Randomness, relativization and Turing degrees

@article{Nies2005RandomnessRA,
  title={Randomness, relativization and Turing degrees},
  author={Andr{\'e} Nies and Frank Stephan and Sebastiaan A. 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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