Maximal extensions of simplification orderings

Abstract

Several well-founded syntactic orderings have been proposed in the literature for proving the termination of rewrite systems. Recursive path orderings (RPO) and their extensions are the most widely used in theorem proving systems such as RRL, REVE, LP. While these orderings can be total (up to equivalence) on ground terms, they are not maximal. That is, when used to compare non-ground terms, there can be terms such that for all ground substitutions, the first term is bigger than the second term, but these orderings declare the two terms as not comparable. A new family of orderings induced by precedence on function symbols, much like RPO, is developed in this paper. Terms are compared by comparing their paths. These ordering are shown to be maximal, and are hence called maximal path orderings. The maximal extension of RPO can be defined using symbolic constraint solving procedures. Such a decision procedure can check, given two terms s and t, whether there is a ground substitution σ that makes σ(s) bigger than σ(t) using RPO. A new decision procedure for the existential fragment of ordering constraints expressed using RPO is given based on the idea of depth bounds. It is shown that given two terms s and t, if there is a ground substitution σ which makes σ(s) bigger than σ(t) using RPO, then there is a ground substitution within depth k * d+k which is also a solution, where k is the number of variables in s and t, and d is the maximum of the depths of s and t.