The Hausdorff-Kuratowski Hierarchy of omega-Regular Languages and a Hierarchy of Muller Automata

R. Barua, The Hausdorff-Kuratowski hierarchy of o-regular languages and a hierarchy of Muller automata, Theoretical Computer Science 96 (1992) 345-360. Given a finite alphabet Z, we give a simple characterization of those Gd subsets of Z” which are deterministic w-regular (i.e. accepted by Biichi automata) over Z and then characterize the w-regular languages in terms of these (rational) G6 sets. Our characterization yields a hierarchy ofw-regular languages similar to the classical difference hierarchy of Hausdorff and Kuratowski for 4 i seti (i.e. the class of sets which are both Fo, and GA.). We then prove that the Hausdorff-Kuratowski difference hierarchy of d i when restricted to o-regular languages coincides with our hierarchy. We obtain this by showing that if an w-regular language K can be separated from another w-regular language L by the union of alternate differences of a decreasing sequence of G6 sets of length n, then there is a decreasing sequence (of length n) of rational Cd sets such that the union of alternate differences separates K from L. Our results not only generalize a result of Landweber (1969), but also yield an effective procedure for determining the complexity of a given Muller automaton. We also show that our hierarchy does not collapse, thus, giving a fine classification of w-regular languages and of Muller automata.