Fixed Point Logics and Complexity Classes

  • Leonid Libkin
Part of the Texts in Theoretical Computer Science book series (TTCS)

Abstract

Most logics we have seen so far are not well suited for expressing many tractable graph properties, such as graph connectivity, reachability, and so on. The limited expressiveness of FO and counting logics is due to the fact that they lack mechanisms for expressing fixed point computations. Other logics we have seen, such as MSO, ∃SO, and ∀SO, can express intractable graph properties.

Keywords

Expense Hull Sine Pebble Weinstein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographic Notes

  1. 65.
    E.A. Emerson. Model checking and the nm-calculus. In [134], pages 185–214.Google Scholar
  2. 33.
    A. Chandra and D. Harel. Structure and complexity of relational queries. Journal of Computer and System Sciences, 25 (1982), 99–128.MATHCrossRefGoogle Scholar
  3. 60.
    H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer-Verlag, 1995.Google Scholar
  4. 133.
    N. Immerman. Descriptive Complexity. Springer-Verlag, 1998.Google Scholar
  5. 106.
    M. Grohe. The structure of fixed-point logics. PhD Thesis, University of Freiburg, 1994.MATHGoogle Scholar
  6. 51.
    A. Dawar and Y. Gurevich. Fixed point logics. Bulletin of Symbolic Logic, 8 (2002), 65–88.MathSciNetMATHCrossRefGoogle Scholar
  7. 148.
    Ph. Kolaitis. On the expressive power of logics on finite models. In [99].Google Scholar
  8. 85.
    M. Frick and M. Grohe. The complexity of first-order and monadic second-order logic revisited. In IEEE Symp. on Logic in Computer Science, 2002, pages 215–224.Google Scholar
  9. 185.
    Y. Moschovakis. Elementary Induction on Abstract Structures. North-Holland, 1974.Google Scholar
  10. 119.
    Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32 (1986), 265–280.MathSciNetMATHCrossRefGoogle Scholar
  11. 165.
    D. Leivant. Inductive definitions over finite structures. Information and Computation 89 (1990), 95–108.MathSciNetMATHCrossRefGoogle Scholar
  12. 129.
    N. Immerman. Relational queries computable in polynomial time (extended abstract). In ACM Symp. on Theory of Computing,1982, ACM Press, pages 147–152.Google Scholar
  13. 212.
    V. Sazonov. Polynomial computability and recursivity in finite domains. Elektronische Informationsverarbeitung and Kybernetik, 16 (1980), 319–323.MathSciNetMATHGoogle Scholar
  14. 244.
    M.Y. Vardi. The complexity of relational query languages. In Proc. ACM Symap. on Theory of Computing, 1982, 137–146.Google Scholar
  15. 172.
    A.B. Livchak (A. B. JIKnuax). Languages for polynomial-time queries (SkbixH R.m noJIHaoMxa.nhIMx 3anpocon). In Computer-based Modeling and Optimization of Heat-power and Electrochemical Objects (Pacnëm u Onmu.Mu3aeyu.n Teuutornexuuneexux u E•texmpoxu.wunecxux O63exmoe c Hoinaoaywo 9BM), Sverdlovsk, 1982, page 41.Google Scholar
  16. 3.
    S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases, Addison-Wesley, 1995.Google Scholar
  17. 194.
    C. Papadimitriou. A note on the expressive power of Prolog. Bulletin of the EATCS, 26 (1985), 21–23.MathSciNetGoogle Scholar
  18. 132.
    N. Immerman. Nondeterministic space is closed under complementation. SIAM Journal on Computing, 17 (1988), 935–938.MathSciNetMATHCrossRefGoogle Scholar
  19. 226.
    R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Informatica, 26 (1988), 279 284.Google Scholar
  20. 191.
    M. Otto. Bounded Variable Logics and Counting: A Study in Finite Models. Springer-Verlag, 1997.MATHGoogle Scholar
  21. 147.
    Ph. Kolaitis. Languages for polynomial-time queries–an ongoing quest. In Proc. 5th Int. Conf. on Database Theory, Springer-Verlag, 1995, pages 38–39.Google Scholar
  22. 30.
    J. Cai, M. Fürer, and N. Immerman. On optimal lower bound on the number of variables for graph identification. Combinatorica, 12 (1992), 389–410.MathSciNetMATHCrossRefGoogle Scholar
  23. 91.
    F. Gire and H. K. Hoang. A more expressive deterministic query language with efficient symmetry-based choice construct. In Logic in Databases, Int. Workshop LID’96, Springer-Verlag, 1996, pages 475–495.Google Scholar
  24. 52.
    A. Dawar and L. Hella. The expressive power of finitely many generalized quantifiers. Information and Computation, 123 (1995), 172–184.MathSciNetMATHCrossRefGoogle Scholar
  25. 130.
    N. Immerman. Relational queries computable in polynomial time. Information and Control, 68 (1986), 86–104.MathSciNetMATHCrossRefGoogle Scholar
  26. 97.
    E. Grädel. Capturing complexity classes by fragments of second order logic. Theoretical Computer Science, 101 (1992), 35–57.MathSciNetMATHCrossRefGoogle Scholar
  27. 101.
    E. Grädel and G. McColm. On the power of deterministic transitive closures. Information and Computation, 119 (1995), 129–135.MathSciNetMATHCrossRefGoogle Scholar
  28. 131.
    N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing, 16 (1987), 760–778.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Leonid Libkin
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations