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<H2><A NAME="sec:A.8"><SPAN class="sec-nr">A.8</SPAN> <SPAN class="sec-title">library(
clpqr ): Constraint Logic Programming over Rationals and Reals</SPAN></A></H2>
<A NAME="clpqr"></A>
<A NAME="sec:lib:clpqr"></A>
<BLOCKQUOTE> Author: <EM>Leslie De Koninck</EM>, K.U. Leuven
</BLOCKQUOTE>
<P>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.<SUP class="fn">85<SPAN class="fn-text">http://www.ai.univie.ac.at/cgi-bin/tr-online?number+95-09</SPAN></SUP>
This manual is roughly based on the manual of the above mentioned
CLP(Q,R) implementation.
<P>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
<A class="pred" href="clpqr.html#bb_inf/4">bb_inf/4</A> in CLP(Q) and <A class="pred" href="clpqr.html#bb_inf/5">bb_inf/5</A>
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.
<P>Please note that the <CODE>library(clpqr)</CODE> library is <EM>not</EM>
an
<EM>autoload</EM> library and therefore this library must be loaded
explicitely before using it:
<PRE class="code">
:- use_module(library(clpq)).
</PRE>
<P>or
<PRE class="code">
:- use_module(library(clpr)).
</PRE>
<H3><A NAME="sec:A.8.1"><SPAN class="sec-nr">A.8.1</SPAN> <SPAN class="sec-title">Solver
predicates</SPAN></A></H3>
The following predicates are provided to work with constraints:
<DL>
<DT class="pubdef"><A NAME="{}/1"><STRONG>{}</STRONG>(<VAR>+Constraints</VAR>)</A></DT>
<DD class="defbody">
Adds the constraints given by <VAR>Constraints</VAR> to the constraint
store.</DD>
<DT class="pubdef"><A NAME="entailed/1"><STRONG>entailed</STRONG>(<VAR>+Constraint</VAR>)</A></DT>
<DD class="defbody">
Succeeds if <VAR>Constraint</VAR> is necessarily true within the current
constraint store. This means that adding the negation of the constraint
to the store results in failure.</DD>
<DT class="pubdef"><A NAME="inf/2"><STRONG>inf</STRONG>(<VAR>+Expression,
-Inf</VAR>)</A></DT>
<DD class="defbody">
Computes the infimum of <VAR>Expression</VAR> within the current state
of the constraint store and returns that infimum in <VAR>Inf</VAR>. This
predicate does not change the constraint store.</DD>
<DT class="pubdef"><A NAME="sup/2"><STRONG>sup</STRONG>(<VAR>+Expression,
-Sup</VAR>)</A></DT>
<DD class="defbody">
Computes the supremum of <VAR>Expression</VAR> within the current state
of the constraint store and returns that supremum in <VAR>Sup</VAR>.
This predicate does not change the constraint store.</DD>
<DT class="pubdef"><A NAME="minimize/1"><STRONG>minimize</STRONG>(<VAR>+Expression</VAR>)</A></DT>
<DD class="defbody">
Minimizes <VAR>Expression</VAR> within the current constraint store.
This is the same as computing the infimum and equation the expression to
that infimum.</DD>
<DT class="pubdef"><A NAME="maximize/1"><STRONG>maximize</STRONG>(<VAR>+Expression</VAR>)</A></DT>
<DD class="defbody">
Maximizes <VAR>Expression</VAR> within the current constraint store.
