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<HTML><HEAD><TITLE>5.7 Generic Propagators</TITLE><LINK href="ozdoc.css" rel="stylesheet" type="text/css"></HEAD><BODY><TABLE align="center" border="0" cellpadding="6" cellspacing="6" class="nav"><TR bgcolor="#DDDDDD"><TD><A href="node20.html#section.fd.watching">&lt;&lt; Prev</A></TD><TD><A href="node14.html">- Up -</A></TD><TD><A href="node22.html#section.fd.nonlinear">Next &gt;&gt;</A></TD></TR></TABLE><DIV id="section.fd.arithmetic"><H2><A name="section.fd.arithmetic">5.7 Generic Propagators</A></H2><P>The generic propagators <CODE>FD<SPAN class="keyword">.</SPAN>sum</CODE>, <CODE>FD<SPAN class="keyword">.</SPAN>sumC</CODE> and <CODE>FD<SPAN class="keyword">.</SPAN>sumCN</CODE> do interval propagation. The propagators <CODE>FD<SPAN class="keyword">.</SPAN>sumAC</CODE> and <CODE>FD<SPAN class="keyword">.</SPAN>sumACN</CODE> do interval propagation but may also cut holes into domains. For example, </P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>dom&nbsp;0<SPAN class="keyword">#</SPAN>10&nbsp;[X&nbsp;Y]}<BR>{FD<SPAN class="keyword">.</SPAN>sumAC&nbsp;[1&nbsp;<SPAN class="keyword">~</SPAN>1]&nbsp;[X&nbsp;Y]&nbsp;<SPAN class="string">'&gt;:'</SPAN>&nbsp;8}</CODE></BLOCKQUOTE><P> will reduce the domains of <CODE>X</CODE> and <CODE>Y</CODE> to <IMG alt="\{0,1,9,10\}" src="latex24.gif">. Except for propagators <CODE>FD<SPAN class="keyword">.</SPAN>sumCN</CODE> and <CODE>FD<SPAN class="keyword">.</SPAN>sumACN</CODE>, equality is exploited, e.&nbsp;g. <CODE>{FD<SPAN class="keyword">.</SPAN>sumC&nbsp;[2&nbsp;3]&nbsp;[A&nbsp;A]&nbsp;<SPAN class="string">'=:'</SPAN>&nbsp;10}</CODE> is equivalent to <CODE>{FD<SPAN class="keyword">.</SPAN>sumC&nbsp;[5]&nbsp;[A]&nbsp;<SPAN class="string">'=:'</SPAN>&nbsp;10}</CODE>. </P><P class="margin"><IMG alt="\underline{x}" src="latex29.gif">,<IMG alt="\overline{x}" src="latex30.gif"></P><P> Let <I>S</I> denote the current constraint store and <I>x</I> a finite domain integer.<IMG alt="\underline{x}" src="latex29.gif"> denotes the largest integer such that <IMG alt="S \models x \geq \underline{x}" src="latex31.gif"> holds. Analogously, <IMG alt="\overline{x}" src="latex30.gif"> denotes the smallest integer such that <IMG alt="S \models x \leq \overline{x}" src="latex32.gif"> holds. </P><P class="margin"><IMG alt="\lfloor n \rfloor$, $\lceil n \rceil" src="latex33.gif"></P><P> Let <I>n</I> denote a real number. <IMG alt="\lfloor n \rfloor" src="latex34.gif"> denotes the largest integer which is equal or smaller than <I>n</I>. Analogously, <IMG alt="\lceil n \rceil" src="latex35.gif"> denotes the smallest integer which is equal or larger than <I>n</I>. </P><DL><DT><A name="label166"></A> <CODE>sum</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sum&nbsp;</CODE><CODE>*<I>Dv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator for </P><P><IMG alt="1*{\tt D}_1+\ldots+1*{\tt D}_n+(-1)*{\tt D}\;\sim_{{\tt A}}\;0" src="latex36.gif"></P><P>For the operational semantics see <CODE>FD<SPAN class="keyword">.