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Book
Algorithmic Information Theory
"The aim of this book is to present the strongest possible version of
G¨odel's incompleteness theorem, using an information-theoretic approach
based on the size of computer programs."
Real-time Traffic| Added | 08 Feb 2009 |
| Updated | 12 Feb 2009 |
| Authors | G J Chaitin |
Table of Contents1 Introduction 13I Formalisms for Computation: Register Machines,Exponential Diophantine Equations, &Pure LISP 192 Register Machines 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Pascal's TriangleMod 2 . . . . . . . . . . . . . . . . . . 26 2.3 LISP RegisterMachines . . . . . . . . . . . . . . . . . . 30 2.4 Variables Used in Arithmetization . . . . . . . . . . . . . 45 2.5 An Example of Arithmetization . . . . . . . . . . . . . . 49 2.6 A Complete Example of Arithmetization . . . . . . . . . 59 2.7 Expansion of =>'s . . . . . . . . . . . . . . . . . . . . . . 63 2.8 Left-Hand Side . . . . . . . . . . . . . . . . . . . . . . . 71 2.9 Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . 753 A Version of Pure LISP 79 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Definition of LISP. . . . . . . . . . . . . . . . . . . . . . 81 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 LISP in LISP I . . . . . . . . . . . . . . . . . . . . . . . 93 3.5 LISP in LISP II . . . . . . . . . . . . . . . . . . . . . . . 94 3.6 LISP in LISP III . . . . . . . . . . . . . . . . . . . . . . 984 The LISP Interpreter EVAL 103 4.1 RegisterMachine Pseudo-Instructions . . . . . . . . . . . 103 4.2 EVAL in RegisterMachine Language . . . . . . . . . . . 106 4.3 The Arithmetization of EVAL . . . . . . . . . . . . . . . 123 4.4 Start of Left-Hand Side . . . . . . . . . . . . . . . . . . . 129 4.5 End of Right-Hand Side . . . . . . . . . . . . . . . . . . 131II Program Size, Halting Probabilities, Randomness, & Metamathematics 1355 Conceptual Development 139 5.1 Complexity via LISP Expressions . . . . . . . . . . . . . 139 5.2 Complexity via Binary Programs . . . . . . . . . . . . . 145 5.3 Self-Delimiting Binary Programs . . . . . . . . . . . . . . 146 5.4 Omega in LISP . . . . . . . . . . . . . . . . . . . . . . . 1496 Program Size 157 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.3 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . 162 6.4 RandomStrings . . . . . . . . . . . . . . . . . . . . . . . 1747 Randomness 179 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.2 RandomReals . . . . . . . . . . . . . . . . . . . . . . . . 1848 Incompleteness 197 8.1 Lower Bounds on Information Content . . . . . . . . . . 197 8.2 RandomReals: First Approach . . . . . . . . . . . . . . 200 8.3 Random Reals: |Axioms| . . . . . . . . . . . . . . . . . . 202 8.4 RandomReals: H(Axioms) . . . . . . . . . . . . . . . . . 2099 Conclusion 21310 Bibliography 215A Implementation Notes 221B S-expressions of Size N 223
Table of Contents1 Introduction 13I Formalisms for Computation: Register Machines,Exponential Diophantine Equations, &Pure LISP 192 Register Machines 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Pascal's TriangleMod 2 . . . . . . . . . . . . . . . . . . 26 2.3 LISP RegisterMachines . . . . . . . . . . . . . . . . . . 30 2.4 Variables Used in Arithmetization . . . . . . . . . . . . . 45 2.5 An Example of Arithmetization . . . . . . . . . . . . . . 49 2.6 A Complete Example of Arithmetization . . . . . . . . . 59 2.7 Expansion of =>'s . . . . . . . . . . . . . . . . . . . . . . 63 2.8 Left-Hand Side . . . . . . . . . . . . . . . . . . . . . . . 71 2.9 Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . 753 A Version of Pure LISP 79 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Definition of LISP. . . . . . . . . . . . . . . . . . . . . . 81 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 LISP in LISP I . . . . . . . . . . . . . . . . . . . . . . . 93 3.5 LISP in LISP II . . . . . . . . . . . . . . . . . . . . . . . 94 3.6 LISP in LISP III . . . . . . . . . . . . . . . . . . . . . . 