From ee32e4952825f0b6e06ca6e01d2ce5665a12882c Mon Sep 17 00:00:00 2001 From: Markus Triska Date: Wed, 16 Oct 2019 06:53:34 +0200 Subject: [PATCH] ADDED: CLP(B), Constraint Logic Programming over Boolean Variables MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit library(clpb) provides CLP(B), Constraint Logic Programming over Boolean variables. It is a SAT solver that seamlessly integrates into Prolog in the sense that logic variables are used to state constraints and report solutions. This library can be used to model and solve many combinatorial problems such as verification, allocation and covering tasks. CLP(B) is an instance of the general CLP(X) scheme, 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: https://github.com/triska/clpb For more information, benchmarks and publications visit: https://www.metalevel.at/clpb/ The interface of this library is consciously kept compatible with the CLP(B) solver of SICStus Prolog, which served as the main inspiration of this library. Many thanks to Mats Carlsson for his elegant example! It is my hope that library(clpb) will allow a port of cTI, and — eventually — of Ulrich Neumerkel's GUPU to Scryer Prolog. Boolean expressions =================== 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 ==================== 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 ======== 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 ============== By default, CLP(B) residual goals appear in (approximately) algebraic normal form (ANF). This projection is often computationally expensive. Assert the fact clpb:clpb_residuals(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: ?- asserta(clpb: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). Monotonic CLP(B) ================ 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 techniques inapplicable (see https://www.metalevel.at/prolog/debugging for more information). Assert the fact clpb:monotonic to make CLP(B) monotonic. If this mode is enabled, then you must wrap CLP(B) variables with the functor v/1. For example: ?- asserta(clpb:monotonic). true. ?- sat(v(X)=:=1#1). X = 0. Enjoy! --- src/prolog/lib/clpb.pl | 1775 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1775 insertions(+) create mode 100644 src/prolog/lib/clpb.pl diff --git a/src/prolog/lib/clpb.pl b/src/prolog/lib/clpb.pl new file mode 100644 index 00000000..0ddfc986 --- /dev/null +++ b/src/prolog/lib/clpb.pl @@ -0,0 +1,1775 @@ +/* CLP(B): Constraint Logic Programming over Boolean Variables + + Copyright (C): 2019 Markus Triska + All rights reserved. + + E-mail: triska@metalevel.at + WWW: http://www.metalevel.at + +*/ + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + 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(non_iso)). +:- use_module(library(dcgs)). +%:- 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(ground, _, Term) :- !, + functor(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). + + + + + +goal_expansion(get_attr(Var, Module, Value), (var(Var),get_atts(Var, Access))) :- + Access =.. [Module,Value]. + +goal_expansion(put_attr(Var, Module, Value), put_atts(Var, Access)) :- + Access =.. [Module,Value]. + +goal_expansion(del_attr(Var, Module), (var(Var) -> put_atts(Var, -Access);true)) :- + Access =.. [Module,_]. + + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + 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(AB) :- 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). +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +same_length([], []). +same_length([_|As], [_|Bs]) :- + same_length(As, Bs). + +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). + + +del_attrs(Var) :- + ( var(Var) -> + put_atts(Var, [ + -clpb(_), + -clpb_bdd(_), + -clpb_atom(_), + -clpb_hash(_), + -clpb_max(_), + -clpb_omit_boolean(_), + -clpb_visited(_) + ]) + ; true + ). + + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + 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). +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +verify_attributes(Var, Other, Gs) :- + % format("~w = ~w\n", [Var,Other]), + ( get_attr(Var, clpb, index_root(I,Root)) -> + ( integer(Other) -> + ( between(0, 1, Other) -> + root_get_formula_bdd(Root, Sat, BDD0), + bdd_restriction(BDD0, I, Other, BDD), + root_put_formula_bdd(Root, Sat, BDD), + Gs = [satisfiable_bdd(BDD)] + ; no_truth_value(Other) + ) + ; atom(Other) -> + root_get_formula_bdd(Root, Sat0, _), + Gs = [root_rebuild_bdd(Root, Sat0)] + ; % due to variable aliasing, any BDDs may become unordered, + % so we need to rebuild the new BDD from the conjunction + % after the unification is in place + root_get_formula_bdd(Root, Sat0, _), + Sat = Sat0*OtherSat, + parse_sat(Other, OtherSat), + sat_roots(Sat, Roots), + phrase(formulas_(Roots), [F|Fs]), + foldl(and, Fs, F, And), + maplist(del_bdd, Roots), + maplist(=(NewRoot), Roots), + Gs = [root_rebuild_bdd(NewRoot, And)] + ) + ; Gs = [] + ). + + +formulas_([]) --> []. +formulas_([Root|Roots]) --> + ( { root_get_formula_bdd(Root, F, _) } -> + [F] + ; [] + ), + formulas_(Roots). + +root_rebuild_bdd(Root, Formula) :- + parse_sat(Formula, Sat), + sat_bdd(Sat, BDD), + is_bdd(BDD), + root_put_formula_bdd(Root, Formula, BDD), + satisfiable_bdd(BDD). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Support for project_attributes/2. + + This is called by the toplevel as + + project_attributes(+QueryVars, +AttrVars) + + in order to project all remaining constraints onto QueryVars. + + All CLP(B) variables that do not occur in QueryVars + need to be existentially quantified, so that they do not occur in + residual goals. This is very easy to do in the case of CLP(B). +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +project_attributes(QueryVars0, _) :- + include(clpb_variable, QueryVars0, QueryVars), + maplist(var_index_root, QueryVars, _, Roots0), + sort(Roots0, Roots), + maplist(remove_hidden_variables(QueryVars), Roots). + +clpb_variable(Var) :- var_index(Var, _). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + All CLP(B) variables occurring in BDDs but not in query variables + become existentially quantified. This must also be reflected in the + formula. In addition, an attribute is attached to these variables + to suppress superfluous sat(V=:=V) goals. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +remove_hidden_variables(QueryVars, Root) :- + root_get_formula_bdd(Root, Formula, BDD0), + maplist(put_visited, QueryVars), + bdd_variables(BDD0, HiddenVars0), + exclude(universal_var, HiddenVars0, HiddenVars), + maplist(unvisit, QueryVars), + foldl(existential, HiddenVars, BDD0, BDD), + foldl(quantify_existantially, HiddenVars, Formula, ExFormula), + root_put_formula_bdd(Root, ExFormula, BDD). + +quantify_existantially(E, E0, E^E0) :- put_attr(E, clpb_omit_boolean, true). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + BDD restriction. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +bdd_restriction(Node, VI, Value, Res) :- + empty_assoc(G0), + phrase(bdd_restriction_(Node, VI, Value, Res), [G0], _), + is_bdd(Res). + +bdd_restriction_(Node, VI, Value, Res) --> + ( { integer(Node) } -> { Res = Node } + ; { node_var_low_high(Node, Var, Low, High) } -> + ( { integer(Var) } -> + ( { Var =:= 0 } -> bdd_restriction_(Low, VI, Value, Res) + ; { Var =:= 1 } -> bdd_restriction_(High, VI, Value, Res) + ; { no_truth_value(Var) } + ) + ; { var_index(Var, I0), + node_id(Node, ID) }, + ( { I0 =:= VI } -> + ( { Value =:= 0 } -> { Res = Low } + ; { Value =:= 1 } -> { Res = High } + ) + ; { I0 > VI } -> { Res = Node } + ; state(G0), { get_assoc(ID, G0, Res) } -> [] + ; bdd_restriction_(Low, VI, Value, LRes), + bdd_restriction_(High, VI, Value, HRes), + make_node(Var, LRes, HRes, Res), + state(G0, G), + { put_assoc(ID, G0, Res, G) } + ) + ) + ; { domain_error(node, Node) } + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Relating a BDD to its elements (nodes and variables). + + Note that BDDs can become quite big (easily millions of nodes), and + memory space is a major bottleneck for many problems. If possible, + we therefore do not duplicate the entire BDD in memory (as in + bdd_ites/2), but only extract its features as needed. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +bdd_nodes(BDD, Ns) :- bdd_nodes(ignore_node, BDD, Ns). + +ignore_node(_). + +% VPred is a unary predicate that is called for each node that has a +% branching variable (= each inner node). + +bdd_nodes(VPred, BDD, Ns) :- + phrase(bdd_nodes_(VPred, BDD), Ns), + maplist(with_aux(unvisit), Ns). + +bdd_nodes_(VPred, Node) --> + ( { node_visited(Node) } -> [] + ; { call(VPred, Node), + with_aux(put_visited, Node), + node_var_low_high(Node, _, Low, High) }, + [Node], + bdd_nodes_(VPred, Low), + bdd_nodes_(VPred, High) + ). + +node_visited(Node) :- integer(Node). +node_visited(Node) :- with_aux(is_visited, Node). + +bdd_variables(BDD, Vs) :- + bdd_nodes(BDD, Nodes), + nodes_variables(Nodes, Vs). + +nodes_variables(Nodes, Vs) :- + phrase(nodes_variables_(Nodes), Vs), + maplist(unvisit, Vs). + +nodes_variables_([]) --> []. +nodes_variables_([Node|Nodes]) --> + { node_var_low_high(Node, Var, _, _) }, + ( { integer(Var) } -> [] + ; { is_visited(Var) } -> [] + ; { put_visited(Var) }, + [Var] + ), + nodes_variables_(Nodes). + +unvisit(V) :- del_attr(V, clpb_visited). + +is_visited(V) :- get_attr(V, clpb_visited, true). + +put_visited(V) :- put_attr(V, clpb_visited, true). + +with_aux(Pred, Node) :- + node_aux(Node, Aux), + call(Pred, Aux). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Internal consistency checks. + + To enable these checks, assert clpb:validation/0. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +:- dynamic(validation/0). + +is_bdd(BDD) :- + ( validation -> + bdd_ites(BDD, ITEs), + pairs_values(ITEs, Ls0), + sort(Ls0, Ls1), + ( same_length(Ls0, Ls1) -> true + ; domain_error(reduced_ites, (ITEs,Ls0,Ls1)) + ), + ( member(ITE, ITEs), \+ registered_node(ITE) -> + domain_error(registered_node, ITE) + ; true + ), + ( member(I, ITEs), \+ clpb_ordered(I) -> + domain_error(ordered_node, I) + ; true + ) + ; true + ). + +clpb_ordered(_-ite(Var,High,Low)) :- + ( var_index(Var, VI) -> + greater_varindex_than(High, VI), + greater_varindex_than(Low, VI) + ; true + ). + +greater_varindex_than(Node, VI) :- + ( integer(Node) -> true + ; node_var_low_high(Node, Var, _, _), + ( var_index(Var, OI) -> + OI > VI + ; true + ) + ). + +registered_node(Node-ite(Var,High,Low)) :- + ( var(Var) -> + low_high_key(Low, High, Key), + lookup_node(Var, Key, Node0), + Node == Node0 + ; true + ). + +bdd_ites(BDD, ITEs) :- + bdd_nodes(BDD, Nodes), + maplist(node_ite, Nodes, ITEs). + +node_ite(Node, Node-ite(Var,High,Low)) :- + node_var_low_high(Node, Var, Low, High). + +%% labeling(+Vs) is multi. +% +% Enumerate concrete