</SPAN>sumC</CODE>. For the relation symbol <CODE><SPAN class="string">'\=:'</SPAN></CODE>, the propagator waits until at most one non-determined variable is left. Then the appropriate value is discarded from the variable's domain. For the other relations, the propagator does interval propagation. </P></DD><DT><A name="label168"></A> <CODE>sumC</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumC&nbsp;</CODE><CODE>+<I>Iv</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>Dv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator for the scalar product of the vectors <CODE><I>Iv</I></CODE> and <CODE><I>Dv</I></CODE>: </P><P><IMG alt="{\tt I}_1*{\tt D}_1+\ldots+{\tt I}_n*{\tt D}_n+(-1)*{\tt D}\;\sim_{{\tt A}}\;0" src="latex37.gif"></P><P> Let <IMG alt="{\tt D}_{n+1}" src="latex38.gif"> be <CODE><I>D</I></CODE> and <IMG alt="{\tt I}_{n+1}" src="latex39.gif"> be -1. Then, the operational semantics is defined as follows. For each product <IMG alt="{\tt I}_k*{\tt D}_k" src="latex40.gif">, an isolation (projection) is computed:</P><P><IMG alt=" {\tt I}_k*{\tt D}_k \;\sim_{{\tt A}}\; \underbrace{- \sum_{i = 1, i \neq k}^{n+1}
{\tt I}_i * {\tt D}_i}_{{\mbox{\sl RHS}}_k}." src="latex41.gif"></P><P>For the right hand side <IMG alt="{\mbox{\sl RHS}}_k" src="latex42.gif">, the upper <IMG alt="\overline{\mbox{\mbox{\sl RHS}}_k}" src="latex43.gif"> and lower limit <IMG alt="\underline{\mbox{\mbox{\sl RHS}}_k}" src="latex44.gif"> are defined as follows.</P><P><IMG alt=" ~$~
\begin{eqnarray*}
\overline{\mbox{\mbox{\sl RHS}}_k} \ =\
-\sum_{i =1, i\neq k, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\underline{{\tt D}}_i \;-\; 
\sum_{i =1, i\neq k, {\tt I}_i < 0}^{n+1} {\tt I}_i *
\overline{{\tt D}}_i 
\\
\underline{\mbox{\mbox{\sl RHS}}_k} \ = \
-\sum_{i =1, i\neq k, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\overline{{\tt D}}_i \;-\; 
\sum_{i =1, i\neq k, {\tt I}_i < 0}^{n+1} {\tt I}_i *
\underline{{\tt D}}_i
\end{eqnarray*}
~$~" src="latex45.gif"></P><P>These values are used to narrow the domain of&nbsp;<IMG alt="{\tt D}_k" src="latex46.gif"> until a fixed point is reached. We describe the propagation for the different possible values of <CODE>A</CODE>. </P><DL><DT><CODE><SPAN class="keyword">=&lt;:</SPAN></CODE></DT><DD><P>Narrowing is done according to the following inequalities.</P><P><IMG alt="{\tt D}_k \leq \left \lfloor  \frac{\overline{{\mbox{\sl RHS}}_k}}{{\tt I}_k}
\right \rfloor \quad 
\mbox{ if\ } {\tt I}_k > 0" src="latex47.gif"></P><P><IMG alt="{\tt D}_k \geq \left \lceil  \frac{\overline{{\mbox{\sl RHS}}_k}}{{\tt I}_k} \right \rceil \quad
\mbox{ if\ } {\tt I}_k < 0" src="latex48.gif"></P><P>Here <IMG alt="x\leq n" src="latex49.gif"> denotes the basic constraint <IMG alt="x \in
\{0,\ldots,n\}" src="latex50.gif"> and <IMG alt="x \geq n" src="latex51.gif"> denotes the basic constraint <IMG alt="x \in
\{n,\ldots,\codeinline{oz}{FD.sup}\}" src="latex52.gif">. </P><P>The propagator ceases to exist, if the following condition holds. </P><P><IMG alt="\sum_{i =1, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\overline{{\tt D}}_i \;+\;
\sum_{i =1, {\tt I}_i < 0}^{n+1} {\tt I}_i *
\underline{{\tt D}}_i 
\leq 0" src="latex53.gif"></P><P>As an example consider </P><BLOCKQUOTE class="code"><CODE>X&nbsp;<SPAN class="keyword">-</SPAN>&nbsp;Y&nbsp;<SPAN class="keyword">=&lt;:</SPAN>&nbsp;Z&nbsp;<SPAN class="keyword">-</SPAN>&nbsp;V</CODE></BLOCKQUOTE><P> We have the following narrowing.</P><P><IMG alt="\codeinline{oz}{X} \leq \overline{\codeinline{oz}{Z}} - \underline{\codeinline{oz}{V}} + \overline{\codeinline{oz}{Y}}
