-
Notifications
You must be signed in to change notification settings - Fork 77
/
Copy pathclpb.pl
2020 lines (1643 loc) · 67.1 KB
/
clpb.pl
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
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/* CLP(B) port to SICStus Prolog
Copyright (C) 2017 Markus Triska (triska@metalevel.at)
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
More information about CLP(B) is available from:
https://www.metalevel.at/clpb/
==============================
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
CLP(B): Constraint Logic Programming over Boolean variables.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Public operators.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
:- op(300, fy, ~).
:- op(500, yfx, #).
:- module(clpb, [
sat/1,
taut/2,
labeling/1,
sat_count/2,
weighted_maximum/3,
random_labeling/2
]).
:- use_module(library(assoc)).
:- use_module(library(between)).
:- use_module(library(atts)).
:- use_module(library(lists)).
:- use_module(library(types)).
:- attribute
clpb/1,
clpb_bdd/1,
clpb_atom/1,
clpb_hash/1,
clpb_max/1,
clpb_omit_boolean/1,
clpb_visited/1.
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Compatibility predicates.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
group_pairs_by_key([], []).
group_pairs_by_key([M-N|T0], [M-[N|TN]|T]) :-
same_key(M, T0, TN, T1),
group_pairs_by_key(T1, T).
same_key(M0, [M-N|T0], [N|TN], T) :-
M0 == M,
!,
same_key(M, T0, TN, T).
same_key(_, L, [], L).
must_be(What, Term) :- must_be(What, unknown(Term)-1, Term).
must_be(acyclic, Where, Term) :- !,
( acyclic_term(Term) ->
true
; domain_error(acyclic_term, Term, Where)
).
must_be(list, Where, Term) :- !,
( acyclic_term(Term), clpz_list(Term, Where) -> true
; type_error(list, Term, Where)
).
must_be(list(What), Where, Term) :- !,
must_be(list, Where, Term),
maplist(must_be(What, Where), Term).
must_be(Type, Goal-Arg, Term) :-
must_be(Term, Type, Goal, Arg).
clpz_list(Nil, _) :- Nil == [].
clpz_list(Ls, Where) :-
( var(Ls) ->
instantiation_error(Ls, Where)
; Ls = [_|Rest],
clpz_list(Rest, Where)
).
instantiation_error(Term) :- instantiation_error(Term, unknown(Term)-1).
instantiation_error(_, Goal-Arg) :-
throw(error(instantiation_error, instantiation_error(Goal, Arg))).
domain_error(Expectation, Term) :-
domain_error(Expectation, Term, unknown(Term)-1).
domain_error(Expectation, Term, Goal-Arg) :-
throw(error(domain_error(Expectation, Term), domain_error(Goal, Arg, Expectation, Term))).
type_error(Expectation, Term) :-
type_error(Expectation, Term, unknown(Term)-1).
type_error(Expectation, Term, Goal-Arg) :-
throw(error(type_error(Expectation, Term), type_error(Goal, Arg, Expectation, Term))).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
foldl/4
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
foldl(Goal_3, Ls, A0, A) :-
foldl_(Ls, Goal_3, A0, A).
foldl_([], _, A, A).
foldl_([L|Ls], G_3, A0, A) :-
call(G_3, L, A0, A1),
foldl_(Ls, G_3, A1, A).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
foldl/5
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
foldl(Goal_4, Xs, Ys, A0, A) :-
foldl_(Xs, Ys, Goal_4, A0, A).
foldl_([], [], _, A, A).
foldl_([X|Xs], [Y|Ys], G_4, A0, A) :-
call(G_4, X, Y, A0, A1),
foldl_(Xs, Ys, G_4, A1, A).
partition(Pred, Ls0, As, Bs) :-
include(Pred, Ls0, As),
exclude(Pred, Ls0, Bs).
sum_list(Ls, S) :- sumlist(Ls, S).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Pairs.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
pairs_keys_values([], [], []).
pairs_keys_values([A-B|ABs], [A|As], [B|Bs]) :-
pairs_keys_values(ABs, As, Bs).
pairs_keys(Ps, Ks) :- pairs_keys_values(Ps, Ks, _).
pairs_values(Ps, Vs) :- pairs_keys_values(Ps, _, Vs).
map_list_to_pairs(Pred, Ls, Ps) :-
map_list_to_pairs2(Ls, Pred, Ps).
map_list_to_pairs2([], _, []).
map_list_to_pairs2([H|T0], Pred, [K-H|T]) :-
call(Pred, H, K),
map_list_to_pairs2(T0, Pred, T).
