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Journal of the London Mathematical Society Advance Access originally published online on July 11, 2007
Journal of the London Mathematical Society 2007 75(3):610-622; doi:10.1112/jlms/jdm041
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© 2007 London Mathematical Society

Using random sets as oracles

Denis R. Hirschfeldt1,, André Nies2 and Frank Stephan3

1 Department of Mathematics
University of Chicago
5734 S. University Ave.
Chicago, IL 60637
USA
2 Department of Computer Science
University of Auckland
Private Bag 92019 Auckland
New Zealand
andre{at}cs.auckland.ac.nz
3 Department of Mathematics and School of Computing
National University of Singapore
2 Science Drive 2
Singapore, 117543
fstephan{at}comp.nus.edu.sg

Let R be a notion of algorithmic randomness for individual subsets of N. A set B is a base for R randomness if there is a Z ≥T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets, and discuss several consequences of this result. On the other hand, the bases for computable randomness include every {Delta}20 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, an n-c.e. set is a base for computable randomness if and only if it is Turing incomplete.

drh{at}math.uchicago.edu

2000 Mathematics Subject Classification 68Q30, 03D80.

The first author was partially supported by the National Science Foundation of the USA, grants DMS-02-00465 and DMS-05-00590; and by the Marsden Fund of New Zealand, grant 03-UOA-130. The second author was partially supported by the Marsden Fund of New Zealand, grant 03-UOA-130. The third author was partially supported by NUS grant R252-000-212-112 and, for a visit to Auckland, by the Marsden Fund of New Zealand, grant 03-UOA-130.

Received February 24, 2006; revised October 12, 2006; published online July 11, 2007.


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