## Technical Reports (CIS)

#### Document Type

Technical Report

#### Subject Area

GRASP

#### Date of this Version

July 1992

#### 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*〉 | 1 ≤

_{i}, v_{i}*i*≤

*n*} a set of

*n*pairs of terms, is there a substitution

*θ*such that, treating each set

*θ(E*) as a set of

_{i}*ground*equations (i.e. holding the variables in

*θ*(

*E*) "rigid"),

_{i}*θ(u*) and

_{i}*θ(v*are provably equal from

_{i})*θ(E*) for

_{i}*i*= 1, ... ,

*n*?

Equivalently, is there a substitution *θ* such that *θ(u _{i}*) and

*θ(v*) can be shown congruent from

_{i}*θ(E*) by the congruence closure method for

_{i}*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.

#### Recommended Citation

Jean H. Gallier, Paliath Narendran, Stan Raatz, and Wayne Snyder, "Theorem Proving Using Equational Matings and Rigid E-Unifications", . July 1992.

**Date Posted:** 25 September 2007

## Comments

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