UniMoK: A System for Combining Equational Unification Algorithms

Abstract

Equational unification algorithms can be used in resolution based theorem provers [9] and rewriting engines [6] to improve their handling of equality. Originally, the requirements of these theorem provers and rewrite engines were such that the unification algorithms had to compute complete sets of unifiers. But with the advent of constraint based approaches to theorem proving [4] and rewriting [8] the interest in unification algorithm that worked merely as decision procedures grew because minimal complete sets of unifiers can be very large – e.g., doubly exponential in the number of variables of the problem in the case of the theory AC – and are hence costly to compute. Because actual unification problems usually contain function symbols from several different signatures, the following combination problem is an important task in unification theory: Given unification algorithms for equational theories E1, E2, . . . , En over pairwise disjoint signatures, provide a general method that gives a unification algorithm for the union E1 ∪ E2 ∪ . . . ∪ En of these theories. Solutions for this problem were provided by Schmidt-Schaus [10] and Boudet [3] for the combination of algorithms calculating complete sets of unifiers and by Baader and Schulz [1] for combining decision procedures. The combination algorithm presented in [1] is mostly of theoretical interest, it contains many non-deterministic decisions, thus the search space that this algorithm spans is so huge, that it is unusable for practical implementations. Therefore the authors developed optimisation methods [7] for the combination algorithm by Baader and Schulz to gain an implementation that can be used in practise. This implementation is UniMoK. UniMoK stands for Unification Module for Keim. It contains algorithms for unification in certain equational theories and it provides several combination methods for them. All combination algorithms in UniMoK are extensions and optimisations of the combination method by Baader and Schulz [1].