984 The LISP Interpreter EVAL 103 4.1 RegisterMachine Pseudo-Instructions . . . . . . . . . . . 103 4.2 EVAL in RegisterMachine Language . . . . . . . . . . . 106 4.3 The Arithmetization of EVAL . . . . . . . . . . . . . . . 123 4.4 Start of Left-Hand Side . . . . . . . . . . . . . . . . . . . 129 4.5 End of Right-Hand Side . . . . . . . . . . . . . . . . . . 131II Program Size, Halting Probabilities, Randomness, & Metamathematics 1355 Conceptual Development 139 5.1 Complexity via LISP Expressions . . . . . . . . . . . . . 139 5.2 Complexity via Binary Programs . . . . . . . . . . . . . 145 5.3 Self-Delimiting Binary Programs . . . . . . . . . . . . . . 146 5.4 Omega in LISP . . . . . . . . . . . . . . . . . . . . . . . 1496 Program Size 157 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.3 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . 162 6.4 RandomStrings . . . . . . . . . . . . . . . . . . . . . . . 1747 Randomness 179 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.2 RandomReals . . . . . . . . . . . . . . . . . . . . . . . . 1848 Incompleteness 197 8.1 Lower Bounds on Information Content . . . . . . . . . . 197 8.2 RandomReals: First Approach . . . . . . . . . . . . . . 200 8.3 Random Reals: |Axioms| . . . . . . . . . . . . . . . . . . 202 8.4 RandomReals: H(Axioms) . . . . . . . . . . . . . . . . . 2099 Conclusion 21310 Bibliography 215A Implementation Notes 221B S-expressions of Size N 223
1 Introduction 13
I Formalisms for Computation: Register Machines,
Exponential Diophantine Equations, &
Pure LISP 19
2 Register Machines 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Pascal's TriangleMod 2 . . . . . . . . . . . . . . . . . . 26
2.3 LISP RegisterMachines . . . . . . . . . . . . . . . . . . 30
2.4 Variables Used in Arithmetization . . . . . . . . . . . . . 45
2.5 An Example of Arithmetization . . . . . . . . . . . . . . 49
2.6 A Complete Example of Arithmetization . . . . . . . . . 59
2.7 Expansion of =>'s . . . . . . . . . . . . . . . . . . . . . . 63
2.8 Left-Hand Side . . . . . . . . . . . . . . . . . . . . . . . 71
2.9 Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . 75
3 A Version of Pure LISP 79
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Definition of LISP. . . . . . . . . . . . . . . . . . . . . . 81
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 LISP in LISP I . . . . . . . . . . . . . . . . . . . . . . . 93
3.5 LISP in LISP II . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 LISP in LISP III . . . . . . . . . . . . . . . . . . . . . . 98
4 The LISP Interpreter EVAL 103
4.1 RegisterMachine Pseudo-Instructions . . . . . . . . . . . 103
4.2 EVAL in RegisterMachine Language . . . . . . . . . . . 106
4.3 The Arithmetization of EVAL . . . . . . . . . . . . . . . 123
4.4 Start of Left-Hand Side . . . . . . . . . . . . . . . . . . . 129
4.5 End of Right-Hand Side . . . . . . . . . . . . . . . . . . 131
II Program Size, Halting Probabilities, Randomness, & Metamathematics 135
5 Conceptual Development 139
5.1 Complexity via LISP Expressions . . . . . . . . . . . . . 139
5.2 Complexity via Binary Programs . . . . . . . . . . . . . 145
5.3 Self-Delimiting Binary Programs . . . . . . . . . . . . . . 146
5.4 Omega in LISP . . . . . . . . . . . . . . . . . . . . . . . 149
6 Program Size 157
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . 162
6.4 RandomStrings . . . . . . . . . . . . . . . . . . . . . . . 174
7 Randomness 179
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.2 RandomReals . . . . . . . . . . . . . . . . . . . . . . . . 184
8 Incompleteness 197
8.1 Lower Bounds on Information Content . . . . . . . . . . 197
8.2 RandomReals: First Approach . . . . . . . . . . . . . . 200
8.3 Random Reals: |Axioms| . . . . . . . . . . . . . . . . . . 202
8.4 RandomReals: H(Axioms) . . . . . . . . . . . . . . . . . 209
9 Conclusion 213
10 Bibliography 215
A Implementation Notes 221
B S-expressions of Size N 223
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