Computability and Randomness [Paperback]

A monograph on the interface of computational complexity and randomness of sets of natural numbers.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.Preface1. The complexity of sets2. The descriptive complexity of strings3. Martin-L?f randomness and its variants4. Diagonally noncomputable functions5. Lowness Properties and REPLACELESSTHANiREPLACEGREATERTHANKREPLACELESSTHAN/iREPLACEGREATERTHAN-triviality6. Some advanced computability theory7. Randomness and betting strategies8. Classes of computational complexity9. Higher computability and randomnessSolutions to exercisesReferencesIndexPhD, Mathematics, Univ. of Heidelberg, Germany, 1992Univ of Wisconsin, Madison 1994Cornell University 1995Univ of Chicago 1995-2001Habilitation, Univ. of Heidelberg, 1998Univ of Auckland 2002-present.60 journal and conference publications.Invited Speaker, International Congress of Mathematicians, Hyderabad 2010A monograph on the interface of computational complexity and ralóª