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== Table of contents  ==
 
== Table of contents  ==
 
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . vii
 
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . vii
 +
 
Preface to the Second Edition . . . . . . . . . . . . . . . . . . xi
 
Preface to the Second Edition . . . . . . . . . . . . . . . . . . xi
 +
 
Preface to the Third Edition . . . . . . . . . . . . . . . . . . . xii
 
Preface to the Third Edition . . . . . . . . . . . . . . . . . . . xii
 +
 
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . xii
 
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . xii
 +
 
Outlines of One-Semester Courses . . . . . . . . . . . . . . . . xv
 
Outlines of One-Semester Courses . . . . . . . . . . . . . . . . xv
 +
 
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
 
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
 +
 
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
 
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
 +
 
1 Preliminaries
 
1 Preliminaries
 +
 
1
 
1
 +
 
1.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . 1
 
1.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . 1
 +
 
1.2 Prerequisites and Notation . . . . . . . . . . . . . . . . . 7
 
1.2 Prerequisites and Notation . . . . . . . . . . . . . . . . . 7
 +
 
1.3 Numbers and Combinatorics . . . . . . . . . . . . . . . . 8
 
1.3 Numbers and Combinatorics . . . . . . . . . . . . . . . . 8
 +
 
1.4 Binary Strings . . . . . . . . . . . . . . . . . . . . . . . . 12
 
1.4 Binary Strings . . . . . . . . . . . . . . . . . . . . . . . . 12
 +
 
1.5 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . 15
 
1.5 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . 15
 +
 
1.6 Basics of Probability Theory . . . . . . . . . . . . . . . . 18
 
1.6 Basics of Probability Theory . . . . . . . . . . . . . . . . 18
 +
 
1.7 Basics of Computability Theory . . . . . . . . . . . . . . 24
 
1.7 Basics of Computability Theory . . . . . . . . . . . . . . 24
 +
 
1.8 The Roots of Kolmogorov Complexity . . . . . . . . . . . 47
 
1.8 The Roots of Kolmogorov Complexity . . . . . . . . . . . 47
 +
 
1.9 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . 49
 
1.9 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . 49
 +
 
1.10 Prediction and Probability . . . . . . . . . . . . . . . . . 59
 
1.10 Prediction and Probability . . . . . . . . . . . . . . . . . 59
−
xviii
+
 
−
Contents
 
 
1.11 Information Theory and Coding . . . . . . . . . . . . . . 65
 
1.11 Information Theory and Coding . . . . . . . . . . . . . . 65
 +
 
1.12 State × Symbol Complexity . . . . . . . . . . . . . . . . 90
 
1.12 State × Symbol Complexity . . . . . . . . . . . . . . . . 90
 +
 
1.13 History and References . . . . . . . . . . . . . . . . . . .
 
1.13 History and References . . . . . . . . . . . . . . . . . . .
−
2 Algorithmic Complexity
+
 
−
92
+
2 Algorithmic Complexity 92
−
101
+
 
 
2.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 104
 
2.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 104
 +
 
2.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 116
 
2.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 116
 +
 
2.3 C as an Integer Function . . . . . . . . . . . . . . . . . . 126
 
2.3 C as an Integer Function . . . . . . . . . . . . . . . . . . 126
 +
 
2.4 Random Finite Sequences . . . . . . . . . . . . . . . . . . 133
 
2.4 Random Finite Sequences . . . . . . . . . . . . . . . . . . 133
 +
 
2.5 *Random Infinite Sequences . . . . . . . . . . . . . . . . 143
 
2.5 *Random Infinite Sequences . . . . . . . . . . . . . . . . 143
 +
 
