Monadic NP and built-in trees

  • Martin Kreidler
  • Detlef Seese
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


In our paper, we prove that Graph Connectivity is not in Monadic NP even in the presence of a built-in relation of arbitrary degree that is cycle-free. We obtain our result by giving a winning strategy for the duplicator in the Ajtai-Fagin Ehrenfeucht-Fraïssé Game. The result can be strengthened to obtain nondefinability for a larger class of graphs.


Monadic Second-Order Logic Descriptive Complexity Theory Ehrenfeucht-Fraïssé Games Finite Model Theory 


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  1. [Ajt]
    M. Ajtai: Σ 11-formulae on finite structures, Annals of Pure and Applied Logic, vol. 24, 1983, pp. 1–48.Google Scholar
  2. [AjFa]
    M. Ajtai and R. Fagin: Reachability is harder for directed than for undirected finite graphs, Journal of Symbolic Logic, 55 (1), 1990, pp. 113–150.Google Scholar
  3. [Bör]
    E. Börger: Berechenbarkeit, Komplexität, Logik, Vieweg 1992.Google Scholar
  4. [BoSi]
    R. Boppana and M. Sipser: The Complexity of Finite Functions. In Handbook of Theoretical Computer Science vol. A (ed. J. van Leeuwen), Elsevier Science Publishers, pp. 757–804, 1990.Google Scholar
  5. [ArFa]
    S. Arora and R. Fagin: On winning strategies in Ehrenfeucht-Fraïssé Games. Preprint, 1995.Google Scholar
  6. [Cou]
    B. Courcelle: Recognizability and second-order definability for sets of finite graphs, Information and Comput. 85, 1990, pp. 12–75.Google Scholar
  7. [DLS]
    A. Durand, C. Lautemann and T. Schwentick: Fragments of Binary NP, In Proc. IEEE Structures in Complexity Theory, 1995, also presented at CSL'95.Google Scholar
  8. [EbFu]
    H.-D. Ebbinghaus, J. Flum: Finite Model Theory, Springer-Verlag, New York, 1995.Google Scholar
  9. [EFT]
    H.-D. Ebbinghaus, J. Flum and W. Thomas: Mathematical Logic, Second Edition, Springer-Verlag, New York, 1993.Google Scholar
  10. [Ehr]
    A. Ehrenfeucht: An application of games to the completeness problem for formalized theories, Fund. Math., 49, 1961, pp. 129–141.Google Scholar
  11. [Fag74]
    R. Fagin: Generalized First-Order spectra and polynomial-time recognizable sets, In Complexity of Computation, (ed. R. Karp) SIAM-AMS Proc. 7, 1974, pp. 27–41.Google Scholar
  12. [Fag75]
    R. Fagin: Monadic generalized spectra, in Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, 1975, pp. 89–96.Google Scholar
  13. [Fag94]
    R. Fagin: Comparing the Power of Monadic NP Games, in Proc. LCC'94, LNCS 960, Springer-Verlag, 1995 pp. 414–425.Google Scholar
  14. [Fra]
    R. Fraïssé: Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sr. A, 1, 1954, pp. 35–182.Google Scholar
  15. [FSV]
    R. Fagin, L. Stockmeyer and M. Vardi: On Monadic NP vs. Monadic co-NP, In Information and Computation, 120, 1995, pp. 18–92.Google Scholar
  16. [Gai]
    H. Gaifman: On local and nonlocal properties, In J. Stern, editor, Logic Colloquium'81, North-Holland, 1982, pp. 105–135.Google Scholar
  17. [Gur]
    Y. Gurevich: Logic and the challenge of computer science, Univ. Michigan Tech. Report CRL-TR-10-85, Sept. 1985.Google Scholar
  18. [Han]
    W. Hanf: Model-theoretic methods in the study of elementary logic, In J. Addison, L. Henkin and A. Tarski, editors, The Theory of Models, North Holland, 1965, pp. 132–145.Google Scholar
  19. [Imm87]
    N. Immerman: Languages that capture complexity classes, SIAM J. of Computing, Vol. 16, No. 4, August 1987, pp. 760–778.Google Scholar
  20. [Imm89]
    N. Immerman: Descriptive and computational complexity, Computational Complexity Theory, Proc. Symp. Applied Math. Vol. 38 (J. Hartmanis, ed.), 1989, pp. 75–91.Google Scholar
  21. [Lau]
    C. Lautemann: Logical Definability of NP-Optimisation Problems with Monadic Auxiliary Predicates, in Proc. CSL'92, San Miniato, Italy, LNCS 702, pp. 327–339, Springer-Verlag, 1993.Google Scholar
  22. [Meh]
    K. Mehlhorn: Graph Algorithms and NP-Completeness, Springer-Verlag, Berlin, 1984.Google Scholar
  23. [Pap]
    C. H. Papadimitriou: Computational Complexity, Addison-Wesley Publishing Company, Reading, Massachusetts, 1994.Google Scholar
  24. [Rei]
    R. Reischuk: Einführung in die Komplexitätstheorie, B.G. Teubner Stuttgart, 1990.Google Scholar
  25. [dRo]
    M. de Rougemont: Second-order and inductive definability on finite structures, Zeitschrift f. math. Logik und Grundlagen d. Math., Bd. 33, 1987, pp. 47–63.Google Scholar
  26. [Sch96a]
    T. Schwentick: On Bijections vs. Unary Functions, In Proc. STACS'96, LNCS 1046, Springer-Verlag, 1996.Google Scholar
  27. [Sch96b]
    T. Schwentick: On Winning Ehrenfeucht Games and Monadic NP, Annals of Pur and Applied Logic vol. 79, pp. 61–92, 1996.Google Scholar
  28. [See96]
    D. Seese: Linear time computable problems and first-order descriptions, in Math. Struct. in Comp. Sc. vol. 6, pp. 1–22, 1996 (to appear).Google Scholar
  29. [She]
    S. Shelah: The Monadic Theory of Order, Annals of Mathematics, vol. 102, pp. 379–419, 1975.Google Scholar
  30. [Spe]
    J. Spencer: Threshold spectra via the Ehrenfeucht game, Disc. Appl. Math. 30, pp. 235–252, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Martin Kreidler
    • 1
  • Detlef Seese
    • 1
  1. 1.Institut AIFBUniversität KarlsruheKarlsruheGermany

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