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The book presents in a mathematical clear way the fundamentals of algorithmic information theory and a few selected applications. This 2nd edition presents new and important results obtained in recent years: the characterization of computable enumerable random reals, the construction of an Omega Number for which ZFC cannot determine any digits, and the first successful attempt to compute the exact values of 64 bits of a specific Omega Number. Finally, the book contains a discussion of some interesting philosophical questions related to randomness and mathematical knowledge.
"Professor Calude has produced a first-rate exposition of up-to-date work in information and randomness." D.S. Bridges, Canterbury University, co-author, with Errett Bishop, of Constructive Analysis;
"The second edition of this classic work is highly recommended to anyone interested in algorithmic information and randomness." G.J. Chaitin, IBM Research Division, New York, author of Conversations with a Mathematician;
"This book is a must for a comprehensive introduction to algorithmic information theory and for anyone interested in its applications in the natural sciences." K. Svozil, Technical University of Vienna, author of Randomness & Undecidability in Physics
Editors Foreword by A. Salomaa, Foreword by G.J. Chaitin; 1. Mathematical Background; 2. Noiseless Coding; 3. Program Size; 4. Computably Enumerable Instantaneous Codes; 5. Random Strings; 6. Random Sequences; 7. Computably Enumerable Random Reals; 8. Randomness and Incompleteness; 9. Applications; 10. Open Problems; Bibliography; Index
Anonymous
Posted October 9, 2004
I stumbled over this (lovely) book a little by accident. As I kept reading, my enthusiasm for the book gradually increased. While the book is addressed perhaps more to students in computation and in CS, it is very attractive also as a text to be used in mainstream mathematics, and in probability theory. It begins with a new look at the classical Kolmogorov construction of measures on infinite product spaces, and asks for explicit ways of labeling them with a class of certain concrete numerical functions. Then it moves onto noiseless coding theory (from communications science), but it stays rooted firmly in classical ideas from Shannon-Kolmogorov communication and information theory. It is indeed pleasing to see that God still plays dice, not only in quantum theory, but also in such classical areas of math as in number theory. From the foreword: ¿¿putting Shannon¿s information theory and Turing¿s computability theory into a cocktail shaker, and shaking vigorously¿¿ The book is a second edition 2002, with a number of attractive additions to the first edition from 1994. It will likely work equally well in a course, as for self-study. The main portion in the book focuses on classical and modern topics in computability, and its connections to randomness; covering concrete halting problems, chaos, cellular automata, algorithms, and their complexity. Palle Jorgensen, October 2004.
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Overview
The book presents in a mathematical clear way the fundamentals of algorithmic information theory and a few selected applications. This 2nd edition presents new and important results obtained in recent years: the characterization of computable enumerable random reals, the construction of an Omega Number for which ZFC cannot determine any digits, and the first successful attempt to compute the exact values of 64 bits of a specific Omega Number. Finally, the book contains a discussion of some interesting ...