Journal of Symbolic Logic

Randomness, relativization and Turing degrees

André Nies, Frank Stephan, and Sebastiaan A. Terwijn

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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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J. Symbolic Logic Volume 70, Issue 2 (2005), 515-535.

First available in Project Euclid: 1 July 2005

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Primary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32] 03D15: Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17] 03D28: Other Turing degree structures 03D80: Applications of computability and recursion theory 28E15: Other connections with logic and set theory


Nies, André; Stephan, Frank; Terwijn, Sebastiaan A. Randomness, relativization and Turing degrees. J. Symbolic Logic 70 (2005), no. 2, 515--535. doi:10.2178/jsl/1120224726.

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