Matching and Uni cation in Rewrite Theories

\Semantic uni cation" is the process of generating a basis set of substitutions (of terms for variables) that makes two given terms equal in a speci ed theory. Semantic uni cation is an important component of some theorem provers. \Semantic matching," a simpler variant of uni cation, where the substitution is made in only one of the terms, has potential usage in programming language interpreters. Decidable matching is required for pattern application in patterndirected languages, while decidable uni cation is useful for theorem proving modulo an equational theory. In this paper we restrict ourselves to matching and uni cation problems in theories that can be presented as convergent rewrite systems, that is, nite sets of equations that compute unique output values when applied (from left-to-right) to input values. The new results presented here, together with existing results, provide a much ner characterization of decidable matching and uni cation than was available before.