Finitary Languages

The class of ω-regular languages provides a robust specification language in verification. Every ω-regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens “eventually”. Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness [AH98]. It requires that there exists an unknown, fixed bound b such that something good happens within b transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Büchi, parity and Streett languages. We give their topological complexity of acceptance conditions, and present a regular-expression characterization of the languages they express. We provide a classification of finitary and classical automata with respect to the expressive power, and give optimal algorithms for classical decisions questions on finitary automata. We (a) show that the finitary languages are Σ2-complete; (b) present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; (c) show that the languages defined by finitary parity automata exactly characterize the star-free fragment of ωB-regular languages; and (d) show that emptiness is NLOGSPACE-complete and universality as well as language inclusion are PSPACE-complete for finitary automata.