Star height of reversible languages and universal automata.
Résumé
The star height of a regular language is an invariant that has been shown to be effectively computable some thirteen years ago by Hashiguchi. But the algorithm that corresponds to his proof is not only of very high complexity but also 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 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 plays the same role with respect to any automaton which recognizes the language as the role played by the minimal automaton with respect to any deterministic automaton. We show that the universal automaton of a reversible language contains a subautomaton where the star height can be computed. Besides the definition of the universal automaton of a language, the main ingredient of the proof is the definition of the subset expansion of an automaton, which, in turn, is a key to proving properties of the universal automaton.