BibTeX
@MISC{Nies_computabilityand,
author = {André Nies},
title = {Computability and Randomness},
year = {}
}
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Abstract
2008 To my parents Christel and Otfrid with gratitudePREFACE The complexity and randomness aspects of sets 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. This book is about the two aspects of sets of natural numbers and about their interplay. Sets of natural numbers are identified with infinite sequences of zeros and ones, and simply called sets. Chapters 1 and 6 are mostly about the complexity aspect. We introduce lowness and highness properties of sets. Chapters 2, 3, and 7 are mostly about the randomness aspect. Firstly we study randomness of finite objects. Then we proceed to sets. We establish a hierarchy of mathematical randomness notions. Each notion matches our intuition of
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