solutions. Assigns truth values to the Boolean +% variables Vs such that all stated constraints are satisfied. + +labeling(Vs0) :- + must_be(list, Vs0), + maplist(labeling_var, Vs0), + variables_in_index_order(Vs0, Vs), + maplist(indomain, Vs). + +labeling_var(V) :- var(V), !. +labeling_var(V) :- V == 0, !. +labeling_var(V) :- V == 1, !. +labeling_var(V) :- domain_error(clpb_variable, V). + +variables_in_index_order(Vs0, Vs) :- + maplist(var_with_index, Vs0, IVs0), + keysort(IVs0, IVs), + pairs_values(IVs, Vs). + +var_with_index(V, I-V) :- + ( var_index_root(V, I, _) -> true + ; I = 0 + ). + +indomain(0). +indomain(1). + + +%% sat_count(+Expr, -Count) is det. +% +% Count the number of admissible assignments. Count is the number of +% different assignments of truth values to the variables in the +% Boolean expression Expr, such that Expr is true and all posted +% constraints are satisfiable. +% +% A common form of invocation is `sat_count(+[1|Vs], Count)`: This +% counts the number of admissible assignments to `Vs` without imposing +% any further constraints. +% +% Examples: +% +% == +% ?- sat(A =< B), Vs = [A,B], sat_count(+[1|Vs], Count). +% Vs = [A, B], +% Count = 3, +% sat(A=:=A*B). +% +% ?- length(Vs, 120), +% sat_count(+Vs, CountOr), +% sat_count(*(Vs), CountAnd). +% Vs = [...], +% CountOr = 1329227995784915872903807060280344575, +% CountAnd = 1. +% == + +sat_count(Sat0, N) :- + catch((parse_sat(Sat0, Sat), + sat_bdd(Sat, BDD), + sat_roots(Sat, Roots), + roots_and(Roots, _-BDD, _-BDD1), + % we mark variables that occur in Sat0 as visited ... + term_variables(Sat0, Vs), + maplist(put_visited, Vs), + % ... so that they do not appear in Vs1 ... + bdd_variables(BDD1, Vs1), + partition(universal_var, Vs1, Univs, Exis), + % ... and then remove remaining variables: + foldl(universal, Univs, BDD1, BDD2), + foldl(existential, Exis, BDD2, BDD3), + variables_in_index_order(Vs, IVs), + foldl(renumber_variable, IVs, 1, VNum), + bdd_count(BDD3, VNum, Count0), + var_u(BDD3, VNum, P), + % Do not unify N directly, because we are not prepared + % for propagation here in case N is a CLP(B) variable. + N0 is 2^(P - 1)*Count0, + % reset all attributes and Aux variables + throw(count(N0))), + count(N0), + N = N0). + +universal(V, BDD, Node) :- + var_index(V, Index), + bdd_restriction(BDD, Index, 0, NA), + bdd_restriction(BDD, Index, 1, NB), + apply(*, NA, NB, Node). + +renumber_variable(V, I0, I) :- + put_attr(V, clpb, index_root(I0,_)), + I is I0 + 1. + +bdd_count(Node, VNum, Count) :- + ( integer(Node) -> Count = Node + ; node_aux(Node, Count), + ( integer(Count) -> true + ; node_var_low_high(Node, V, Low, High), + bdd_count(Low, VNum, LCount), + bdd_count(High, VNum, HCount), + bdd_pow(Low, V, VNum, LPow), + bdd_pow(High, V, VNum, HPow), + Count0 is LPow*LCount + HPow*HCount, + Count = Count0 + ) + ). + + +bdd_pow(Node, V, VNum, Pow) :- + var_index(V, Index), + var_u(Node, VNum, P), + Pow is 2^(P - Index - 1). + +var_u(Node, VNum, Index) :- + ( integer(Node) -> Index = VNum + ; node_varindex(Node, Index) + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Pick a solution in such a way that each solution is equally likely. