Exploring RANDOMNESS

I Introduction. - Historical introductionA century of controversy over the foundations of mathematics. - What is LISP? Why do I like it?. - How to program my universal Turing machine in LISP. - II Program Size. - A self-delimiting Turing machine considered as a set of (program output) pairs. - How to construct self-delimiting Turing machines: the Kraft inequality. - The connection between program-size complexity and algorithmic probability: H(x) = ? log2P(x) +O(1). Occam's razor: there are few minimum-size programs. - The basic result on relative complexity: H(y?x) = H(xy)-H(x)+O(1). - III Randomness. - Theoretical interludeWhat is randomness? My definitions. - Proof that Martin-Löf randomness is equivalent to Chaitin randomness. - Proof that Solovay randomness is equivalent to Martin-Löf randomness. - Proof that Solovay randomness is equivalent to strong Chaitin randomness. - IV Future Work. - Extending AIT to the size of programs for computing infinite sets and to computations with oracles. - PostscriptLetter to a daring young reader. Language: English