:- discontiguous clpb:goal_expansion/5.
goal_expansion(get_attr(Var, Module, Value), _, clpb, (var(Var),get_atts(Var, [+Access])), []) :-
Access =.. [Module,Value].
goal_expansion(put_attr(Var, Module, Value), _, clpb, put_atts(Var, [+Access]), []) :-
Access =.. [Module,Value].
goal_expansion(del_attr(Var, Module), _, clpb, (var(Var) -> put_atts(Var, [-Access]);true), []) :-
Access =.. [Module,_].
/** <module> CLP(B): Constraint Logic Programming over Boolean Variables
## Introduction {#clpb-intro}
This library provides CLP(B), Constraint Logic Programming over
Boolean variables. It can be used to model and solve combinatorial
problems such as verification, allocation and covering tasks.
CLP(B) is an instance of the general [CLP(_X_) scheme](<#clp>),
extending logic programming with reasoning over specialised domains.
The implementation is based on reduced and ordered Binary Decision
Diagrams (BDDs).
Usage examples of this library are available in a public git
repository: [**github.com/triska/clpb**](https://github.com/triska/clpb)
We recommend the following reference (PDF:
[metalevel.at/swiclpb.pdf](https://www.metalevel.at/swiclpb.pdf))
for citing this library in scientific publications:
==
@inproceedings{Triska2016,
author = "Markus Triska",
title = "The {Boolean} Constraint Solver of {SWI-Prolog}:
System Description",
booktitle = "FLOPS",
series = "LNCS",
volume = 9613,
year = 2016,
pages = "45--61"
}
==
and the following URL to link to its documentation:
http://eu.swi-prolog.org/man/clpb.html
## Boolean expressions {#clpb-exprs}
A _Boolean expression_ is one of:
| `0` | false |
| `1` | true |
| _variable_ | unknown truth value |
| _atom_ | universally quantified variable |
| ~ _Expr_ | logical NOT |
| _Expr_ + _Expr_ | logical OR |
| _Expr_ * _Expr_ | logical AND |
| _Expr_ # _Expr_ | exclusive OR |
| _Var_ ^ _Expr_ | existential quantification |
| _Expr_ =:= _Expr_ | equality |
| _Expr_ =\= _Expr_ | disequality (same as #) |
| _Expr_ =< _Expr_ | less or equal (implication) |
| _Expr_ >= _Expr_ | greater or equal |
| _Expr_ < _Expr_ | less than |
| _Expr_ > _Expr_ | greater than |
| card(Is,Exprs) | _see below_ |
| `+(Exprs)` | _see below_ |
| `*(Exprs)` | _see below_ |
where _Expr_ again denotes a Boolean expression.
The Boolean expression card(Is,Exprs) is true iff the number of true
expressions in the list `Exprs` is a member of the list `Is` of
integers and integer ranges of the form `From-To`.
`+(Exprs)` and `*(Exprs)` denote, respectively, the disjunction and
conjunction of all elements in the list `Exprs` of Boolean
expressions.
Atoms denote parametric values that are universally quantified. All
universal quantifiers appear implicitly in front of the entire
expression. In residual goals, universally quantified variables always
appear on the right-hand side of equations. Therefore, they can be
used to express functional dependencies on input variables.
## Interface predicates {#clpb-interface}
The most frequently used CLP(B) predicates are:
* sat(+Expr)
True iff the Boolean expression Expr is satisfiable.
* taut(+Expr, -T)
If Expr is a tautology with respect to the posted constraints, succeeds
with *T = 1*. If Expr cannot be satisfied, succeeds with *T = 0*.
Otherwise, it fails.
* labeling(+Vs)
Assigns truth values to the variables Vs such that all constraints
are satisfied.