2.6 Statistical Properties of Finite Sequences . . . . . . . . . 165
 
2.6 Statistical Properties of Finite Sequences . . . . . . . . . 165
 +
 
2.7 Algorithmic Properties of C
 
2.7 Algorithmic Properties of C
 +
 
2.8 Algorithmic Information Theory . . . . . . . . . . . . . . 186
 
2.8 Algorithmic Information Theory . . . . . . . . . . . . . . 186
 +
 
2.9 History and References . . . . . . . . . . . . . . . . . . . 192
 
2.9 History and References . . . . . . . . . . . . . . . . . . . 192
−
. . . . . . . . . . . . . . . . 174
+
 
−
3 Algorithmic Prefix Complexity
+
3 Algorithmic Prefix Complexity 197
−
197
+
 
 
3.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 200
 
3.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 200
 +
 
3.2 *Sizes of the Constants . . . . . . . . . . . . . . . . . . . 206
 
3.2 *Sizes of the Constants . . . . . . . . . . . . . . . . . . . 206
 +
 
3.3 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 211
 
3.3 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 211
 +
 
3.4 K as an Integer Function . . . . . . . . . . . . . . . . . . 216
 
3.4 K as an Integer Function . . . . . . . . . . . . . . . . . . 216
 +
 
3.5 Random Finite Sequences . . . . . . . . . . . . . . . . . . 218
 
3.5 Random Finite Sequences . . . . . . . . . . . . . . . . . . 218
 +
 
3.6 *Random Infinite Sequences . . . . . . . . . . . . . . . . 220
 
3.6 *Random Infinite Sequences . . . . . . . . . . . . . . . . 220
 +
 
3.7 Algorithmic Properties of K . . . . . . . . . . . . . . . . 239
 
3.7 Algorithmic Properties of K . . . . . . . . . . . . . . . . 239
 +
 
3.8 *Complexity of Complexity . . . . . . . . . . . . . . . . . 241
 
3.8 *Complexity of Complexity . . . . . . . . . . . . . . . . . 241
 +
 
3.9 *Symmetry of Algorithmic Information . . . . . . . . . . 244
 
3.9 *Symmetry of Algorithmic Information . . . . . . . . . . 244
 +
 
3.10 History and References . . . . . . . . . . . . . . . . . . . 255
 
3.10 History and References . . . . . . . . . . . . . . . . . . . 255
−
4 Algorithmic Probability
+
 
−
259
+
4 Algorithmic Probability 259
 +
 
 
4.1 Semicomputable Functions Revisited . . . . . . . . . . . . 260
 
4.1 Semicomputable Functions Revisited . . . . . . . . . . . . 260
 +
 
4.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . 262
 
4.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . 262
 +
 
4.3 Discrete Sample Space . . . . . . . . . . . . . . . . . . . . 265
 
4.3 Discrete Sample Space . . . . . . . . . . . . . . . . . . . . 265
 +
 
4.4 Universal Average-Case Complexity . . . . . . . . . . . . 290
 
4.4 Universal Average-Case Complexity . . . . . . . . . . . . 290
 +
 
4.5 Continuous Sample Space . . . . . . . . . . . . . . . . . . 294
 
4.5 Continuous Sample Space . . . . . . . . . . . . . . . . . . 294
 +
 
4.6 Universal Average-Case Complexity, Continued . . . . . . 330
 
4.6 Universal Average-Case Complexity, Continued . . . . . . 330
 +
 
4.7 History and References . . . . . . . . . . . . . . . . . . . 331
 
4.7 History and References . . . . . . . . . . . . . . . . . . . 331
−
Contents
+
 
−
5 Inductive Reasoning
+
5 Inductive Reasoning xix 339
−
xix
+
 
−
339
 
 
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 339
 
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 339
 +
 
5.2 Solomonoff’s Theory of Prediction . . . . . . . . . . . . . 348
 
5.2 Solomonoff’s Theory of Prediction . . . . . . . . . . . . . 348
 +
 
5.3 Simple Pac-Learning . . . . . . . . . . . . . . . . . . . . . 370
 
5.3 Simple Pac-Learning . . . . . . . . . . . . . . . . . . . . . 370
 +
 
5.4 Hypothesis Identification by MDL . . . . . . . . . . . . . 382
 
