Computability and Randomness

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OUP Oxford, Mar 29, 2012 - Mathematics - 456 pages
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.
 

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Contents

Indices and approximations 1 2 Relative computational complexity of sets
A Turing machinereads andwrites symbols from a finite alphabetwhich includes thesymbols 0 and1onthese tapes The input tapes are readonly whilet...
0
k1
ψx
if P halts on inputs x
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About the author (2012)

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

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