Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 74, Issue 2 (2009), 665-678.
Lowness for Kurtz randomness
We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu , 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.
J. Symbolic Logic Volume 74, Issue 2 (2009), 665-678.
First available in Project Euclid: 2 June 2009
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Greenberg, Noam; Miller, Joseph S. Lowness for Kurtz randomness. J. Symbolic Logic 74 (2009), no. 2, 665--678. doi:10.2178/jsl/1243948333. http://projecteuclid.org/euclid.jsl/1243948333.