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Randomness, relativization and Turing degrees

Published online by Cambridge University Press:  12 March 2014

André Nies,
Frank Stephan and
Sebastiaan A. Terwijn
André Nies
Affiliation:
Department of Computer Science, University of Auckland, 38 Princes ST, New Zealand, E-mail: anie010@cs.auckland.ac.nz
Frank Stephan
Affiliation:
Departments of Computer Science and Mathematics, National University of Singapore, 3 Science Drive 2, Singapore 117543, Republic of Singapore, E-mail: fstephan@comp.nus.edu.sg
Sebastiaan A. Terwijn
Affiliation:
Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Wiedner Hauptstrasse 8–10/E104, A-1040 Vienna, Austria, E-mail: terwijn@logic.at
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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 l-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.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005, pp. 515 - 535
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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