This is the same as computing the supremum and equating the expression
to that supremum.</DD>
<DT class="pubdef"><A NAME="bb_inf/5"><STRONG>bb_inf</STRONG>(<VAR>+Ints,
+Expression, -Inf, -Vertex, +Eps</VAR>)</A></DT>
<DD class="defbody">
This predicate is offered in CLP(R) only. It computes the infimum of
<VAR>Expression</VAR> within the current constraint store, with the
additional constraint that in that infimum, all variables in <VAR>Ints</VAR>
have integral values. <VAR>Vertex</VAR> will contain the values of <VAR>Ints</VAR>
in the infimum. <VAR>Eps</VAR> denotes how much a value may differ from
an integer to be considered an integer. E.g. when
<VAR>Eps</VAR> = 0.001, then X = 4.999 will be considered as an integer
(5 in this case). <VAR>Eps</VAR> should be between 0 and 0.5.</DD>
<DT class="pubdef"><A NAME="bb_inf/4"><STRONG>bb_inf</STRONG>(<VAR>+Ints,
+Expression, -Inf, -Vertex</VAR>)</A></DT>
<DD class="defbody">
This predicate is offered in CLP(Q) only. It behaves the same as
<A class="pred" href="clpqr.html#bb_inf/5">bb_inf/5</A> but does not use
an error margin.</DD>
<DT class="pubdef"><A NAME="bb_inf/3"><STRONG>bb_inf</STRONG>(<VAR>+ints,
+Expression, -Inf</VAR>)</A></DT>
<DD class="defbody">
The same as <A class="pred" href="clpqr.html#bb_inf/5">bb_inf/5</A> or <A class="pred" href="clpqr.html#bb_inf/4">bb_inf/4</A>
but without returning the values of the integers. In CLP(R), an error
margin of 0.001 is used.</DD>
<DT class="pubdef"><A NAME="dump/3"><STRONG>dump</STRONG>(<VAR>+Target,
+Newvars, -CodedAnswer</VAR>)</A></DT>
<DD class="defbody">
Returns the constraints on <VAR>Target</VAR> in the list <VAR>CodedAnswer</VAR>
where all variables of <VAR>Target</VAR> have veen replaced by <VAR>NewVars</VAR>.
This operation does not change the constraint store. E.g. in
<PRE class="code">
dump([X,Y,Z],[x,y,z],Cons)
</PRE>
<P>Cons will contain the constraints on X, Y and Z where these variables
have been replaced by atoms x, y and z.
<P></DD>
</DL>
<H3><A NAME="sec:A.8.2"><SPAN class="sec-nr">A.8.2</SPAN> <SPAN class="sec-title">Syntax
of the predicate arguments</SPAN></A></H3>
The arguments of the predicates defined in the subsection above are
defined in <A class="tab" href="clpqr.html#tab:clpqrbnf">table 9</A>.
Failing to meet the syntax rules will result in an exception.
<P>
<CENTER>
<TABLE BORDER=2 FRAME=box RULES=groups>
<TR VALIGN=top><TD><Constraints> </TD><TD ALIGN=right>::=</TD><TD><<VAR>Constraint</VAR>> </TD><TD>single
constraint </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Constraint</VAR>>
, <<VAR>Constraints</VAR>> </TD><TD>conjunction </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Constraint</VAR>>
; <<VAR>Constraints</VAR>> </TD><TD>disjunction </TD></TR>
<TR VALIGN=top><TD>
<P><Constraint> </TD><TD ALIGN=right>::=</TD><TD><<VAR>Expression</VAR>> <CODE><</CODE> <<VAR>Expression</VAR>> </TD><TD>less
than </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>> <CODE>></CODE> <<VAR>Expression</VAR>> </TD><TD>greater
than </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>> <CODE>=<</CODE> <<VAR>Expression</VAR>> </TD><TD>less
or equal </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><CODE><=</CODE>(<<VAR>Expression</VAR>>, <<VAR>Expression</VAR>>)</TD><TD>less
or equal </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>> <CODE>>=</CODE> <<VAR>Expression</VAR>> </TD><TD>greater
or equal </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>> <CODE>=\=</CODE> <<VAR>Expression</VAR>> </TD><TD>not
equal </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
=:= <<VAR>Expression</VAR>> </TD><TD>equal </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
= <<VAR>Expression</VAR>> </TD><TD>equal </TD></TR>
<TR VALIGN=top><TD>
<P><Expression> </TD><TD ALIGN=right>::=</TD><TD><<VAR>Variable</VAR>> </TD><TD>Prolog
variable </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Number</VAR>> </TD><TD>Prolog
number (float, integer) </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>+<<VAR>Expression</VAR>> </TD><TD>unary
