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Название: Information and Randomness: An Algorithmic Perspective
Автор: Calude C.S.
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.