0–1 Laws for Fragments of Second-Order Logic: An Overview
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.
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- [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
- [Bo85]Bollobas, B: Random Graphs. Academic Press, 1985.Google Scholar
- [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
- [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
- [DG79]Dreben, D., and Goldfarb, W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley,1979.Google Scholar
- [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
- [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
- [Im83]Immerman, N. : Languages which capture complexity classes. Proc. 15th ACM Symp. on Theory of Computing, Boston, 1983, pp. 347–354. Google Scholar
- [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
- [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
- [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
- [Le79]Lewis, H.R.: Unsolvable Classes of Quantificational Formulas. Addison- Wesley, 1979.Google Scholar
- [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
- [PS90]Pacholski, L., Szwast, W.: A counterexample to the 0–1 law for existential second-order minimal Gödel sentences. Unpublished, 1990.Google Scholar
- [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
- [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