Degrees of autostability relative to strong constructivizations

Article

Abstract

The spectra of the Turing degrees of autostability of computable models are studied. For almost prime decidable models, it is shown that the autostability spectrum relative to strong constructivizations of such models always contains a certain recursively enumerable Turing degree; moreover, it is shown that for any recursively enumerable Turing degree, there exist prime models in which this degree is the least one in the autostability spectrum relative to strong constructivizations.

References

  1. 1.
    A. I. Mal’tsev, “On Recursive Abelian Groups,” Dokl. Akad. Nauk SSSR 146(5), 1009–1012 (1962) [Sov. Math., Dokl. 32, 1431–1434 (1963)].MathSciNetGoogle Scholar
  2. 2.
    A. I. Mal’tsev, “Constructive Algebras. I,” Usp. Mat. Nauk 16(3), 3–60 (1961) [Russ. Math. Surv. 16 (3), 77–129 (1961)].Google Scholar
  3. 3.
    A. Fröhlich and J. C. Shepherdson, “Effective Procedures in Field Theory,” Philos. Trans. R. Soc. London, Ser. A 248, 407–432 (1956).MATHCrossRefGoogle Scholar
  4. 4.
    S. I. Adyan, “Unsolvability of Some Algorithmic Problems in the Theory of Groups,” Tr. Mosk. Mat. Obshch. 6, 231–298 (1957).MathSciNetMATHGoogle Scholar
  5. 5.
    S. I. Adyan, “The Problem of Identity in Associative Systems of a Special Form,” Dokl. Akad. Nauk SSSR 135(6), 1297–1300 (1960) [Sov. Math., Dokl. 1, 1360–1363 (1960)].Google Scholar
  6. 6.
    P. S. Novikov and S. I. Adyan, “Das Wortproblem für Halbgruppen mit einseitiger Kürzungsregel,” Z. Math. Log. Grundlagen Math. 4, 66–88 (1958).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A. A. Markov, “Impossibility of Certain Algorithms in the Theory of Associative Systems. II,” Dokl. Akad. Nauk SSSR 58(3), 353–356 (1947).MATHGoogle Scholar
  8. 8.
    B. L. van der Waerden, “Eine Bemerkung über die Unzerlegbarkeit von Polynomen,” Math. Ann. 102, 738–739 (1930).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    S. S. Goncharov, “Constructivizability of Superatomic Boolean Algebras,” Algebra Logika 12(1), 31–40 (1973) [Algebra Logic 12, 17–22 (1973)].MATHGoogle Scholar
  10. 10.
    S. Wehner, “Enumerations, Countable Structures and Turing Degrees,” Proc. Am. Math. Soc. 126, 2131–2139 (1998).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    T. A. Slaman, “Relative to Any Nonrecursive Set,” Proc. Am. Math. Soc. 126, 2117–2122 (1998).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    S. S. Goncharov, “Limit Equivalent Constructivizations,” in Mathematical Logic and the Theory of Algorithms (Nauka, Novosibirsk, 1982), Tr. Inst. Mat., Sib. Otd. Akad. Nauk SSSR 2, pp. 4–12.Google Scholar
  13. 13.
    S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller, and R. Solomon, “Enumerations in Computable Structure Theory,” Ann. Pure Appl. Logic 136(3), 219–246 (2005).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    S. S. Goncharov, “Computability and Computable Models,” in Mathematical Problems from Applied Logic. II, Ed. by D. Gabbay, S. S. Goncharov, and M. Zakharyashev (Springer, New York, 2007), Int. Math. Ser. 5, pp. 99–216.CrossRefGoogle Scholar
  15. 15.
    C. J. Ash, “Categoricity in Hyperarithmetical Degrees,” Ann. Pure Appl. Logic 34(1), 1–14 (1987).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    C. J. Ash and J. F. Khight, Computable Structures and the Hyperarithmetical Hierarchy (Elsevier, Amsterdam, 2000).MATHGoogle Scholar
  17. 17.
    C. Ash, J. Knight, M. Manasse, and T. Slaman, “Generic Copies of Countable Structures,” Ann. Pure Appl. Logic 42(3), 195–205 (1989).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    J. Chisholm, E. B. Fokina, S. S. Goncharov, V. S. Harizanov, J. F. Knight, and S. Quinn, “Intrinsic Bounds on Complexity and Definability at Limit Levels,” J. Symb. Logic 74(3), 1047–1060 (2009).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    E. B. Fokina, I. Kalimullin, and R. Miller, “Degrees of Categoricity of Computable Structures,” Arch. Math. Logic 49, 51–67 (2010).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Yu. L. Ershov, “Constructive Models,” in Selected Topics of Algebra and Logic (Nauka, Novosibirsk, 1973), pp. 111–130 [in Russian].Google Scholar
