A.7 library(clpb): Constraint Logic Programming over Boolean Variables

author
Markus Triska

A.7.1 Introduction

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(.) scheme, extending logic programming with reasoning over specialised domains.

The implementation is based on reduced and ordered Binary Decision Diagrams (BDDs).

A.7.2 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.

A.7.3 Interface predicates

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).

A.7.4 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.

A.7.5 Obtaining BDDs

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 is generally faster, and 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.

[semidet]sat(+Expr)
True iff Expr is a satisfiable Boolean expression.
[semidet]taut(+Expr, -T)
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.
[multi]labeling(+Vs)
Assigns truth values to the Boolean variables Vs such that all stated constraints are satisfied.
[det]sat_count(+Expr, -N)
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.
[det]random_labeling(+Seed, +Vs)
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.
[multi]weighted_maximum(+Weights, +Vs, -Maximum)
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.