\quad \quad
\codeinline{oz}{Y} \geq \underline{\codeinline{oz}{X}} - \overline{\codeinline{oz}{Z}} + \overline{\codeinline{oz}{V}}
\quad \quad
\codeinline{oz}{Z} \geq \underline{\codeinline{oz}{X}} - \overline{\codeinline{oz}{Y}} + \underline{\codeinline{oz}{V}}
\quad\quad
\codeinline{oz}{V} \leq \overline{\codeinline{oz}{Z}} - \underline{\codeinline{oz}{X}} + \overline{\codeinline{oz}{Y}}" src="latex54.gif"></P><P>The propagator ceases to exist if <IMG alt="\overline{\codeinline{oz}{X}} - \underline{\codeinline{oz}{Y}}
\leq \underline{\codeinline{oz}{Z}} - \overline{\codeinline{oz}{V}}" src="latex55.gif"> holds. </P></DD><DT><CODE><SPAN class="keyword">&gt;=:</SPAN></CODE></DT><DD><P>This case can be reduced to <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE> due to the observation that </P><P><IMG alt="{\tt I}_1*{\tt D}_1+\ldots+{\tt I}_n*{\tt D}_n+(-1)*{\tt D}\;\geq\;0" src="latex56.gif"></P><P>is equivalent to </P><P><IMG alt="(-{\tt I}_1)*{\tt D}_1+\ldots+(-{\tt I}_n)*{\tt D}_n+1*{\tt D}\;\leq\;0" src="latex57.gif"></P><P>Alternatively, <IMG alt="\underline{\mbox{\mbox{\sl RHS}}_k}" src="latex44.gif"> can be used for the definition. </P></DD><DT><CODE><SPAN class="keyword">&lt;:</SPAN></CODE> </DT><DD><P>Analogous to <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE></P></DD><DT><CODE><SPAN class="keyword">&gt;:</SPAN></CODE> </DT><DD><P>Analogous to <CODE><SPAN class="keyword">&gt;=:</SPAN></CODE></P></DD><DT><CODE><SPAN class="keyword">=:</SPAN></CODE></DT><DD><P>In this case, the operational semantics is defined by conjunction of the formulas given for <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE> and <CODE><SPAN class="keyword">&gt;=:</SPAN></CODE>. Furthermore, coreferences are realized in that, e.&nbsp;g. the propagator <CODE>3<SPAN class="keyword">*</SPAN>X<SPAN class="keyword">=:</SPAN>3<SPAN class="keyword">*</SPAN>Y</CODE> tells the basic constraint&nbsp;<CODE>X=Y</CODE>. </P></DD><DT><CODE><SPAN class="keyword">\=:</SPAN></CODE></DT><DD><P>In this case, the propagator waits until at most one non-determined variable is left, say <IMG alt="{\tt D}_k" src="latex46.gif">. Then, <IMG alt="{\mbox{\sl RHS}}_k" src="latex42.gif"> denotes a unique integer value which is discarded from the domain of <IMG alt="{\tt D}_k" src="latex46.gif">.</P></DD></DL><P></P><P>Additional propagation is achieved through the realization of coreferences, i.&nbsp;e. equality between variables. If the store <CODE><I>S</I></CODE> entails (without loss of generality) <IMG alt="{\tt D}_1 = {\tt D}_2" src="latex58.gif">, the generic propagator evolves into:</P><P><IMG alt="({\tt I}_1 + {\tt I}_2) * {\tt D}_2 + \ldots +
{\tt I}_n*{\tt D}_n+(-1)*{\tt D}\;\sim_{{\tt A}}\;0  " src="latex59.gif"></P><P></P></DD><DT><A name="label170"></A> <CODE>sumCN</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumCN&nbsp;</CODE><CODE>+<I>Iv</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>Dvv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator for </P><P><IMG alt="{\tt I}_1*{\tt D}_{11}*\ldots*{\tt D}_{1m_1}+\ldots+{\tt I}_n*{\tt D}_{n1}*\ldots*{\tt D}_{nm_n}+
(-1)*{\tt D}\;\sim_{{\tt A}}\;0" src="latex60.gif"></P><P> Let <IMG alt="{\tt D}_{(n+1)1}" src="latex61.gif"> be <CODE><I>D</I></CODE>, <IMG alt="{\tt I}_{n+1}" src="latex39.gif"> be -1, and <IMG alt="m_{n+1}" src="latex62.gif"> be 1. Then, the operational semantics is defined as follows. For <IMG alt="k, 1
