### Randomness, relativization and Turing degrees

André Nies, Frank Stephan, and Sebastiaan A. Terwijn
Source: J. Symbolic Logic Volume 70, Issue 2 (2005), 515-535.

#### 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 ∅(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.

First Page:
Primary Subjects: 68Q30, 03D15, 03D28, 03D80, 28E15
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1120224726
Digital Object Identifier: doi:10.2178/jsl/1120224726
Mathematical Reviews number (MathSciNet): MR2140044
Zentralblatt MATH identifier: 02247425

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