Lowness for Demuth Randomness

Abstract

Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

Characterizing Lowness for Demuth Randomness

“…The next proposition, which completes the proof of the theorem, is an analog of a result of Downey's and Ng's [15], which we mentioned and used above, that lowness for Demuth randomness implies being computably dominated. This proposition uses the power of the full relativization of weak Demuth randomness.…”

“…We hope that these techniques will be useful for eventually giving a unifying explanation for the coincidence of lowness for randomness and lowness for tests. We use the characterization to show the existence of non-computable oracles that are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). …”

“…The first step toward Theorem 1.4 was taken by Downey and Ng [15], who showed that every oracle which is low for Demuth randomness is computably dominated. As we shall shortly see, we make use of their result in the proof of Theorem 1.4.…”

“…We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability.…”

On the Interplay Between Effective Notions of Randomness and Genericity

“…It is already known that a Demuth random cannot compute a pb-generic: indeed, Downey et al [DJS96] proved that X ∈ 2 ω computes a pb-generic if and only if it has array noncomputable degree, meaning that X can compute a total function f : N → N which is dominated by no ω-c.a. function g (where g is said to dominate f if f (n) ≤ g(n) for almost all n), and Downey and Ng [DN09] showed that all Demuth randoms have array computable degree. The next theorem improves this.…”