1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"
"http://www.w3.org/TR/REC-html40/loose.dtd">
<HTML>
<HEAD>
<META http-equiv="Content-Type" content="text/html; charset=ISO-8859-1">
<META name="GENERATOR" content="hevea 1.08">
<LINK rel="stylesheet" type="text/css" href="gprolog.css">
<TITLE>
Boolean and reified constraints
</TITLE>
</HEAD>
<BODY TEXT=black BGCOLOR=white>
<A HREF="gprolog060.html"><IMG SRC ="previous_motif.gif" ALT="Previous"></A>
<A HREF="gprolog054.html"><IMG SRC ="contents_motif.gif" ALT="Up"></A>
<A HREF="gprolog062.html"><IMG SRC ="next_motif.gif" ALT="Next"></A>
<HR>
<H3 CLASS="subsection"><A NAME="htoc326">8.7</A> Boolean and reified constraints</H3><UL>
<LI><A HREF="gprolog061.html#toc267">Boolean FD expressions</A>
<LI><A HREF="gprolog061.html#toc268"><TT>(#\)/1</TT> - constraint NOT,
<TT>(#<=>)/2</TT> - constraint equivalent,<BR>
<TT>(#\<=>)/2</TT> - constraint different,
<TT>(##)/2</TT> - constraint XOR,<BR>
<TT>(#==>)/2</TT> - constraint imply,
<TT>(#\==>)/2</TT> - constraint not imply,<BR>
<TT>(#/\)/2</TT> - constraint AND,
<TT>(#\/\)/2</TT> - constraint NAND,<BR>
<TT>(#\/)/2</TT> - constraint OR,
<TT>(#\\/)/2</TT> - constraint NOR</A>
<LI><A HREF="gprolog061.html#toc269"><TT>fd_cardinality/2</TT>,
<TT>fd_cardinality/3</TT>,
<TT>fd_at_least_one/1</TT>,
<TT>fd_at_most_one/1</TT>,<BR>
<TT>fd_only_one/1</TT></A>
</UL>
<A NAME="Boolean-and-reified-constraints"></A>
<A NAME="toc267"></A>
<H4 CLASS="subsubsection"><A NAME="htoc327">8.7.1</A> Boolean FD expressions</H4>
<A NAME="Boolean-FD-expressions"></A>
An boolean FD expression is a Prolog term built from integers (0 for false,
1 for true), variables (Prolog or FD variables), partial AC arithmetic
constraints (section <A HREF="gprolog060.html#Partial-AC:-(:=)/2">8.6.2</A>), full AC arithmetic constraints
(section <A HREF="gprolog060.html#Full-AC:-(:=:)/2">8.6.3</A>) and functors (or operators) that represent
boolean functions. When a sub-expression of a boolean expression is an
arithmetic constraint <I><TT>c</TT></I>, it is reified. Namely, as soon as the
solver detects that <I><TT>c</TT></I> is true (i.e. <EM>entailed</EM>) the
sub-expression has the value <TT>1</TT>. Similarly when the solver detects
that <I><TT>c</TT></I> is false (i.e. <EM>disentailed</EM>) the sub-expression
evaluates as <TT>0</TT>. While neither the entailment nor the disentailment
can be detected the sub-expression is evaluated as a domain <TT>0..1</TT>.
The following table details the components of an FD boolean expression:<BR>
<TABLE BORDER=1 CELLSPACING=0 CELLPADDING=1>
<TR><TD ALIGN=left NOWRAP>FD Expression</TD>
<TD ALIGN=left NOWRAP>Result</TD>
</TR>
<TR><TD ALIGN=left NOWRAP>Prolog variable</TD>
<TD ALIGN=left NOWRAP>domain <TT>0..1</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP>FD variable <TT>X</TT></TD>
<TD ALIGN=left NOWRAP>domain of <TT>X</TT>, <TT>X</TT> is constrained to
be in <TT>0..1</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>0</TT> (integer)</TD>
<TD ALIGN=left NOWRAP><TT>0</TT> (false)</TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>1</TT> (integer)</TD>
<TD ALIGN=left NOWRAP><TT>1</TT> (true)</TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>#\ E</TT></TD>
<TD ALIGN=left NOWRAP>not <TT>E</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #<=> E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> equivalent to <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #\<=> E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> not equivalent to
<TT>E2</TT> (i.e. <TT>E1</TT> different from <TT>E2)</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 ## E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> exclusive OR <TT>E2</TT> (i.e. <TT>E1</TT>
not equivalent to <TT>E2)</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #==> E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> implies <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #\==> E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> does not imply <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #/\ E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> AND <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #\/\ E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> NAND <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #\/ E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> OR <TT>E2</TT></TD>
