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EXACT PAIRS FOR THE IDEAL OF THE K-TRIVIAL SEQUENCES IN THE TURING DEGREES

Published online by Cambridge University Press:  18 August 2014

GEORGE BARMPALIAS  and
ROD G. DOWNEY
GEORGE BARMPALIAS
Affiliation:
STATE KEY LAB OF COMPUTER SCIENCE, INSTITUTE OF SOFTWARE, CHINESE ACADEMY OF SCIENCES, BEIJING 100190, CHINA SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH, VICTORIA UNIVERSITY, WELLINGTON, NEW ZEALAND, E-mail: [email protected], URL: http://www.barmpalias.net
ROD G. DOWNEY
Affiliation:
SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH, VICTORIA UNIVERSITY, P.O. BOX 600, WELLINGTON, NEW ZEALAND, E-mail: [email protected], URL: http://homepages.ecs.vuw.ac.nz/~downey
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Abstract

The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].

We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a K-trivial degree d such that for all degrees a, b which are not K-trivial and a > d, b > d there exists a degree v which is not K-trivial and a > v, b > v. This work sheds light to the question of the definability of the K-trivial degrees in the c.e. degrees.

Keywords

03D2503D3268Q30Computably enumerableTuring degreesKolmogorov complexityK-trivial setsexact pairs
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 3 , September 2014 , pp. 676 - 692
Copyright
Copyright © The Association for Symbolic Logic 2014 

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