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<h2 id="sec:clpqr"><a id="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 id="sec:clpqr"></a>
<a id="sec:lib:clpqr"></a>
<blockquote> Author: <em>Christian Holzbaur</em>, ported to SWI-Prolog 
by <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">123<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 
explicitly before using it:

<pre class="code">
:- use_module(library(clpq)).
</pre>

<p>or

<pre class="code">
:- use_module(library(clpr)).
</pre>

<p><h3 id="sec:clpqr-predicates"><a id="sec:A.8.1"><span class="sec-nr">A.8.1</span> <span class="sec-title">Solver 
predicates</span></a></h3>

<a id="sec:clpqr-predicates"></a> The following predicates are provided 
to work with constraints:

<dl class="latex">
<dt class="pubdef"><a id="{}/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 id="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 id="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 id="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 id="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 equating the expression to 
that infimum.</dd>
<dt class="pubdef"><a id="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 id="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 id="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 id="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 id="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 been 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><code>Cons</code> will contain the constraints on X, Y and Z, where 
these variables have been replaced by atoms x, y and z.

<p></dd>
</dl>

<p><h3 id="sec:clpqr-arg-syntax"><a id="sec:A.8.2"><span class="sec-nr">A.8.2</span> <span class="sec-title">Syntax 
of the predicate arguments</span></a></h3>

<a id="sec:clpqr-arg-syntax"></a> 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.
<div style="text-align:center">
<table class="latex frame-box">
<tr><td>&lt;<var>Constraints</var>&gt; </td><td align=right>::=</td><td>&lt;<var>Constraint</var>&gt; </td><td>single 
constraint </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Constraint</var>&gt; , &lt;<var>Constraints</var>&gt; </td><td>conjunction </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Constraint</var>&gt; ; &lt;<var>Constraints</var>&gt; </td><td>disjunction </td></tr>
<tr><td>

<p>&lt;<var>Constraint</var>&gt; </td><td align=right>::=</td><td>&lt;<var>Expression</var>&gt; <code>&lt;</code> &lt;<var>Expression</var>&gt; </td><td>less 
than </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; <code>&gt;</code> &lt;<var>Expression</var>&gt; </td><td>greater 
than </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; <code>=&lt;</code> &lt;<var>Expression</var>&gt; </td><td>less 
or equal </td></tr>
<tr><td></td><td align=right>|</td><td><code>&lt;=</code>(&lt;<var>Expression</var>&gt;, &lt;<var>Expression</var>&gt;)</td><td>less 
or equal </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; <code>&gt;=</code> &lt;<var>Expression</var>&gt; </td><td>greater 
or equal </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; <code>=\=</code> &lt;<var>Expression</var>&gt; </td><td>not 
equal </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; =:= &lt;<var>Expression</var>&gt; </td><td>equal </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; = &lt;<var>Expression</var>&gt; </td><td>equal </td></tr>
<tr><td>

<p>&lt;<var>Expression</var>&gt; </td><td align=right>::=</td><td>&lt;<var>Variable</var>&gt; </td><td>Prolog 
variable </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Number</var>&gt; </td><td>Prolog 
number </td></tr>
<tr><td></td><td align=right>|</td><td>+&lt;<var>Expression</var>&gt; </td><td>unary 
plus </td></tr>
<tr><td></td><td align=right>|</td><td>-&lt;<var>Expression</var>&gt; </td><td>unary 
minus </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; + &lt;<var>Expression</var>&gt; </td><td>addition </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; - &lt;<var>Expression</var>&gt; </td><td>substraction </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; * &lt;<var>Expression</var>&gt; </td><td>multiplication </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; / &lt;<var>Expression</var>&gt; </td><td>division </td></tr>
<tr><td></td><td align=right>|</td><td>abs(&lt;<var>Expression</var>&gt;)</td><td>absolute 
value </td></tr>
<tr><td></td><td align=right>|</td><td>sin(&lt;<var>Expression</var>&gt;)</td><td>sine </td></tr>
<tr><td></td><td align=right>|</td><td>cos(&lt;<var>Expression</var>&gt;)</td><td>cosine </td></tr>
<tr><td></td><td align=right>|</td><td>tan(&lt;<var>Expression</var>&gt;)</td><td>tangent </td></tr>
<tr><td></td><td align=right>|</td><td>exp(&lt;<var>Expression</var>&gt;)</td><td>exponent </td></tr>
<tr><td></td><td align=right>|</td><td>pow(&lt;<var>Expression</var>&gt;)</td><td>exponent </td></tr>
<tr><td></td><td align=right>|</td><td>&lt;<var>Expression</var>&gt; <code>^</code> &lt;<var>Expression</var>&gt; </td><td>exponent </td></tr>
<tr><td></td><td align=right>|</td><td>min(&lt;<var>Expression</var>&gt;, &lt;<var>Expression</var>&gt;)</td><td>minimum </td></tr>
<tr><td></td><td align=right>|</td><td>max(&lt;<var>Expression</var>&gt;, &lt;<var>Expression</var>&gt;)</td><td>maximum </td></tr>
</table>
<div class="caption"><b>Table 9 : </b>CLP(Q,R) constraint BNF</div>
<a id="tab:clpqrbnf"></a>
</div>

