Automatic Groups and Knuth-Bendix with Infinitely Many Rules

Abstract

It is shown how to use a small finite state automaton in two variables in order to carry out part of the Knuth--Bendix process for rewriting words in a group. The main objective is to provide a substitute for the most space-demanding module of the existing software which attempts to find a shortlex-automatic structure for a group. The two-variable automaton can be used to store an infinite set of rules and to carry out fast reduction of arbitrary words using this infinite set. We introduce a new operation, which we call welding, which applies to an arbitrary finite state automaton. In our context this operation is vital. We point out a small potential improvement in the subset algorithm for making a non-deterministic automaton deterministic.