FM-Representability and Beyond

  • Marcin Mostowski
  • Konrad Zdanowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


This work concerns representability of arithmetical notions in finite models. It follows the paper by Marcin Mostowski [1], where the notion of FM–representability has been defined. We discuss how far this notion captures the methodological idea of representing infinite sets in finite but potentially infinite domains.

We consider mainly some weakenings of the notion of FM–representability. We prove that relations weakly FM–representable are exactly those being \(\Sigma_{\rm 2}^{\rm 0}\)–definable. Another weakening of the notion, namely statistical representability, turns out to be equivalent to the original one. Additionally, we consider the complexity of sets of formulae naturally defined in finite models. We state that the set of sentences true in almost all finite arithmetical models is \(\Sigma_{\rm 2}^{\rm 0}\)–complete and that the set of formulae FM–representing some relations is \(\Pi^{0}_{3}\)–complete.


Turing Machine Free Variable Finite Model Statistical Representability Arithmetical Formula 
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  1. 1.
    Mostowski, M.: On representing concepts in finite models. Mathematical Logic Quarterly 47, 513–523 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aristotle: Physics. The Internet Classics Archive (written 350 B.C.) translated by Hardie, R.P., Gaye, R.K., available at:
  3. 3.
    Mycielski, J.: Analysis without actual infinity. Journal of Symbolic Logic 46, 625–633 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Mostowski, M.: On representing semantics in finite models. In: Rojszczak, A., Cachro, J., Kurczewski, G. (eds.) Philosophical Dimensions of Logic and Science, pp. 15–28. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  5. 5.
    Mostowski, M.: Potential infinity and the Church Thesis. In: manuscript, see also an extended abstract in the electronic proceedings of Denis Richard 60-th Birthday Conference, Clermont–Rerrand (2000)Google Scholar
  6. 6.
    Mostowski, M.: Truth definitions in finite models (1993) (manuscript)Google Scholar
  7. 7.
    Kołodziejczyk, L.: Truth definitions in finite models. The Journal of Symbolic Logic 69, 183–200 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kołodziejczyk, L.: A finite model-theoretical proof of a property of bounded query classes within PH. The Journal of Symbolic Logic 69, 1105–1116 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ebbinghaus, H.D., Flum, J.: Finite model theory. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  10. 10.
    Soare, R.I.: Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer, Heidelberg (1987)Google Scholar
  11. 11.
    Krynicki, M., Zdanowski, K.: Theories of arithmetics in finite models. Journal of Symbolic Logic 70(1), 1–28 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mostowski, M., Wasilewska, A.: Arithmetic of divisibility in finite models. Mathematical Logic Quarterly 50(2), 169–174 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Zdanowski, K.: Arithmetics in finite but potentially infinite worlds. PhD thesis, Warsaw University (2005) (in preparation)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Marcin Mostowski
    • 1
  • Konrad Zdanowski
    • 2
  1. 1.Department of Logic, Institute of PhilosophyWarsaw UniversityWarszawaPoland
  2. 2.Institute of MathematicsPolish Academy of Science00-956 Warszawa 10Poland

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