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This essential companion to Chaitin's successful books The Unknowable and The Limits of Mathematics, presents the technical core of his theory of program-size complexity. The two... Read more
This essential companion to Chaitin's successful books The Unknowable and The Limits of Mathematics, presents the technical core of his theory of program-size complexity. The two previous volumes are more concerned with applications to meta-mathematics. LISP is used to present the key algorithms and to enable computer users to interact with the authors proofs and discover for themselves how they work. The LISP code for this book is available at the author's Web site together with a Java applet LISP interpreter. "No one has looked deeper and farther into the abyss of randomness and its role in mathematics than Greg Chaitin. This book tells you everything hes seen. Don miss it." John Casti, Santa Fe Institute, Author of Goedel: A Life of Logic.'
Exploring RANDOMNESS Hardback edition by Gregory J. Chaitin
I Introduction.- Historical introduction-A 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(x,y)-H(x)+O(1).- III Randomness.- Theoretical interlude-What is randomness? My definitions.- Proof that Martin-Loef randomness is equivalent to Chaitin randomness.- Proof that Solovay randomness is equivalent to Martin-Loef 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.- Postscript-Letter to a daring young reader.
Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein's general relativity, as he proved that Einstein's theory allows for time machines....
The author of Meta Math! explains how evolution works on a mathematical level, arguing that mathematical theory is an essential part of evolution while highlighting mathematical principles in the biological world. 25,000 first printing.
Considering combination and focusing of information as the relevant operations leads to a generic algebraic structure for information. Information algebras provide a natural framework to define and study uncertain information. Uncertain information is represented by random variables that naturally form information algebras.
This essential companion to Chaitins highly successful The Limits of Mathematics, gives a brilliant historical survey of important work on the foundations of mathematics. The Unknowable is a very readable introduction to Chaitins ideas, and includes software (on the authors website) that will enable users to interact with the authors proofs.
Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of Goedel's incompleteness theorem, using an information theoretic approach based on the size of computer programs.
Contains 23 non-technical papers. This book includes essays which summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of Godel and Turing on the limits of mathematical methods, both in logic and in computation.
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