\libdoc{clpqr}{Constraint Logic Programming over Rationals and Reals}
\label{sec:lib:clpqr}

\begin{quote}
Author: \emph{Christian Holzbaur}, ported to SWI-Prolog by \emph{Leslie De Koninck}, K.U. Leuven
\end{quote}

This CLP(Q,R) system is a port of the CLP(Q,R) system of Sicstus
Prolog by Christian Holzbaur: Holzbaur C.: OFAI clp(q,r) Manual,
Edition 1.3.3, Austrian Research Institute for Artificial Intelligence,
Vienna, TR-95-09, 1995.%
	\footnote{http://www.ai.univie.ac.at/cgi-bin/tr-online?number+95-09}
This manual is roughly based on the manual of the above mentioned CLP(Q,R)
implementation.

The CLP(Q,R) system consists of two components: the CLP(Q) library for handling
constraints over the rational numbers and the CLP(R) library for handling
constraints over the real numbers (using floating point numbers as
representation). Both libraries offer the same predicates (with exception of
\predref{bb_inf}{4} in CLP(Q) and \predref{bb_inf}{5} in CLP(R)). It is allowed
to use both libraries in one program, but using both CLP(Q) and CLP(R)
constraints on the same variable will result in an exception.

Please note that the \pllib{clpqr} library is \emph{not} an
\jargon{autoload} library and therefore this library must be loaded
explicitly before using it:

\begin{code}
:- use_module(library(clpq)).
\end{code}

\noindent
or

\begin{code}
:- use_module(library(clpr)).
\end{code}

\noindent
\subsection{Solver predicates}
\label{sec:clpqr-predicates}
%=============================

The following predicates are provided to work with constraints:

\begin{description}
	\predicate{\Scurl}{1}{+Constraints}
Adds the constraints given by \arg{Constraints} to the constraint store.

	\predicate{entailed}{1}{+Constraint}
Succeeds if \arg{Constraint} is necessarily true within the current
constraint store. This means that adding the negation of the constraint
to the store results in failure.

	\predicate{inf}{2}{+Expression, -Inf}
Computes the infimum of \arg{Expression} within the current state of the
constraint store and returns that infimum in \arg{Inf}. This predicate
does not change the constraint store.

	\predicate{sup}{2}{+Expression, -Sup}
Computes the supremum of \arg{Expression} within the current state of
the constraint store and returns that supremum in \arg{Sup}. This
predicate does not change the constraint store.

	\predicate{minimize}{1}{+Expression}
Minimizes \arg{Expression} within the current constraint store. This is
the same as computing the infimum and equating the expression to that
infimum.

	\predicate{maximize}{1}{+Expression}
Maximizes \arg{Expression} within the current constraint store. This is
the same as computing the supremum and equating the expression to that
supremum.

	\predicate{bb_inf}{5}{+Ints, +Expression, -Inf, -Vertex, +Eps}
This predicate is offered in CLP(R) only. It computes the infimum of
\arg{Expression} within the current constraint
store, with the additional constraint that in that infimum, all
variables in \arg{Ints} have integral values. \arg{Vertex} will contain
the values of \arg{Ints} in the infimum. \arg{Eps} denotes how much a
value may differ from an integer to be considered an integer. E.g.\ when
\arg{Eps} = 0.001, then X = 4.999 will be considered as an integer (5 in
this case). \arg{Eps} should be between 0 and 0.5.

	\predicate{bb_inf}{4}{+Ints, +Expression, -Inf, -Vertex}
This predicate is offered in CLP(Q) only. It behaves the same as
\predref{bb_inf}{5} but does not use an error margin.

	\predicate{bb_inf}{3}{+Ints, +Expression, -Inf}
The same as \predref{bb_inf}{5} or \predref{bb_inf}{4} but without returning
the values of the integers. In CLP(R), an error margin of 0.001 is used.

	\predicate{dump}{3}{+Target, +Newvars, -CodedAnswer}
Returns the constraints on \arg{Target} in the list \arg{CodedAnswer}
where all variables of \arg{Target} have been replaced by \arg{NewVars}.
This operation does not change the constraint store. E.g.\ in

\begin{code}
dump([X,Y,Z],[x,y,z],Cons)
\end{code}

\noindent
\verb$Cons$ will contain the constraints on X, Y and Z, where these variables
have been replaced by atoms x, y and z.

\end{description}

\subsection{Syntax of the predicate arguments}
\label{sec:clpqr-arg-syntax}
%=============================================

The arguments of the predicates defined in the subsection above are
defined in \tabref{clpqrbnf}. Failing to meet the syntax rules will
result in an exception.

\begin{table}
\begin{center}
\begin{tabular}{|lrl|l|}
\hline
\bnfmeta{Constraints} \isa \bnfmeta{Constraint}				& single constraint \\
	      \ora \bnfmeta{Constraint} , \bnfmeta{Constraints}		& conjunction \\
	      \ora \bnfmeta{Constraint} ; \bnfmeta{Constraints}		& disjunction \\

