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Definability of Languages by Generalized First-Order Formulas over (N,+)

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STACS 2006 (STACS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3884))

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Abstract

We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC 1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.

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Authors and Affiliations

  1. Computer Science Department, Boston College, Chestnut Hill, Massachusetts, 02476, USA

    Amitabha Roy & Howard Straubing

Authors
  1. Amitabha Roy
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  2. Howard Straubing
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Editor information

Editors and Affiliations

  1. LIF, CNRS & Univ. de Provence,, Marseille, France

    Bruno Durand

  2. RWTH Aachen University,, Lehrstuhl Informatik 7,, 52056,, Aachen,, Germany

    Wolfgang Thomas

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© 2006 Springer-Verlag Berlin Heidelberg

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Roy, A., Straubing, H. (2006). Definability of Languages by Generalized First-Order Formulas over (N,+). In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_40

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  • DOI: https://doi.org/10.1007/11672142_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32301-3

  • Online ISBN: 978-3-540-32288-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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