\leq k \leq n+1" src="latex63.gif">, an isolation (projection) is computed:</P><P><IMG alt="{\tt I}_{k}*{\tt D}_{k1}*\ldots*{\tt D}_{km_k} \;\sim_{{\tt A}}\; \underbrace{- {\sum_{i = 1, i \neq k}^{n+1}
{\tt I}_i * \prod_{j=1}^{m_i}{\tt D}_{ij}}}_{{\mbox{\sl RHS}}_k}" src="latex64.gif"></P><P> For the right hand side <IMG alt="{\mbox{\sl RHS}}_k" src="latex42.gif">, the upper <IMG alt="\overline{\mbox{\mbox{\sl RHS}}_k}" src="latex43.gif"> and lower limit <IMG alt="\underline{\mbox{\mbox{\sl RHS}}_k}" src="latex44.gif"> are defined as follows.</P><P><IMG alt="~$~
\begin{eqnarray*}
\overline{\mbox{\mbox{\sl RHS}}_k} \ =\
-\sum_{i =1, i\neq k, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\underline{{\tt D}}_{ij} \;-\;
\sum_{i =1, i\neq k, {\tt I}_i < 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\overline{{\tt D}}_{ij}
\\
\underline{\mbox{\mbox{\sl RHS}}_k} \ = \ 
-\sum_{i =1, i\neq k, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\overline{{\tt D}}_{ij} \;-\;
\sum_{i =1, i\neq k, {\tt I}_i < 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\underline{{\tt D}}_{ij}
\end{eqnarray*}
~$~" src="latex65.gif"></P><P>These values are used to narrow the domain of&nbsp;<IMG alt="{\tt D}_{kl}, 1 \leq l
\leq m_k," src="latex66.gif"> until a fixed point is reached. We describe the propagation for the different possible values of <CODE>A</CODE>. </P><DL><DT><CODE><SPAN class="keyword">=&lt;:</SPAN></CODE></DT><DD><P>The narrowing is done according to the following inequalities.</P><P><IMG alt="{\tt D}_{kl} \leq \left \lfloor  \frac{\overline{\mbox{\sl RHS}}_k}{{\tt I}_k * \prod_{j=1,
j\neq l}^{m_k} \underline{{\tt D}}_{kj}} \right \rfloor \quad
\mbox{ if\ } {\tt I}_k > 0" src="latex67.gif"></P><P></P><P><IMG alt="{\tt D}_{kl} \geq \left \lceil  \frac{\overline{\mbox{\sl RHS}}_k}{{\tt I}_k * \prod_{j=0,
j\neq l}^{m_k} \overline{{\tt D}}_{kj}} \right \rceil \quad
\mbox{ if\ } {\tt I}_k < 0" src="latex68.gif"></P><P> Here <IMG alt="x\leq n" src="latex49.gif"> denotes the basic constraint <IMG alt="x \in
\{0,\ldots,n\}" src="latex50.gif"> and <IMG alt="x \geq n" src="latex51.gif"> denotes the basic constraint <IMG alt="x \in
\{n,\ldots,\codeinline{oz}{FD.sup}\}" src="latex52.gif">. </P><P>The propagator ceases to exist, if the following condition holds. </P><P><IMG alt="\sum_{i =1, {\tt I}_i > 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\overline{{\tt D}}_{ij} + 
\sum_{i =1, {\tt I}_{i} < 0}^{n+1} {\tt I}_i *
\prod_{j=1}^{m_i}\underline{{\tt D}}_i 
\leq 0" src="latex69.gif"></P><P> As an example consider </P><BLOCKQUOTE class="code"><CODE>3<SPAN class="keyword">*</SPAN>X<SPAN class="keyword">*</SPAN>Y&nbsp;<SPAN class="keyword">-</SPAN>&nbsp;Z&nbsp;<SPAN class="keyword">=&lt;:</SPAN>&nbsp;A</CODE></BLOCKQUOTE><P> We have the following formulas.</P><P><IMG alt="\codeinline{oz}{X} \leq \left\lfloor \frac{\overline{\codeinline{oz}{A}} +
\overline{\codeinline{oz}{Z}}}{3*\underline{\codeinline{oz}{Y}}} \right\rfloor 
\quad \quad
\codeinline{oz}{Y} \leq \left\lfloor \frac{\overline{\codeinline{oz}{A}} +
\overline{\codeinline{oz}{Z}}}{3*\underline{\codeinline{oz}{X}}} \right\rfloor
\quad \quad
\codeinline{oz}{Z} \geq \underline{\codeinline{oz}{X}} *\overline{\codeinline{oz}{Y}} - \overline{\codeinline{oz}{A}}