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +%% random_labeling(+Seed, +Vs) is det. +% +% Select a single random solution. An admissible assignment of truth +% values to the Boolean variables in Vs is chosen in such a way that +% each admissible assignment is equally likely. Seed is an integer, +% used as the initial seed for the random number generator. + +single_bdd(Vars0) :- + maplist(monotonic_variable, Vars0, Vars), + % capture all variables with a single BDD + sat(+[1|Vars]). + +random_labeling(Seed, Vars) :- + must_be(list, Vars), + set_random(seed(Seed)), + ( ground(Vars) -> true + ; catch((single_bdd(Vars), + once((member(Var, Vars),var(Var))), + var_index_root(Var, _, Root), + root_get_formula_bdd(Root, _, BDD), + bdd_variables(BDD, Vs), + variables_in_index_order(Vs, IVs), + foldl(renumber_variable, IVs, 1, VNum), + phrase(random_bindings(VNum, BDD), Bs), + maplist(del_attrs, Vs), + % reset all attribute modifications + throw(randsol(Vars, Bs))), + randsol(Vars, Bs), + true), + maplist(call, Bs), + % set remaining variables to 0 or 1 with equal probability + include(var, Vars, Remaining), + maplist(maybe_zero, Remaining) + ). + +maybe_zero(Var) :- + ( maybe -> Var = 0 + ; Var = 1 + ). + +random_bindings(_, Node) --> { Node == 1 }, !. +random_bindings(VNum, Node) --> + { node_var_low_high(Node, Var, Low, High), + bdd_count(Node, VNum, Total), + bdd_count(Low, VNum, LCount) }, + ( { maybe(LCount, Total) } -> + [Var=0], random_bindings(VNum, Low) + ; [Var=1], random_bindings(VNum, High) + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Find solutions with maximum weight. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +%% weighted_maximum(+Weights, +Vs, -Maximum) is multi. +% +% Enumerate weighted optima over admissible assignments. Maximize a +% linear objective function over Boolean variables Vs with integer +% coefficients Weights. This predicate assigns 0 and 1 to the +% variables in Vs such that all stated constraints are satisfied, and +% Maximum is the maximum of sum(Weight_i*V_i) over all admissible +% assignments. On backtracking, all admissible assignments that +% attain the optimum are generated. +% +% This predicate can also be used to _minimize_ a linear Boolean +% program, since negative integers can appear in Weights. +% +% Example: +% +% == +% ?- sat(A#B), weighted_maximum([1,2,1], [A,B,C], Maximum). +% A = 0, B = 1, C = 1, Maximum = 3. +% == + +weighted_maximum(Ws, Vars, Max) :- + must_be(list(integer), Ws), + must_be(list(var), Vars), + single_bdd(Vars), + Vars = [Var|_], + var_index_root(Var, _, Root), + root_get_formula_bdd(Root, _, BDD0), + bdd_variables(BDD0, Vs), + % existentially quantify variables that are not considered + maplist(put_visited, Vars), + exclude(is_visited, Vs, Unvisited), + maplist(unvisit, Vars), + foldl(existential, Unvisited, BDD0, BDD), + maplist(var_with_index, Vars, IVs), + pairs_keys_values(Pairs0, IVs, Ws), + keysort(Pairs0, Pairs1), + pairs_keys_values(Pairs1, IVs1, WeightsIndexOrder), + pairs_values(IVs1, VarsIndexOrder), + % Pairs is a list of Var-Weight terms, in index order of Vars + pairs_keys_values(Pairs, VarsIndexOrder, WeightsIndexOrder), + bdd_maximum(BDD, Pairs, Max), + max_labeling(BDD, Pairs). + +max_labeling(1, Pairs) :- max_upto(Pairs, _, _). +max_labeling(node(_,Var,Low,High,Aux), Pairs0) :- + max_upto(Pairs0, Var, Pairs), + get_attr(Aux, clpb_max, max(_,Dir)), + direction_labeling(Dir, Var, Low, High, Pairs). + +max_upto([], _, _). +max_upto([Var0-Weight|VWs0], Var, VWs) :- + ( Var == Var0 -> VWs = VWs0 + ; Weight =:= 0 -> + ( Var0 = 0 ; Var0 = 1 ), + max_upto(VWs0, Var, VWs) + ; Weight < 0 -> Var0 = 0, max_upto(VWs0, Var, VWs) + ; Var0 = 1, max_upto(VWs0, Var, VWs) + ). + +direction_labeling(low, 0, Low, _, Pairs) :- max_labeling(Low, Pairs). +direction_labeling(high, 1, _, High, Pairs) :- max_labeling(High, Pairs). + +bdd_maximum(1, Pairs, Max) :- + pairs_values(Pairs, Weights0), + include(<(0), Weights0, Weights), + sum_list(Weights, Max). +bdd_maximum(node(_,Var,Low,High,Aux), Pairs0, Max) :- + ( get_attr(Aux, clpb_max, max(Max,_)) -> true + ; ( skip_to_var(Var, Weight, Pairs0, Pairs), + ( Low == 0 -> + bdd_maximum_(High, Pairs, MaxHigh, MaxToHigh), + Max is MaxToHigh + MaxHigh + Weight, + Dir = high + ; High == 0 -> + bdd_maximum_(Low, Pairs, MaxLow, MaxToLow), + Max is MaxToLow + MaxLow, + Dir = low + ; bdd_maximum_(Low, Pairs, MaxLow, MaxToLow), + bdd_maximum_(High, Pairs, MaxHigh, MaxToHigh), + Max0 is MaxToLow + MaxLow, + Max1 is MaxToHigh + MaxHigh + Weight, + Max is max(Max0,Max1), + ( Max0 =:= Max1 -> Dir = _Any + ; Max0 < Max1 -> Dir = high + ; Dir = low + ) + ), + store_maximum(Aux, Max, Dir) + ) + ). + +bdd_maximum_(Node, Pairs, Max, MaxTo) :- + bdd_maximum(Node, Pairs, Max), + between_weights(Node, Pairs, MaxTo). + +store_maximum(Aux, Max, Dir) :- put_attr(Aux, clpb_max, max(Max,Dir)). + +between_weights(Node, Pairs0, MaxTo) :- + ( Node == 1 -> MaxTo = 0 + ; node_var_low_high(Node, Var, _, _), + phrase(skip_to_var_(Var, _, Pairs0, _), Weights0), + include(<(0), Weights0, Weights), + sum_list(Weights, MaxTo) + ). + +skip_to_var(Var, Weight, Pairs0, Pairs) :- + phrase(skip_to_var_(Var, Weight, Pairs0, Pairs), _). + +skip_to_var_(Var, Weight, [Var0-Weight0|VWs0], VWs) --> + ( { Var == Var0 } -> + { Weight = Weight0, VWs0 = VWs } + ; ( { Weight0 =< 0 } -> [] + ; [Weight0] + ), + skip_to_var_(Var, Weight, VWs0, VWs) + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Projection to residual goals. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + + +:- dynamic(clpb_residuals/1). + +attribute_goals(Var) --> + { var_index_root(Var, _, Root) }, + ( { root_get_formula_bdd(Root, Formula, BDD) } -> + { del_bdd(Root) }, + ( { clpb_residuals(bdd) } -> + { bdd_nodes(BDD, Nodes), + phrase(nodes(Nodes), Ns) }, + [clpb:'$clpb_bdd'(Ns)] + ; { prepare_global_variables(BDD), + phrase(sat_ands(Formula), Ands0), + ands_fusion(Ands0, Ands), + maplist(formula_anf, Ands, ANFs0), + sort(ANFs0, ANFs1), + exclude(eq_1, ANFs1, ANFs2), + variables_separation(ANFs2, ANFs) }, + sats(ANFs) + ), + ( { get_attr(Var, clpb_atom, Atom) } -> + [clpb:sat(Var=:=Atom)] + ; [] + ), + % formula variables not occurring in the BDD should be booleans + { bdd_variables(BDD, Vs), + maplist(del_clpb, Vs), + term_variables(Formula, RestVs0), + include(clpb_variable, RestVs0, RestVs) }, + booleans(RestVs) + ; boolean(Var) % the variable may have occurred only in taut/2 + ). + +del_clpb(Var) :- + del_attr(Var, clpb), + del_attr(Var, clpb_hash). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + To make residual projection work with recorded constraints, the + global counters must be adjusted so that new variables and nodes + also get new IDs. Also, clpb_next_id/2 is used to actually create + these counters, because creating them with b_setval/2 would make + them [] on backtracking, which is quite unfortunate in itself. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +b_setval(K, T) :- bb_b_put(K, T). +nb_setval(K, T) :- bb_put(K, T). +b_getval(K, T) :- bb_get(K, T). + +prepare_global_variables(BDD) :- + clpb_next_id('$clpb_next_var', V0), + clpb_next_id('$clpb_next_node', N0), + bdd_nodes(BDD, Nodes), + foldl(max_variable_node, Nodes, V0-N0, MaxV0-MaxN0), + MaxV is MaxV0 + 1, + MaxN is MaxN0 + 1, + b_setval('$clpb_next_var', MaxV), + b_setval('$clpb_next_node', MaxN). + +max_variable_node(Node, V0-N0, V-N) :- + node_id(Node, N1), + node_varindex(Node, V1), + N is max(N0,N1), + V is max(V0,V1). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Fuse formulas that share the same variables into single conjunctions. