Relativizations of randomness and genericity notions

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.