Generalized Quantifiers, an Introduction

  • Jouko Väänänen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1754)


We review recent work in the field of generalized quantifiers on finite models. We give an idea of the methods that are available in this area. Main emphasis is on definability issues, such as whether there is a logic for the PTIME properties of unordered finite models


Turing Machine Order Logic Point Query Winning Strategy Point Logic 
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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  1. 1.
    A.V. Aho and J.D. Ullman, Universality of data retrieval languages, Sixth ACM Symposium on Principles of Programming Languages, 1979, 110–117.Google Scholar
  2. 2.
    J. Cai, M. Fürer and N. Immerman, An Optimal Lower Bound on the Number of Variables for Graph Identification, Combinatorica 12:4 (1992), 389–410.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Chandra and D. Harel. Structure and complexity of relational queries. Journal of Computer and System Sciences, 25:99–128, 1982.zbMATHCrossRefGoogle Scholar
  4. 4.
    E.F. Codd, Relational completeness of database sublanguages, in: Database Systems (R. Rustin, ed.), Prentice-Hall, 1972, 65–98.Google Scholar
  5. 5.
    E. Dahlhaus, Skolem normal forms concerning the least fixpoint, in: Computation theory and logic (E. Börger, ed.), 101–106, Lecture Notes in Comput. Sci., 270, Springer, Berlin-New York, 1987.Google Scholar
  6. 6.
    A. Dawar, Generalized quantifiers and logical reducibilities, Journal of Logic and Computation, 5(1995), 213–226.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Dawar and E. Grädel, Generalized quantifiers and 0–1 laws, Proc. 10th IEEE Symp. on Logic in Computer Science, 1995, 54–64.Google Scholar
  8. 8.
    R. Fagin, The number of finite relational structures, Discrete Mathematics 19(1977), 17–21.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in: Complexity of Computation (R. Karp, ed.) SIAM-AMS Proc. 7, 1974, 27–41.Google Scholar
  10. 10.
    M. Grohe and L. Hella, A double arity hierarchy theorem for transitive closure logic, Archive for Mathematical Logic, 35(3): 157–172, 1996.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Y. Gurevich. Toward logic tailored for computational complexity. In M. M. Richter et al., editor, Computation and Proof Theory, Lecture Notes in Mathematics 1104, pages 175–216. Springer-Verlag, 1984.Google Scholar
  12. 12.
    Y. Gurevich. Logic and the challenge of computer science. In E. Börger, editor, Current trends in theoretical computer science, pages 1–57. Computer Science Press, 1988.Google Scholar
  13. 13.
    Y. Gurevich, Zero-one laws, EATCS Bulletin 46 (1992), 90–106.zbMATHGoogle Scholar
  14. 14.
    L. Hella, Logical hierarchies in PTIME, Information and Computation 129: 1–19, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    L. Hella and G. Sandu, Partially ordered connectives and finite graphs, in: Quantifiers: Logics, models and computation (M. Krynicki, M. Mostowski and L. Szczerba, eds.), vol II, Kluwer Academic Publishers 1995, 79–88.Google Scholar
  16. 16.
    L. Hella, Ph. Kolaitis and K. Luosto, Almost everywhere equivalence of logics infinite model theory, Bulletin of Symbolic Logic 2(4), 1996, 422–443.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. Hella, K. Luosto and J. Väänänen, The hierarchy theorem for generalized quantifiers. Journal Symbolic Logic 61 (1996), no. 3, 802–817.zbMATHCrossRefGoogle Scholar
  18. 18.
    L. Hella, J. Väänänen and D. Westerståhl, Definability of polyadic lifts of generalized quantifiers, Journal of Logic, Language and Information 6 (1997), no. 3, 305–335.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1986), 86–104.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    N. Immerman, Languages that capture complexity classes, SIAM J. Comput. 16, No. 4 (1987), 760–778.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Lindström, First order predicate logic with generalized quantifiers, Theoria 32 (1966), 186–195.MathSciNetGoogle Scholar
  22. 22.
    A. Mostowski, On a generalization of quantifiers. Fundamenta Mathematicae 44 (1957) 12–36.MathSciNetGoogle Scholar
  23. 23.
    J. Nurmonen, On winning strategies with unary quantifiers, Journal of Logic and Computation, 6(6): 779–798, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    M. Vardi, Complexity of relational query languages, 14th ACM STOC Symposium (1982), 137–146.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jouko Väänänen
    • 1
  1. 1.Department of MathematicsUniversity of HelsinkiYliopistonkatu 5Finland

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