5.4 Hypothesis Identification by MDL . . . . . . . . . . . . . 382
 +
 
5.5 Nonprobabilistic Statistics . . . . . . . . . . . . . . . . . 401
 
5.5 Nonprobabilistic Statistics . . . . . . . . . . . . . . . . . 401
 +
 
5.6 History and References . . . . . . . . . . . . . . . . . . . 431
 
5.6 History and References . . . . . . . . . . . . . . . . . . . 431
−
6 The Incompressibility Method
+
 
−
441
+
6 The Incompressibility Method 441
 +
 
 
6.1 Three Examples . . . . . . . . . . . . . . . . . . . . . . . 442
 
6.1 Three Examples . . . . . . . . . . . . . . . . . . . . . . . 442
 +
 
6.2 High-Probability Properties . . . . . . . . . . . . . . . . . 448
 
6.2 High-Probability Properties . . . . . . . . . . . . . . . . . 448
 +
 
6.3 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 451
 
6.3 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 451
 +
 
6.4 Kolmogorov Random Graphs . . . . . . . . . . . . . . . . 461
 
6.4 Kolmogorov Random Graphs . . . . . . . . . . . . . . . . 461
 +
 
6.5 Compact Routing . . . . . . . . . . . . . . . . . . . . . . 469
 
6.5 Compact Routing . . . . . . . . . . . . . . . . . . . . . . 469
 +
 
6.6 Average-Case Analysis of Sorting . . . . . . . . . . . . . . 476
 
6.6 Average-Case Analysis of Sorting . . . . . . . . . . . . . . 476
 +
 
6.7 Longest Common Subsequence . . . . . . . . . . . . . . . 486
 
6.7 Longest Common Subsequence . . . . . . . . . . . . . . . 486
 +
 
6.8 Formal Language Theory . . . . . . . . . . . . . . . . . . 490
 
6.8 Formal Language Theory . . . . . . . . . . . . . . . . . . 490
 +
 
6.9 Online CFL Recognition . . . . . . . . . . . . . . . . . . 497
 
6.9 Online CFL Recognition . . . . . . . . . . . . . . . . . . 497
 +
 
6.10 Turing Machine Time Complexity . . . . . . . . . . . . . 502
 
6.10 Turing Machine Time Complexity . . . . . . . . . . . . . 502
−
6.11 Communication Complexity
+
 
−
. . . . . . . . . . . . . . . . 516
+
6.11 Communication Complexity. . . . . . . . . . . . . . . . 516
 +
 
 
6.12 Circuit Complexity . . . . . . . . . . . . . . . . . . . . . 521
 
6.12 Circuit Complexity . . . . . . . . . . . . . . . . . . . . . 521
 +
 
6.13 History and References . . . . . . . . . . . . . . . . . . . 524
 
6.13 History and References . . . . . . . . . . . . . . . . . . . 524
−
7 Resource-Bounded Complexity
+
 
−
531
+
7 Resource-Bounded Complexity 531
 +
 
 
7.1 Mathematical Theory . . . . . . . . . . . . . . . . . . . . 532
 
7.1 Mathematical Theory . . . . . . . . . . . . . . . . . . . . 532
 +
 
7.2 Language Compression . . . . . . . . . . . . . . . . . . . 550
 
7.2 Language Compression . . . . . . . . . . . . . . . . . . . 550
 +
 
7.3 Computational Complexity . . . . . . . . . . . . . . . . . 562
 
7.3 Computational Complexity . . . . . . . . . . . . . . . . . 562
 +
 
7.4 Instance Complexity . . . . . . . . . . . . . . . . . . . . . 571
 
7.4 Instance Complexity . . . . . . . . . . . . . . . . . . . . . 571
 +
 
7.5 Kt and Universal Search . . . . . . . . . . . . . . . . . . . 577
 
7.5 Kt and Universal Search . . . . . . . . . . . . . . . . . . . 577
 +
 
7.6 Time-Limited Universal Distributions . . . . . . . . . . . 582
 
7.6 Time-Limited Universal Distributions . . . . . . . . . . . 582
 +
 
7.7 Logical Depth . . . . . . . . . . . . . . . . . . . . . . . . 589
 
7.7 Logical Depth . . . . . . . . . . . . . . . . . . . . . . . . 589
 +
 
7.8 History and References . . . . . . . . . . . . . . . . . . . 596
 
7.8 History and References . . . . . . . . . . . . . . . . . . . 596
−
xx
+
 
−
Contents
+
8 Physics, Information, and Computation 601
−
8 Physics, Information, and Computation
+
 
