Advertisement

Siberian Mathematical Journal

, Volume 49, Issue 3, pp 512–523 | Cite as

Estimation of the algorithmic complexity of classes of computable models

  • E. N. Pavlovskii
Article

Abstract

We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ 2 0 -complete), computable models with ω-categorical theories (Δ ω 0 -complex Π ω+2 0 -set), prime models (Δ ω 0 -complex Π ω+2 0 -set), models with ω 1-categorical theories (Δ ω 0 -complex Σ ω+1 0 -set. We obtain a universal lower bound for the model-theoretic properties preserved by Marker’s extensions (Δ ω 0 .

Keywords

computable model index set 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Goncharov S. S. and Knight J. F., “Computable structure and non-structure theorems,” Algebra and Logic, 41, No. 6, 351–373 (2002).CrossRefMathSciNetGoogle Scholar
  2. 2.
    Nurtazin A. T., Computable Classes and Algebraic Criteria of Autostability [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Inst. Mat. Mekh., Alma-Ata (1974).Google Scholar
  3. 3.
    Goncharov S. S. and Ershov Yu. L., Constructive Models [in Russian], Nauchnaya Kniga, Novosibirsk (1999).zbMATHGoogle Scholar
  4. 4.
    Goncharov S. S., “Computability and computable models, mathematical problems from applied logic. II,” in: Logics for the XXIst Century. Edited by D. M. Gabbay, S. S. Goncharov, and M. Zakharyaschev, Springer-Verlag, New York, 2006, pp. 99–216 (International Mathematical Series, New York).Google Scholar
  5. 5.
    Calvert W., “The isomorphism problem for classes of computable fields,” Archive Math. Logic, 34, No. 3, 327–336 (2004).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Calvert W., “The isomorphism problem for computable abelian p-groups of bounded length,” J. Symbolic Logic, 70, No. 1, 331–345 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Calvert W., Cummins D., Knight J. F., and Miller S., “Comparing classes of finite structures,” Algebra and Logic, 43, No. 6, 666–701 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Calvert W., Harizanov V., Knight J. F., and Miller S., “Index sets of computable structures,” Algebra and Logic, 45, No. 5, 306–325 (2006).CrossRefMathSciNetGoogle Scholar
  9. 9.
    Csima B. F., Montalbán A., and Shore R. A., “Boolean algebras, Tarski invariants, and index sets,” Notre Dame J. Formal Logic, 47, No. 1, 1–23 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dobritsa V. P., “Complexity of the index set of a constructive model,” Algebra and Logic, 22, No. 4, 269–276 (1983).CrossRefMathSciNetGoogle Scholar
  11. 11.
    White W., “On the complexity of categoricity in computable structures,” Math. Logic Quart., 49, No. 6, 603–614 (2003).zbMATHCrossRefGoogle Scholar
  12. 12.
    White W., Characterizations for Computable Structures: Ph.D. thesis, Cornell Univ. (2000).Google Scholar
  13. 13.
    Fokina E. B., “Index sets of decidable models,” Siberian Math. J., 48, No. 5, 939–948 (2007).CrossRefMathSciNetGoogle Scholar
  14. 14.
    Marker D., “Non-Σn-axiomatizable almost strongly minimal theories,” J. Symbolic Logic, 54, No. 3, 921–927 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Goncharov S. S. and Khoussainov B., “Complexity of theories of computable categorical models,” Algebra and Logic, 43, No. 6, 365–373 (2004).CrossRefMathSciNetGoogle Scholar
  16. 16.
    Rogers H., Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Comp., New York; St. Louis; San Francisco; Toronto; London; Sydney (1972).zbMATHGoogle Scholar
  17. 17.
    Chang C. C. and Keisler H. J., Model Theory [Russian translation], Mir, Moscow (1977).Google Scholar
  18. 18.
    Erimbetov M. M., “Complete theories with 1-cardinal formulas,” Algebra i Logika, 14, No. 3, 245–257 (1975).zbMATHMathSciNetGoogle Scholar
  19. 19.
    Marker D., Model Theory: An Introduction, Springer-Verlag, New York (2002) (Graduate Texts in Mathematics; 217).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations