Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures

Abstract

Most of the work on the combination of uni cation algorithms for the union of disjoint equational theories has been restricted to algorithms which compute nite complete sets of uni ers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of uni cation problems without computing uni ers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called uni cation problems with constant restrictions|a slight generalization of uni cation problems with constants|is decidable for these theories. As a consequence of this new method, we can for example show that general A-uni ability, i.e., solvability of A-uni cation problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute nite complete sets of uni ers. Manfred Schmidt-Schau ' combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that uni cation in the union of disjoint equational theories is nitary, if general uni cation|i.e., uni cation of terms with additional free function symbols|is nitary in the single theories.

E-unification with Constants vs. General E-unification

“…He has not restricted theories to have any particular type, instead he showed that all equational theories E 1 , E 2 over disjoint signatures that have decidable constant elimination problems admit a combination procedure. This result was improved by Baader and Schulz in [3,4]. They showed that all equational theories E 1 , E 2 over disjoint signatures having decidable E 1 -and E 2 -unification problems with linear constant restrictions admit a combination procedure.…”

“…They showed that all equational theories E 1 , E 2 over disjoint signatures having decidable E 1 -and E 2 -unification problems with linear constant restrictions admit a combination procedure. Since Eunification with linear constant restrictions is equivalent to general E-unification, the result from [3,4] cannot be improved.…”

“…This problem can be also stated in terms of positive equational logic (however the logic formulation will not be used in this paper). The proofs that certain problems are equivalent can be found in [3]. In this paper we give a construction of an equational theory, which shows that the answer to Problem #66 is positive.…”

“…The above question (in different variants) was stated in [3,4,12] and it is also on the RTA open problem list [11] as Problem #66. This problem can be also stated in terms of positive equational logic (however the logic formulation will not be used in this paper).…”

Unification Modulo ACUI Plus Distributivity Axioms

“…These linear constraints are similar to the linear constant restrictions of [7]. The following two theorems relate the unification problem S over ACUID l to its h-image T over ACUIH (recall that H denotes the set of homomorphisms H (S)).…”