Journal of Symbolic Logic

Lowness for Kurtz randomness

Noam Greenberg and Joseph S. Miller

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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.

Article information

J. Symbolic Logic Volume 74, Issue 2 (2009), 665-678.

First available in Project Euclid: 2 June 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D80: Applications of computability and recursion theory
Secondary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]


Greenberg, Noam; Miller, Joseph S. Lowness for Kurtz randomness. J. Symbolic Logic 74 (2009), no. 2, 665--678. doi:10.2178/jsl/1243948333.

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