\quad \quad
\codeinline{oz}{A} \geq \underline{\codeinline{oz}{X}} *\overline{\codeinline{oz}{Y}} - \overline{\codeinline{oz}{Z}}" src="latex70.gif"></P><P>The propagator ceases to exist if <IMG alt="3*\overline{\codeinline{oz}{X}}*\overline{\codeinline{oz}{Y}} -
\underline{\codeinline{oz}{Z}} \leq \underline{\codeinline{oz}{A}}" src="latex71.gif"> holds. </P></DD><DT><CODE><SPAN class="keyword">&gt;=:</SPAN></CODE></DT><DD><P>This case can be reduced to <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE> due to the observation that </P><P><IMG alt="{\tt I}_1*{\tt D}_{11}*\ldots*{\tt D}_{1k_1}+\ldots+{\tt I}_n*{\tt D}_{n1}*\ldots*{\tt D}_{nk_n}+(-1)*{\tt D}_{(n+1)1}\;\leq\;0" src="latex72.gif"></P><P>is equivalent to </P><P><IMG alt="(-{\tt I}_1)*{\tt D}_{11}*\ldots*{\tt D}_{1k_1}+\ldots+(-{\tt I}_n)*{\tt D}_{n1}*\ldots*{\tt D}_{nk_n}+1*{\tt D}_{(n+1)1}\;\geq\;0" src="latex73.gif"></P><P>Alternatively, <IMG alt="\underline{\mbox{\mbox{\sl RHS}}_k}" src="latex44.gif"> can be used for the definition. </P></DD><DT><CODE><SPAN class="keyword">&lt;:</SPAN></CODE></DT><DD><P>Analogous to <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE></P></DD><DT><CODE><SPAN class="keyword">&gt;:</SPAN></CODE></DT><DD><P>Analogous to <CODE><SPAN class="keyword">&gt;=:</SPAN></CODE></P></DD><DT><CODE><SPAN class="keyword">=:</SPAN></CODE></DT><DD><P>In this case, the operational semantics is defined by conjunction of the formulas given for <CODE><SPAN class="keyword">=&lt;:</SPAN></CODE> and <CODE><SPAN class="keyword">&gt;=:</SPAN></CODE>.</P></DD><DT><CODE><SPAN class="keyword">\=:</SPAN></CODE></DT><DD><P>In this case, the propagator waits until at most one non-determined variable is left, say <IMG alt="D_{kl}" src="latex74.gif">. Then, <IMG alt="{\mbox{\sl RHS}}_k" src="latex42.gif"> denotes a unique integer, and if </P><P><IMG alt="\frac{{\mbox{\sl RHS}}_k}{{\tt I}_k * \prod_{j=1,
j\neq l}^{m_k} {{\tt D}}_{kj}} " src="latex75.gif"></P><P> denotes an integer value, this value is discarded from the domain of <IMG alt="{\tt D}_{kl}" src="latex76.gif">.</P></DD></DL><P> Coreferences are not exploited for nonlinear generic constraints. The only exception is the expression </P><BLOCKQUOTE class="code"><CODE>X&nbsp;<SPAN class="keyword">*</SPAN>&nbsp;X&nbsp;<SPAN class="keyword">=:</SPAN>&nbsp;Y</CODE></BLOCKQUOTE><P> which has the same operational semantics as <CODE>{FD<SPAN class="keyword">.</SPAN>times&nbsp;X&nbsp;X&nbsp;Y}</CODE> (but note that the occurring variables are not automatically constrained to finite domain integers). </P></DD><DT><A name="label172"></A> <CODE>sumAC</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumAC&nbsp;</CODE><CODE>+<I>Iv</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>Dv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator for the absolute value of the scalar product of the vectors <CODE><I>Iv</I></CODE> and <CODE><I>Dv</I></CODE>: </P><P><IMG alt="|{\tt Iv} * {\tt Dv}| = |{\tt I}_1*{\tt D}_1 + \ldots +
{\tt I}_n*{\tt D}_n|\;\sim_{{\tt A}}\;{\tt D}" src="latex77.gif"></P><P> The operational semantics is as follows. If <CODE><I>A</I></CODE> is <CODE><SPAN class="string">'&lt;:'</SPAN></CODE>, <CODE><SPAN class="string">'=&lt;:'</SPAN></CODE> or <CODE><SPAN class="string">'\=:'</SPAN></CODE>, the following definition holds. </P><P><IMG alt=" {\tt Iv} * {\tt Dv} \sim_{{\tt A}} D \;\wedge\; (-{\tt Iv})*{\tt Dv}
