Algorithmic Randomness and Complexity

• Main Authors: Downey, Rodney G. (Author), Hirschfeldt, Denis R. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: New York, NY : Springer New York, 2010.
• Edition: 1.
• Series: Theory and Applications of Computability, In cooperation with the association Computability in Europe,
• Subjects: Mathematics.; Information theory.; Computer science.; Computer software.; Algorithms.; Algorithm Analysis and Problem Complexity.; Theory of Computation.; Computation by Abstract Devices.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-0-387-68441-3

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 obje...

Table of Contents:

• Preface
• Acknowledgments
• Introduction
• I. Background
• Preliminaries
• Computability Theory
• Kolmogorov Complexity of Finite Strings
• Relating Plain and Prefix-Free Complexity
• Effective Reals
• II. Randomness of Sets
• Martin-Lf̲ Randomness
• Other Notions of Effective Randomness
• Algorithmic Randomness and Turing Reducibility
• III. Relative Randomness
• Measures of Relative Randomness
• The Quantity of K- and Other Degrees
• Randomness-Theoretic Weakness
• Lowness for Other Randomness Notions
• Effective Hausdorff Dimension
• IV. Further Topics
• Omega as an Operator
• Complexity of C.E. Sets
• References
• Index.