<p><h3 id="sec:clpqr-unification"><a id="sec:A.8.3"><span class="sec-nr">A.8.3</span> <span class="sec-title">Use 
of unification</span></a></h3>

<a id="sec:clpqr-unification"></a>

<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 class="latex">
<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>

<p><h3 id="sec:clpqr-non-linear"><a id="sec:A.8.4"><span class="sec-nr">A.8.4</span> <span class="sec-title">Non-linear 
constraints</span></a></h3>

<a id="sec:clpqr-non-linear"></a> 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>.
<div style="text-align:center">
<table class="latex frame-box">
<tr><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><td></td><td>A and (B or C) are ground</td><td>20 = 5 * C or 20 = B 
* 4 </td></tr>
<tr class="hline"><td><var>A = B / C</var> </td><td>C is ground</td><td>A 
= B / 3 </td></tr>
<tr><td></td><td>A and B are ground</td><td>4 = 12 / C </td></tr>
<tr class="hline"><td><var>X = min(Y,Z)</var> </td><td>Y and Z are 
ground</td><td>X = min(4,3) </td></tr>
<tr><td><var>X = max(Y,Z)</var> </td><td>Y and Z are ground</td><td>X = 
max(4,3) </td></tr>
<tr><td><var>X = abs(Y)</var> </td><td>Y is ground</td><td>X = abs(-7) </td></tr>
<tr class="hline"><td><var>X = pow(Y,Z)</var> </td><td>X and Y are 
ground</td><td>8 = 2 <code>^</code> Z </td></tr>
<tr><td><var>X = exp(Y,Z)</var> </td><td>X and Z are ground</td><td>8 = 
Y <code>^</code> 3 </td></tr>
<tr><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>
<tr class="hline"><td><var>X = sin(Y)</var> </td><td>X is ground</td><td>1 
= sin(Y) </td></tr>
<tr><td><var>X = cos(Y)</var> </td><td>Y is ground</td><td>X = 
sin(1.5707) </td></tr>
<tr><td><var>X = tan(Y)</var> </td><td></td></tr>
</table>
<div class="caption"><b>Table 10 : </b>CLP(Q,R) isolating axioms</div>
<a id="tab:clpqraxioms"></a>
</div>

<p><h3 id="sec:clpqr-status"><a id="sec:A.8.5"><span class="sec-nr">A.8.5</span> <span class="sec-title">Status 
and known problems</span></a></h3>

<a id="sec:clpqr-status"></a>

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

<p>
<ul class="latex">
<li><i>Top-level output</i><br>
The top-level output may contain variables not present in the original 
query:

<pre class="code">
?- {X+Y&gt;=1}.
{Y=1-X+_G2160, _G2160&gt;=0}.

?-
</pre>

<p>Nonetheless, for linear constraints this kind of answer means 
unconditional satisfiability.

<p>
<li><i>Dumping constraints</i><br>
The first argument of <a id="idx:dump3:1808"></a><a class="pred" href="clpqr.html#dump/3">dump/3</a> 
has to be a list of free variables at call-time:

<pre class="code">
?- {X=1},dump([X],[Y],L).
ERROR: Unhandled exception: Unknown message:
       instantiation_error(dump([1],[_G11],_G6),1)
?-
</pre>

<p>
</ul>
</body></html>
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