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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
Article

Abstract

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.

Keywords

Class Versus Computable Function Partial Relativization Minimal Pair Random Real 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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