Exploring RANDOMNESS

Authors:

ISBN: 978-1-4471-1085-9 (Print) 978-1-4471-0307-3 (Online)

Table of contents (13 chapters)

  1. Front Matter

    Pages i-x

  2. Introduction

    1. Front Matter

      Pages 1-1

    2. No Access

      Book Chapter

      Pages 3-28

      Historical introduction — A century of controversy over the foundations of mathematics

    3. No Access

      Book Chapter

      Pages 29-43

      What is LISP? Why do I like it?

    4. No Access

      Book Chapter

      Pages 45-59

      How to program my universal Turing machine in LISP

  3. Program Size

    1. Front Matter

      Pages 61-61

    2. No Access

      Book Chapter

      Pages 63-72

      A self-delimiting Turing machine considered as a set of (program, output) pairs

    3. No Access

      Book Chapter

      Pages 73-83

      How to construct self-delimiting Turing machines: the Kraft inequality

    4. No Access

      Book Chapter

      Pages 85-93

      The connection between program-size complexity and algorithmic probability: H(x)=-log2 P(x)+O(1). Occam’s razor: there are few minimum-size programs

    5. No Access

      Book Chapter

      Pages 95-108

      The basic result on relative complexity:H(y∣x)=H(x,y)-H(x)+O(1)

  4. Randomness

    1. Front Matter

      Pages 109-109

    2. No Access

      Book Chapter

      Pages 111-127

      Theoretical interlude—What is randomness? My definitions

    3. No Access

      Book Chapter

      Pages 129-137

      Proof that Martin-Löf randomness is equivalent to Chaitin randomness

    4. No Access

      Book Chapter

      Pages 139-142

      Proof that Solovay randomness is equivalent to Martin-Löf randomness

    5. No Access

      Book Chapter

      Pages 143-146

      Proof that Solovay randomness is equivalent to strong Chaitin randomness

  5. Future Work

    1. Front Matter

      Pages 147-147

    2. No Access

      Book Chapter

      Pages 149-159

      Extending AIT to the size of programs for computing infinite sets and to computations with oracles

    3. No Access

      Book Chapter

      Pages 161-164

      Postscript—Letter to a daring young reader