Combination techniques and decision problems for disunification

Abstract

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1,…, E n in order to obtain a unification algorithm for the union E 1 ∪ ⋯ ∪ E n of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E 1 ∪ ⋯ ∪ E n . Our first result says that solvability of disunification problems in the free algebra of the combined theory E 1 ∪ ⋯ ∪ E n is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i ( i = 1,…, n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 ∪ ⋯ ∪ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not E i -equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory E i is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories E i are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent.