Abstract
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.
Similar content being viewed by others
References
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.
A. R. Day and J. S. Miller, Randomness for non-computable measures, Transactions of the American Mathematical Society 365 (2013), 3575–3591.
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.
P. Gács, Uniform test of algorithmic randomness over a general space, Theoretical Computer Science 341 (2005), 91–137.
T. Grubba, M. Schröder and K. Weihrauch, Computable metrization, Mathematical Logic Quarterly 53 (2007), 381–395.
I. Sh. Kalimullin, Definability of the jump operator in the enumeration degrees, Journal of Mathematical Logic 3 (2003), 257–267.
T. Kihara and A. Pauly, Point degree spectra of represented spaces, https://arxiv.org/abs/1405.6866.
S. C. Kleene, Introduction to Metamathematics, D. Van Nostrand, New York, 1952.
L. A. Levin, Uniform tests for randomness, Dokladi Akademii Nauk SSSR 227 (1976), 33–35.
J. S. Miller, Degrees of unsolvability of continuous functions, Journal of Symbolic Logic 69 (2004), 555–584.
J. Myhill, Note on degrees of partial functions, Proceedings of the American Mathematical Society 12 (1961), 519–521.
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.
A. L. Selman, Arithmetical reducibilities. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 17 (1971), 335–350.
A. Tychonoff, Über einen Metrisationssatz von P. Urysohn, Mathematische Annalen 95 (1926), 139–142.
P. Urysohn, Zum Metrisationsproblem, Mathematische Annalen 94 (1925), 309–315.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by NSF grant DMS1600228.
The third author was partially supported by grant #358043 from the Simons Foundation.
The fourth author was partially supported by National Science Fund of Bulgaria grant #01/18 from 23.07.2017 and by National Science Foundation grant DMS1762648.
Rights and permissions
About this article
Cite this article
Andrews, U., Igusa, G., Miller, J.S. et al. Characterizing the continuous degrees. Isr. J. Math. 234, 743–767 (2019). https://doi.org/10.1007/s11856-019-1943-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1943-x