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On Boolean Closed Full Trios and Rational Kripke Frames

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Abstract

We study what languages can be constructed from a non-regular language L using Boolean operations and synchronous or non-synchronous rational transductions. If all rational transductions are allowed, one can construct the whole arithmetical hierarchy relative to L. In the case of synchronous rational transductions, we present non-regular languages that allow constructing languages arbitrarily high in the arithmetical hierarchy and we present non-regular languages that allow constructing only recursive languages. A consequence of the results is that aside from the regular languages, no full trio generated by a single language is closed under complementation. Another consequence is that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.

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Notes

  1. These latter closure properties are needed in order to realize projection and cylindrification of relations.

  2. This is the only point were we need the recursiveness of L.

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Authors and Affiliations

  1. Fachbereich Informatik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Georg Zetzsche

  2. Institut für Theoretische Informatik, Technische Universität Ilmenau, Ilmenau, Germany

    Dietrich Kuske

  3. Department für Elektrotechnik und Informatik, Universität Siegen, Siegen, Germany

    Markus Lohrey

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  1. Georg Zetzsche
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  2. Dietrich Kuske
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  3. Markus Lohrey
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Correspondence to Georg Zetzsche.

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Zetzsche, G., Kuske, D. & Lohrey, M. On Boolean Closed Full Trios and Rational Kripke Frames. Theory Comput Syst 60, 438–472 (2017). https://doi.org/10.1007/s00224-016-9694-0

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