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0–1 Laws for Fragments of Second-Order Logic: An Overview

  • Phokion G. Kolaitis
  • Moshe Y. Vardi
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 21)

Abstract

The probability of a property on the collection of all finite relational structures is the limit as n → ∞ of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that the 0–1 law holds for any property expressible in first-order logic, i.e., the probability of any such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable.

We investigate here fragments of existential second-order logic in which we restrict the patterns of first-order quantifiers. We focus on fragments in which the first-order part belongs to a prefix class. We show that the classifications of prefix classes of first-order logic with equality according to the solvability of the finite satisfiability problem and according to the 0–1 law for the corresponding ∑ 1 1 fragment are identical.

Keywords

Satisfying Assignment Existential Quantifier Extension Axiom Asymptotic Probability Transfer Theorem 
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. [AH78]
    Abramson, F.D., Harrington, L.A.: Models without indiscernibles. J. Symbolic Logic 43(1978), pp. 572–600.MathSciNetMATHCrossRefGoogle Scholar
  2. [Aj83]
    Ajtai, M. : ∑ 1 1-formulas on finite structures. Ann. of Pure and Applied Logic 2443(1983), pp. 1–48. MathSciNetCrossRefGoogle Scholar
  3. [AU79]
    Aho, V.H., Ullman, J.D.: Universality of data retrieval languages. Proceedings 6th ACM Symp. on Principles of Programming Languages, 1979, pp. 110–117.Google Scholar
  4. [BH79]
    Blass, A., Harary, F.: Properties of almost all graphs and complexes. J. Graph Theory 3(1979), pp. 225–240.MathSciNetMATHCrossRefGoogle Scholar
  5. [Bo85]
    Bollobas, B: Random Graphs. Academic Press, 1985.Google Scholar
  6. [Ch52]
    Chernoff, H. : A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observation. Ann. Math. Stat. 23(1952), pp. 493–509. MathSciNetMATHCrossRefGoogle Scholar
  7. [CH82]
    Chandra, A., Harel, D.: Structure and Complexity of Relational Queries. J. Computer and Systems Sciences 25(1982), pp. 99–128.MATHCrossRefGoogle Scholar
  8. [CK81]
    Chandra, A., Kozen, D., Stockmeyer, L.: Alternation. J. ACM 28(1981), pp. 114–133. MathSciNetMATHCrossRefGoogle Scholar
  9. [Co88]
    Compton, K.J.: 0–1 laws in logic and combinatorics, in NATO Adv. Study Inst, on Algorithms and Order (I. Rival, ed.), D. Reidel, 1988, pp. 353–383.Google Scholar
  10. [Da84]
    Dalhlaus, E. : Reductions to NP-complete problems by interpretations. Logic and Machines: Decision Problems and Complexity (E. Börger et al., eds.), Springer-Verlag, Lecture Notes in Computer Science 171, 1984, pp. 357–365. Google Scholar
  11. [DG79]
    Dreben, D., and Goldfarb, W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley,1979.Google Scholar
  12. [Fa74]
    Fagin, R. : Generalized first-order spectra and polynomial time recognizable sets. Complexity of Computations (R. Karp, ed.), SIAM-AMS Proc. 7(1974), pp. 43–73. Google Scholar
  13. [Fa76]
    Fagin, R. : Probabilities on finite models. J. Symbolic Logic 41(1976), pp. 50–58. MathSciNetMATHCrossRefGoogle Scholar
  14. [Ga64]
    Gaifman, H.: Concerning measures in first-order calculi. Israel J. Math. 2(1964). pp. 1–18. MathSciNetMATHCrossRefGoogle Scholar
  15. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability - A Guide to the Theory of NP-Completeness, W.H. Freeman and Co., 1979.Google Scholar
  16. [GKLT69]
    Glebskii, Y.V., Kogan, D.I., Liogonkii. M.I., Talanov, V.A.: Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics 5(1969), pp. 142–154. CrossRefGoogle Scholar
  17. [Go84]
    Goldfarb, W.D. : The Gödel class with equality is unsolvable. Bull. Amer. Math. Soc. (New Series) 10(1984), pp. 113–115.MathSciNetMATHCrossRefGoogle Scholar
