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Counting and Locality over Finite Structures A Survey

  • Leonid Libkin
  • Juha Nurmonen
Conference paper
  • 222 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1754)

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.

Keywords

Linear Order Transitive Closure Expressive Power Winning Strategy Isomorphism Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    S. Abiteboul, R. Hull and V. Vianu. Foundations of Databases, Addison Wesley, 1995.Google Scholar
  2. 2.
    S. Abiteboul and V. Vianu. Computing with first-order logic. Journal of Computer and System Sciences 50 (1995), 309–335.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Agrawal, E. Allender and S. Datta. On TC0, AC0, and arithmetic circuits. In Proc. 12th IEEE Conf. on Computational Complexity, 1997.Google Scholar
  4. 4.
    E. Allender. Circuit complexity before the dawn of the new millennium. In Proc. 16th Conf. on Foundations of Software Technology and Theoretical Computer Science (FST&TCS’96), Springer LNCS vol. 1180, 1996, 1–18.Google Scholar
  5. 5.
    D.A. Barrington, N. Immerman and H. Straubing. On uniformity within NC1. J. Comput. and Syst. Sci., 41:274–306,1990.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Benedikt, H.J. Keisler. On expressive power of unary counters. Proc. Intl. Conf. on Database Theory (ICDT’97), Springer LNCS 1186, January 1997.Google Scholar
  7. 7.
    M. Benedikt and L. Libkin. Relational queries over interpreted structures. J. ACM, to appear. Extended abstract in PODS’97, pages 87–98.Google Scholar
  8. 8.
    R.B. Boppana and M. Sipser. The Complexity of Finite Functions. In “Handbook of Theoretical Computer Science,” Volume A, Chapter 14, pages 759–804, North Holland, 1990.Google Scholar
  9. 9.
    J. Cai, M. Fürer and N. Immerman. On optimal lower bound on the number of variables for graph identification. Combinatorica, 12 (1992), 389–410.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Consens and A. Mendelzon. Low complexity aggregation in GraphLog and Datalog, Theoretical Computer Science 116 (1993), 95–116. Extended abstract in ICDT’90.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    G. Dong, L. Libkin and L. Wong. Local properties of query languages. Theoretical Computer Science, to appear. Extended abstract in ICDT’97, LNCS vol. 1186, pages 140–154.Google Scholar
  12. 12.
    H.-D. Ebbinghaus. Extended logics: the general framework. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, Springer-Verlag, 1985, pages 25–76.Google Scholar
  13. 13.
    H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer Verlag, 1995.Google Scholar
  14. 14.
    K. Etessami. Counting quantifiers, successor relations, and logarithmic space, In Proc. Structure in Complexity Theory, 1995.Google Scholar
  15. 15.
    K. Etessami. Counting quantifiers, successor relations, and logarithmic space, Journal of Computer and System Sciences, 54 (1997), 400–411.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Descriptive Complexity and Finite Models, AMS 1997.Google Scholar
  17. 17.
    R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78–92.CrossRefMathSciNetGoogle Scholar
  18. 18.
    H. Gaifman. On local and non-local properties, Proceedings of the Herbrand Symposium, Logic Colloquium’ 81, North Holland, 1982.Google Scholar
  19. 19.
    M. Grohe and T. Schwentick. Locality of order-invariant first-order formulas. In MFCS’98, pages 437–445.Google Scholar
  20. 20.
    S Grumbach and T. Milo. Towards tractable algebras for bags. Journal of Computer and System Sciences, 52 (1996), 570–588.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    S. Grumbach, L. Libkin, T. Milo and L. Wong. Query languages for bags: expressive power and complexity. SIGACT News, 27 (1996), 30–37.CrossRefGoogle Scholar
  22. 22.
    W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132–145.Google Scholar
  23. 23.
    L. Hella. Logical hierarchies in PTIME. Information and Computation, 129 (1996), 1–19.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, to appear.Google Scholar
  25. 25.
    L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In Proc. 14th IEEE Symp. on Logic in Computer Science (LICS’99), Trento, Italy, July 1999, pages 35–44.Google Scholar
  26. 26.
    L. Hella and J. Nurmonen. Vectorization hierarchies of some graph quantifiers. Archive for Mathematical Logic, to appear.Google Scholar
  27. 27.
    L. Hella and G. Sandu. Partially ordered connectives and finite graphs. In M. Krynicki, M. Mostowski and L. Szczerba, eds, Quantifiers: Logics, Models and Computation II, Kluwer Academic Publishers, 1995, pages 79–88.Google Scholar
  28. 28.
    N. Immerman. Descriptive Complexity. Springer-Varlag, 1999.Google Scholar
  29. 29.
    N. Immerman. Languages that capture complexity classes. SIAM J. Comput. 16 (1987), 760–778.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    N. Immerman and E. Lander. Describing graphs: A first order approach to graph canonization. In “Complexity Theory Retrospective”, Springer Verlag, Berlin, 1990.Google Scholar
  31. 31.
    Ph. Kolaitis and J. Väänänen. Generalized quantifiers and pebble games on finite structures. Annals of Pure and Applied Logic, 74 (1995), 23–75.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS’97), Warsaw, Poland, June–July 1997, pages 204–215.Google Scholar
  33. 33.
    L. Libkin. On counting logics and local properties. In Proc. 13th IEEE Symp. on Logic in Computer Science (LICS’98), Indianapolis, June 1998, pages 501–512.Google Scholar
  34. 34.
    L. Libkin. Logics with counting, auxiliary relations, and lower bounds for invariant queries. In Proc. 14th IEEE Symp. on Logic in Computer Science (LICS’99), Trento, Italy, July 1999, pages 316–325.Google Scholar
  35. 35.
    L. Libkin and L. Wong. Some properties of query languages for bags. In Proc. Database Programming Languages 1993, Springer Verlag, 1994, pages 97–114.Google Scholar
  36. 36.
    L. Libkin and L. Wong. Query languages for bags and aggregate functions. Journal of Computer and System Sciences 55 (1997), 241–272. Extended abstract in PODS’94, pages 155–166.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    L. Libkin and L. Wong. On the power of aggregation in relational query languages. In Proc. Database Programming Languages 1997, Springer LNCS 1369, pages 260–280.Google Scholar
  38. 38.
    L. Libkin and L. Wong. Unary quantifiers, transitive closure, and relations of large degree. In Proc. 15th Symp. on Theoretical Aspects of Computer Science (STACS’98), Springer LNCS 1373, pages 183–193.Google Scholar
  39. 39.
    K. Luosto. Hierarchies of monadic generalized quantifiers. Journal of Symbolic Logic, to appear.Google Scholar
  40. 40.
    J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779–798.zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, to appear. Extended abstract in Proc. 11th IEEE Symp. on Logic in Computer Science (LICS’96), New Brunswick, NJ, July 1996, pages 484–493.Google Scholar
  42. 42.
    J. Nurmonen. Unary quantifiers and finite structures. PhD Thesis, University of Helsinki, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Leonid Libkin
    • 1
  • Juha Nurmonen
    • 2
  1. 1.Bell LaboratoriesMurray HillUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of LeicesterLeicesterUK

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