Characterizing the strongly jump-traceable sets via randomness

Abstract

We show that if a set A is computable from every superlow 1random set, then A is strongly jump-traceable. Together with a result from [16], this theorem shows that the computably enumerable (c.e.) strongly jumptraceable sets are exactly the c.e. sets computable from every superlow 1random set.We also prove the analogous result for superhighness: a c.e. set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set.Finally, we show that for each cost function c with the limit condition there is a 1-random ∆ 0 2 set Y such that every c.e. set A T Y obeys c. To do so, we connect cost function strength and the strength of randomness notions. Together with a theorem from [16], this result gives a full correspondence between obedience of cost functions and being computable from ∆ 0 2 1-random sets.

Demuth randomness and computational complexity

“…As in [10], we define a Turing functional to be a partial computable function Γ : 2 <ω × ω → ω, such that for all x < ω, the domain of Γ (−, x) is an antichain of 2 <ω (in other words, that domain is prefix-free). The idea is that the functional is the collection of minimal oracle computations of an oracle Turing machine.…”

“…In [10,Section 7] notions are studied that interpolate between limit randomness and Demuth randomness. The idea is to restrict the number of changes of the m-th component by counting down along a computable well-order such as ω 2 .…”

“…There is multiple evidence [10] that the strongly jump-traceable c.e. sets, introduced in [7], form a very small subclass of the c.e.…”

Upper bounds on ideals in the computably enumerable Turing degrees

“…sets below all superlow ML-random sets, and also with the degrees of c.e. sets below all superhigh ML-random sets [18,7]. Some Σ 0 3 ideals lie strictly between S and K. For instance, let Y be a superlow Martin-Loef random set, and let B be a c.e.…”

Strong jump-traceability II: K-triviality

“…Greenberg, Hirschfeldt, and Nies [10] proved that a c.e. set is strongly jump-traceable if and only if it is computable from every superlow random sets, if and only if it is computable from every superhigh random set.…”