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|Title:||Relativizations of randomness and genericity notions||Authors:||Franklin, J.N.Y.
|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|
|Appears in Collections:||Staff Publications|
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