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Level Two of the Quantifier Alternation Hierarchy over Infinite Words

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Computer Science – Theory and Applications (CSR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9691))

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Abstract

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment \(\varSigma _2\) over infinite words. Here, \(\varSigma _2\) consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is \(\mathbb {B}\varSigma _2\). Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an \(\omega \)-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun’s recent decidability result for \(\mathbb {B}\varSigma _2\) from finite to infinite words.

This work was supported by the German Research Foundation (DFG) under grants DI 435/5-2 and DI 435/6-1.

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Notes

  1. 1.

    During the preparation of this submission, we learned that Pierron, Place and Zeitoun [11] independently found another proof for the decidability of \(\mathbb {B}\varSigma _2\) over infinite words. For documenting the independency of the two proofs, we also include the technical report of our submission in the list of references [6].

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Authors and Affiliations

  1. FMI, Universität Stuttgart, Stuttgart, Germany

    Manfred Kufleitner & Tobias Walter

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  1. Manfred Kufleitner
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  2. Tobias Walter
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Correspondence to Tobias Walter .

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Editors and Affiliations

  1. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

  2. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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Kufleitner, M., Walter, T. (2016). Level Two of the Quantifier Alternation Hierarchy over Infinite Words. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_16

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  • DOI: https://doi.org/10.1007/978-3-319-34171-2_16

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