Relativizations of randomness and genericity notions

Abstract

A set A is a basis for Schnorr randomness if and only if it is Turing reducible to a set R which is Schnorr random relative to A. One can define a basis for weak 1-genericity similarly. It is shown that A is a basis for Schnorr randomness if and only if A is a basis for weak 1-genericity if and only if the halting problem K is not Turing reducible to A. Furthermore, call a set A high for Schnorr randomness versus Martin-Löf randomness if and only if every set which is Schnorr random relative to A is also Martin-Löf random unrelativized. It is shown that A is high for Schnorr randomness versus Martin-Löf randomness if and only if K is Turing reducible to A. Other results concerning highness for other pairs of randomness notions are also included.

Schnorr Triviality and Its Equivalent Notions

“…It should be noted that Franklin, Stephan and Yu [11] studied a base for Schnorr randomness, which is a notion different from the one considered in our study.…”

Defining a randomness notion via another

“…Proposition For and , the second relation in holds. Moreover, letting PA denote the set of all (codes of) completions of Peano Arithmetic, we have Proof Franklin, Stephan and Yu [, Proposition 4.1] proved that High(CR, MLR) includes PA. Downey, Hirschfelt, Miller and Nies [, Proposition 7.4] showed that any Martin‐Löf random real is Martin‐Löf relative to some element in PA. Since , it is known that any Martin‐Löf random real is computably random relative to some element in PA.…”

“…Proposition The second relation in is false when is (KR, SR), (KR, CR), (KR, MLR), (KR, W2R), , (SR, CR), (SR, MLR) or (SR, W2R). In fact, is empty if S is SR, CR, MLR, W2R or , and holds. Proof Franklin, Stephan and Yu [, Theorem 2.2] proved that the equalities hold, and they [, Remark 1.4] showed High(KR, SR) is empty. Our theorem follows from these facts.…”

“…The lowness and highness notions for pairs of related classes have been extensively studied. Cf., e.g., [1,2,6,8,17] and others. By the definitions of Low(R, S) and High(S, R), we have that…”

“…The lowness and highness notions has been given substantial attention in the literature. Cf., e.g, . For two randomness notions R and S , let and denote the sets of all reals X such that and , respectively.…”