise description of the problem. We describe what we the solution, we give a concise description of the problem. We describe what we
already know, the \parameters{}, what we wish to know, the \variables{}, and the already know, the \parameters{}, what we wish to know, the \variables{}, and
relationships that should exist between them, the \constraints{}. the relationships that should exist between them, the \constraints{}.
This type of combinatorial problem is typically called a \gls{csp}. The goal of This type of combinatorial problem is typically called a \gls{csp}. The goal of
a \gls{csp} is to find values for the \variables{} that satisfy the a \gls{csp} is to find values for the \variables{} that satisfy the
\constraints{} or prove that no such assignment exists. Many \cmls\ also support \constraints{} or prove that no such assignment exists. Many \cmls\ also
the modelling of \gls{cop}, where a \gls{csp} is augmented with a support the modelling of \gls{cop}, where a \gls{csp} is augmented with a
\gls{objective} \(z\). In this case the goal is to find a solution that \gls{objective} \(z\). In this case the goal is to find a solution that
satisfies all \constraints{} while minimising (or maximising) \(z\). satisfies all \constraints{} while minimising (or maximising) \(z\).
@ -37,12 +38,12 @@ requirements of the model.
\begin{example}% \begin{example}%
\label{ex:back-knapsack} \label{ex:back-knapsack}
Let us consider the following scenario: Packing for a weekend trip, I have to 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 decide which toys to bring for my dog, Audrey. We only have a small amount of
space left in the car, so we cannot bring all the toys. Since Audrey gets enjoys space left in the car, so we cannot bring all the toys. Since Audrey gets
playing with some toys more than others, we can now try and pick the toys that enjoys playing with some toys more than others, we can now try and pick the
bring Audrey the most amount of joy, but still fit in the car. The following set toys that bring Audrey the most amount of joy, but still fit in the car. The
of equations describe this knapsack problem as a \gls{cop}: following set of equations describe this knapsack problem as a \gls{cop}:
\begin{equation*} \begin{equation*}
\text{maximise}~z~\text{subject to}~ \text{maximise}~z~\text{subject to}~
@ -53,22 +54,22 @@ requirements of the model.
\end{cases} \end{cases}
\end{equation*} \end{equation*}
In these equations \(S\) is set \variable{}. It contains the selection of toys In these equations \(S\) is set \variable{}. It contains the selection of
that will be packed for the trip. \(z\) is the objective \variable{} that is toys that will be packed for the trip. \(z\) is the objective \variable{}
maximised to find the optimal selections of toys to pack. The \parameter{} that is maximised to find the optimal selections of toys to pack. The
\(T\) is the set of all the toys. The \(joy\) and \(space\) functions are \parameter{} \(T\) is the set of all the toys. The \(joy\) and \(space\)
\parameters{} used to map toys, \( t \in T\), to a value depicting the amount functions are \parameters{} used to map toys, \( t \in T\), to a value
of enjoyment and space required respectively. Finally, the \parameter{} \(C\) depicting the amount of enjoyment and space required respectively. Finally,
is that depicts the total space that is left in the car before packing the the \parameter{} \(C\) is that depicts the total space that is left in the
toys. car before packing the toys.
This constraint model gives an abstract mathematical definition of the This constraint model gives an abstract mathematical definition of the
\gls{cop} that would be easy to adjust to changes in the requirements. To \gls{cop} that would be easy to adjust to changes in the requirements. To
solve instances of this problem, however, these instances have to be solve instances of this problem, however, these instances have to be
transformed into input accepted by a \solver{}. \cmls{} are designed to allow transformed into input accepted by a \solver{}. \cmls{} are designed to allow
the modeller to express combinatorial problems similar to the above the modeller to express combinatorial problems similar to the above
mathematical definition and generate a definition that can be used by mathematical definition and generate a definition that can be used by
dedicated solvers. dedicated solvers.
\end{example} \end{example}