The unification of a CLP(B) variable _X_ with a term _T_ is equivalent
to posting the constraint sat(X=:=T).
## Examples {#clpb-examples}
Here is an example session with a few queries and their answers:
==
?- use_module(library(clpb)).
true.
?- sat(X*Y).
X = Y, Y = 1.
?- sat(X * ~X).
false.
?- taut(X * ~X, T).
T = 0,
sat(X=:=X).
?- sat(X^Y^(X+Y)).
sat(X=:=X),
sat(Y=:=Y).
?- sat(X*Y + X*Z), labeling([X,Y,Z]).
X = Z, Z = 1, Y = 0 ;
X = Y, Y = 1, Z = 0 ;
X = Y, Y = Z, Z = 1.
?- sat(X =< Y), sat(Y =< Z), taut(X =< Z, T).
T = 1,
sat(X=:=X*Y),
sat(Y=:=Y*Z).
?- sat(1#X#a#b).
sat(X=:=a#b).
==
The pending residual goals constrain remaining variables to Boolean
expressions and are declaratively equivalent to the original query.
The last example illustrates that when applicable, remaining variables
are expressed as functions of universally quantified variables.
## Obtaining BDDs {#clpb-residual-goals}
By default, CLP(B) residual goals appear in (approximately) algebraic
normal form (ANF). This projection is often computationally expensive.
You can set the Prolog flag `clpb_residuals` to the value `bdd` to see
the BDD representation of all constraints. This results in faster
projection to residual goals, and is also useful for learning more
about BDDs. For example:
==
?- set_prolog_flag(clpb_residuals, bdd).
true.
?- sat(X#Y).
node(3)- (v(X, 0)->node(2);node(1)),
node(1)- (v(Y, 1)->true;false),
node(2)- (v(Y, 1)->false;true).
==
Note that this representation cannot be pasted back on the toplevel,
and its details are subject to change. Use copy_term/3 to obtain
such answers as Prolog terms.
The variable order of the BDD is determined by the order in which the
variables first appear in constraints. To obtain different orders,
you can for example use:
==
?- sat(+[1,Y,X]), sat(X#Y).
node(3)- (v(Y, 0)->node(2);node(1)),
node(1)- (v(X, 1)->true;false),
node(2)- (v(X, 1)->false;true).
==
## Enabling monotonic CLP(B) {#clpb-monotonic}
In the default execution mode, CLP(B) constraints are _not_ monotonic.
This means that _adding_ constraints can yield new solutions. For
example:
==
?- sat(X=:=1), X = 1+0.
false.
?- X = 1+0, sat(X=:=1), X = 1+0.
X = 1+0.
==
This behaviour is highly problematic from a logical point of view, and
it may render [**declarative
debugging**](https://www.metalevel.at/prolog/debugging)
techniques inapplicable.
Set the flag `clpb_monotonic` to `true` to make CLP(B) *monotonic*. If
this mode is enabled, then you must wrap CLP(B) variables with the
functor `v/1`. For example:
==
?- set_prolog_flag(clpb_monotonic, true).
true.
?- sat(v(X)=:=1#1).
X = 0.
==
@author [Markus Triska](https://www.metalevel.at)
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Each CLP(B) variable belongs to exactly one BDD. Each CLP(B)
variable gets an attribute (in module "clpb") of the form:
index_root(Index,Root)
where Index is the variable's unique integer index, and Root is the
root of the BDD that the variable belongs to.
Each CLP(B) variable also gets an attribute in module clpb_hash: an
association table node(LID,HID) -> Node, to keep the BDD reduced.
The association table of each variable must be rebuilt on occasion
to remove nodes that are no longer reachable. We rebuild the
association tables of involved variables after BDDs are merged to
build a new root. This only serves to reclaim memory: Keeping a
node in a local table even when it no longer occurs in any BDD does
not affect the solver's correctness. However, apply_shortcut/4
relies on the invariant that every node that occurs in the relevant
BDDs is also registered in the table of its branching variable.
