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 (section 8), 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:
0false 1true variable unknown truth value atom universally quantified variable ~Exprlogical NOT Expr + Expr logical OR Expr * Expr logical AND Expr # Expr exclusive OR Var ^Exprexistential quantification Expr =:=Exprequality Expr =\=Exprdisequality (same as #) Expr =<Exprless or equal (implication) Expr >=Exprgreater 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.
We 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, we 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.
In this example, we are attempting to place I pigeons into J holes in such a way that each hole contains at most one pigeon. One interesting property of this task is that it can be formulated using only cardinality constraints (card/2). Another interesting aspect is that this task has no short resolution refutations in general.
In the following, we use Prolog DCG notation to describe a list Cs of CLP(B) constraints that must all be satisfied.
:- use_module(library(clpb)).
:- use_module(library(clpfd)).
pigeon(I, J, Rows, Cs) :-
length(Rows, I), length(Row, J),
maplist(same_length(Row), Rows),
transpose(Rows, TRows),
phrase((all_cards(Rows,[1]),all_cards(TRows,[0,1])), Cs).
all_cards([], _) --> [].
all_cards([Ls|Lss], Cs) --> [card(Cs,Ls)], all_cards(Lss, Cs).
Example queries:
?- pigeon(9, 8, Rows, Cs), sat(*(Cs)). false. ?- pigeon(2, 3, Rows, Cs), sat(*(Cs)), append(Rows, Vs), labeling(Vs), maplist(portray_clause, Rows). [0, 0, 1]. [0, 1, 0]. etc.
Consider a Boolean circuit that express the Boolean function XOR
with 4 NAND gates. We can model such a circuit with CLP(B)
constraints as follows:
:- use_module(library(clpb)).
nand_gate(X, Y, Z) :- sat(Z =:= ~(X*Y)).
xor(X, Y, Z) :-
nand_gate(X, Y, T1),
nand_gate(X, T1, T2),
nand_gate(Y, T1, T3),
nand_gate(T2, T3, Z).
Using universally quantified variables, we can show that the circuit
does compute XOR as intended:
?- xor(x, y, Z). sat(Z=:=x#y).
The interface predicates of this library follow the example of SICStus Prolog.
Use SICStus Prolog for higher performance in many cases.
In the following, each CLP(B) predicate is described in more detail.
We recommend the following link to refer to this manual:
http://eu.swi-prolog.org/man/clpb.html
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.