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 allows 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). 1.

Abstract. We introduce some new variations of the notions of being Martin-Löf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability. 1

Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an X-recursive sequence of X-recursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLR-low if every function which is boundedly limit recursive in X is boundedly limit recursive in 0. This is a lowness property in the sense of Nies. These notions were introduced by Joshua A. Cole and the speaker in a recently submitted paper on mass problems and hyperarithmeticity. The purpose of this talk is to compare BLR-lowness to similar properties which have been considered in the recursion-theoretic literature. Among the properties discussed are: K-triviality, superlowness, jump-traceability, weak jump-traceability, total ω-recursive enumerability, array recursiveness, array jump-recursiveness, and strong jump-traceability. 2 Definition. If X is a Turing oracle, let BLR(X) betheset of number-theoretic functions f: ω → ω which are boundedly limit recursive in X. This means that there exist an X-recursive approximating function ˜ f(n, s) and a recursive bounding function ̂ f(n) such that and for all n. f(n) = lims ˜ f(n, s) |{s | ˜ f(n, s) ̸ = ˜ f(n, s +1)} | < ̂ f(n) In particular, BLR(0) = {f | f ≤ wtt 0 ′}. The BLR operator was introduced in Mass problems and hyperarithmeticity, by Joshua A. Cole and Stephen G. Simpson, 20 pages, submitted 2006 to JML. 3 Cole and Simpson used the BLR operator to construct a natural embedding of the hyperarithmetical hierarchy into P w. Namely, we proved that the Muchnik degrees inf(h ∗ α, 1) forα<ωCK 1 are distinct ∈Pw.