It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, (...) we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts (...) and methods originating from randomness enrich computability theory.The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability. Both the basics and recent research results are desribed, providing a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory. (shrink)

We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of (...) Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. (shrink)

We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA, and the number of values enumerated is at most h. A′ (...) is well-approximable if can be effectively approximated with less than h changes at input x, for each order function h. We prove that there is a strongly jump-traceable set which is not computable, and that if A′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and strong jump-traceability in terms of Kolmogorov complexity. We also investigate other properties of these lowness properties. (shrink)

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the sets of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, and b3 be the degrees of unsolvability of the mass problems associated with AED, MLR × AED, and MLR ∩ AED, respectively. Let [MATHEMATICAL SCRIPT CAPITAL P]w be the lattice of degrees of unsolvability of mass problems associated with nonempty Π01 subsets of 2ω. Let 1 (...) and 0 be the top and bottom elements of [MATHEMATICAL SCRIPT CAPITAL P]w. We show that inf, inf, and inf belong to [MATHEMATICAL SCRIPT CAPITAL P]w and 0 < inf < inf < inf < 1. Under the natural embedding of the recursively enumerable Turing degrees into [MATHEMATICAL SCRIPT CAPITAL P]w, we show that inf and inf but not inf 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 theorems due to Hirschfeldt, Miller, Nies, and Stephan. (shrink)

We prove that there is a ${\mathrm{\Delta }}_{2}^{0}$ , 1-random set Y such that every computably enumerable set which is computable from Y is strongly jump-traceable. We also show that for every order function h there is an ω-c.e. random set Y such that every computably enumerable set which is computable from Y is h-jump-traceable. This establishes a correspondence between rates of jump-traceability and computability from ω-c.e. random sets.