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Topological complexity of locally finite ω-languages

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Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOC ω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally finite ω-languages which are Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet (Automates et Théorie Descriptive, Ph.D. Thesis. Université Paris 7, March 1992) and of Duparc et al. [Computer science and the fine structure of Borel sets. Theor Comput Sci 257(1–2):85–105, 2001].

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  1. Equipe Modèles de Calcul et Complexité, Laboratoire de l’Informatique du Parallélisme,, UMR 5668, CNRS, ENS Lyon, UCB Lyon, INRIA, CNRS et Ecole Normale Supérieure de Lyon, 46, Allée d’Italie, 69364, Lyon Cedex 07, France

    Olivier Finkel

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Finkel, O. Topological complexity of locally finite ω-languages. Arch. Math. Logic 47, 625–651 (2008). https://doi.org/10.1007/s00153-008-0101-7

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