The Hausdorff-Kuratowski hierarchy of ω-regular languages and a hierarchy of Muller automata

Abstract

Given a finite alphabet Σ, we give a simple characterization of those Gδ subsets of Σω which are deterministic ω-regular (i.e. accepted by Büchi automata) over Σ and then characterize the ω-regular languages in terms of these (rational) Gδ sets. Our characterization yields a hierarchy of ω-regular languages similar to the classical difference hierarchy of Hausdorff and Kuratowski for Δ03 sets (i.e. the class of sets which are both Fσδ and Gδσ). We then prove that the Hausdorff-Kuratowski difference hierarchy of Δ03 when restricted to ω-regular languages coincides with our hierarchy. We obtain this by showing that if an ω-regular language K can be separated from another ω-regular language L by the union of alternate differences of a decreasing sequence of Gδ sets of length n, then there is a decreasing sequence (of length n) of rational Gδ 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 ω-regular languages and of Muller automata. © 1992.