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Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

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Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7966))

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Abstract

A Presburger formula is a Boolean formula with variables in ℕ that can be written using addition, comparison (≤, =, etc.), Boolean operations (and, or, not), and quantifiers (∀ and ∃). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p 1,…,p n ) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. In the full version of this paper, we also translate known computational complexity results into this setting and discuss open directions.

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Authors and Affiliations

  1. Oberlin College, Oberlin, Ohio, USA

    Kevin Woods

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  1. Kevin Woods
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, Wolfson Building, Parks Road, University of Oxford, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

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Woods, K. (2013). Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_37

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  • DOI: https://doi.org/10.1007/978-3-642-39212-2_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39211-5

  • Online ISBN: 978-3-642-39212-2

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