Notre Dame Journal of Formal Logic

Categoricity Spectra for Rigid Structures

Ekaterina Fokina, Andrey Frolov, and Iskander Kalimullin

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For a computable structure M, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of M. If the spectrum has a least degree, this degree is called the degree of categoricity of M. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.

Article information

Notre Dame J. Formal Logic, Volume 57, Number 1 (2016), 45-57.

Received: 14 February 2013
Accepted: 26 August 2013
First available in Project Euclid: 30 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C57: Effective and recursion-theoretic model theory [See also 03D45] 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

computable structure rigid structure computably categorical categoricity spectrum degree of categoricity


Fokina, Ekaterina; Frolov, Andrey; Kalimullin, Iskander. Categoricity Spectra for Rigid Structures. Notre Dame J. Formal Logic 57 (2016), no. 1, 45--57. doi:10.1215/00294527-3322017.

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  • [1] Anderson, B. A., and B. F. Csima, “Degrees that are not degrees of categoricity,” to appera in Notre Dame Journal of Formal Logic, preprint, arXiv:1210.4220v2 [math.LO].
  • [2] Csima, B. F., J. N. Y. Franklin, and R. A. Shore, “Degrees of categoricity and the hyperarithmetic hierarchy,” Notre Dame Journal of Formal Logic, vol. 54 (2013), pp. 215–31.
  • [3] Fokina, E. B., I. Kalimullin, and R. Miller, “Degrees of categoricity of computable structures,” Archive for Mathematical Logic, vol. 49 (2010), pp. 51–67.
  • [4] Fröhlich, A., and J. C. Shepherdson, “Effective procedures in field theory,” Philosophical Transactions of the Royal Society of London Series A, vol. 248 (1956), pp. 407–32.
  • [5] Goncharov, S. S., “Autostable models and algorithmic dimensions,” pp. 261–87 in Handbook of Recursive Mathematics, Vol. 1, vol. 138 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998.
  • [6] Hirschfeldt, D. R., B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures,” Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.
  • [7] Mal’cev, A. I., “Constructive algebras, I,” Uspekhi Mathematicheskikh Nauk, vol. 16 (1961), pp. 3–60.
  • [8] Mal’cev, A. I., “On recursive Abelian groups,” Doklady Akademiya Nauk SSSR, vol. 146 (1962), pp. 1009–12.
  • [9] Miller, R., “$d$-computable categoricity for algebraic fields,” Journal of Symbolic Logic, vol. 74 (2009), pp. 1325–51.
  • [10] Richter, L. J., “Degrees of structures,” Journal of Symbolic Logic, vol. 46 (1981), pp. 723–31.
  • [11] Rodgers, H., Theory of Recursive Functions and Effective Computability, Massachusetts Institute of Technology Press, Cambridge, Mass., 1987.