Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 10
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    PATEY, LUDOVIC 2016. OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS. The Bulletin of Symbolic Logic, Vol. 22, Issue. 02, p. 151.

    Kihara, Takayuki and Miyabe, Kenshi 2015. Unified characterizations of lowness properties via Kolmogorov complexity. Archive for Mathematical Logic, Vol. 54, Issue. 3-4, p. 329.

    Patey, Ludovic 2015. Ramsey-type graph coloring and diagonal non-computability. Archive for Mathematical Logic, Vol. 54, Issue. 7-8, p. 899.

    Higuchi, Kojiro and Peng, Ningning 2014. Defining a randomness notion via another. Mathematical Logic Quarterly, Vol. 60, Issue. 4-5, p. 280.

    Kihara, T. and Miyabe, K. 2014. Uniform Kurtz randomness. Journal of Logic and Computation, Vol. 24, Issue. 4, p. 863.

    Barmpalias, George Miller, Joseph S. and Nies, André 2012. Randomness notions and partial relativization. Israel Journal of Mathematics, Vol. 191, Issue. 2, p. 791.

    Bienvenu, Laurent and Miller, Joseph S. 2012. Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, Vol. 163, Issue. 5, p. 506.

    Diamondstone, David and Kjos-Hanssen, Bjørn 2012. Martin-Löf randomness and Galton–Watson processes. Annals of Pure and Applied Logic, Vol. 163, Issue. 5, p. 519.

    KJOS-HANSSEN, BJØRN 2011. A STRONG LAW OF COMPUTATIONALLY WEAK SUBSETS. Journal of Mathematical Logic, Vol. 11, Issue. 01, p. 1.

    Kjos-Hanssen, Bjørn Nies, André Stephan, Frank and Yu, Liang 2010. Higher Kurtz randomness. Annals of Pure and Applied Logic, Vol. 161, Issue. 10, p. 1280.


Lowness for Kurtz randomness

  • Noam Greenberg (a1) and Joseph S. Miller (a2)
  • DOI:
  • Published online: 01 March 2014

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *