E-Book, Englisch, 164 Seiten, eBook
Reihe: Discrete Mathematics and Theoretical Computer Science
Chaitin Exploring RANDOMNESS
2001
ISBN: 978-1-4471-0307-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, 164 Seiten, eBook
Reihe: Discrete Mathematics and Theoretical Computer Science
ISBN: 978-1-4471-0307-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
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.'
Zielgruppe
Research
Autoren/Hrsg.
Weitere Infos & Material
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-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.- Postscript—Letter to a daring young reader.