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +ands_fusion(Ands0, Ands) :- + maplist(with_variables, Ands0, Pairs0), + keysort(Pairs0, Pairs), + group_pairs_by_key(Pairs, Groups), + pairs_values(Groups, Andss), + maplist(list_to_conjunction, Andss, Ands). + +with_variables(F, Vs-F) :- + term_variables(F, Vs0), + variables_in_index_order(Vs0, Vs). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + If possible, separate variables into different sat/1 goals. + A formula F can be split in two if for two of its variables A and B, + taut((A^F)*(B^F) =:= F, 1) holds. In the first conjunct, A does not + occur, and in the second, B does not occur. We separate variables + until that is no longer possible. There may be a better way to do this. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +variables_separation(Fs0, Fs) :- separation_fixpoint(Fs0, [], Fs). + +separation_fixpoint(Fs0, Ds0, Fs) :- + phrase(variables_separation_(Fs0, Ds0, Rest), Fs1), + partition(anf_done, Fs1, Ds1, Fs2), + maplist(arg(1), Ds1, Ds2), + maplist(arg(1), Fs2, Fs3), + append(Ds0, Ds2, Ds3), + append(Rest, Fs3, Fs4), + sort(Fs4, Fs5), + sort(Ds3, Ds4), + ( Fs5 == [] -> Fs = Ds4 + ; separation_fixpoint(Fs5, Ds4, Fs) + ). + +anf_done(done(_)). + +variables_separation_([], _, []) --> []. +variables_separation_([F0|Fs0], Ds, Rest) --> + ( { member(Done, Ds), F0 == Done } -> + variables_separation_(Fs0, Ds, Rest) + ; { sat_rewrite(F0, F), + sat_bdd(F, BDD), + bdd_variables(BDD, Vs0), + exclude(universal_var, Vs0, Vs), + maplist(existential_(BDD), Vs, Nodes), + phrase(pairs(Nodes), Pairs), + group_pairs_by_key(Pairs, Groups), + phrase(groups_separation(Groups, BDD), ANFs) }, + ( { ANFs = [_|_] } -> + list(ANFs), + { Rest = Fs0 } + ; [done(F0)], + variables_separation_(Fs0, Ds, Rest) + ) + ). + + +existential_(BDD, V, Node) :- existential(V, BDD, Node). + +groups_separation([], _) --> []. +groups_separation([BDD1-BDDs|Groups], OrigBDD) --> + { phrase(separate_pairs(BDDs, BDD1, OrigBDD), Nodes) }, + ( { Nodes = [_|_] } -> + nodes_anfs([BDD1|Nodes]) + ; [] + ), + groups_separation(Groups, OrigBDD). + +separate_pairs([], _, _) --> []. +separate_pairs([BDD2|Ps], BDD1, OrigBDD) --> + ( { apply(*, BDD1, BDD2, And), + And == OrigBDD } -> + [BDD2] + ; [] + ), + separate_pairs(Ps, BDD1, OrigBDD). + +nodes_anfs([]) --> []. +nodes_anfs([N|Ns]) --> { node_anf(N, ANF) }, [anf(ANF)], nodes_anfs(Ns). + +pairs([]) --> []. +pairs([V|Vs]) --> pairs_(Vs, V), pairs(Vs). + +pairs_([], _) --> []. +pairs_([B|Bs], A) --> [A-B], pairs_(Bs, A). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Set the Prolog flag clpb_residuals to bdd to obtain the BDD nodes + as residuals. Note that they cannot be used as regular goals. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +nodes([]) --> []. +nodes([Node|Nodes]) --> + { node_var_low_high(Node, Var0, Low, High), + var_or_atom(Var0, Var), + maplist(node_projection, [Node,High,Low], [ID,HID,LID]), + var_index(Var0, VI) }, + [ID-(v(Var,VI) -> HID ; LID)], + nodes(Nodes). + + +node_projection(Node, Projection) :- + node_id(Node, ID), + ( integer(ID) -> Projection = node(ID) + ; Projection = ID + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + By default, residual goals are sat/1 calls of the remaining formulas, + using (mostly) algebraic normal form. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +sats([]) --> []. +sats([A|As]) --> [clpb:sat(A)], sats(As). + +booleans([]) --> []. +booleans([B|Bs]) --> boolean(B), { del_clpb(B) }, booleans(Bs). + +boolean(Var) --> + ( { get_attr(Var, clpb_omit_boolean, true) } -> [] + ; [clpb:sat(Var =:= Var)] + ). