Lowness and Triviality for Other Randomness Notions

Abstract

We now turn to lowness notions for other notions of randomness. We begin with Schnorr randomness. Since there is no universal Schnorr test, it is not clear that the notions of a set A being low for Schnorr randomness (i.e.,every Schnorr random set is Schnorr random relative to A) and being low for Schnorr tests (i.e., every Schnorr test relative to A can be covered by an unrelativized Schnorr test1) should be the same. In fact, this was an open question in Ambos-Spies and Kucera [9]. As we will see, it was solved by Kjos-Hanssen, Nies, and Stephan [207].