Please use this identifier to cite or link to this item: https://doi.org/10.1112/blms/bdr007
Title: Relativizations of randomness and genericity notions
Authors: Franklin, J.N.Y.
Stephan, F. 
Yu, L.
Issue Date: Aug-2011
Citation: Franklin, J.N.Y., Stephan, F., Yu, L. (2011-08). Relativizations of randomness and genericity notions. Bulletin of the London Mathematical Society 43 (4) : 721-733. ScholarBank@NUS Repository. https://doi.org/10.1112/blms/bdr007
Abstract: A set A is a base for Schnorr randomness if it is Turing reducible to a set R that is Schnorr random relative to A, and the notion of a base for weak 1-genericity can be defined similarly. We show that A is a base for Schnorr randomness if and only if A is a base for weak 1-genericity if and only if the halting set K is not Turing reducible to A. Furthermore, we define a set A to be high for Schnorr randomness versus Martin-Löf randomness if and only if every set that is Schnorr random relative to A is also Martin-Löf random unrelativized, and we show that A is high for Schnorr randomness versus Martin-Löf randomness if and only if K is Turing reducible to A. Results concerning highness for other pairs of randomness notions are also presented. © 2011 London Mathematical Society.
Source Title: Bulletin of the London Mathematical Society
URI: http://scholarbank.nus.edu.sg/handle/10635/104046
ISSN: 00246093
DOI: 10.1112/blms/bdr007
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