</TR>
<TR><TD ALIGN=left NOWRAP><TT>E1 #\\/ E2</TT></TD>
<TD ALIGN=left NOWRAP><TT>E1</TT> NOR <TT>E2</TT></TD>
</TR></TABLE><BR>
<TT>#<=></TT>, <TT>#\<=></TT>, <TT>##</TT>,
<TT>#==></TT>, <TT>#\==></TT>, <TT>#/\</TT>,
<TT>#\/\</TT>, <TT>#\/</TT> and <TT>#\\/</TT> are
predefined infix operators. <TT>#\</TT> is a predefined prefix operator
(section <A HREF="gprolog037.html#op/3:(Term-input/output)">7.14.10</A>).<BR>
<BR>
<B>Errors</B><BR>
<TABLE CELLSPACING=2 CELLPADDING=0>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>a sub-expression <TT>E</TT> is neither a variable nor an integer (0
or 1) nor an FD boolean functor nor reified constraint</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(fd_bool_evaluable, E)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>an expression is too complex</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>resource_error(too_big_fd_constraint)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>a sub-expression is an invalid reified constraint</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left>an arithmetic constraint error (section <A HREF="gprolog060.html#FD-arithmetic-expressions">8.6.1</A>)</TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR></TABLE><BR>
<A NAME="toc268"></A>
<H4 CLASS="subsubsection"><A NAME="htoc328">8.7.2</A> <TT>(#\)/1</TT> - constraint NOT,
<TT>(#<=>)/2</TT> - constraint equivalent,<BR>
<TT>(#\<=>)/2</TT> - constraint different,
<TT>(##)/2</TT> - constraint XOR,<BR>
<TT>(#==>)/2</TT> - constraint imply,
<TT>(#\==>)/2</TT> - constraint not imply,<BR>
<TT>(#/\)/2</TT> - constraint AND,
<TT>(#\/\)/2</TT> - constraint NAND,<BR>
<TT>(#\/)/2</TT> - constraint OR,
<TT>(#\\/)/2</TT> - constraint NOR</H4>
<B>Templates</B>
<DL CLASS="list" COMPACT="compact"><DT CLASS="dt-list"><DD CLASS="dd-list"><TT>
#\(?fd_bool_evaluable)<BR>
#<=>(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#\<=>(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
##(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#==>(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#\==>(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#/\(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#\/\(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#\/(?fd_bool_evaluable, ?fd_bool_evaluable)<BR>
#\\/(?fd_bool_evaluable, ?fd_bool_evaluable)</TT></DL>
<B>Description</B><BR>
<BR>
<TT>#\= FdBoolExpr1</TT> constraints <TT>FdBoolExpr1</TT> to be
false.<BR>
<BR>
<TT>FdBoolExpr1 #<=> FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> to be equivalent to <TT>FdBoolExpr2</TT>.<BR>
<BR>
<TT>FdBoolExpr1 #\<=> FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> to be equivalent to not <TT>FdBoolExpr2</TT>.<BR>
<BR>
<TT>FdBoolExpr1 ## FdBoolExpr2</TT> constrains <TT>FdBoolExpr1</TT> XOR
<TT>FdBoolExpr2</TT> to be true<BR>
<BR>
<TT>FdBoolExpr1 #==> FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> to imply <TT>FdBoolExpr2</TT>.<BR>
<BR>
<TT>FdBoolExpr1 #\==> FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> to not imply <TT>FdBoolExpr2</TT>.<BR>
<BR>
<TT>FdBoolExpr1 #/\ FdBoolExpr2</TT> constrains <TT>FdBoolExpr1</TT>
AND <TT>FdBoolExpr2</TT> to be true.<BR>
<BR>
<TT>FdBoolExpr1 #\/\ FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> AND <TT>FdBoolExpr2</TT> to be false.<BR>
<BR>
<TT>FdBoolExpr1 #\/ FdBoolExpr2</TT> constrains <TT>FdBoolExpr1</TT>
OR <TT>FdBoolExpr2</TT> to be true.<BR>
<BR>
<TT>FdBoolExpr1 #\\/ FdBoolExpr2</TT> constrains
<TT>FdBoolExpr1</TT> OR <TT>FdBoolExpr2</TT> to be false.<BR>
<BR>
<TT>FdBoolExpr1</TT> and <TT>FdBoolExpr2</TT> are boolean FD expressions
(section <A HREF="#Boolean-FD-expressions">8.7.1</A>).<BR>
<BR>
Note that <TT>#\<=></TT> (not equivalent) and <TT>##</TT>
(exclusive or) are synonymous.<BR>
<BR>
These predicates post boolean constraints that are managed by the FD solver
using a partial arc-consistency algorithm to reduce the domain of involved
variables. The (dis)entailment of reified constraints is detected using
either the bounds (for partial AC arithmetic constraints) or the full domain
(for full AC arithmetic constraints).<BR>
<BR>
