Israel Journal of Mathematics

, Volume 191, Issue 2, pp 791–816 | Cite as

Randomness notions and partial relativization

  • George Barmpalias
  • Joseph S. Miller
  • André Nies


We study the computational complexity of an oracle set using a number of notions of randomness that lie between Martin-Löf randomness and 2-randomness in terms of strength. These notions are weak 2-randomness, weak randomness relative to ∅′, Demuth randomness and Schnorr randomness relative to ∅′. We characterize the oracles A such that ML[A] ⊆ C, where C is such a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A, using a new meta-concept called partial relativization. We study the reducibility associated with weak 2-randomness and relate it with LR-reducibility.


Class Versus Computable Function Partial Relativization Minimal Pair Random Real 
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Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  • George Barmpalias
    • 1
  • Joseph S. Miller
    • 2
  • André Nies
    • 3
  1. 1.School of Mathematics, Statistics, and Computer ScienceVictoria UniversityWellingtonNew Zealand
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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