Mathematical Logic Quarterly, 53, 2007, pp. 483–492. We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω.Let1 and 0 be the top and bottom elements of Pw. We show that inf(b1, 1) andinf(b2, 1) andinf(b3, 1) belongtoPw and that 0 < inf(b1, 1) < inf(b2, 1) < inf(b3, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, weshow that inf(b1, 1) andinf(b3, 1) but not inf(b2, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more self-contained, we exposit the proofs of some recent

Research supported by NSF grants DMS-0600823 and DMS-0652637.

Recall that D T is the upper semilattice of all

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.

Recall that E T is the upper semilattice of recursively enumerable Turing degrees. Two basic, classical, unresolved issues concerning E T are: Issue 1: To find a specific, natural, r.e. Turing degree a ∈ E T which is> 0 and < 0 ′. Issue 2: To find a “smallness property ” of an infinite co-r.e. set A ⊆ ω which insures that deg T(A) = a ∈ E T is> 0 and < 0 ′. These unresolved issues go back to Post’s 1944 paper, Recursively enumerable sets of positive integers and their decision problems. Mass Problems to the Rescue! We address Issues 1 and 2 by passing from decision problems to mass problems. 2 Outline of this talk: We embed E T into a slightly larger structure, Pw, which is much better behaved. In the Pw context, we obtain satisfactory, positive answers to Issues 1 and 2. What is this wonderful structure Pw? Briefly, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. In order to explain Pw, we must first explain: • mass problems, • weak degrees, and • nonempty Π 0 1 subsets of 2ω. 3 Mass problems (informal discussion): A “decision problem ” is the problem of deciding whether a given n ∈ ω belongs to a fixed set A ⊆ ω or not. To compare decision problems, we use Turing reducibility. A ≤ T B means that A can be computed using an oracle for B. A “mass problem ” is a problem with a not necessarily unique solution. (By contrast, a “decision problem ” has only one solution.) The “mass problem ” associated with a set P ⊆ ω ω is the “problem ” of computing an element of P. The “solutions ” of P are the elements of P. One mass problem is said to be “reducible” to another if, given any solution of the second problem, we can use it as an oracle to compute a solution of the first problem. 4 Rigorous definition: Let P and Q be subsets of ω ω. We view P and Q as mass problems. We say that P is weakly reducible to Q if (∀Y ∈ Q) (∃X ∈ P) (X ≤ T Y). This is abbreviated P ≤w Q.