Computing Exact Approximations of a Chaitin Omega Number

Editora:

KS OmniScriptum Publishing

Código:

491_9783639135077

Vendido e entregue por Um Livro

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In this monograph,the research aimed to compute some exact bits of a Chaitin Omega number. A Chaitin Omega numbers are halting probabilities of a specificmathematical model of the ubiquitous PC called self-delimiting Turing machine. In 1936,Turing showed that no mechanical procedure and therefore no formal axiomatic theory can solve Turings halting problem,the question of whether a given computer program willeventually halt. An Omega number combines allinstances of Turings halting problem into aparadoxical real number. Its binary digits or bitsare algorithmically random and cannot bedistinguished from the the result of independent tossof a fair coin. Omega has a simple mathematical definition,but itdoes not enable us to determine more than finitelymany of its digits and no other definition can do itbetter. Furthermore,as nobody before was able tocompute any exact bit of a natural Omega number,the carrying on the computation is much moredemanding than solving Turings halting problem.We reviewed the properties of Omega numbers leadingto the computation of approximations to obtaininitial exact 64 bits of a Chaitin Omega number.

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