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clpb.pl -- CLP(B): Constraint Logic Programming over Boolean Variables |
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, 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/
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 and https://www.metalevel.at/boolean.pdf respectively.
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.
The most frequently used CLP(B) predicates are:
The unification of a CLP(B) variable X with a term T is equivalent
to posting the constraint sat(X=:=T)
.
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.
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).
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.
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.
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.
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.