ScholarlyCommons: Repository

Title

Theorem Proving Using Equational Matings and Rigid E-Unifications

Author(s)

Jean H. Gallier, University of PennsylvaniaFollow
Paliath Narendran, State University of New York
Stan Raatz, Rutgers University
Wayne Snyder, Boston University

Document Type

Technical Report

Date of this Version

July 1992

Comments

University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-88-15.

Abstract

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 = {E_i | 1 ≤ i ≤ n} a family of n finite sets of equations and S = {〈u_i, v_i〉 | 1 ≤ i ≤ n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(E_i) as a set of ground equations (i.e. holding the variables in θ(E_i) "rigid"), θ(u_i) and θ(v_i) are provably equal from θ(E_i) for i = 1, ... ,n?

Equivalently, is there a substitution θ such that θ(u_i) and θ(v_i) can be shown congruent from θ(E_i) 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.