Advertisement

Query languages with counters

  • Stéphane Grumbach
  • Christophe Tollu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 646)

Abstract

We investigate the expressive power of query languages with counting ability. We define a LOGSPACE extension of first order logic and a PTIME extension of fixpoint logic with counters. We develop specific techniques, such as games, for dealing with languages with counters and therefore integers. We prove in particular that the arity of the tuples which are counted induces a strict expressivity hierarchy. We also establish results about the asymptotic probabilities of sentences with counters. In particular we show that first order logic with comparison of the cardinalities of relations has a. 0/1 law.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Abiteboul and V. Vianu. Generic computation and its complexity. In Proc. 23rd ACM Symp. on Theory of Computing, pages 209–219, 1991.Google Scholar
  2. [2]
    A. Blass, Y. Gurevich, and D. Kozen. A zero-one law for logic with a fixed-point operator. Information and Control, 67:70–90, 1985.Google Scholar
  3. [3]
    J. Cai, M. Furer, and N. Immerman. An optimal lower bound on the number of variables for graph identification. In Proc 30th IEEE Symp. on Foundations of Computer Science, pages 612–617, 1989.Google Scholar
  4. [4]
    A. Chandra and D. Harel. Computable queries for relational data bases. Journal of Computer and System Sciences, 21(2):156–178, Oct. 1980.CrossRefGoogle Scholar
  5. [5]
    A. Chandra and D. Harel. Structure and complexity of relational queries. Journal of Computer and System Sciences, 25(1):99–128, Aug. 1982.CrossRefGoogle Scholar
  6. [6]
    P. Dublish and S. Maheshwari. Expressibility of bounded-arity fixed-point query hierarchies. In Proc. 8th ACM Symp. on Principles of Database Systems, pages 324–335, 1989.Google Scholar
  7. [7]
    A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Mathematica, 49:128–141, 1961.Google Scholar
  8. [8]
    R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50–58, 1976.Google Scholar
  9. [9]
    R. Fraissé. Sur les classifications des systèmes de relations. Publ. Sci. Univ Alger, I:1, 1954.Google Scholar
  10. [10]
    H. Gaifman. On local and non local properties. In J. Stern, editor, Proc. Herbrand Symposium, Logic Colloquium, pages 105–135. North Holland, 1981.Google Scholar
  11. [11]
    Y. Glebskii, D. Kogan, M. Liogon'kii, and V. Talanov. Range and degree of realizability of formulas in the restricted predicate calculus. Kibernetica, 2:17–28, 1969. English translation in Cybernetics, 5:142–154, 1969.Google Scholar
  12. [12]
    E. Grandjean. Complexity of the first order theory of almost all structures. Information and Control, 52:180–204, 1983.Google Scholar
  13. [13]
    Y. Gurevich and S. Shelah. Fixed-point extensions of first order logic. In Proc 26th IEEE Symp. on Foundations of Computer Science, 1985.Google Scholar
  14. [14]
    S. Grumbach and C. Tollu. Asymptotic probabilities of query languages with counting. (In preparation)Google Scholar
  15. [15]
    Y. Gurevich. Algebras of feasible functions. In Proc 24th IEEE Symp. on Foundations of Computer Science, pages 210–214, 1983.Google Scholar
  16. [16]
    N. Immerman and E. Lander. Complexity Theory Retrospective, chapter Describing Graphs: A First-Order Approach to Graph Canonization, pages 59–81. Springer Verlag, Ed. A. Selman, 1990.Google Scholar
  17. [17]
    N. Immerman. Relational queries computable in polynomial time. Inf. and Control, 68:86–104, 1986.Google Scholar
  18. [18]
    V. Knyazev. Zero-one law for an extension of first-order predicate language. Kybernetika, 2:110–113, 1990. English translation in Cybernetics, 26:292–296, 1990.Google Scholar
  19. [19]
    P. Kolaitis and M.Y. Vardi. The decision problem for the probabilities of higher properties. In Proc. 19th ACM Symp. on Theory of Computing, pages 425–435, 1987.Google Scholar
  20. [20]
    P. Kolaitis and M.Y. Vardi. 0/1 laws and decision problems for fragments of second-order logic. In Proc. 3rd IEEE Symp. of Logic in Computer Science, pages 2–11, 1988.Google Scholar
  21. [21]
    P. Kolaitis and M.Y. Vardi. 0/1 laws for infinitary logic. In Proc. 5th IEEE Symp. of Logic in Computer Science, pages 156–167, 1990.Google Scholar
  22. [22]
    S. Lindell. An analysis of fixed-point queries on binary trees. Theorical Computer Science, 85:75–95, 1991.Google Scholar
  23. [23]
    J. Lynch. Almost sure theories. Annals of Mathematical Logic, 18:91–135, 1980.Google Scholar
  24. [24]
    M. Otto. The expressive power of fixed-point logic with counting. Draft, 1992.Google Scholar
  25. [25]
    R. Rado. Universal graphs and universal functions. Acta Arithmetica, 9:331–340, 1964.Google Scholar
  26. [26]
    M. Vardi. Relational queries computable in polynomial time. In Proc. 14th ACM Symp. on Theory of Computing, pages 137–146, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Stéphane Grumbach
    • 1
  • Christophe Tollu
    • 2
  1. 1.I.N.R.I.A.Le ChesnayFrance
  2. 2.LIPN-Institut GaliléeUniversité Paris-NordVilletaneuseFrance

Personalised recommendations