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 report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness notions (...) going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

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)

Kuera, A., On relative randomness, Annals of Pure and Applied Logic 63 61–67. It is the aim of the paper to answer a question raised by M. van Lambalgen and D. Zambella whether there can be a nonrecursive set A having the property that there is a set B such that B is 1-random relative to A and simultaneously A is recursive in B. We give apositive answer to this question as well as further information about relative randomness.