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. Covering the basics as well as recent research results, this book provides 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.

André Nies received his PhD in Mathematics form the University of Heidelberg, Germany, in 1992. From 1994 to 1995 he was a Visiting Professor at the University of Wisconsin at Madison and Cornell. In 1995 he took the post of Assistant Professor at the University of Chicago. Since 2002 he has been Senior Lecturer in the Department of Computer Science, University of Aukland.