Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Abstract

Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every left-c.e. real of that degree is reducible in a computable Lipschitz way to a random left-c.e. real (an Ω-number).

Characterizing the strongly jump-traceable sets via randomness

“…Assuming that e > 0, this run of R e,η was called by some S e,η x where again h(x) |γ|. By Claim 6.15 (2), P = P (v) , where v r(e − 1, y) for some y < x. Thus, effectively in γ, we get a bound on q(v) for the V -index v of P, and so with the aid of Lemma 6.5 (and again Claim 6.15(1)), a bound on the number of times a change as in (ii) may happen.…”

“…So M (0, x) = B(x, q(0)) is an upper bound as desired. Now assume that e > 0 and that (1) and (2) have been computed for all pairs (e ′ , x ′ ) that lexicographically precede the pair (e, x). By Claim 6.14, we may let M (e, x) be the product of (i) a bound on the number of runs of R e,η that are called by some S e−1,η y with parameter h(y) < h(x), and (ii) a bound on the number of times a single run R e,η can call S e,γ…”

TOTALLY Ω-Computably ENUMERABLE DEGREES AND BOUNDING CRITICAL TRIPLES

“…if it is the limit of a computable nondecreasing sequence of rationals, and that a c.e., prefix free set of strings A presents α if α = σ∈A 2 −|σ| . Theorem 1.14 (Downey and Greenberg [7] In yet another sequel to the present paper, Barmpalias, Downey and Greenberg [3] demonstrate that the class of totally ω-c.e. degrees captures a number of constructions.…”

A Hierarchy of Computably Enumerable Degrees

“…Let us recall some of the results. [13]). (7) it contains a set which is not reducible to the halting problem with tiny use (Franklin, Greenberg, Stephan, and Wu [46]).…”