Journal of Symbolic Logic 70 (2):515-535 (2005)

Abstract
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 ∅. 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|-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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DOI http://projecteuclid.org/euclid.jsl/1120224726
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References found in this work BETA

On Relative Randomness.Antonín Kučera - 1993 - Annals of Pure and Applied Logic 63 (1):61-67.

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Citations of this work BETA

Calibrating Randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
Randomness and Computability: Open Questions.Joseph S. Miller & André Nies - 2006 - Bulletin of Symbolic Logic 12 (3):390-410.
Demuth Randomness and Computational Complexity.Antonín Kučera & André Nies - 2011 - Annals of Pure and Applied Logic 162 (7):504-513.
The K -Degrees, Low for K Degrees,and Weakly Low for K Sets.Joseph S. Miller - 2009 - Notre Dame Journal of Formal Logic 50 (4):381-391.

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