Automata for Pro-V Topologies

Abstract

In this paper, we give an automata theoretic version of several algorithms dealing with profinite topologies. The profinite topology was first introduced for the free group by M. Hall, Jr. and by Reutenauer for the free monoid. It is the initial topology defined by all the monoid morphisms from the free monoid into a discrete finite group. For a variety of finite groups V, the pro-V topology is defined in the same way by replacing “group” by “group in V” in the definition. Recently, by a geometric approach, Steinberg developed an efficient algorithm to compute the closure, for some pro-V topologies (including the profinite one), of a rational language given by a finite automaton. In this paper we show that these algorithms can be obtained by an automata theoretic approach by using a result of Pin and Reutenauer. We also analyze precisely the complexity of these algorithms.