\bnfmeta{Constraint} \isa \bnfmeta{Expression} {\Slt} \bnfmeta{Expression}		& less than \\
	     \ora \bnfmeta{Expression} {\Sgt} \bnfmeta{Expression}		& greater than \\
	     \ora \bnfmeta{Expression} {\Sle} \bnfmeta{Expression}	& less or equal \\
	     \ora {\Sel}(\bnfmeta{Expression}, \bnfmeta{Expression})	& less or equal \\
	     \ora \bnfmeta{Expression} {\Sge} \bnfmeta{Expression}	& greater or equal \\
	     \ora \bnfmeta{Expression} {\Sane} \bnfmeta{Expression}	& not equal \\
	     \ora \bnfmeta{Expression} =:= \bnfmeta{Expression}		& equal \\
	     \ora \bnfmeta{Expression} = \bnfmeta{Expression}		& equal \\

\bnfmeta{Expression} \isa \bnfmeta{Variable}				& Prolog variable \\
	     \ora \bnfmeta{Number}				& Prolog number \\
	     \ora +\bnfmeta{Expression}				& unary plus \\
	     \ora -\bnfmeta{Expression}				& unary minus \\
	     \ora \bnfmeta{Expression} + \bnfmeta{Expression}		& addition \\
	     \ora \bnfmeta{Expression} - \bnfmeta{Expression}		& substraction \\
	     \ora \bnfmeta{Expression} * \bnfmeta{Expression}		& multiplication \\
	     \ora \bnfmeta{Expression} / \bnfmeta{Expression}		& division \\
	     \ora abs(\bnfmeta{Expression})			& absolute value \\
	     \ora sin(\bnfmeta{Expression})			& sine \\
	     \ora cos(\bnfmeta{Expression})			& cosine \\
	     \ora tan(\bnfmeta{Expression})			& tangent \\
	     \ora exp(\bnfmeta{Expression})			& exponent \\
	     \ora pow(\bnfmeta{Expression})			& exponent \\
	     \ora \bnfmeta{Expression} {\Shat} \bnfmeta{Expression}		& exponent \\
	     \ora min(\bnfmeta{Expression}, \bnfmeta{Expression})	& minimum \\
	     \ora max(\bnfmeta{Expression}, \bnfmeta{Expression})	& maximum \\
\hline
\end{tabular}
    \caption{CLP(Q,R) constraint BNF}
    \label{tab:clpqrbnf}
\end{center}
\end{table}


\subsection{Use of unification}
\label{sec:clpqr-unification}

Instead of using the \predref{\Scurl}{1} predicate, you can also use the standard
unification mechanism to store constraints. The following code samples
are equivalent:

\begin{itemlist}

\item [Unification with a variable]

\begin{code}
{X =:= Y}
{X = Y}
X = Y
\end{code}

\noindent
\item [Unification with a number]

\begin{code}
{X =:= 5.0}
{X = 5.0}
X = 5.0
\end{code}

\noindent
\end{itemlist}

\subsection{Non-linear constraints}
\label{sec:clpqr-non-linear}
%==================================

The CLP(Q,R) system deals only passively with non-linear constraints. They
remain in a passive state until certain conditions are satisfied. These
conditions, which are called the isolation axioms, are given in
\tabref{clpqraxioms}.

\begin{table}
\begin{center}
\begin{tabular}{|l|p{0.3\linewidth}|p{0.3\linewidth}|}
\hline
$A = B * C$ & B or C is ground			& A = 5 * C or A = B * 4 \\
	    & A and (B or C) are ground		& 20 = 5 * C or 20 = B * 4 \\
\hline
$A = B / C$ & C is ground			& A = B / 3 \\
	    & A and B are ground		& 4 = 12 / C \\
\hline
$X = min(Y,Z)$ & Y and Z are ground		& X = min(4,3) \\
$X = max(Y,Z)$ & Y and Z are ground		& X = max(4,3) \\
$X = abs(Y)$   & Y is ground			& X = abs(-7) \\
\hline
$X = pow(Y,Z)$ & X and Y are ground		& 8 = 2 {\Shat} Z \\
$X = exp(Y,Z)$ & X and Z are ground		& 8 = Y {\Shat} 3 \\
$X = Y$ {\Shat} $Z$  & Y and Z are ground		& X = 2 {\Shat} 3 \\
\hline
$X = sin(Y)$	& X is ground			& 1 = sin(Y) \\
$X = cos(Y)$	& Y is ground			& X = sin(1.5707) \\
$X = tan(Y)$    && \\
\hline
\end{tabular}
	\caption{CLP(Q,R) isolating axioms}
	\label{tab:clpqraxioms}
\end{center}
\end{table}

\subsection{Status and known problems}
\label{sec:clpqr-status}

The clpq and clpr libraries are `orphaned', i.e., they currently have
no maintainer.

\begin{itemlist}
    \item [Top-level output]
The top-level output may contain variables not present in the original
query:

\begin{code}
?- {X+Y>=1}.
{Y=1-X+_G2160, _G2160>=0}.

?-
\end{code}

\noindent
Nonetheless, for linear constraints this kind of answer means
unconditional satisfiability.

    \item [Dumping constraints]
The first argument of \index{dump/3}\predref{dump}{3} has to be a list of free variables at
call-time:

\begin{code}
?- {X=1},dump([X],[Y],L).
ERROR: Unhandled exception: Unknown message:
       instantiation_error(dump([1],[_G11],_G6),1)
?-
\end{code}

\noindent
\end{itemlist}