/* Part of SWI-Prolog Author: Markus Triska E-mail: triska@gmx.at WWW: http://www.swi-prolog.org Copyright (C): 2014, 2015 Markus Triska This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this library; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA As a special exception, if you link this library with other files, compiled with a Free Software compiler, to produce an executable, this library does not by itself cause the resulting executable to be covered by the GNU General Public License. This exception does not however invalidate any other reasons why the executable file might be covered by the GNU General Public License. */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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 ]). :- use_module(library(error)). :- use_module(library(assoc)). :- use_module(library(apply_macros)). /** Constraint Logic Programming over Boolean Variables ### Introduction {#clpb-intro} Constraint programming is a declarative formalism that lets you state relations between terms. 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. The implementation is based on reduced and ordered Binary Decision Diagrams (BDDs). ### Boolean expressions {#clpb-exprs} A _Boolean expression_ is one of: | `0` | false | | `1` | true | | _variable_ | unknown truth value | | ~ _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. ### Interface predicates {#clpb-interface} Important interface predicates of CLP(B) 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(1#X#X*Y), sat(1#Y#Y*Z). == The pending residual goals constrain remaining variables to Boolean expressions and are declaratively equivalent to the original query. @author Markus Triska */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Type checking. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ is_sat(V) :- var(V), !. is_sat(I) :- integer(I), between(0, 1, I). 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). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Rewriting to canonical expressions. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ % elementary sat_rewrite(V, V) :- var(V), !. sat_rewrite(I, I) :- integer(I). 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) :- ( 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), 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 % % 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), sat_roots(Sat, Roots), catch((roots_and(Roots, _-1, _-Ands), ( T = 0, unsatisfiable_conjunction(Sat, Ands) -> true ; T = 1, unsatisfiable_conjunction(1#Sat, Ands) -> true ; false ), % reset all attributes throw(truth(T))), truth(T), true). unsatisfiable_conjunction(Sat, Ands) :- sat_bdd(Sat, BDD), bdd_and(BDD, Ands, B), B == 0. satisfiable_bdd(BDD) :- ( BDD == 0 -> false ; BDD == 1 -> true ; bdd_variables(BDD, Vs), node_var_low_high(BDD, Var, _, _), var_index(Var, Lowest), maplist(variable_definite_value(BDD,Lowest), Vs, Values), ( maplist(var, Values) -> true % nothing to propagate ; Vs = Values % propagate all assignments at once ) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - We walk the BDD and check for the given variable if only a single value can make the formula true. This means: 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. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ variable_definite_value(BDD, Lowest, Var, Value) :- var_index(Var, VI), ( VI < Lowest -> Value = _ % either value is possible ; ( bdd_nodes(var_always_0(VI), BDD, _) -> Value = 0 ; bdd_nodes(var_always_1(VI), BDD, _) -> Value = 1 ; Value = _ % either value is possible ) ). var_always_0(VI, Node) :- node_var_low_high(Node, OVar, Low, High), single_truth_value(VI, OVar, High, Low). % note reverse order! var_always_1(VI, Node) :- node_var_low_high(Node, OVar, Low, High), single_truth_value(VI, OVar, Low, High). single_truth_value(VI, OVar, Child1, Child2) :- % If any variable is instantiated, the following fails, as intended. % This only means that we do not perform any propagation for now. var_index(OVar, OVI), ( VI =:= OVI -> Child1 == 0 ; OVI > VI -> true ; % OVI < VI -> \+ index_skipped(VI, Child1), \+ index_skipped(VI, Child2) ). index_skipped(VI, ChildNode) :- ( ChildNode == 0 -> false ; ChildNode == 1 -> true ; node_var_low_high(ChildNode, ChildVar, _, _), var_index(ChildVar, ChildIndex), VI < ChildIndex ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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), % 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. formulas_variables([], []) --> []. formulas_variables([F|Fs], [V|Vs]) --> ( { var(F) } -> { V = 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 they are unified with anything. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ attr_unify_hook(index_root(I,Root), Other) :- ( 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), satisfiable_bdd(BDD) ; no_truth_value(Other) ) ; parse_sat(Other, OtherSat), root_get_formula_bdd(Root, Sat0, _), Sat = Sat0*OtherSat, sat_roots(Sat, Roots), maplist(root_rebuild_bdd, Roots), roots_and(Roots, 1-1, And-BDD1), maplist(del_bdd, Roots), maplist(=(NewRoot), Roots), root_put_formula_bdd(NewRoot, And, BDD1), is_bdd(BDD1), satisfiable_bdd(BDD1) ). root_rebuild_bdd(Root) :- ( root_get_formula_bdd(Root, F0, _) -> parse_sat(F0, Sat), sat_bdd(Sat, BDD), root_put_formula_bdd(Root, F0, BDD) ; true ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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 or AttrVars 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, AttrVars) :- append(QueryVars0, AttrVars, QueryVars1), include(clpb_variable, QueryVars1, 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, 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) --> ( { integer(Node) ; with_aux(is_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) ). 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, set the flag clpb_validation to true. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ is_bdd(BDD) :- ( current_prolog_flag(clpb_validation, true) -> 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 ) ; true ). registered_node(Node-ite(Var,High,Low)) :- low_high_key(Low, High, Key), lookup_node(Var, Key, Node0), Node == Node0. 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 nondet. % % Assigns truth values to the Boolean variables Vs such that all % stated constraints are satisfied. labeling(Vs0) :- must_be(list, Vs0), variables_in_index_order(Vs0, Vs), maplist(indomain, Vs). 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, -N) is det. % % N 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. % % Example: % % == % ?- 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), % ... and then remove remaining variables: foldl(existential, Vs1, BDD1, BDD2), variables_in_index_order(Vs, IVs), foldl(renumber_variable, IVs, 1, VNum), bdd_count(BDD2, VNum, Count0), var_u(BDD2, 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). 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), Count is LPow*LCount + HPow*HCount ) ). 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) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Projection to residual goals. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ attribute_goals(Var) --> { var_index_root(Var, _, Root) }, ( { root_get_formula_bdd(Root, Formula, BDD) } -> { del_bdd(Root), phrase(sat_ands(Formula), Ands), maplist(formula_anf, Ands, ANFs0), sort(ANFs0, ANFs1), exclude(eq_1, ANFs1, ANFs), % 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) }, sats(ANFs), booleans(RestVs) ; boolean(Var) % the variable may have occurred only in taut/2 ). del_clpb(Var) :- del_attr(Var, clpb). sats([]) --> []. sats([A|As]) --> [sat(A)], sats(As). booleans([]) --> []. booleans([B|Bs]) --> boolean(B), booleans(Bs). boolean(Var) --> ( { get_attr(Var, clpb_omit_boolean, true) } -> [] ; [sat(Var =:= Var)] ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Relate a formula to its algebraic normal form (ANF). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ formula_anf(Formula0, ANF) :- sat_rewrite(Formula0, Formula), sat_bdd(Formula, Node), node_xors(Node, Xors), maplist(list_to_conjunction, Xors, [Conj|Conjs]), foldl(xor, Conjs, Conj, ANF). 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, Var, Low, High), 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. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ make_clpb_var('$clpb_next_var') :- nb_setval('$clpb_next_var', 0). make_clpb_var('$clpb_next_node') :- nb_setval('$clpb_next_node', 0). :- multifile user:exception/3. user:exception(undefined_global_variable, Name, retry) :- make_clpb_var(Name), !. clpb_next_id(Var, ID) :- b_getval(Var, ID), Next is ID + 1, b_setval(Var, Next). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The variable attributes below are not used as constraints by this library. Project remaining attributes to empty lists of residuals. Because accessing these hooks is basically a cross-module call, we must declare them public. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- public clpb_hash:attr_unify_hook/2, clpb_bdd:attribute_goals//1, clpb_hash:attribute_goals//1, clpb_omit_boolean:attribute_goals//1. clpb_hash:attr_unify_hook(_,_). % this unification is always admissible clpb_bdd:attribute_goals(_) --> []. clpb_hash:attribute_goals(_) --> []. clpb_omit_boolean:attribute_goals(_) --> []. % clpb_hash:attribute_goals(Var) --> % { get_attr(Var, clpb_hash, Assoc), % assoc_to_list(Assoc, List0), % maplist(node_portray, List0, List) }, [Var-List]. % node_portray(Key-Node, Key-Node-ite(Var,High,Low)) :- % node_var_low_high(Node, Var, Low, High). :- multifile sandbox:safe_global_variable/1. sandbox:safe_global_variable('$clpb_next_var'). sandbox:safe_global_variable('$clpb_next_node').