Star height of reversible languages and universal automata

Abstract

The star height of a regular language is an invariant that has been shown to be effectively computable in 1988 by Hashiguchi. But the algorithm that corresponds to his proof leads to impossible computations even for very small instances. Here we solve the problem (of computing star height) for a special class of regular languages, called reversible languages, that have attracted much attention in various areas of formal language and automata theory in the past few years. These reversible languages also strictly extend the classes of languages considered by McNaughton, Cohen, and Hashiguchi for the same purpose, and with different methods. Our method is based upon the definition (inspired by the reading of Conway’s book) of an automaton that is effectively associated to every language — which we call the universal automaton of the language — and that contains the image of any automaton that accepts the language. We show that the universal automaton of a reversible language contains a sub automaton where the star height can be computed.