An Introduction to Kolmogorov Complexity and Its Applications

Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Preface to the Fourth Edition
How to Use This Book
Outlines of One-Semester Courses
Contents
List of Figures
List of Tables
1 Preliminaries
	1.1 A Brief Introduction
	1.2 Prerequisites and Notation
	1.3 Numbers and Combinatorics
	1.4 Binary Strings
	1.5 Asymptotic Notation
	1.6 Basics of Probability Theory
	1.7 Basics of Computability Theory
	1.8 The Roots of Kolmogorov Complexity
	1.9 Randomness
	1.10 Prediction and Probability
	1.11 Information Theory and Coding
	1.12 State x Symbol Complexity
	1.13 History and References
2 Algorithmic Complexity
	2.1 The Invariance Theorem
	2.2 Incompressibility
	2.3 C as an Integer Function
	2.4 Random Strings
	2.5 *Random Sequences
	2.6 Statistical Properties of Strings
	2.7 Algorithmic Properties of C
	2.8 Algorithmic Information Theory
	2.9 History and References
3 Algorithmic Prefix Complexity
	3.1 The Invariance Theorem
	3.2 Incompressibility
	3.3 K as an Integer Function
	3.4 Random Strings
	3.5 *Random Sequences
	3.6 Algorithmic Properties of K
	3.7 *Complexity of Complexity
	3.8 *Symmetry of Algorithmic Information
	3.9 *Sizes of the Constants
	3.10 History and References
4 Algorithmic Probability
	4.1 Semicomputable Functions
	4.2 Measure Theory
	4.3 Discrete Sample Space
	4.4 Universal Average-Case Complexity
	4.5 Continuous Sample Space
	4.6 Universal Average-Case Complexity, Continued
	4.7 History and References
5 Inductive Reasoning
	5.1 Introduction
	5.2 Solomonoff’s Theory of Prediction
	5.3 Simple Pac-Learning
	5.4 Hypothesis Identification by MDL
	5.5 Nonprobabilistic Statistics
	5.6 History and References
6 The Incompressibility Method
	6.1 Three Examples
	6.2 High-Probability Properties
	6.3 Combinatorics
	6.4 Kolmogorov Random Graphs
	6.5 Compact Routing
	6.6 Average-Case Analysis of Sorting
	6.7 Longest Common Subsequence
	6.8 Formal Language Theory
	6.9 Online CFL Recognition
	6.10 Turing Machine Time Complexity
	6.11 Communication Complexity
	6.12 Circuit Complexity
	6.13 Lov´asz Local Lemma
	6.14 History and References
7 Resource-Bounded Complexity
	7.1 Mathematical Theory
	7.2 Language Compression
	7.3 Computational Complexity
	7.4 Instance Complexity
	7.5 Kt and Universal Search
	7.6 Time-Limited Universal Distributions
	7.7 Logical Depth
	7.8 History and References
8 Physics, Information, and Computation
	8.1 Information Theory
	8.2 Reversible Computation
	8.3 Information Distance
	8.4 Normalization
	8.5 Information Diameter
	8.6 Thermodynamics
	8.7 Entropy Revisited
	8.8 Quantum Kolmogorov Complexity
	8.9 Compression in Nature
	8.10 History and References
References
Index