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

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Abstract

Given a finite alphabet Σ, we give a simple characterization of those Gδsubsets of Σωwhich are deterministic ω-regular (i.e. accepted by Büchi automata) over Σ and then characterize the ω-regular languages in terms of these (rational) Gδsets. Our characterization yields a hierarchy of ω-regular languages similar to the classical difference hierarchy of Hausdorff and Kuratowski for Δ03sets (i.e. the class of sets which are both Fσδand Gδσ). We then prove that the Hausdorff-Kuratowski difference hierarchy of Δ03when 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 Gδsets of length n, then there is a decreasing sequence (of length n) of rational Gδsets 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.

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Barua, R. (1992). The Hausdorff-Kuratowski hierarchy of ω-regular languages and a hierarchy of Muller automata. Theoretical Computer Science, 96(2), 345–360. https://doi.org/10.1016/0304-3975(92)90342-D

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