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

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

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

ORDER-SORTED RIGID E-UNIFICATION

Rigid E-Unification is a special type of unification which arises naturally when extending Andrew's method of matings to logic with equality. We study the rigid E-Unification problem in the presence of subsorts. We present an order sorted method for the computation of order sorted rigid-E-unifiers. The method is based on an unsorted one which we refine and extend to handle sort information. Our approach is to incorporate the sort information within the method so as to leverage it. We show via examples how the order sorted method is able to detect failures due to sort conflicts at an early stage in the construction of potential rigid E Unifiers. The algorithm presented here is NP-complete, as is the unsorted one. This is significant, specially due to the complications presented by the sort information.Information Systems Working Papers Serie

Equality elimination for the inverse method and extension procedures

We demonstrate how to handle equality in the inverse method using equality elimination. In the equality elimination method, proofs consist of two parts. In the first part we try to solve equations obtaining so called solution clauses. Solution clauses are obtained by a very refined strategy — basic superposition with selection function. In the second part, we perform the usual sequent proof search by the inverse method. Our approach is called equality elimination because we eliminate all occurrences of equality in the first part of the proof. Unlike the previous approach proposed by Maslov, our method uses most general substitutions, orderin

The undecidability of simultaneous rigid E-unification with two variables

Abstract. Recently it was proved that the problem of simultaneous rigid E-unification, or SREU, is undecidable. Here we show that 4 rigid equations with ground left-hand sides and 2 variables already imply undecidability. As a corollary we improve the undecidability result of the 3*-fragment of intuitionistic logic with equality. Our proof shows undecidability of a very restricted subset of the 33-fragment. Together with other results, it contributes to a complete characterization of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. 1 I n t r o d u c t i o n Recently it was proved that the problem of simultaneous rigid E-unification (SREU) is undecidable Background of S R E U Simultaneous rigid E-unification was proposed by Ga~er, Raatz and Snyder 1 It has been noted by Gurevich and Veanes that 3 rigid equations suffice

ORDER-SORTED RIGID E-UNIFICATION

Rigid E-Unification is a special type of unification which arises naturally when extending Andrew's method of matings to logic with equality. We study the rigid E-Unification problem in the presence of subsorts. We present an order sorted method for the computation of order sorted rigid-E-unifiers. The method is based on an unsorted one which we refine and extend to handle sort information. Our approach is to incorporate the sort information within the method so as to leverage it. We show via examples how the order sorted method is able to detect failures due to sort conflicts at an early stage in the construction of potential rigid E Unifiers. The algorithm presented here is NP-complete, as is the unsorted one. This is significant, specially due to the complications presented by the sort information.Information Systems Working Papers Serie

Preprints of Proceedings of GWAI-92

This is a preprint of the proceedings of the German Workshop on Artificial Intelligence (GWAI) 1992. The final version will appear in the Lecture Notes in Artificial Intelligence

Unification in the Description Logic EL w.r.t. Cycle-Restricted TBoxes

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive Description Logic EL is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using EL. On the other hand, unification in EL has recently been shown to be NP-complete, and thus of significantly lower complexity than unification in other DLs of similarly restricted expressive power. However, the unification algorithms for EL developed so far cannot deal with general concept inclusion axioms (GCIs). This paper makes a considerable step towards addressing this problem, but the GCIs our new unification algorithm can deal with still need to satisfy a certain cycle restriction