Randomness, relativization and Turing degrees

  title={Randomness, relativization and Turing degrees},
  author={Andr{\'e} Nies and Frank Stephan and Sebastiaan A. Terwijn},
  journal={Journal of Symbolic Logic},
  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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TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets