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The Ehrenfeucht-Fraïssé games for transitive closure

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Logical Foundations of Computer Science — Tver '92 (LFCS 1992)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 620))

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Abstract

We present in this paper Ehrenfeucht-Fraïssé games for the transitive closure logics DTC, TC and ATC. Although the existence of such games for logics with generalized quantifiers was known before, the exact formulation of these games for the predicate transformers arising from various transitive-closure-like operations is new. The games are sound and complete both for finite and infinite models. Combined with well known theorems of Fagin and Immerman these games shed new light on a model theoretic interpretation of the separability of the complexity classes L, NL, P and NP. We discuss this perspective in detail with two examples CLIQUE and HAMILTONIAN by showing that considered as ordered graphs neither of these is definable in monadic second order logic, and hence also not in any transitive closure logic where the predicate transformers are restricted to binary relations.

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References

  1. M. Ajtai and R. Fagin. Reachability is harder for directed than for undirected finite graphs. Journal of Symbolic Logic, 55.1:113–150, 1990.

    Google Scholar 

  2. J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

    Google Scholar 

  3. Y. Bargury and J.A. Makowsky. The expressive power of transitive closure and 2-way multihead automata. to appear in CSL'91, 1991.

    Google Scholar 

  4. A. Caló. The expressive power of transitive closure. M.Sc. Thesis, Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel, 1989.

    Google Scholar 

  5. B. Courcelle. The monadic second-order theory of graphs VI: On several representations of graphs by logical structures to appear in Discrete and Applied Mathematics.

    Google Scholar 

  6. M. de Rougemont. Second-order and inductive definability on finite structures. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 33:47–63, 1987.

    Google Scholar 

  7. H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1980.

    Google Scholar 

  8. R. Fagin. Generalized first-order spectra and polynomial time recognizable sets. In R. Karp, editor, Complexity of Computation, pages 27–41. American Mathematical Society Proc, 7, Society for Industrial and Applied Mathematics, 1974.

    Google Scholar 

  9. R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41.1:50–58, 1976.

    Google Scholar 

  10. J. Flum. On bounded theories. to appear in CSL'91, 1991.

    Google Scholar 

  11. Haim Gaifman. On local and non-local properties. In J. Stern, editor, Logic Colloquium '81, pages 105–135. North-Holland publishing Company, 1982.

    Google Scholar 

  12. D. Gabbay, A. Pnueli, S. Shelah, and J. Stavi. On temporal analysis of fairness. In Proceedings of the 7th ACM-POPL, pages 163–173, 1980.

    Google Scholar 

  13. J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley Series in Computer Science. Addison-Wesley, 1980.

    Google Scholar 

  14. N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing, 16(4):760–778, Aug 1987. Also appeared as a preliminary report in Proceedings of the 15th Annual ACM Symposium on the Theory of Computing.

    Article  Google Scholar 

  15. N. Immerman. Nondeterministic space is closed under complement. SIAM Journal on Computing, 17:935–938, 1988.

    Article  Google Scholar 

  16. R.E. Ladner. Apllication of model theoretic games to discrete linera orders and finite automata. Information and Control, 33:281–303, 1977.

    Article  Google Scholar 

  17. J.A. Makowsky. The impact of model theory on theoretical computer science. To appear in the Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science, Uppsala 1991, 1992.

    Google Scholar 

  18. J.A. Makowsky. Model theory and computer science: An appetizer. To appear in the Handbook of Logic in Computer Science, S. Abramski, D. Gabbay and T. Maibaum eds., Oxford University Press, 1992.

    Google Scholar 

  19. J.A. Makowsky and D. Mundici. Abstract equivalence relations. In Model-Theoretic Logics, Perspectives in Mathematical Logic, chapter 19. Springer Verlag, 1985.

    Google Scholar 

  20. J. A. Makowsky and M. Ziegler. Topological model theory with an interior operator: Consistency properties and back and forth arguments. Archiv für mathematische Logik und Grundlagenforschung, 20:37–54, 1980.

    Google Scholar 

  21. Enshao Shen. Manuscript. unpublished, Department of Mathematics, University of Freiburg, Germany, 1991.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel

    A. Calò & J. A. Makowsky

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  1. A. Calò
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  2. J. A. Makowsky
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Anil Nerode Mikhail Taitslin

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© 1992 Springer-Verlag Berlin Heidelberg

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Calò, A., Makowsky, J.A. (1992). The Ehrenfeucht-Fraïssé games for transitive closure. In: Nerode, A., Taitslin, M. (eds) Logical Foundations of Computer Science — Tver '92. LFCS 1992. Lecture Notes in Computer Science, vol 620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023863

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  • DOI: https://doi.org/10.1007/BFb0023863

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55707-4

  • Online ISBN: 978-3-540-47276-6

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