  18. [Gr83]
    Grandjean, E. : Complexity of the first-order theory of almost all structures. Information and Control 52(1983), pp. 180–204. MathSciNetCrossRefGoogle Scholar
  19. [Gu69]
    Gurevich, Y. : The decision problem for logic of predicates and operations. Algebra and Logic 8(1969), pp. 160–174.MATHCrossRefGoogle Scholar
  20. [Gu76]
    Gurevich, Y. : The decision problem for standard classes. J. Symbolic Logic 41(1976), pp. 460–464. MathSciNetMATHCrossRefGoogle Scholar
  21. [Gu84]
    Gurevich, Y. : Toward logic tailored for computational complexity. Computation and Proof Theory (M.M. Riether et al., eds.), Springer-Verlag, Lecture Notes in Math. 1104, 1984, pp. 175–216. CrossRefGoogle Scholar
  22. [GS83]
    Gurevich, Y., and Shelah, S.: Random models and the Gödel case of the decision problem. J. of Symbolic Logic 48(1983), pp. 1120–1124. MathSciNetMATHCrossRefGoogle Scholar
  23. [HIS85]
    Hartmanis, J., Immerman, N., Sewelson, J.:Sparse sets in NP-P - EXPTIME vs. NEXPTIME. Information and Control 65(1985), pp. 159–181.MathSciNetCrossRefGoogle Scholar
  24. [Im83]
    Immerman, N. : Languages which capture complexity classes. Proc. 15th ACM Symp. on Theory of Computing, Boston, 1983, pp. 347–354. Google Scholar
  25. [Im86]
    Immerman, N. : Relational queries computable in polynomial time. Information and Control 68(1986), pp. 86–104. MathSciNetMATHCrossRefGoogle Scholar
  26. [Ka87]
    Kaufmann, M.: A counterexample to the 0–1 law for existential monadic second-order logic. CLI Internal Note 32, Computational Logic Inc., Dec. 1987.Google Scholar
  27. [KS85]
    Kauffman, M., Shelah, S: On random models of finite power and monadic logic. Discrete Mathematics 54(1985), pp. 285–293.MathSciNetCrossRefGoogle Scholar
  28. [KV87]
    Kolaitis, P., Vardi, M.Y.: The decision problem for the probabilities of higher-order properties. Proc. 19th ACM Symp. on Theory of Computing, New York, May 1987, pp. 425T435.Google Scholar
  29. [KV90]
    Kolaitis, P., Vardi, M.Y.: 0–1 laws and decision problems for fragments of second-order logic. Information and Computation, in press.Google Scholar
  30. [Le79]
    Lewis, H.R.: Unsolvable Classes of Quantificational Formulas. Addison- Wesley, 1979.Google Scholar
  31. [NR77]
    Nešetřil, J., Rödl, V.: Partitions of finite relational and set systems. J. Combinatorial Theory A 22(1977), pp. 289–312.MATHCrossRefGoogle Scholar
  32. [NR83]
    Nešetřil, J., Rödl, V.: Ramsey classes of set systems. J. Combinatorial Theory A 34(1983), pp. 183–201. CrossRefGoogle Scholar
  33. [Po76]
    Pósa, L. : Hamiltonian circuits in random graphs. Discrete Math. 14(1976), pp. 359–364. MathSciNetMATHCrossRefGoogle Scholar
  34. [PS89]
    Pacholski, L., Szwast, W.: The 0–1 law fails for the class of existential second-order Gödel sentences with equality. Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989, pp. 160–163.Google Scholar
  35. [PS90]
    Pacholski, L., Szwast, W.: A counterexample to the 0–1 law for existential second-order minimal Gödel sentences. Unpublished, 1990.Google Scholar
  36. [Ra28]
    Ramsey, F.P. : On a problem in formal logic. Proc. London Math. Soc. 30(1928). pp. 264–286.CrossRefGoogle Scholar
  37. [Tr50]
    Trakhtenbrot, B.A. : The impossibilty of an algorithm for the decision problem for finite models. Doklady Akademii Nauk SSR 70(1950), PP. 569–572. Google Scholar
  38. [Va82]
    Vardi. M.Y. : The complexity of relational query languages. Proc. 14th ACM Symp. on Theory of Computing, San Francisco, 1982, pp. 137–146. Google Scholar

Copyright information

© Springer-Verlag New York, Inc 1992

Authors and Affiliations

  • Phokion G. Kolaitis
    • 1
  • Moshe Y. Vardi
    • 2
  1. 1.University of CaliforniaSanta CruzUSA
  2. 2.IBM Almaden Research CenterUSA

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