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June 2005 Randomness, relativization and Turing degrees
André Nies, Frank Stephan, Sebastiaan A. Terwijn
J. Symbolic Logic 70(2): 515-535 (June 2005). DOI: 10.2178/jsl/1120224726


We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅(n-1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ |x|-c. The ‘only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees.


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André Nies. Frank Stephan. Sebastiaan A. Terwijn. "Randomness, relativization and Turing degrees." J. Symbolic Logic 70 (2) 515 - 535, June 2005.


Published: June 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1090.03013
MathSciNet: MR2140044
Digital Object Identifier: 10.2178/jsl/1120224726

Primary: 03D15 , 03D28 , 03D80 , 28E15 , 68Q30

Rights: Copyright © 2005 Association for Symbolic Logic


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Vol.70 • No. 2 • June 2005
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