STACS 1998: STACS 98 pp 444-454 | Cite as

Local normal forms for first-order logic with applications to games and automata

  • Thomas Schwentick
  • Klaus Barthelmann
Logic II
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)

Abstract

Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃χ1...,χl∀yϑ where ϑ is r-local around y, i. e. quantification in ϑ is restricted to elements of the universe of distance at most r from y. From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

Keywords

Normal Form Free Variable Atomic Formula Winning Strategy Isomorphism Type 
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. [AF97]
    Sanjeev Arora and Ronald Fagin. On winning strategies in EhrenfeuchtFraïssé games. Theoretical Computer Science, 174(1–2):97–121, 1997.MATHCrossRefMathSciNetGoogle Scholar
  2. [Com83]
    K. Compton. Some useful preservation theorems. Journal of Symbolic Logic, 48:427–440, 1983.MATHCrossRefMathSciNetGoogle Scholar
  3. [DLW97]
    G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proc. Int. Conf. on Database Theory, LNCS, pages 140–154. Springer-Verlag, 1997.Google Scholar
  4. [EF95]
    H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer-Verlag, 1995.Google Scholar
  5. [Ehr61]
    A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fund. Math., 49:129–141, 1961.MATHMathSciNetGoogle Scholar
  6. [Fag74]
    R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In R. M. Karp, editor, Complexity of Computation, SIAM-AMS Proceedings, Vol. 7, pages 43–73, 1974.Google Scholar
  7. [Fag97]
    R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.Google Scholar
  8. [Fra54]
    R. Fraïssé. Sur quelques classifications des systèmes de relations. Publ. Sci. Univ. Alger. Sér. A, 1:35–182, 1954.Google Scholar
  9. [FSV95]
    R. Fagin, L. Stockmeyer, and M. Vardi. On monadic NP vs. monadic coNP. Information and Computation, 120:78–92, 1995. Preliminaxy version appeared in 1993 IEEE Conference on Structure in Complexity Theory, pp. 19–30.MATHCrossRefMathSciNetGoogle Scholar
  10. [Gai82]
    H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105–135. North Holland, 1982.Google Scholar
  11. [Han65]
    W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132–145. North Holland, 1965.Google Scholar
  12. [Lib97]
    L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.Google Scholar
  13. [Sch96]
    T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61–92, 1996.MATHCrossRefMathSciNetGoogle Scholar
  14. [Tho91]
    W. Thomas. On logics, tilings and automata. In Proc. ICALP'91, number 510 in Lecture Notes in Computer Science, pages 441-453. Springer, 1991. [Tho97] W. Thomas. Automata theory on trees and partial orders. In Proc. TAPSOFT'97, number 1214 in Lecture Notes in Computer Science, pages 20–38. Springer, 1997.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Thomas Schwentick
    • 1
  • Klaus Barthelmann
    • 1
  1. 1.Johannes Gutenberg-Universität MainzMainz

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