plus </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>-<<VAR>Expression</VAR>> </TD><TD>unary
minus </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
+ <<VAR>Expression</VAR>> </TD><TD>addition </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
- <<VAR>Expression</VAR>> </TD><TD>substraction </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
* <<VAR>Expression</VAR>> </TD><TD>multiplication </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>>
/ <<VAR>Expression</VAR>> </TD><TD>division </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>abs(<<VAR>Expression</VAR>>)</TD><TD>absolute
value </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>sin(<<VAR>Expression</VAR>>)</TD><TD>sine </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>cos(<<VAR>Expression</VAR>>)</TD><TD>cosine </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>tan(<<VAR>Expression</VAR>>)</TD><TD>tangent </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>exp(<<VAR>Expression</VAR>>)</TD><TD>exponent </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>pow(<<VAR>Expression</VAR>>)</TD><TD>exponent </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD><<VAR>Expression</VAR>> <CODE>^</CODE> <<VAR>Expression</VAR>> </TD><TD>exponent </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>min(<<VAR>Expression</VAR>>, <<VAR>Expression</VAR>>)</TD><TD>minimum </TD></TR>
<TR VALIGN=top><TD></TD><TD ALIGN=right>|</TD><TD>max(<<VAR>Expression</VAR>>, <<VAR>Expression</VAR>>)</TD><TD>maximum </TD></TR>
</TABLE>
<TABLE ALIGN=center WIDTH="75%"><TR><TD>
<B>Table 9 : </B>CLP(Q,R) constraint BNF</TABLE>
<A NAME="tab:clpqrbnf"></A>
</CENTER>
<H3><A NAME="sec:A.8.3"><SPAN class="sec-nr">A.8.3</SPAN> <SPAN class="sec-title">Use
of unification</SPAN></A></H3>
<P>Instead of using the <A class="pred" href="clpqr.html#{}/1">{}/1</A>
predicate, you can also use the standard unification mechanism to store
constraints. The following code samples are equivalent:
<P>
<UL>
<LI><I>Unification with a variable</I><BR>
<PRE class="code">
{X =:= Y}
{X = Y}
X = Y
</PRE>
<P>
<LI><I>Unification with a number</I><BR>
<PRE class="code">
{X =:= 5.0}
{X = 5.0}
X = 5.0
</PRE>
<P>
</UL>
<H3><A NAME="sec:A.8.4"><SPAN class="sec-nr">A.8.4</SPAN> <SPAN class="sec-title">Non-linear
constraints</SPAN></A></H3>
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
<A class="tab" href="clpqr.html#tab:clpqraxioms">table 10</A>.
<P>
<CENTER>
<TABLE BORDER=2 FRAME=box RULES=groups>
<TR VALIGN=top><TD><VAR>A = B * C</VAR> </TD><TD>B or C is ground</TD><TD>A
= 5 * C or A = B * 4 </TD></TR>
<TR VALIGN=top><TD></TD><TD>A and (B or C) are ground</TD><TD>20 = 5 * C
or 20 = B * 4 </TD></TR>
<TBODY>
<TR VALIGN=top><TD><VAR>A = B / C</VAR> </TD><TD>C is ground</TD><TD>A =
B / 3 </TD></TR>
<TR VALIGN=top><TD></TD><TD>A and B are ground</TD><TD>4 = 12 / C </TD></TR>
<TBODY>
<TR VALIGN=top><TD><VAR>X = min(Y,Z)</VAR> </TD><TD>Y and Z are ground</TD><TD>X
= min(4,3) </TD></TR>
<TR VALIGN=top><TD><VAR>X = max(Y,Z)</VAR> </TD><TD>Y and Z are ground</TD><TD>X
= max(4,3) </TD></TR>
<TR VALIGN=top><TD><VAR>X = abs(Y)</VAR> </TD><TD>Y is ground</TD><TD>X
= abs(-7) </TD></TR>
<TBODY>
<TR VALIGN=top><TD><VAR>X = pow(Y,Z)</VAR> </TD><TD>X and Y are ground</TD><TD>8
= 2 <CODE>^</CODE> Z </TD></TR>
<TR VALIGN=top><TD><VAR>X = exp(Y,Z)</VAR> </TD><TD>X and Z are ground</TD><TD>8
= Y <CODE>^</CODE> 3 </TD></TR>
<TR VALIGN=top><TD><VAR>X = Y</VAR> <CODE>^</CODE> <VAR>Z</VAR> </TD><TD>Y
and Z are ground</TD><TD>X = 2 <CODE>^</CODE> 3 </TD></TR>
<TBODY>
<TR VALIGN=top><TD><VAR>X = sin(Y)</VAR> </TD><TD>X is ground</TD><TD>1
= sin(Y) </TD></TR>
<TR VALIGN=top><TD><VAR>X = cos(Y)</VAR> </TD><TD>Y is ground</TD><TD>X
= sin(1.5707) </TD></TR>
<TR VALIGN=top><TD><VAR>X = tan(Y)</VAR> </TD><TD></TD></TR>
</TABLE>
<TABLE ALIGN=center WIDTH="75%"><TR><TD>
<B>Table 10 : </B>CLP(Q,R) isolating axioms</TABLE>
<A NAME="tab:clpqraxioms"></A>
</CENTER>
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