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Lowness for Kurtz randomness

Published online by Cambridge University Press:  12 March 2014

Noam Greenberg  and
Joseph S. Miller
Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand, E-mail: [email protected]
Joseph S. Miller
Affiliation:
Department of Mathematics, University of Wisconsin, Madison. Wi 53706-1388., USA, E-mail: [email protected]
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Abstract

We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity.

We also consider Low(ℳ, Kurtz), the class of degrees a such that every element of ℳ is a-Kurtz random. These are characterised when ℳ is the class of Martin-Löf random, computably random, or Schnorr random reals. We show that Low(ML, Kurtz) coincides with the non-DNR degrees, while both Low(CR, Kurtz) and Low(Schnorr, Kurtz) are exactly the non-high, non-DNR degrees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 665 - 678
Copyright
Copyright © Association for Symbolic Logic 2009

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