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

Four classes of sets have been introduced independently by various researchers: low for K, low for ML-randomness, basis for ML-randomness and K-trivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is K-trivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for Martin-Löf randomness if each Martin-Löf random set is already Martin-Löf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for ML-randomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with K-triviality. All

We consider the hardness of enumerating k possible values for the Kolmogorov complexity function C(x) so that one of them is correct. We show several results including Any computable enumerator for C(x) must enumerate n) possibilities. If a k-enumerator (xed k) for C is reducible to an r.e. set A then A is Turing-equivalent to the halting problem. Every nonrecursive set is not weak-truth-table reducible to some 2-enumerator for C(x) or any other recursively-bounded function. The time-bounded enumeration question gives a new characterization of the class symP dened by Russell and Sundaram. Enumerating O(log n) values of the space-bounded Kolmogorov function remains hard for PSPACE. 1 Introduction The Kolmogorov complexity of a binary string x, C(x), is the size of the smallest program that outputs x. Kolmogorov complexity has its roots in the study of randomness: it is one way to measure randomness in a string. It has had a vast area of applications, including i...

Abstract. We show that every strongly jump-traceable set is K-trivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1.