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Benign cost functions and lowness properties

Published online by Cambridge University Press:  12 March 2014

Noam Greenberg  and
André Nies
Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, Wellington, New Zealand, E-mail: greenberg@msor.vuw.ac.nz
André Nies
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand, E-mail: andre@cs.auckland.ac.nz
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Abstract

We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 289 - 312
Copyright
Copyright © Association for Symbolic Logic 2011

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