Theorem Proving Using Equational Matings and Rigid E-Unifications

In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (first-order) languages with equality. A decidable version of E-unification called rigid E-unification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid E-unification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of E-unification. Problem: Given →/E = {Ei | 1 ≤ i ≤ n} a family of n finite sets of equations and S = {〈ui, vi〉 | 1 ≤ i ≤ n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e. holding the variables in θ(Ei) rigid ), θ(ui) and θ(vi) are provably equal from θ(Ei) for i = 1, ... ,n? Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i 1, ... , n? A substitution θ solving the above problem is called a rigid →/E-unifier of S, and a pair (→/E, S) such that S has some rigid →/E-unifier is called an equational premating. It is shown that deciding whether a pair 〈→/E, S〉 is an equational premating is an NP-complete problem

Unification Procedures in Automated Deduction Methods Based on Matings: A Survey

Unification procedures arising in methods for automated theorem proving based on matings are surveyed. We begin by reviewing some fundamentals of automated deduction, including the Skolem form and the Skolem-Herbrand-Gödel theorem. Next, the method of matings for first-order languages without equality due to Andrews and Bibel is presented. Standard unification is described in terms of transformations on systems (following the approach of Martelli and Montanari, anticipated by Herbrand). Some fast unification algorithms are also sketched, in particular, a unification closure algorithm inspired by Paterson and Wegman\u27s method. The method of matings is then extended to languages with equality. This extention leads naturally to a generalization of standard unification called rigid E-unification (due to Gallier, Narendran, Plaisted, and Snyder). The main properties of rigid E-unification, decidability, NP-completeness, and finiteness of complete sets, are discussed

Rigid E-unification: NP-completeness and applications to equational matings

AbstractRigid E-unification is a restricted kind of unification modulo equational theories, or E-unification, that arises naturally in extending Andrew's theorem proving method of matings to first-order languages with equality. This extension was first presented by J. H. Gallier, S. Raatz, and W. Snyder, who conjectured that rigid E-unification is decidable. In this paper, it is shown that rigid E-unification is NP-complete and that finite complete sets of rigid E-unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are also discussed

Rigid E-Unification: NP-Completeness and Applications to Equational Matings

Rigid E-unification is a restricted kind of unification modulo equational theories, or E-unification, that arises naturally in extending Andrews\u27s theorem proving method of matings to first-order languages with equality. This extension was first presented in Gallier, Raatz, and Snyder, where it was conjectured that rigid E-unification is decidable. In this paper, it is shown that rigid E-unification is NP-complete and that finite complete sets of rigid E-unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are also discussed