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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: ganchev@fmi.uni-sofia.bg
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: Iskander.Kalimullin@kpfu.ru
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON 480 LINCOLN DR., MADISON, WI53706, USAE-mail: jmiller@math.wisc.eduE-mail: msoskova@math.wisc.edu
MARIYA I. SOSKOVA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON 480 LINCOLN DR., MADISON, WI53706, USAE-mail: jmiller@math.wisc.eduE-mail: msoskova@math.wisc.edu
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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
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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