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Relate a formula to its algebraic normal form (ANF). +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +formula_anf(Formula0, ANF) :- + parse_sat(Formula0, Formula), + sat_bdd(Formula, Node), + node_anf(Node, ANF). + +node_anf(Node, ANF) :- + node_xors(Node, Xors0), + maplist(maplist(monotonic_variable), Xors0, Xors), + maplist(list_to_conjunction, Xors, Conjs), + ( Conjs = [Var,C|Rest], clpb_var(Var) -> + foldl(xor, Rest, C, RANF), + ANF = (Var =\= RANF) + ; Conjs = [One,Var,C|Rest], One == 1, clpb_var(Var) -> + foldl(xor, Rest, C, RANF), + ANF = (Var =:= RANF) + ; Conjs = [C|Cs], + foldl(xor, Cs, C, ANF) + ). + +monotonic_variable(Var0, Var) :- + ( var(Var0), monotonic -> + Var = v(Var0) + ; Var = Var0 + ). + +clpb_var(Var) :- var(Var), !. +clpb_var(v(_)). + +list_to_conjunction([], 1). +list_to_conjunction([L|Ls], Conj) :- foldl(and, Ls, L, Conj). + +xor(A, B, B # A). + +eq_1(V) :- V == 1. + +node_xors(Node, Xors) :- + phrase(xors(Node), Xors0), + % we remove elements that occur an even number of times (A#A --> 0) + maplist(sort, Xors0, Xors1), + pairs_keys_values(Pairs0, Xors1, _), + keysort(Pairs0, Pairs), + group_pairs_by_key(Pairs, Groups), + exclude(even_occurrences, Groups, Odds), + pairs_keys(Odds, Xors2), + maplist(exclude(eq_1), Xors2, Xors). + +even_occurrences(_-Ls) :- length(Ls, L), L mod 2 =:= 0. + +xors(Node) --> + ( { Node == 0 } -> [] + ; { Node == 1 } -> [[1]] + ; { node_var_low_high(Node, Var0, Low, High), + var_or_atom(Var0, Var), + node_xors(Low, Ls0), + node_xors(High, Hs0), + maplist(with_var(Var), Ls0, Ls), + maplist(with_var(Var), Hs0, Hs) }, + list(Ls0), + list(Ls), + list(Hs) + ). + +list([]) --> []. +list([L|Ls]) --> [L], list(Ls). + +with_var(Var, Ls, [Var|Ls]). + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + Global variables for unique node and variable IDs and atoms. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + +make_clpb_var('$clpb_next_var') :- nb_setval('$clpb_next_var', 0). + +make_clpb_var('$clpb_next_node') :- nb_setval('$clpb_next_node', 0). + +make_clpb_var('$clpb_atoms') :- + empty_assoc(E), + nb_setval('$clpb_atoms', E). + +:- initialization((make_clpb_var(_),false;true)). + +clpb_next_id(Var, ID) :- + b_getval(Var, ID), + Next is ID + 1, + b_setval(Var, Next). + +clpb_atom_var(Atom, Var) :- + b_getval('$clpb_atoms', A0), + ( get_assoc(Atom, A0, Var) -> true + ; put_attr(Var, clpb_atom, Atom), + put_attr(Var, clpb_omit_boolean, true), + put_assoc(Atom, A0, Var, A), + b_setval('$clpb_atoms', A) + ). + + + +/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +Compatibility predicates. +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ + + +include(Goal, List, Is) :- + include_(List, Goal, Is). + +include_([], _, []). +include_([X1|Xs1], P, Is) :- + ( call(P, X1) + -> Is = [X1|Is1] + ; Is = Is1 + ), + include_(Xs1, P, Is1). + + +exclude(Goal, List, Is) :- + exclude_(List, Goal, Is). + +exclude_([], _, []). +exclude_([X1|Xs1], P, Is) :- + ( call(P, X1) + -> Is = Is1 + ; Is = [X1|Is1] + ), + exclude_(Xs1, P, Is1). + + +partition(Pred, List, Less, Equal, Greater) :- + partition_(List, Pred, Less, Equal, Greater). + +partition_([], _, [], [], []). +partition_([H|T], Pred, L, E, G) :- + call(Pred, H, Diff), + partition_(Diff, H, Pred, T, L, E, G). + +partition_(<, H, Pred, T, [H|Rest], E, G) :- + partition_(T, Pred, Rest, E, G). +partition_(=, H, Pred, T, L, [H|Rest], G) :- + partition_(T, Pred, L, Rest, G). +partition_(>, H, Pred, T, L, E, [H|Rest]) :- + partition_(T, Pred, L, E, Rest). -- 2.54.0