Tractable and intractable instances of combination problems for unification and disunification

Abstract

The algorithms for combining decision procedures for (dis)unification in the union of arbitrary disjoint equational theories known from the literature generate an exponential number of 'pure' output problems over the component theories. Despite several optimization techniques, no general results are known that describe in which cases a polynomial-time combination procedure is (im)possible. In the first part of this paper we characterize a large class of equational theories E where E-unification with free function symbols is NP-hard and where a polynomial-time optimization of the combination algorithm is impossible (unless P = NP). This supports the conjecture that for the theories in this class there is no polynomial-time algorithm for combining E-unification with constants with free unification. In the second part of the paper we introduce another class of equational theories where (dis)unification algorithms can be combined deterministically and in polynomial time. Under reasonable assumptions on the form of the given unification procedure, each unitary, regular and collapse-free equational theory with a polynomial-time unification algorithm belongs to this class.