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Exploring RANDOMNESS, Softcover reprint of the original 1st ed. 2001 Discrete Mathematics and Theoretical Computer Science Series

Langue : Anglais

Auteur :

Couverture de l’ouvrage Exploring RANDOMNESS
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.'
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.
From the reviews:"In this book on algorithmic information theory, the author compares his concept of randomness (for recursive functions) which is based on the complexity (length) of the generating algorithm (program) with other concepts (by Martin-Löw, Solovay) and discusses its relation to incompleteness and the halting problem. Algorithms (needed for proof) are described in a (small) dialect of LISP. The style mostly is that of a lecture, lively and readable." (P. Schmitt, Monatshefte für Mathematik, Vol. 141 (1), 2004)"Chaitin is the main architect of a new branch of mathematics called algorithmic information theory, or 'AIT'. ... in Exploring Randomness, he develops algorithmic theory, further revealing its technical core. This is important work, with implications that go far beyond the arcane arguments of one branch of mathematics. ... As one gets to the substance ... it is difficult to resist Chaitin's enthusiastic style and obvious intelligence. Beyond the technicalities of the argument, the reader is quickly drawn into a fundamental new landscape of ideas." (Jacques F. Vallee, Journal of Scientific Exploration, Vol. 16 (4), 2002)"Chaitin's latest three books form a nice triangular base to support and explore the concepts underlying algorithmic information theory (AIT) - a clever blend of Gödel, Turing, and Shannon that Chaitin developed in his late teens ... . this set of three volumes packages the material in a nice, quite digestible fashion ... . Chaitin's results demonstrate that not only there is no structure to foundation of mathematics, the foundation is in fact random." (The Mathematica Journal, April, 2002)"The book is devoted to a Lisp formalism for exploring the basic ideas, concepts and results on program-size complexity and random sequences. The book contains a wealth of exercises, ranging from the 'mathematical equivalent of finger warm-ups for pianists' to substantial programming projects, from open questions to questions the author cannot eve

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Ouvrage de 164 p.

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Date de parution :

Ouvrage de 164 p.

15.5x23.5 cm

Sous réserve de disponibilité chez l'éditeur.

105,49 €

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