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Lowness for Kurtz randomness

  • Noam Greenberg (a1) and Joseph S. Miller (a2)

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(, Kurtz), the class of degrees a such that every element of is a-Kurtz random. These are characterised when 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.

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[3] Rodney G. Downey , Evan J. Griffiths , and Stephanie Reid , On Kurtz randomness. Theoretical Computer Science, vol. 321 (2004), no. 2-3, pp. 249270.

[7] Bjørn Kjos-Hanssen , André Nies , and Frank Stephan , Lowness for the class of Schnorr random reals, SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647657 (electronic).

[9] André Nies , Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), no. 1. pp. 274305.

[10] André Nies , Computability and randomness, Oxford University Press, 2009, in preparation.

[14] Theodore A. Slaman and Robert Solovay , When oracles do not help, COLT '91: Proceedings of the fourth annual workshop on Computational Learning Theory ( Leslie G. Valiant and Manfred K. Warmuth , editors), Morgan Kaufmann Publishers Inc., San Francisco. CA, USA. 1991, pp. 379383.

[15] Clifford Spector , On degrees of recursive unsolvability, Annals of Mathematics, (2), vol. 64 (1956), pp. 581592.

[16] Frank Stephan and Liang Yu , Lowness for weakly 1-generic and Kurtz-random. Theory and applications of models of computation ( Jin yi Cai , S. Barry Cooper , and Angsheng Li , editors), Lecture Notes in Computer Science, vol. 3959, Springer, Berlin, 2006, pp. 756764.

[18] Liang Yu , Lowness for genericity, Archive for Mathematical Logic, vol. 45 (2006), no. 2, pp. 233238.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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