A root is a logical variable with a single attribute ("clpb_bdd")
of the form:
Sat-BDD
where Sat is the SAT formula (in original form) that corresponds to
BDD. Sat is necessary to rebuild the BDD after variable aliasing,
and to project all remaining constraints to a list of sat/1 goals.
Finally, a BDD is either:
*) The integers 0 or 1, denoting false and true, respectively, or
*) A node of the form
node(ID, Var, Low, High, Aux)
Where ID is the node's unique integer ID, Var is the
node's branching variable, and Low and High are the
node's low (Var = 0) and high (Var = 1) children. Aux
is a free variable, one for each node, that can be used
to attach attributes and store intermediate results.
Variable aliasing is treated as a conjunction of corresponding SAT
formulae.
You should think of CLP(B) as a potentially vast collection of BDDs
that can range from small to gigantic in size, and which can merge.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Type checking.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
is_sat(V) :- var(V), !, non_monotonic(V).
is_sat(v(V)) :- var(V), !.
is_sat(v(I)) :- integer(I), between(0, 1, I).
is_sat(I) :- integer(I), between(0, 1, I).
is_sat(A) :- atom(A).
is_sat(~A) :- is_sat(A).
is_sat(A*B) :- is_sat(A), is_sat(B).
is_sat(A+B) :- is_sat(A), is_sat(B).
is_sat(A#B) :- is_sat(A), is_sat(B).
is_sat(A=:=B) :- is_sat(A), is_sat(B).
is_sat(A=\=B) :- is_sat(A), is_sat(B).
is_sat(A=<B) :- is_sat(A), is_sat(B).
is_sat(A>=B) :- is_sat(A), is_sat(B).
is_sat(A<B) :- is_sat(A), is_sat(B).
is_sat(A>B) :- is_sat(A), is_sat(B).
is_sat(+(Ls)) :- must_be(list, Ls), maplist(is_sat, Ls).
is_sat(*(Ls)) :- must_be(list, Ls), maplist(is_sat, Ls).
is_sat(X^F) :- var(X), is_sat(F).
is_sat(card(Is,Fs)) :-
must_be(list(ground), Is),
must_be(list, Fs),
maplist(is_sat, Fs).
:- dynamic monotonic/0.
non_monotonic(X) :-
( var_index(X, _) ->
% OK: already constrained to a CLP(B) variable
true
; monotonic ->
instantiation_error(X)
; true
).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Rewriting to canonical expressions.
Atoms are converted to variables with a special attribute.
A global lookup table maintains the correspondence between atoms and
their variables throughout different sat/1 goals.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
% elementary
sat_rewrite(V, V) :- var(V), !.
sat_rewrite(I, I) :- integer(I), !.
sat_rewrite(A, V) :- atom(A), !, clpb_atom_var(A, V).
sat_rewrite(v(V), V).
sat_rewrite(P0*Q0, P*Q) :- sat_rewrite(P0, P), sat_rewrite(Q0, Q).
sat_rewrite(P0+Q0, P+Q) :- sat_rewrite(P0, P), sat_rewrite(Q0, Q).
sat_rewrite(P0#Q0, P#Q) :- sat_rewrite(P0, P), sat_rewrite(Q0, Q).
sat_rewrite(X^F0, X^F) :- sat_rewrite(F0, F).
sat_rewrite(card(Is,Fs0), card(Is,Fs)) :-
maplist(sat_rewrite, Fs0, Fs).
% synonyms
sat_rewrite(~P, R) :- sat_rewrite(1 # P, R).
sat_rewrite(P =:= Q, R) :- sat_rewrite(~P # Q, R).
sat_rewrite(P =\= Q, R) :- sat_rewrite(P # Q, R).
sat_rewrite(P =< Q, R) :- sat_rewrite(~P + Q, R).
sat_rewrite(P >= Q, R) :- sat_rewrite(Q =< P, R).
sat_rewrite(P < Q, R) :- sat_rewrite(~P * Q, R).
sat_rewrite(P > Q, R) :- sat_rewrite(Q < P, R).
sat_rewrite(+(Ls), R) :- foldl(or, Ls, 0, F), sat_rewrite(F, R).
sat_rewrite(*(Ls), R) :- foldl(and, Ls, 1, F), sat_rewrite(F, R).
