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clpb.pl
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clpb.pl
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/* Part of SWI-Prolog
Author: Markus Triska
E-mail: triska@metalevel.at
WWW: http://www.swi-prolog.org
Copyright (C): 2014-2018 Markus Triska
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in
the documentation and/or other materials provided with the
distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
CLP(B): Constraint Logic Programming over Boolean variables.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
:- module(clpb, [
op(300, fy, ~),
op(500, yfx, #),
sat/1,
taut/2,
labeling/1,
sat_count/2,
weighted_maximum/3,
random_labeling/2
]).
:- use_module(library(error)).
:- use_module(library(assoc)).
:- use_module(library(apply_macros)).
:- create_prolog_flag(clpb_monotonic, false, []).
:- create_prolog_flag(clpb_residuals, default, []).
/** <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).
Benchmarks and usage examples of this library are available from:
[__https://www.metalevel.at/clpb/__](https://www.metalevel.at/clpb/)
We recommend the following references 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"
}
@article{Triska2018,
title = "Boolean constraints in {SWI-Prolog}:
A comprehensive system description",
journal = "Science of Computer Programming",
volume = "164",
pages = "98 - 115",
year = "2018",
note = "Special issue of selected papers from FLOPS 2016",
issn = "0167-6423",
doi = "https://doi.org/10.1016/j.scico.2018.02.001",
url = "http://www.sciencedirect.com/science/article/pii/S0167642318300273",
author = "Markus Triska",
keywords = "CLP(B), Boolean unification, Decision diagrams, BDD"
}
==
These papers are available from
[https://www.metalevel.at/swiclpb.pdf](https://www.metalevel.at/swiclpb.pdf)
and
[https://www.metalevel.at/boolean.pdf](https://www.metalevel.at/boolean.pdf)
respectively.
## 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) | cardinality constraint (_see below_) |
| `+(Exprs)` | n-fold disjunction (_see below_) |
| `*(Exprs)` | n-fold conjunction (_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`. For example, to
state that precisely two of the three variables `X`, `Y` and `Z` are
`true`, you can use `sat(card([2],[X,Y,Z]))`.
`+(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).
non_monotonic(X) :-
( var_index(X, _) ->
% OK: already constrained to a CLP(B) variable
true
; current_prolog_flag(clpb_monotonic, true) ->
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
% are only used as substitutes for formulas
del_attrs(Ex).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Compute F(NA, NB).
We use a DCG to thread through an implicit argument G0, an
association table F(IDA,IDB) -> Node, used for memoization.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
apply(F, NA, NB, Node) :-
empty_assoc(G0),
phrase(apply(F, NA, NB, Node), [G0], _).
apply(F, NA, NB, Node) -->
( { integer(NA), integer(NB) } -> { once(bool_op(F, NA, NB, Node)) }
; { apply_shortcut(F, NA, NB, Node) } -> []
; { node_id(NA, IDA), node_id(NB, IDB), Key =.. [F,IDA,IDB] },
( state(G0), { get_assoc(Key, G0, Node) } -> []
; apply_(F, NA, NB, Node),
state(G0, G),
{ put_assoc(Key, G0, Node, G) }
)
).
apply_shortcut(+, NA, NB, Node) :-
( NA == 0 -> Node = NB
; NA == 1 -> Node = 1
; NB == 0 -> Node = NA
; NB == 1 -> Node = 1
; false
).
apply_shortcut(*, NA, NB, Node) :-
( NA == 0 -> Node = 0
; NA == 1 -> Node = NB
; NB == 0 -> Node = 0
; NB == 1 -> Node = NA
; false
).
apply_(F, NA, NB, Node) -->
{ var_less_than(NA, NB),
!,
node_var_low_high(NA, VA, LA, HA) },
apply(F, LA, NB, Low),
apply(F, HA, NB, High),
make_node(VA, Low, High, Node).
apply_(F, NA, NB, Node) -->
{ node_var_low_high(NA, VA, LA, HA),
node_var_low_high(NB, VB, LB, HB),
VA == VB },
!,
apply(F, LA, LB, Low),
apply(F, HA, HB, High),
make_node(VA, Low, High, Node).
apply_(F, NA, NB, Node) --> % NB < NA
{ node_var_low_high(NB, VB, LB, HB) },
apply(F, NA, LB, Low),
apply(F, NA, HB, High),
make_node(VB, Low, High, Node).
node_varindex(Node, VI) :-
node_var_low_high(Node, V, _, _),
var_index(V, VI).
var_less_than(NA, NB) :-
( integer(NB) -> true
; node_varindex(NA, VAI),
node_varindex(NB, VBI),
VAI < VBI
).
bool_op(+, 0, 0, 0).
bool_op(+, 0, 1, 1).
bool_op(+, 1, 0, 1).
bool_op(+, 1, 1, 1).
bool_op(*, 0, 0, 0).
bool_op(*, 0, 1, 0).
bool_op(*, 1, 0, 0).
bool_op(*, 1, 1, 1).
bool_op(#, 0, 0, 0).
bool_op(#, 0, 1, 1).
bool_op(#, 1, 0, 1).
bool_op(#, 1, 1, 0).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Access implicit state in DCGs.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
state(S) --> state(S, S).
state(S0, S), [S] --> [S0].
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Unification. X = Expr is equivalent to sat(X =:= Expr).
Current limitation:
===================
The current interface of attributed variables is not general enough
to express what we need. For example,
?- sat(A + B), A = A + 1.
should be equivalent to
?- sat(A + B), sat(A =:= A + 1).
However, attr_unify_hook/2 is only called *after* the unification
of A with A + 1 has already taken place and turned A into a cyclic
ground term, raised an error or failed (depending on the flag
occurs_check), making it impossible to reason about the variable A
in the unification hook. Therefore, a more general interface for
attributed variables should replace the current one. In particular,
unification filters should be able to reason about terms before