Characterizing the continuous degrees

  • Uri Andrews
  • Gregory Igusa
  • Joseph S. Miller
  • Mariya I. SoskovaEmail author


The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are all very close to total: joining a continuous degree with a total degree that is not below it always results in a total degree. We call this property almost totality.

We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of enumeration degrees [1], we see that the continuous degrees are also definable. Applying earlier work on the continuous degrees [10], this shows that the relation “PA above” on the total degrees is definable in the enumeration degrees.

In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly [7]. We prove that the enumeration degree of A is continuous if and only if A is codable, meaning that A is enumeration above the complement of an infinite tree, every path of which enumerates A.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Cai, H. A. Ganchev, S. Lempp, J. S. Miller and M. I. Soskova, Defining totality in the enumeration degrees, Journal of the American Mathematical Society 29 (2016), 1051–1067.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. R. Day and J. S. Miller, Randomness for non-computable measures, Transactions of the American Mathematical Society 365 (2013), 3575–3591.MathSciNetCrossRefGoogle Scholar
  3. [3]
    R. M. Friedberg and H. Rogers, Jr., Reducibility and completeness for sets of integers, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 5 (1959), 117–125.MathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Gács, Uniform test of algorithmic randomness over a general space, Theoretical Computer Science 341 (2005), 91–137.MathSciNetCrossRefGoogle Scholar
  5. [5]
    T. Grubba, M. Schröder and K. Weihrauch, Computable metrization, Mathematical Logic Quarterly 53 (2007), 381–395.MathSciNetCrossRefGoogle Scholar
  6. [6]
    I. Sh. Kalimullin, Definability of the jump operator in the enumeration degrees, Journal of Mathematical Logic 3 (2003), 257–267.MathSciNetCrossRefGoogle Scholar
  7. [7]
    T. Kihara and A. Pauly, Point degree spectra of represented spaces,
  8. [8]
    S. C. Kleene, Introduction to Metamathematics, D. Van Nostrand, New York, 1952.zbMATHGoogle Scholar
  9. [9]
    L. A. Levin, Uniform tests for randomness, Dokladi Akademii Nauk SSSR 227 (1976), 33–35.MathSciNetGoogle Scholar
  10. [10]
    J. S. Miller, Degrees of unsolvability of continuous functions, Journal of Symbolic Logic 69 (2004), 555–584.MathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Myhill, Note on degrees of partial functions, Proceedings of the American Mathematical Society 12 (1961), 519–521.MathSciNetCrossRefGoogle Scholar
  12. [12]
    M. Schröder, Effective metrization of regular spaces, in Computability and Complexity in Analysis, Informatik-Berichte, Vol. 235, FernUniversität in Hagen, Hagen, 1998, pp. 63–80.Google Scholar
  13. [13]
    A. L. Selman, Arithmetical reducibilities. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 17 (1971), 335–350.MathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Tychonoff, Über einen Metrisationssatz von P. Urysohn, Mathematische Annalen 95 (1926), 139–142.MathSciNetCrossRefGoogle Scholar
  15. [15]
    P. Urysohn, Zum Metrisationsproblem, Mathematische Annalen 94 (1925), 309–315.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Uri Andrews
    • 1
  • Gregory Igusa
    • 2
  • Joseph S. Miller
    • 1
  • Mariya I. Soskova
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

Personalised recommendations