Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T22:54:54.787Z Has data issue: false hasContentIssue false
  • English
  • Français

A STRUCTURAL DICHOTOMY IN THE ENUMERATION DEGREES

Part of: Computability and recursion theory

Published online by Cambridge University Press:  10 July 2020

HRISTO A. GANCHEV ,
ISKANDER SH. KALIMULLIN ,
JOSEPH S. MILLER  and
MARIYA I. SOSKOVA
HRISTO A. GANCHEV
Affiliation:
FACULTY OF MATHEMATICS AND INFORMATICS SOFIA UNIVERSITY 5 JAMES BOURCHIER BLVD., SOFIA 1164, BULGARIAE-mail: [email protected]
ISKANDER SH. KALIMULLIN
Affiliation:
N.I. LOBACHEVSKY INSTITUTE OF MATHEMATICS AND MECHANICS KAZAN (VOLGA REGION) FEDERAL UNIVERSITY UL. KREMLEVSKAYA 18, KAZAN, TATARSTAN 420008, RUSSIAN FEDERATIONE-mail: [email protected]
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON 480 LINCOLN DR., MADISON, WI53706, USAE-mail: [email protected]: [email protected]
MARIYA I. SOSKOVA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON 480 LINCOLN DR., MADISON, WI53706, USAE-mail: [email protected]: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half of a nontrivial relativized $\mathcal {K}$ -pair. This leads to a structural dichotomy in the enumeration degrees.

Keywords

enumeration degreesdefinabilitycontinuous degreescomputable metric spaces

MSC classification

Primary: 03D30: Other degrees and reducibilities
Secondary: 03D78: Computation over the reals
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 527 - 544
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, U., Ganchev, H. A., Kuyper, R., Lempp, S., Miller, J. S., Soskova, A. A., and Soskova, M. I., On cototality and the skip operator in the enumeration degrees . Transactions of the American Mathematical Society, vol. 372 (2019), no. 3, pp. 1631–1670.CrossRefGoogle Scholar
Andrews, U., Igusa, G., Miller, J. S., and Soskova, M. I., Characterizing the continuous degrees. Israel Journal of Mathematics , vol. 234 (2019), no. 2, 743–767.Google Scholar
Cai, M., Ganchev, H. A., Lempp, S., Miller, J. S., and Soskova, M. I., Defining totality in the enumeration degrees . Journal of the American Mathematical Society, vol. 29 (2016), no. 4, pp. 1051–1067.10.1090/jams/848CrossRefGoogle Scholar
Day, A. R. and Miller, J. S., Randomness for non-computable measures . Transactions of the American Mathematical Society, vol. 365 (2013), no. 7, pp. 3575–3591.CrossRefGoogle Scholar
Friedberg, R. M. and Rogers, H. Jr, Reducibility and completeness for sets of integers . Zeitschrift für Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 117–125.CrossRefGoogle Scholar
Ganchev, H. and Soskova, M. I., Cupping and definability in the local structure of the enumeration degrees . this Journal, vol. 77 (2012), no. 1, pp. 133–158.Google Scholar
Ganchev, H. A. and Soskova, M. I., Definability via Kalimullin pairs in the structure of the enumeration degrees . Transactions of the American Mathematical Society, vol. 367 (2015), pp. 4873–4893.CrossRefGoogle Scholar
Ganchev, H. A. and Soskova, M. I., The jump hierarchy in the enumeration degrees . Computability, vol. 7 (2018), no. 2–3, pp. 179–188.CrossRefGoogle Scholar
Jockusch, C. G. Jr, Semirecursive sets and positive reducibility . Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420–436.CrossRefGoogle Scholar
Kalimullin, I. S., Definability of the jump operator in the enumeration degrees . Journal of Mathematical Logic, vol. 3 (2003), no. 2, pp. 257–267.CrossRefGoogle Scholar
Kihara, T., Ng, K. M., and Pauly, A., Enumeration degrees and non-metrizable topology, preprint, 2017.Google Scholar
Kihara, T. and Pauly, A., Point degree spectra of represented spaces, submitted.Google Scholar
Levin, L. A., Uniform tests for randomness . Doklady Akademii Nauk SSSR, vol. 227 (1976), no. 1, pp. 33–35.Google Scholar
McCarthy, E., Cototal enumeration degrees and the Turing degree spectra of minimal subshifts . Proceedings of the American Mathematical Society, vol. 146 (2018), pp. 3541–3552.CrossRefGoogle Scholar
Miller, J. S., Degrees of unsolvability of continuous functions . this Journal, vol. 69 (2004), no. 2, pp. 555–584.Google Scholar
Miller, J. S. and Soskova, M. I., Randomness relative to an enumeration oracle, in preparation.Google Scholar
Miller, J. S. and Soskova, M. I., Density of the cototal enumeration degrees . Annals of Pure and Applied Logic, vol. 169 (2018), no. 5, pp. 450–462.CrossRefGoogle Scholar
Odifreddi, P. G., Classical Recursion Theory, vol. II, Studies in Logic and the Foundations of Mathematics, 143, North-Holland Publishing Co., Amsterdam, 1999.Google Scholar
Richter, L. J., Degrees of structures , this Journal, vol. 46 (1981), no. 4, pp. 723–731.Google Scholar
Selman, A. L., Arithmetical reducibilities. I . Zeitschrift für Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 335–350.CrossRefGoogle Scholar
Soskov, I. N., A jump inversion theorem for the enumeration jump . Archive for Mathematical Logic, vol. 39 (2000), pp. 417–437.CrossRefGoogle Scholar