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 G δ subsets of Σ ω 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) G δ sets. Our characterization yields a hierarchy of ω-regular languages similar to the classical difference hierarchy of Hausdorff and Kuratowski for Δ 0 3 sets (i.e. the class of sets which are both F σδ and G δσ). We then prove that the Hausdorff-Kuratowski difference hierarchy of Δ 0 3 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 G δ sets of length n, then there is a decreasing sequence (of length n) of rational G δ 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.