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Theory and Applications of Computability

Algorithmic Randomness and Complexity

Authors: Downey, Rodney G., Hirschfeldt, Denis R.

  • Essential resource for all researchers in theoretical computer science, logic, computability theory, and complexity
  • Written by experts
  • First comprehensive treatment on the subject
  • An essential resource for all researchers and graduate students in theoretical computer science, logic, computability theory and complexity
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eBook $69.99
price for USA (gross)
  • ISBN 978-0-387-68441-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA
  • ISBN 978-0-387-95567-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this Textbook

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. Much of this theory can be seen as exploring the relationships between three fundamental concepts: relative computability, as measured by notions such as Turing reducibility; information content, as measured by notions such as Kolmogorov complexity; and randomness of individual objects, as first successfully defined by Martin-Löf. Although algorithmic randomness has been studied for several decades, a dramatic upsurge of interest in the area, starting in the late 1990s, has led to significant advances. This is the first comprehensive treatment of this important field, designed to be both a reference tool for experts and a guide for newcomers. It surveys a broad section of work in the area, and presents most of its major results and techniques in depth. Its organization is designed to guide the reader through this large body of work, providing context for its many concepts and theorems, discussing their significance, and highlighting their interactions. It includes a discussion of effective dimension, which allows us to assign concepts like Hausdorff dimension to individual reals, and a focused but detailed introduction to computability theory. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

Reviews

From the reviews:

“Develops the prerequisites to algorithmic randomness: computability theory and Kolmogorov complexity. … Studying these … one should be able to proceed in the area with confidence. A draft of the book under review has been circulating for years and the reviewer found it to be the best source when attempting to conduct research in the area … . It is advantageous for the future of the area of algorithmic randomness that these two books were published at the cusp of a period of great activity.” (Bjørn Kjos-Hanssen, Mathematical Reviews, Issue 2012 g)

“A thorough and systematic study of algorithmic randomness, this long-awaited work is an irreplaceable source of well-presented classic and new results for advanced undergraduate and graduate students, as well as researchers in the field and related areas. The book joins a select number of books in this category.” (Hector Zenil, ACM Computing Reviews, October, 2011)


Table of contents (16 chapters)

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-0-387-68441-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA
  • ISBN 978-0-387-95567-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Algorithmic Randomness and Complexity
Authors
Series Title
Theory and Applications of Computability
Copyright
2010
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-0-387-68441-3
DOI
10.1007/978-0-387-68441-3
Hardcover ISBN
978-0-387-95567-4
Series ISSN
2190-619X
Edition Number
1
Number of Pages
XXVIII, 855
Topics