or(A, B, B + A).
and(A, B, B * A).
must_be_sat(Sat) :-
must_be(acyclic, Sat),
( is_sat(Sat) -> true
; no_truth_value(Sat)
).
no_truth_value(Term) :- domain_error(clpb_expr, Term).
parse_sat(Sat0, Sat) :-
must_be_sat(Sat0),
sat_rewrite(Sat0, Sat),
term_variables(Sat, Vs),
maplist(enumerate_variable, Vs).
enumerate_variable(V) :-
( var_index_root(V, _, _) -> true
; clpb_next_id('$clpb_next_var', Index),
put_attr(V, clpb, index_root(Index,_)),
put_empty_hash(V)
).
var_index(V, I) :- var_index_root(V, I, _).
var_index_root(V, I, Root) :- get_attr(V, clpb, index_root(I,Root)).
put_empty_hash(V) :-
empty_assoc(H0),
put_attr(V, clpb_hash, H0).
sat_roots(Sat, Roots) :-
term_variables(Sat, Vs),
maplist(var_index_root, Vs, _, Roots0),
term_variables(Roots0, Roots).
%% sat(+Expr) is semidet.
%
% True iff Expr is a satisfiable Boolean expression.
sat(Sat0) :-
( phrase(sat_ands(Sat0), Ands), Ands = [_,_|_] ->
maplist(sat, Ands)
; parse_sat(Sat0, Sat),
sat_bdd(Sat, BDD),
sat_roots(Sat, Roots),
roots_and(Roots, Sat0-BDD, And-BDD1),
maplist(del_bdd, Roots),
maplist(=(Root), Roots),
root_put_formula_bdd(Root, And, BDD1),
is_bdd(BDD1),
satisfiable_bdd(BDD1)
).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Posting many small sat/1 constraints is better than posting a huge
conjunction (or negated disjunction), because unneeded nodes are
removed from node tables after BDDs are merged. This is not
possible in sat_bdd/2 because the nodes may occur in other BDDs. A
better version of sat_bdd/2 or a proper implementation of a unique
table including garbage collection would make this obsolete and
also improve taut/2 and sat_count/2 in such cases.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
sat_ands(X) -->
( { var(X) } -> [X]
; { X = (A*B) } -> sat_ands(A), sat_ands(B)
; { X = *(Ls) } -> sat_ands_(Ls)
; { X = ~Y } -> not_ors(Y)
; [X]
).
sat_ands_([]) --> [].
sat_ands_([L|Ls]) --> [L], sat_ands_(Ls).
not_ors(X) -->
( { var(X) } -> [~X]
; { X = (A+B) } -> not_ors(A), not_ors(B)
; { X = +(Ls) } -> not_ors_(Ls)
; [~X]
).
not_ors_([]) --> [].
not_ors_([L|Ls]) --> [~L], not_ors_(Ls).
del_bdd(Root) :- del_attr(Root, clpb_bdd).
root_get_formula_bdd(Root, F, BDD) :- get_attr(Root, clpb_bdd, F-BDD).
root_put_formula_bdd(Root, F, BDD) :- put_attr(Root, clpb_bdd, F-BDD).
roots_and(Roots, Sat0-BDD0, Sat-BDD) :-
foldl(root_and, Roots, Sat0-BDD0, Sat-BDD),
rebuild_hashes(BDD).
root_and(Root, Sat0-BDD0, Sat-BDD) :-
( root_get_formula_bdd(Root, F, B) ->
Sat = F*Sat0,
bdd_and(B, BDD0, BDD)
; Sat = Sat0,
BDD = BDD0
).
bdd_and(NA, NB, And) :-
apply(*, NA, NB, And),
is_bdd(And).
