Computability and Randomness Oxford Logic Guides Series, Vol. 51

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.

Preface. 1. The complexity of sets. 2. The descriptive complexity of strings. 3. Martin-Lof randomness and its variants. 4. Diagonally noncomputable functions. 5. Lowness Properties and alt,iagt,Kalt,/iagt,-triviality. 6. Some advanced computability theory. 7. Randomness and betting strategies. 8. Classes of computational complexity. 9. Higher computability and randomness. Solutions to exercises. References. Index.

PhD, Mathematics, Univ. of Heidelberg, Germany, 1992 Univ of Wisconsin, Madison 1994 Cornell University 1995 Univ of Chicago 1995-2001 Habilitation, Univ. of Heidelberg, 1998 Univ of Auckland 2002-present. 60 journal and conference publications. Invited Speaker, International Congress of Mathematicians, Hyderabad 2010