<TT>#<=></TT>, <TT>#\<=></TT>, <TT>##</TT>,
<TT>#==></TT>, <TT>#\==></TT>, <TT>#/\</TT>,
<TT>#\/\</TT>, <TT>#\/</TT> and <TT>#\\/</TT> are
predefined infix operators. <TT>#\</TT> is a predefined prefix operator
(section <A HREF="gprolog037.html#op/3:(Term-input/output)">7.14.10</A>).<BR>
<BR>
<B>Errors</B><BR>
<BR>
Refer to the syntax of boolean FD expressions for possible errors
(section <A HREF="#Boolean-FD-expressions">8.7.1</A>).<BR>
<BR>
<B>Portability</B><BR>
<BR>
GNU Prolog predicates.<BR>
<BR>
<A NAME="toc269"></A>
<H4 CLASS="subsubsection"><A NAME="htoc329">8.7.3</A> <TT>fd_cardinality/2</TT>,
<TT>fd_cardinality/3</TT>,
<TT>fd_at_least_one/1</TT>,
<TT>fd_at_most_one/1</TT>,<BR>
<TT>fd_only_one/1</TT></H4>
<A NAME="fd-cardinality/2"></A>
<B>Templates</B>
<DL CLASS="list" COMPACT="compact"><DT CLASS="dt-list"><DD CLASS="dd-list"><TT>
fd_cardinality(+fd_bool_evaluable_list, ?fd_variable)<BR>
fd_cardinality(+integer, ?fd_variable, +integer)<BR>
fd_at_least_one(+fd_bool_evaluable_list)<BR>
fd_at_most_one(+fd_bool_evaluable_list)<BR>
fd_only_one(+fd_bool_evaluable_list)</TT></DL>
<B>Description</B><BR>
<BR>
<TT>fd_cardinality(List, Count)</TT> unifies <TT>Count</TT> with the number
of constraints that are true in <TT>List</TT>. This is equivalent to post
the constraint <TT>B<SUB>1</SUB> + B<SUB>2</SUB> + ...+ B<SUB><I>n</I></SUB> #= Count</TT>
where each variable <TT>Bi</TT> is a new variable defined by the constraint
<TT>B<SUB><I>i</I></SUB> #<=> C<SUB><I>i</I></SUB></TT> where <TT>C<SUB><I>i</I></SUB></TT> is the
<TT>i</TT><EM>th</EM> constraint of <TT>List</TT>. Each <TT>C<SUB><I>i</I></SUB></TT>
must be a boolean FD expression (section <A HREF="#Boolean-FD-expressions">8.7.1</A>).<BR>
<BR>
<TT>fd_cardinality(Lower, List, Upper)</TT> is equivalent to
<TT>fd_cardinality(List, Count), Lower #=< Count, Count #=<
Upper</TT><BR>
<BR>
<TT>fd_at_least_one(List)</TT> is equivalent to
<TT>fd_cardinality(List, Count), Count #>= 1</TT>.<BR>
<BR>
<TT>fd_at_most_one(List)</TT> is equivalent to
<TT>fd_cardinality(List, Count), Count #=< 1</TT>.<BR>
<BR>
<TT>fd_only_one(List)</TT> is equivalent to <TT>fd_cardinality(List,
1)</TT>.<BR>
<BR>
<B>Errors</B><BR>
<TABLE CELLSPACING=2 CELLPADDING=0>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>List</TT> is a partial list</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>instantiation_error</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>List</TT> is neither a partial list nor a list</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(list, List)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>Count</TT> is neither an FD variable nor an integer</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(fd_variable, Count)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>Lower</TT> is a variable</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>instantiation_error</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>Lower</TT> is neither a variable nor an integer</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(integer, Lower)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>Upper</TT> is a variable</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>instantiation_error</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>Upper</TT> is neither a variable nor an integer</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(integer, Upper)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>an element <TT>E</TT> of the <TT>List</TT> list is an invalid
boolean expression</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left>an FD boolean constraint (section <A HREF="#Boolean-FD-expressions">8.7.1</A>)</TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR></TABLE><BR>
<B>Portability</B><BR>
<BR>
GNU Prolog predicates.<BR>
<BR>
<HR SIZE=2>
Copyright (C) 1999-2007 Daniel Diaz
<BR>
<BR>
Verbatim copying and distribution of this entire article is permitted in any
medium, provided this notice is preserved. <BR>
<BR>
<A HREF="index.html#copyright">More about the copyright</A>
<HR>
<A HREF="gprolog060.html"><IMG SRC ="previous_motif.gif" ALT="Previous"></A>
<A HREF="gprolog054.html"><IMG SRC ="contents_motif.gif" ALT="Up"></A>
<A HREF="gprolog062.html"><IMG SRC ="next_motif.gif" ALT="Next"></A>
</BODY>
</HTML>
|
