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Counting and Locality over Finite Structures A Survey

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Generalized Quantifiers and Computation (ESSLLI 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1754))

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Abstract

We survey recent results on logics with counting and their local properties. We first consider game-theoretic characterizations of first-order logic and its counting extensions provided by unary generalized quantifiers. We then study Gaifman’s and Hanf’s locality theorems, their connection with game characterizations, and examples of their usage in proving expressivity bounds for first-order logic and its extensions. We review the abstract notions of Gaifman’s and Hanf’s locality, and show how they are related. We also consider a closely related bounded degree property, and demonstrate its usefulness in proving expressivity bounds. We discuss two applications. One deals with proving lower bounds for the complexity class TC0. In particular, we use logical characterization of TC0 and locality theorems for first-order with counting quantifiers to provide lower bounds. We then explain how the notions of locality are used in database theory to prove that extensions of relational calculus with aggregate functions and grouping still lack the power to express fixpoint computation.

Supported by EPSRC grant GR/K 96564.

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Author information

Authors and Affiliations

  1. Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ, 07974, USA

    Leonid Libkin

  2. Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Juha Nurmonen

Authors
  1. Leonid Libkin
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  2. Juha Nurmonen
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Helsinki, P.O. Box 4, Yliopistonkatu 5, 00014, Helsinki, Finland

    Jouko Väänänen

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© 1999 Springer-Verlag Berlin Heidelberg

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Libkin, L., Nurmonen, J. (1999). Counting and Locality over Finite Structures A Survey. In: Väänänen, J. (eds) Generalized Quantifiers and Computation. ESSLLI 1997. Lecture Notes in Computer Science, vol 1754. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46583-9_2

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  • DOI: https://doi.org/10.1007/3-540-46583-9_2

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  • Print ISBN: 978-3-540-66993-7

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