−
601
 
 
8.1 Information Theory . . . . . . . . . . . . . . . . . . . . . 602
 
8.1 Information Theory . . . . . . . . . . . . . . . . . . . . . 602
 +
 
8.2 Reversible Computation . . . . . . . . . . . . . . . . . . . 629
 
8.2 Reversible Computation . . . . . . . . . . . . . . . . . . . 629
 +
 
8.3 Information Distance . . . . . . . . . . . . . . . . . . . . 641
 
8.3 Information Distance . . . . . . . . . . . . . . . . . . . . 641
 +
 
8.4 Normalized Information Distance . . . . . . . . . . . . . . 660
 
8.4 Normalized Information Distance . . . . . . . . . . . . . . 660
 +
 
8.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 674
 
8.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 674
 +
 
8.6 Entropy Revisited . . . . . . . . . . . . . . . . . . . . . . 686
 
8.6 Entropy Revisited . . . . . . . . . . . . . . . . . . . . . . 686
 +
 
8.7 Quantum Kolmogorov Complexity . . . . . . . . . . . . . 696
 
8.7 Quantum Kolmogorov Complexity . . . . . . . . . . . . . 696
 +
 
8.8 Compression in Nature . . . . . . . . . . . . . . . . . . . 711
 
8.8 Compression in Nature . . . . . . . . . . . . . . . . . . . 711
 +
 
8.9 History and References . . . . . . . . . . . . . . . . . . . 714
 
8.9 History and References . . . . . . . . . . . . . . . . . . . 714
 +
 
References 723
 
References 723
 +
 
Index 765
 
Index 765
−
 
  
 
== Links ==
 
== Links ==

Version vom 17. November 2014, 16:02 Uhr


Reference

Li, M., Vitányi, P.: An Introduction To Kolmogorov Complexity and its Applications. 3nd edn. Springer, New York (2008).

DOI

http://www.springer.com/mathematics/applications/book/978-0-387-33998-6

Abstract

Develops Kolmogorov theory in detail and outlines the wide range of illustrative applications

Presents recent, major results in the field: topics include Omega numbers, Kolmogorov-Loveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, and Shannon information

Introduces results from prominent researchers in the field, such as those found by Slaman-Kucera, Downey, Hirschfeldt, Nies, Solovay, Miller and many more

Includes applications to hierarchical clustering, phylogeny in genomics, classification of music, Google, and the World Wide Web

Additional exercises on new results

Improved sections on high-probability properties and Kolmogorov’s structure function, information distance

This ongoing bestseller, now in its third edition, is considered the standard reference on Kolmogorov complexity, a modern theory of information that is concerned with information in individual objects.

New key features and topics in the 3rd edition:

  • New results on randomness
  • Kolmogorov's structure function, model selection, and MDL
  • Incompressibility method: counting unlabeled graphs, Shellsort, communication complexity
  • Derandomization
  • Kolmogorov complexity versus Shannon information, rate distortion, lossy compression, denoising
  • Theoretical results on information distance
  • The similarity metric with applications to genomics, phylogeny, clustering, classification, semantic meaning, question-answer systems
  • Quantum Kolmogorov complexity

Written by two experts in the field, this book is ideal for advanced undergraduate students, graduate students, and researchers in all fields of science. It is self-contained: it contains the basic requirements from mathematics, probability theory, statistics, information theory, and computer science. Included are history, theory, new developments, a wide range of applications, numerous (new) problem sets, comments, source references, and hints to solutions of problems. This is the only comprehensive treatment of the central ideas of Kolmogorov complexity and their applications.

Extended Abstract

Reviews

Bibtex

Table of contents

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . vii

Preface to the Second Edition . . . . . . . . . . . . . . . . . . xi

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . xii

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . xii

Outlines of One-Semester Courses . . . . . . . . . . . . . . . . xv

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

1 Preliminaries

1

1.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . 1

1.2 Prerequisites and Notation . . . . . . . . . . . . . . . . . 7

1.3 Numbers and Combinatorics . . . . . . . . . . . . . . . . 8

1.4 Binary Strings . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . 15

1.6 Basics of Probability Theory . . . . . . . . . . . . . . . . 18

1.7 Basics of Computability Theory . . . . . . . . . . . . . . 24

1.8 The Roots of Kolmogorov Complexity . . . . . . . . . . . 47

1.9 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.10 Prediction and Probability . . . . . . . . . . . . . . . . . 59