%% taut(+Expr, -T) is semidet
%
% Tautology check. Succeeds with T = 0 if the Boolean expression Expr
% cannot be satisfied, and with T = 1 if Expr is always true with
% respect to the current constraints. Fails otherwise.
taut(Sat0, T) :-
parse_sat(Sat0, Sat),
( T = 0, \+ sat(Sat) -> true
; T = 1, tautology(Sat) -> true
; false
).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The algebraic equivalence: tautology(F) <=> \+ sat(~F) does NOT
hold in CLP(B) because the quantifiers of universally quantified
variables always implicitly appear in front of the *entire*
expression. Thus we have for example: X+a is not a tautology, but
~(X+a), meaning forall(a, ~(X+a)), is unsatisfiable:
sat(~(X+a)) = sat(~X * ~a) = sat(~X), sat(~a) = X=0, false
The actual negation of X+a, namely ~forall(A,X+A), in terms of
CLP(B): ~ ~exists(A, ~(X+A)), is of course satisfiable:
?- sat(~ ~A^ ~(X+A)).
%@ X = 0,
%@ sat(A=:=A).
Instead, of such rewriting, we test whether the BDD of the negated
formula is 0. Critically, this avoids constraint propagation.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
tautology(Sat) :-
( phrase(sat_ands(Sat), Ands), Ands = [_,_|_] ->
maplist(tautology, Ands)
; catch((sat_roots(Sat, Roots),
roots_and(Roots, _-1, _-Ands),
sat_bdd(1#Sat, BDD),
bdd_and(BDD, Ands, B),
B == 0,
% reset all attributes
throw(tautology)),
tautology,
true)
).
satisfiable_bdd(BDD) :-
( BDD == 0 -> false
; BDD == 1 -> true
; ( bdd_nodes(var_unbound, BDD, Nodes) ->
bdd_variables_classification(BDD, Nodes, Classes),
partition(var_class, Classes, Eqs, Bs, Ds),
domain_consistency(Eqs, Goal),
aliasing_consistency(Bs, Ds, Goals),
maplist(unification, [Goal|Goals])
; % if any variable is instantiated, we do not perform
% any propagation for now
true
)
).
var_class(_=_, <).
var_class(further_branching(_,_), =).
var_class(negative_decisive(_), >).
unification(true).
unification(A=B) :- A = B. % safe_goal/1 detects safety of this call
var_unbound(Node) :-
node_var_low_high(Node, Var, _, _),
var(Var).
universal_var(Var) :- get_attr(Var, clpb_atom, _).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
By aliasing consistency, we mean that all unifications X=Y, where
taut(X=:=Y, 1) holds, are posted.
To detect this, we distinguish two kinds of variables among those
variables that are not skipped in any branch: further-branching and
negative-decisive. X is negative-decisive iff every node where X
appears as a branching variable has 0 as one of its children. X is
further-branching iff 1 is not a direct child of any node where X
appears as a branching variable.
Any potential aliasing must involve one further-branching, and one
negative-decisive variable. X=Y must hold if, for each low branch
of nodes with X as branching variable, Y has high branch 0, and for
each high branch of nodes involving X, Y has low branch 0.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
aliasing_consistency(Bs, Ds, Goals) :-
phrase(aliasings(Bs, Ds), Goals).
aliasings([], _) --> [].
aliasings([further_branching(B,Nodes)|Bs], Ds) -->
{ var_index(B, BI) },
aliasings_(Ds, B, BI, Nodes),
aliasings(Bs, Ds).
aliasings_([], _, _, _) --> [].
aliasings_([negative_decisive(D)|Ds], B, BI, Nodes) -->
{ var_index(D, DI) },
( { DI > BI,
always_false(high, DI, Nodes),
always_false(low, DI, Nodes),
var_or_atom(D, DA), var_or_atom(B, BA) } ->
[DA=BA]
; []
),
aliasings_(Ds, B, BI, Nodes).
var_or_atom(Var, VA) :-
( get_attr(Var, clpb_atom, VA) -> true
; VA = Var
).
always_false(Which, DI, Nodes) :-
phrase(nodes_always_false(Nodes, Which, DI), Opposites),
maplist(with_aux(unvisit), Opposites).
nodes_always_false([], _, _) --> [].
nodes_always_false([Node|Nodes], Which, DI) -->
{ which_node_child(Which, Node, Child),
opposite(Which, Opposite) },
opposite_always_false(Opposite, DI, Child),
nodes_always_false(Nodes, Which, DI).