\sim_{{\tt A}} D" src="latex78.gif"></P><P> If <CODE><I>A</I></CODE> is <CODE><SPAN class="string">'&gt;:'</SPAN></CODE>, <CODE><SPAN class="string">'&gt;=:'</SPAN></CODE> or <CODE><SPAN class="string">'=:'</SPAN></CODE>, the following definition holds. </P><P><IMG alt="{\tt Iv} * {\tt Dv} \sim_{{\tt A}} D \;\vee\; (-{\tt Iv})*{\tt Dv}
\sim_{{\tt A}} D" src="latex79.gif"></P><P>where the disjunction is realized by constructive disjunction. </P></DD><DT><A name="label174"></A> <CODE>sumACN</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumACN&nbsp;</CODE><CODE>+<I>Iv</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>Dvv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator for </P><P><IMG alt="|{\tt I}_1*{\tt D}_{11}*\ldots*{\tt D}_{1k_1}+\ldots+{\tt I}_n*{\tt D}_{n1}*\ldots*{\tt D}_{nk_n}|\;\sim_{{\tt A}}\;{\tt D}" src="latex80.gif"></P><P></P><P>The operational semantics is defined analogously to <CODE>FD<SPAN class="keyword">.</SPAN>sumAC</CODE>. </P></DD><DT><A name="label176"></A> <CODE>sumD</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumD&nbsp;</CODE><CODE>*<I>Dv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator analogous to <CODE>FD<SPAN class="keyword">.</SPAN>sum</CODE> but performs <EM>domain-consistent</EM> propagation. Note that only equality (<CODE><I>A</I></CODE> is <CODE><SPAN class="string">'=:'</SPAN></CODE>) and disequality (<CODE><I>A</I></CODE> is <CODE><SPAN class="string">'\=:'</SPAN></CODE>) are supported, as for inequalities domain and bounds propagation are equivalent. </P></DD><DT><A name="label178"></A> <CODE>sumCD</CODE></DT><DD><BLOCKQUOTE class="synopsis"><P></P><BLOCKQUOTE class="code"><CODE>{FD<SPAN class="keyword">.</SPAN>sumCD&nbsp;</CODE><CODE>+<I>Iv</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>Dv</I></CODE><CODE>&nbsp;</CODE><CODE>+<I>A</I></CODE><CODE>&nbsp;</CODE><CODE>*<I>D</I></CODE><CODE>}</CODE></BLOCKQUOTE><P></P></BLOCKQUOTE></DD><DD><P>creates a propagator analogous to <CODE>FD<SPAN class="keyword">.</SPAN>sumC</CODE> but performs <EM>domain-consistent</EM> propagation. Note that only equality (<CODE><I>A</I></CODE> is <CODE><SPAN class="string">'=:'</SPAN></CODE>) and disequality (<CODE><I>A</I></CODE> is <CODE><SPAN class="string">'\=:'</SPAN></CODE>) are supported, as for inequalities domain and bounds propagation are equivalent.</P></DD></DL><P></P></DIV><TABLE align="center" border="0" cellpadding="6" cellspacing="6" class="nav"><TR bgcolor="#DDDDDD"><TD><A href="node20.html#section.fd.watching">&lt;&lt; Prev</A></TD><TD><A href="node14.html">- Up -</A></TD><TD><A href="node22.html#section.fd.nonlinear">Next &gt;&gt;</A></TD></TR></TABLE><HR><ADDRESS><A href="http://www.ps.uni-sb.de/~duchier/">Denys&nbsp;Duchier</A>, <A href="http://www.ps.uni-sb.de/~kornstae/">Leif&nbsp;Kornstaedt</A>, <A href="http://www.ps.uni-sb.de/~homik/">Martin&nbsp;Homik</A>, <A href="http://www.ps.uni-sb.de/~tmueller/">Tobias&nbsp;Müller</A>, <A href="http://www.ps.uni-sb.de/~schulte/">Christian&nbsp;Schulte</A> and&nbsp;<A href="http://www.info.ucl.ac.be/~pvr">Peter&nbsp;Van Roy</A><BR><SPAN class="version">Version 1.2.3 (20011221)</SPAN></ADDRESS></BODY></HTML>
