# Randomness, relativization and Turing degrees

@article{Nies2005RandomnessRA, title={Randomness, relativization and Turing degrees}, author={Andr{\'e} Nies and Frank Stephan and Sebastiaan Terwijn}, journal={Journal of Symbolic Logic}, year={2005}, volume={70}, pages={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…

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