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|Title:||Randomness, relativization and turing degrees|
|Source:||Nies, A.,Stephan, F.,Terwijn, S.A. (2005). Randomness, relativization and turing degrees. Journal of Symbolic Logic 70 (2) : 515-535. ScholarBank@NUS Repository. https://doi.org/10.2178/jsl/1120224726|
|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 Øn-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 re. 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. © 2005, Association for Symbolic Logic.|
|Source Title:||Journal of Symbolic Logic|
|Appears in Collections:||Staff Publications|
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