A note on confluent Thue systems

Abstract

If one is to use a rewriting system, then it is important to know whether the word problem is decidable. If a finite rewriting system is both confluent and has the property that there are no infinite reduction chains, then the word problem is decidable (see [Hu80] for a discussion). In this note string rewriting is considered and Thue systems are viewed as string rewriting systems, that is, a Thue system is considered to be a rewriting system where the domain consists of the set of all strings over some fixed finite alphabet. Here it is shown that for a finite Thue system T such that (i) T is confluent and (ii) there is a total recursive function R that bounds the lengths of reduction chains of T, there is an algorithm to solve the word problem for T and the algorithm has running time O(R(n)), where n is the length of the input. If a finite Thue system is viewed as a rewriting system and reduction of the length of strings is the basis for defining the notion of reduction for rewriting rules, then for any string w, no sequence of consecutive reductions applied to w can have length (i.e., the number of reductions performed) greater than [w[, where [w[ denotes the length of w. This fact leads to a linear-time algorithm for the word problem for confluent Thue systems [Bo82] where length is the basis for defining reduction. In this paper a more general situation is studied. Let R : N --* N be a function that is strictly increasing. Supppose that T is a Thue system with the property that no sequence of consecutive reductions that begins with an application of a reduction to a string w can have length that is greater than R([wD; such a system is called R-noetherian. Suppose that T is a finite Thue system that is both