Notre Dame Journal of Formal Logic

On the Degree Structure of Equivalence Relations Under Computable Reducibility

Keng Meng Ng and Hongyuan Yu

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We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.

Article information

Notre Dame J. Formal Logic, Volume 60, Number 4 (2019), 733-761.

Received: 5 November 2017
Accepted: 18 July 2018
First available in Project Euclid: 12 September 2019

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Digital Object Identifier

Primary: 03D28: Other Turing degree structures
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

degrees equivalence relations density effective enumerations universality


Ng, Keng Meng; Yu, Hongyuan. On the Degree Structure of Equivalence Relations Under Computable Reducibility. Notre Dame J. Formal Logic 60 (2019), no. 4, 733--761. doi:10.1215/00294527-2019-0028.

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  • [1] Andrews, U., S. Lempp, J. S. Miller, K. M. Ng, L. San Marco, and A. Sorbi, “Universal computably enumerable equivalence relations,” Journal of Symbolic Logic, vol. 79 (2014), pp. 60–88.
  • [2] Bernardi, C., and A. Sorbi, “Classifying positive equivalence relations,” Journal of Symbolic Logic, vol. 48 (1983), pp. 529–38.
  • [3] Calvert, W., D. Cummins, J. F. Knight, and S. Miller, “Comparing classes of finite structures” (in Russian), Algebra i Logika, vol. 43 (2004), pp. 666–701; English translation in Algebra and Logic, vol. 43 (2004), pp. 374–92.
  • [4] Calvert, W., and J. F. Knight, “Classification from a computable viewpoint,” Bulletin of Symbolic Logic, vol. 12 (2006), pp. 191–218.
  • [5] Coskey, S., J. D. Hamkins, and R. Miller, “The hierarchy of equivalence relations on the natural numbers under computable reducibility,” Computability, vol. 1 (2012), pp. 15–38.
  • [6] Ershov, Y. I., “Theory of numberings,” pp. 473–503 in Handbook of Computability Theory, edited by E. R. Griffor, vol. 140 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1999.
  • [7] Fokina, E. B., and S.-D. Friedman, “Equivalence relations on classes of computable structures,” pp. 198–207 in Mathematical Theory and Computational Practice, edited by K. Ambos-Spies, B. Löwe, and W. Merkle, vol. 5635 of Lecture Notes in Computer Science, Springer, Berlin, 2009.
  • [8] Friedman, H., and L. Stanley, “A Borel reducibility theory for classes of countable structures,” Journal of Symbolic Logic, vol. 54 (1989), pp. 894–914.
  • [9] Gao, S., and P. Gerdes, “Computably enumerable equivalence relations,” Studia Logica, vol. 67 (2001), pp. 27–59.
  • [10] Ianovski, E., R. Miller, K. M. Ng, and A. Nies, “Complexity of equivalence relations and preorders from computability theory,” Journal of Symbolic Logic, vol. 79 (2014), pp. 859–81.
  • [11] Lachlan, A. H., “Initial segments of one-one degrees,” Pacific Journal of Mathematics, vol. 29 (1969), pp. 351–66.
  • [12] Miller, R., and K. M. Ng, “Finitary reducibility on equivalence relations,” Journal of Symbolic Logic, vol. 81 (2016), pp. 1225–54.
  • [13] Soare, R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic, Springer, Berlin, 1987.
  • [14] Young, P. R., “Notes on the structure of recursively enumerable sets,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1963.