1.11 Information Theory and Coding . . . . . . . . . . . . . . 65

1.12 State × Symbol Complexity . . . . . . . . . . . . . . . . 90

1.13 History and References . . . . . . . . . . . . . . . . . . .

2 Algorithmic Complexity 92

2.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 104

2.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 116

2.3 C as an Integer Function . . . . . . . . . . . . . . . . . . 126

2.4 Random Finite Sequences . . . . . . . . . . . . . . . . . . 133

2.5 *Random Infinite Sequences . . . . . . . . . . . . . . . . 143

2.6 Statistical Properties of Finite Sequences . . . . . . . . . 165

2.7 Algorithmic Properties of C

2.8 Algorithmic Information Theory . . . . . . . . . . . . . . 186

2.9 History and References . . . . . . . . . . . . . . . . . . . 192

3 Algorithmic Prefix Complexity 197

3.1 The Invariance Theorem . . . . . . . . . . . . . . . . . . 200

3.2 *Sizes of the Constants . . . . . . . . . . . . . . . . . . . 206

3.3 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 211

3.4 K as an Integer Function . . . . . . . . . . . . . . . . . . 216

3.5 Random Finite Sequences . . . . . . . . . . . . . . . . . . 218

3.6 *Random Infinite Sequences . . . . . . . . . . . . . . . . 220

3.7 Algorithmic Properties of K . . . . . . . . . . . . . . . . 239

3.8 *Complexity of Complexity . . . . . . . . . . . . . . . . . 241

3.9 *Symmetry of Algorithmic Information . . . . . . . . . . 244

3.10 History and References . . . . . . . . . . . . . . . . . . . 255

4 Algorithmic Probability 259

4.1 Semicomputable Functions Revisited . . . . . . . . . . . . 260

4.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . 262

4.3 Discrete Sample Space . . . . . . . . . . . . . . . . . . . . 265

4.4 Universal Average-Case Complexity . . . . . . . . . . . . 290

4.5 Continuous Sample Space . . . . . . . . . . . . . . . . . . 294

4.6 Universal Average-Case Complexity, Continued . . . . . . 330

4.7 History and References . . . . . . . . . . . . . . . . . . . 331

5 Inductive Reasoning xix 339

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 339

5.2 Solomonoff’s Theory of Prediction . . . . . . . . . . . . . 348

5.3 Simple Pac-Learning . . . . . . . . . . . . . . . . . . . . . 370

5.4 Hypothesis Identification by MDL . . . . . . . . . . . . . 382

5.5 Nonprobabilistic Statistics . . . . . . . . . . . . . . . . . 401

5.6 History and References . . . . . . . . . . . . . . . . . . . 431

6 The Incompressibility Method 441

6.1 Three Examples . . . . . . . . . . . . . . . . . . . . . . . 442

6.2 High-Probability Properties . . . . . . . . . . . . . . . . . 448

6.3 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 451

6.4 Kolmogorov Random Graphs . . . . . . . . . . . . . . . . 461

6.5 Compact Routing . . . . . . . . . . . . . . . . . . . . . . 469

6.6 Average-Case Analysis of Sorting . . . . . . . . . . . . . . 476

6.7 Longest Common Subsequence . . . . . . . . . . . . . . . 486

6.8 Formal Language Theory . . . . . . . . . . . . . . . . . . 490

6.9 Online CFL Recognition . . . . . . . . . . . . . . . . . . 497

6.10 Turing Machine Time Complexity . . . . . . . . . . . . . 502

6.11 Communication Complexity. . . . . . . . . . . . . . . . 516

6.12 Circuit Complexity . . . . . . . . . . . . . . . . . . . . . 521

6.13 History and References . . . . . . . . . . . . . . . . . . . 524

7 Resource-Bounded Complexity 531

7.1 Mathematical Theory . . . . . . . . . . . . . . . . . . . . 532

7.2 Language Compression . . . . . . . . . . . . . . . . . . . 550

7.3 Computational Complexity . . . . . . . . . . . . . . . . . 562

7.4 Instance Complexity . . . . . . . . . . . . . . . . . . . . . 571

7.5 Kt and Universal Search . . . . . . . . . . . . . . . . . . . 577

7.6 Time-Limited Universal Distributions . . . . . . . . . . . 582

7.7 Logical Depth . . . . . . . . . . . . . . . . . . . . . . . . 589

7.8 History and References . . . . . . . . . . . . . . . . . . . 596

8 Physics, Information, and Computation 601

8.1 Information Theory . . . . . . . . . . . . . . . . . . . . . 602

8.2 Reversible Computation . . . . . . . . . . . . . . . . . . . 629

8.3 Information Distance . . . . . . . . . . . . . . . . . . . . 641

8.4 Normalized Information Distance . . . . . . . . . . . . . . 660

8.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 674

8.6 Entropy Revisited . . . . . . . . . . . . . . . . . . . . . . 686

8.7 Quantum Kolmogorov Complexity . . . . . . . . . . . . . 696

8.8 Compression in Nature . . . . . . . . . . . . . . . . . . . 711

8.9 History and References . . . . . . . . . . . . . . . . . . . 714

References 723

Index 765

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