  21. 21.
    M. D. Morley, “Decidable Models,” Isr. J. Math. 25, 233–240 (1976).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    S. S. Goncharov and A. T. Nurtazin, “Constructive Models of Complete Solvable Theories,” Algebra Logika 12(2), 125–142 (1973) [Algebra Logic 12, 67–77 (1973)].MATHCrossRefGoogle Scholar
  23. 23.
    Yu. Ershov and S. Goncharov, “Elementary Theories and Their Constructive Models,” in Handbook of Recursive Mathematics, Vol. 1: Recursive Model Theory (Elsevier, Amsterdam, 1998), Stud. Logic Found. Math. 138, pp. 115–165.Google Scholar
  24. 24.
    A. T. Nurtazin, “Strong andWeak Constructivization and Computable Families,” Algebra Logika 13(3), 311–323 (1974) [Algebra Logic 13, 177–184 (1974)].MathSciNetMATHGoogle Scholar
  25. 25.
    L. Harrington, “Recursively Presentable Prime Models,” J. Symb. Logic 39, 305–309 (1974).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    T. S. Millar, “Foundations of Recursive Model Theory,” Ann. Math. Logic 13(1), 45–72 (1978).MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    J. Chisholm, “Effective Model Theory vs. Recursive Model Theory,” J. Symb. Logic 55(3), 1168–1191 (1990).MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    S. S. Goncharov, Countable Boolean Algebras and Decidability (Nauchnaya Kniga, Novosibirsk, 1996; Plenum, New York, 1997).MATHGoogle Scholar
  29. 29.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models (Nauchnaya Kniga, Novosibirsk, 1999); Engl. transl.: Yu. L. Ershov and S. S. Goncharov, Constructive Models (Consultants Bureau, New York, 2000).MATHGoogle Scholar
  30. 30.
    S. S. Goncharov, “Autostability of Prime Models under Strong Constructivizations,” Algebra Logika 48(6), 729–740 (2009) [Algebra Logic 48, 410–417 (2009)].MathSciNetGoogle Scholar
  31. 31.
    S. S. Goncharov, “On Autostability of Almost Prime Models Relative to Strong Constructivizations,” Usp. Mat. Nauk 65(5), 107–142 (2010) [Russ. Math. Surv. 65, 901–935 (2010)].MathSciNetGoogle Scholar
  32. 32.
    H. Rogers, Jr., Theory of Recursive Functions and Effective Computability (McGraw-Hill, Maidenhead, Berksh., 1967; Mir, Moscow, 1972).MATHGoogle Scholar
  33. 33.
    C. C. Chang and H. J. Keisler, Model Theory (North-Holland, Amsterdam, 1973; Mir, Moscow, 1977).MATHGoogle Scholar
  34. 34.
    A. I. Mal’cev, Algorithms and Recursive Functions (Nauka, Moscow, 1965; Wolters-Noordhoff, Groningen, 1970).Google Scholar
  35. 35.
    E. A. Palyutin, “The Algebras of Formulae of Countably Categorical Theories,” Colloq. Math. 31(2), 157–159 (1974).MathSciNetMATHGoogle Scholar
  36. 36.
    S. S. Goncharov and J. F. Knight, “Computable Structure and Non-structure Theorems,” Algebra Logika 41(6), 639–681 (2002) [Algebra Logic 41, 351–373 (2002)].MathSciNetMATHGoogle Scholar
  37. 37.
    Yu. L. Ershov, Decision Problems and Constructive Models (Nauka, Moscow, 1980) [in Russian].Google Scholar
  38. 38.
    C. F. D. McCoy, “Δ02 -Categoricity in Boolean Algebras and Linear Orderings,” Ann. Pure Appl. Logic 119(1–3), 85–120 (2003).MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    C. F. D. McCoy, “Partial Results in Δ30-Categoricity in Linear Orderings and Boolean Algebras,” Algebra Logika 41(5), 531–552 (2002) [Algebra Logic 41, 295–305 (2002)].MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Novosibirsk State University and Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia

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