which_node_child(low, Node, Child) :-
node_var_low_high(Node, _, Child, _).
which_node_child(high, Node, Child) :-
node_var_low_high(Node, _, _, Child).
opposite(low, high).
opposite(high, low).
opposite_always_false(Opposite, DI, Node) -->
( { node_visited(Node) } -> []
; { node_var_low_high(Node, Var, Low, High),
with_aux(put_visited, Node),
var_index(Var, VI) },
[Node],
( { VI =:= DI } ->
{ which_node_child(Opposite, Node, Child),
Child == 0 }
; opposite_always_false(Opposite, DI, Low),
opposite_always_false(Opposite, DI, High)
)
).
further_branching(Node) :-
node_var_low_high(Node, _, Low, High),
Low \== 1,
High \== 1.
negative_decisive(Node) :-
node_var_low_high(Node, _, Low, High),
( Low == 0 -> true
; High == 0 -> true
; false
).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Instantiate all variables that only admit a single Boolean value.
This is the case if: The variable is not skipped in any branch
leading to 1 (its being skipped means that it may be assigned
either 0 or 1 and can thus not be fixed yet), and all nodes where
it occurs as a branching variable have either lower or upper child
fixed to 0 consistently.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
domain_consistency(Eqs, Goal) :-
maplist(eq_a_b, Eqs, Vs, Values),
Goal = (Vs = Values). % propagate all assignments at once
eq_a_b(A=B, A, B).
consistently_false_(Which, Node) :-
which_node_child(Which, Node, Child),
Child == 0.
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
In essentially one sweep of the BDD, all variables can be classified:
Unification with 0 or 1, further branching and/or negative decisive.
Strategy: Breadth-first traversal of the BDD, failing (and thus
clearing all attributes) if the variable is skipped in some branch,
and moving the frontier along each time.
A formula is only satisfiable if it is a tautology after all (also
implicitly) existentially quantified variables are projected away.
However, we only need to check this explicitly if at least one
universally quantified variable appears. Otherwise, we know that
the formula is satisfiable at this point, because its BDD is not 0.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
bdd_variables_classification(BDD, Nodes, Classes) :-
nodes_variables(Nodes, Vs0),
variables_in_index_order(Vs0, Vs),
( partition(universal_var, Vs, [_|_], Es) ->
foldl(existential, Es, BDD, 1)
; true
),
phrase(variables_classification(Vs, [BDD]), Classes),
maplist(with_aux(unvisit), Nodes).
variables_classification([], _) --> [].
variables_classification([V|Vs], Nodes0) -->
{ var_index(V, Index) },
( { phrase(nodes_with_variable(Nodes0, Index), Nodes) } ->
( { maplist(consistently_false_(low), Nodes) } -> [V=1]
; { maplist(consistently_false_(high), Nodes) } -> [V=0]
; []
),
( { maplist(further_branching, Nodes) } ->
[further_branching(V, Nodes)]
; []
),
( { maplist(negative_decisive, Nodes) } ->
[negative_decisive(V)]
; []
),
{ maplist(with_aux(unvisit), Nodes) },
variables_classification(Vs, Nodes)
; variables_classification(Vs, Nodes0)
).
nodes_with_variable([], _) --> [].
nodes_with_variable([Node|Nodes], VI) -->
{ Node \== 1 },
( { node_visited(Node) } -> nodes_with_variable(Nodes, VI)
; { with_aux(put_visited, Node),
node_var_low_high(Node, OVar, Low, High),
var_index(OVar, OVI) },
{ OVI =< VI },
( { OVI =:= VI } -> [Node]
; nodes_with_variable([Low,High], VI)
),
nodes_with_variable(Nodes, VI)
).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Node management. Always use an existing node, if there is one.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
make_node(Var, Low, High, Node) :-
( Low == High -> Node = Low
; low_high_key(Low, High, Key),
( lookup_node(Var, Key, Node) -> true
; clpb_next_id('$clpb_next_node', ID),
Node = node(ID,Var,Low,High,_Aux),
register_node(Var, Key, Node)
)
).
make_node(Var, Low, High, Node) -->
% make it conveniently usable within DCGs
{ make_node(Var, Low, High, Node) }.
% The key of a node for hashing is determined by the IDs of its
% children.
low_high_key(Low, High, node(LID,HID)) :-
node_id(Low, LID),
node_id(High, HID).
rebuild_hashes(BDD) :-
bdd_nodes(nodevar_put_empty_hash, BDD, Nodes),
maplist(re_register_node, Nodes).
nodevar_put_empty_hash(Node) :-
node_var_low_high(Node, Var, _, _),
empty_assoc(H0),
put_attr(Var, clpb_hash, H0).
re_register_node(Node) :-
node_var_low_high(Node, Var, Low, High),
low_high_key(Low, High, Key),
register_node(Var, Key, Node).
register_node(Var, Key, Node) :-
get_attr(Var, clpb_hash, H0),
put_assoc(Key, H0, Node, H),
put_attr(Var, clpb_hash, H).
lookup_node(Var, Key, Node) :-
get_attr(Var, clpb_hash, H0),
get_assoc(Key, H0, Node).
node_id(0, false).
node_id(1, true).
node_id(node(ID,_,_,_,_), ID).
node_aux(Node, Aux) :- arg(5, Node, Aux).
node_var_low_high(Node, Var, Low, High) :-
Node = node(_,Var,Low,High,_).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
sat_bdd/2 converts a SAT formula in canonical form to an ordered
and reduced BDD.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
sat_bdd(V, Node) :- var(V), !, make_node(V, 0, 1, Node).
sat_bdd(I, I) :- integer(I), !.
sat_bdd(V^Sat, Node) :- !, sat_bdd(Sat, BDD), existential(V, BDD, Node).
sat_bdd(card(Is,Fs), Node) :- !, counter_network(Is, Fs, Node).
sat_bdd(Sat, Node) :- !,
Sat =.. [F,A,B],
sat_bdd(A, NA),
sat_bdd(B, NB),
apply(F, NA, NB, Node).
existential(V, BDD, Node) :-
var_index(V, Index),
bdd_restriction(BDD, Index, 0, NA),
bdd_restriction(BDD, Index, 1, NB),
apply(+, NA, NB, Node).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Counter network for card(Is,Fs).
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
counter_network(Cs, Fs, Node) :-
same_length([_|Fs], Indicators),
fill_indicators(Indicators, 0, Cs),
phrase(formulas_variables(Fs, Vars0), ExBDDs),
maplist(unvisit, Vars0),
% The counter network is built bottom-up, so variables with
% highest index must be processed first.
variables_in_index_order(Vars0, Vars1),
reverse(Vars1, Vars),
counter_network_(Vars, Indicators, Node0),
foldl(existential_and, ExBDDs, Node0, Node).
% Introduce fresh variables for expressions that are not variables.
% These variables are later existentially quantified to remove them.
% Also, new variables are introduced for variables that are used more
% than once, as in card([0,1],[X,X,Y]), to keep the BDD ordered.
formulas_variables([], []) --> [].
formulas_variables([F|Fs], [V|Vs]) -->
( { var(F), \+ is_visited(F) } ->
{ V = F,
put_visited(F) }
; { enumerate_variable(V),
sat_rewrite(V =:= F, Sat),
sat_bdd(Sat, BDD) },
[V-BDD]
),
formulas_variables(Fs, Vs).
counter_network_([], [Node], Node).
counter_network_([Var|Vars], [I|Is0], Node) :-
foldl(indicators_pairing(Var), Is0, Is, I, _),
counter_network_(Vars, Is, Node).
indicators_pairing(Var, I, Node, Prev, I) :- make_node(Var, Prev, I, Node).
fill_indicators([], _, _).
fill_indicators([I|Is], Index0, Cs) :-
( memberchk(Index0, Cs) -> I = 1
; member(A-B, Cs), between(A, B, Index0) -> I = 1
; I = 0
),
Index1 is Index0 + 1,
fill_indicators(Is, Index1, Cs).
existential_and(Ex-BDD, Node0, Node) :-
bdd_and(BDD, Node0, Node1),
existential(Ex, Node1, Node),
% remove attributes to avoid residual goals for variables that