On bijections vs. unary functions

  • Thomas Schwentick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively.

In this paper it is shown that
  • Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide;

  • the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AF90]
    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. [AF94]
    S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraïssé games. Unpublished manuscript, 1994.Google Scholar
  3. [Ajt83]
    M. Ajtai. Σ 11 formulae on finite structures. Ann. of Pure and Applied Logic, 24:1–48, 1983.CrossRefGoogle Scholar
  4. [Cos93]
    S. Cosmadakis. Logical reducibility and monadic NP. In Proc. 34th IEEE Symp. on Foundations of Computer Science, pages 52–61, 1993.Google Scholar
  5. [DLS95]
    A. Durand, C. Lautemann, and T. Schwentick. Fragments of binary NP. In Annual Conference of the EACSL, 1995.Google Scholar
  6. [dR87]
    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. [EFT92]
    H.-D. Ebbinghaus, J. Flum, and W. Thomas. Einführung in die mathematische Logik. BI, Mannheim, 3rd edition, 1992.Google Scholar
  8. [Ehr61]
    A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fund. Math., 49:129–141, 1961.Google Scholar
  9. [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
  10. [Fag75]
    R. Fagin. Monadic generalized spectra. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 21:89–96, 1975.Google Scholar
  11. [Fra54]
    R. Fraïssé. Sur quelques classifications des systèmes de relations. Publ. Sci. Univ. Alger. Sér. A, 1:35–182, 1954.Google Scholar
  12. [FSV93]
    R. Fagin, L. Stockmeyer, and M. Vardi. On monadic NP vs. monadic co-NP. In The Proceedings of the 8th Annual IEEE Conference on Structure in Complexity Theory, pages 19–30, 1993.Google Scholar
  13. [Gra84]
    E. Grandjean. The spectra of first-order sentences and computational complexity. SIAM Journal on Computing, 13:356–373, 1984.CrossRefGoogle Scholar
  14. [Gra85]
    E. Grandjean. Universal quantifiers and. time complexity of random access machines. Mathematical System Theory, 13:171–187, 1985.CrossRefGoogle Scholar
  15. [Gra90]
    E. Grandjean. First-order spectra with one variable. Journal of Computer and System Sciences, 40:136–153, 1990.Google Scholar
  16. [Loe91]
    B. Loescher. Begründung, Verallgemeinerung und Anwendung der Ehrenfeucht-Spiele in der Relationentheorie von Roland Fraïssé. Informatik-bericht 2/91, Institut für Informatik, Universität Mainz, 1991.Google Scholar
  17. [LS95]
    B. Loescher and A. Sharell. Functions vs. relations on finite structures — a finer hierarchy in existential second order logic. presented at ASL-Logic Colloquium, FMT-23, Haifa, 1995.Google Scholar
  18. [Lyn82]
    J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127–144, 1982.CrossRefGoogle Scholar
  19. [Nur95]
    J. Nurmonen. On winning strategies with unary quantifiers. Preprint 77, Department of mathematics, University of Helsinki, 1995.Google Scholar
  20. [Sch94]
    T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614–622, 1994.Google Scholar
  21. [Sch95]
    T. Schwentick. Graph connectivity, monadic NP and built-in relations of moderate degree. In Proc. 22nd International Colloq. on Automata, Languages, and Programming, pages 405–416, 1995.Google Scholar
  22. [Sek60]
    M. Sekanina. On an ordering of the set of vertices of a connected graph. Spisy Přírod. Fak. Univ. Brno, pages 137–141, 1960.Google Scholar
  23. [Ten75]
    R. Tenney. Second-order Ehrenfeucht games and the decidability of the second-order theory of an equivalence relation. Journal of the Australian Mathematical Society, 20:323–331, 1975.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Thomas Schwentick
    • 1
    • 2
  1. 1.Universität MainzGermany
  2. 2.Institut für InformatikJohannes-Gutenberg Universität MainzGermany

Personalised recommendations