Basics of Term Rewriting

Abstract

Terms and trees are important structured objects that can be found almost everywhere in Computer Science, not only in connection with their mathematical foundations. In any axiomatic definition of structures and equationally defined calculi one will find the combination of axioms, i.e., equations, and a method of deriving new theorems from those by using deduction rules. This situation traditionally can be found in logic and became important in applications such as symbolic algebraic computation, program specification and verification using abstract data types, and automated theorem proving. Term rewriting can be seen as a generalization of string rewriting as studied early in this century by Axel Thue. Terms are structured strings that can well be visualized by rooted, node-labelled, and ordered trees. This was already recognized by Axel Thue in 1910, [Thue10]. In his paper he investigated the formulation of new concepts from already given ones and designed a theory of terms or trees and how to create and use them. He posed what we now call the word problem for equational theories and thereby introduced the idea of a universal algebra and the free algebra over sets of equations. The theorems deduced syntactically from a set of axioms within a calculus are useful only in the context of a semantic interpretation in an algebraic structure that models the theory specified by the axioms and the deduction rules. The semantic equality means that a theorem a = b is true in each model of a given theory whereas syntactic equality a = E b is its deducibility (provability) from the axioms E. From Birkhoff 1935, [Birk35], we know that semantic and syntactic equality coincides and thus the deduction rules specified in the calculus could in general be used for deciding semantic equality.