Schnorr trivial sets and truth-table reducibility

Abstract

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey. Griffiths and LaForte.

Schnorr Triviality and Its Equivalent Notions

“…Kjos-Hanssen et al [14] strengthened this equivalence to the equivalence between two reducibility of ≤ LR and ≤ LK . By the results in Franklin and Stephan [10] and Miyabe [18], lowness for uniformly computable measure machines and lowness for uniform Schnorr randomness are euiqvalent. Then we strengthned this equivalence to the equivalence between reducibilities.…”

“…We refer to [24,3,4,25] for computability from 2 ω to τ , from 2 ω to R and so on. Miyabe and Rute [19] showed that uniform Schnorr randomness is equivalent to tt-Schnorr randomness studied in [10,18].…”

“…However this notion is not equivalent to Schnorr triviality. Actually Franklin and Stephan [10] showed that there exists a Schnorr trivial set that is not truth-table reducible to any Schnorr random set. It should be noted that the following notions are equivalent:…”

“…Franklin and Stephan [10] also showed the equivalence to a kind of being a base for Schnorr randomness that uses new reducibility ≤ snr . We would like to have a notion of being a base for Schnorr randomness that uses truth-table reducibility.…”

“…Franklin and Stephan [10] and Miyabe [18] claimed that truth-table reducibility is more suitable than Turing reducibility when studying a notion of relativized Schnorr randomness and proposed truth-table Schnorr randomness (tt-Schnorr randomness) and showed the equivalence of the following:…”