Equivalence and Isomorphism for Boolean Constraint Satisfaction

Abstract

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomial-time solvable or coNPcomplete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNP-hard if the corresponding equivalence problem is coNP-hard, and polynomial-time many-one reducible to the graph isomorphism problem in all other cases.

On the Expression Complexity of Equivalence and Isomorphism of Primitive Positive Formulas

“…Although this relation may not be in B, it is clearly pp-definable over B and hence is preserved by all polymorphisms of B. The resulting quantifier-free ppformulas may be reduced to GI by [9,Claim 22]. We can clearly carry this out in polynomial time if CSP(I(A B )) is in P.…”

“…We develop some relationships between the complexity of the problems studied here and the complexity of the CSP, and in some concrete cases make use of the complexity results on the boolean CSP due to Allender et al [2]. Böhler et al [9,8] have studied equivalence and isomorphism for boolean constraint satisfaction. The problems that they studied have a formulation similar to ours, with the key difference that the formulas that they studied are not permitted to contain existential quantification.…”

Isomorphic Implication

“…The approach taken in [BHRV02,BHRV04,BHRV03], which examine the complexity of the isomorphism problem for Boolean constraints, is to first prove lower bounds for the case with constants, and then to show that all the hardness reductions can be modified to obtain reductions for the cases without constants.…”

“…O i is a set of constraint applications of D, since there exists a constraint application A(x, y) of D that is equivalent to x → y (see [BHRV02,Claim 14]). …”

“…For the case of most interest for this paper, the Boolean isomorphism problem for constraints, Böhler et al [BHRV02,BHRV04,BHRV03] have shown that this problem is in P, GI-complete, or GI-hard, coNP-hard, and in P NP || (the class of problems solvable in polynomial time with one round of parallel queries to NP). As in Schaefer's theorem, simple properties of the set of constraints determine the complexity.…”