On the Unification of Rational Terms.

Abstract

First-order unification algorithms on finite terms are formalized by inference rules, and their termination, soundness and completeness have been presented in many papers. Unification algorithms on rational terms have also been presented in several papers. However, it has not been shown that the solution of the set obtained by inference rules from a given set of equations coincides with the rational substitution which is the most general unifier of the original set. Furthermore, the precise proofs of termination, soundness and completeness are not clear in many cases. In this paper, we revisit the foundations of unification algorithm on rational terms within the framework of term rewriting systems. First, we show that the solution of a regular system is the most general unifier of the set of equations corresponding to the regular system. Next, we reformulate the unification algorithm on rational terms by inference rules, and then give the proofs of its termination, soundness and completeness, respectively. Finally, we show that a unifiable set of equations has a most general unifier as a rational substitution.