A Gandy Theorem for Abstract Structures and Applications to First-Order Definability

  • Oleg V. Kudinov
  • Victor L. Selivanov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5635)

Abstract

We establish a Gandy theorem for a class of abstract structures and deduce some corollaries, in particular the maximal definability result for arithmetical structures in the class. We also show that the arithmetical structures under consideration are biinterpretable (without parameters) with the standard model of arithmetic. As an example we show that for any k ≥ 3 a predicate on the quotient structure of the h-quasiorder of finite k-labeled forests is definable iff it is arithmetical and invariant under automorphisms.

Keywords

Gandy theorem definability least fixed point biinterpretability labeled forest h-quasiorder 

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References

  1. [Ba75]
    Barwise, J.: Admissible Sets and Structures. Springer, Berlin (1975)CrossRefMATHGoogle Scholar
  2. [Er96]
    Ershov, Y.L.: Definability and Computability. Plenum, New-York (1996)MATHGoogle Scholar
  3. [He93]
    Hertling, P.: Topologische Komplexitätsgrade von Funktionen mit endlichem Bild. Informatik-Berichte 152, 34 pages, Fernuniversität Hagen (1993)Google Scholar
  4. [KS07]
    Kudinov, O.V., Selivanov, V.L.: Definability in the homomorphic quasiorder of finite labeled forests. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 436–445. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. [KS07a]
    Kudinov, O.V., Selivanov, V.L.: Undecidability in the homomorphic quasiorder of finite labeled forests. Journal of Logic and Computation 17, 1135–1151 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. [KSZ08]
    Kudinov, O.V., Selivanov, V.L., Zhukov, A.V.: Definability in the h-quasiorder of labeled forests. Annals of Pure and Applied Logic (2008), doi: 10.1016/ j.apal. 2008.09.026Google Scholar
  7. [KSZ09]
    Kudinov, O.V., Selivanov, V.L., Zhukov, A.V.: Definability of closure operations in the h-quasiorder of labeled forests. In: Local volume of this conferenceGoogle Scholar
  8. [Ku06]
    Kuske, D.: Theories of orders on the set of words. RAIRO Theoretical Informatics and Applications 40, 53–74 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. [Se04]
    Selivanov, V.L.: Boolean hierarchy of partitions over reducible bases. Algebra and Logic 43(1), 44–61 (2004)MathSciNetCrossRefGoogle Scholar
  10. [Se07]
    Selivanov, V.L.: Hierarchies of \(\Delta^0_2\)-measurable k-partitions. Mathematical Logic Quarterly 53, 446–461 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Oleg V. Kudinov
    • 1
  • Victor L. Selivanov
    • 2
  1. 1.S.L. Sobolev Institute of MathematicsSiberian Division Russian Academy of SciencesRussia
  2. 2.A.P. Ershov Institute of Informatics SystemsSiberian Division Russian Academy of SciencesRussia

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