Problema da parada: diferenças entre revisões
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O problema da parada é um problema de decisão sobre as propriedades de programas de computadores em um determinado modelo [[Turing-completo]] de computação, por exemplo todos os programas que podem ser escritos em uma [[linguagem de programação]] que é geral o suficiente para ser equivalente a uma máquina de Turing. O problema é determinar para uma dada entrada se o programa irá parar com ela. Nesta área de trabalho abstrata não há limitações de memória ou tempo necessário para a execução de um programa, pois poderão ser necessários tempo e espaço arbitrários para o programa parar. A questão é se o programa simplesmente poderá parar com uma certa entrada.
Por exemplo, em [[pseudocódigo]], o programa:
:<tt>enquanto Verdadeiro: continue</tt>
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Um [[problema de decisão]] é um conjunto de números naturais; o "problema" é determinar se um número em particular pertence ao conjunto.
Dada uma [[enumeração de Gödel]] <math>\varphi</math> de uma [[função computável]] (como os [[números de descrição]] de Turing) e uma [[função de pareamento]] <math>\langle i, x \rangle</math>, '''o problema da parada''' é o problema de decisão para o conjunto:
:: <math>K_{\varphi}^{0} := \{ \langle i, x \rangle | \varphi_i(x) \ \mathrm{existe} \}</math>
com <math>\varphi_i</math> a ''i''-ésima função computável na enumeração de Gödel <math>\varphi</math>.
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A prova segue estabelecendo que toda função parcial computável com dois argumentos se diferencia da função necessária ''h''. Com esta finalidade, dada qualquer função parcial computável binária ''f'', a seguinte [[função parcial]] ''g'' também é computável por um certo programa ''e'':
:<math>g(i) =
\begin{cases}
0 & \text{se } f(i,i) = 0,\\
\text{indefinida} & \text{caso contrario.}
\end{cases}</math>
A verificação de que ''g'' é computável depende das construções seguintes:
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==Referências Bibliográficas==
* {{
* [[Alan Turing]], ''On computable numbers, with an application to the Entscheidungsproblem'', Proceedings of the London Mathematical Society, Series 2, 42 (1936), pp 230–265. [http://www.turingarchive.org/browse.php/B/12 online version] This is the epochal paper where Turing defines [[Turing machine]]s, formulates the halting problem, and shows that it (as well as the [[Entscheidungsproblem]]) is unsolvable.
* [[c2:HaltingProblem]]
* [[B. Jack Copeland]] ed. (2004), ''The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma,'' Clarendon Press (Oxford University Press), Oxford UK, ISBN 0-19-825079-7.
* {{
* {{
* [[Alfred North Whitehead]] and [[Bertrand Russell]], ''Principia Mathematica'' to *56, Cambridge at the University Press, 1962. Re: the problem of paradoxes, the authors discuss the problem of a set not be an object in any of its "determining functions", in particular "Introduction, Chap. 1 p. 24 "...difficulties which arise in formal logic", and Chap. 2.I. "The Vicious-Circle Principle" p.37ff, and Chap. 2.VIII. "The Contradictions" p. 60ff.
* [[Martin Davis]], "What is a computation", in ''Mathematics Today'', Lynn Arthur Steen, Vintage Books (Random House), 1980. A wonderful little paper, perhaps the best ever written about Turing Machines for the non-specialist. Davis reduces the Turing Machine to a far-simpler model based on Post's model of a computation. Discusses [[Chaitin]] proof. Includes little biographies of [[Emil Post]], [[Julia Robinson]].
* [[Marvin Minsky]], ''Computation, Finite and Infinite Machines'', Prentice-Hall, Inc., N.J., 1967. See chapter 8, Section 8.2 "The Unsolvability of the Halting Problem." Excellent, i.e. readable, sometimes fun. A classic.
* [[Roger Penrose]], ''The Emperor's New Mind: Concerning computers, Minds and the Laws of Physics'', Oxford University Press, Oxford England, 1990 (with corrections). Cf: Chapter 2, "Algorithms and Turing Machines". An overly-complicated presentation (see Davis's paper for a better model), but a thorough presentation of Turing machines and the halting problem, and Church's Lambda Calculus.
* [[John Hopcroft]] and [[Jeffrey Ullman]], ''Introduction to Automata Theory, Languages and Computation'', Addison-Wesley, Reading Mass, 1979. See Chapter 7 "Turing Machines." A book centered around the machine-interpretation of "languages", NP-Completeness, etc.
* [[Andrew Hodges]], ''Alan Turing: The Enigma'', Simon and Schuster, New York. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
* [[Constance Reid]], ''Hilbert'', Copernicus: Springer-Verlag, New York, 1996 (first published 1970). Fascinating history of German mathematics and physics from 1880s through 1930s. Hundreds of names familiar to mathematicians, physicists and engineers appear in its pages. Perhaps marred by no overt references and few footnotes: Reid states her sources were numerous interviews with those who personally knew Hilbert, and Hilbert's letters and papers.
* [[Edward Beltrami]], ''What is Random? Chance and order in mathematics and life'', Copernicus: Springer-Verlag, New York, 1999. Nice, gentle read for the mathematically-inclined non-specialist, puts tougher stuff at the end. Has a Turing-machine model in it. Discusses the [[Chaitin]] contributions.
* [[Ernest Nagel]] and [[James R. Newman]], ''Godel’s Proof'', New York University Press, 1958. Wonderful writing about a very difficult subject. For the mathematically-inclined non-specialist. Discusses [[Gentzen]]'s proof on pages 96–97 and footnotes. Appendices discuss the [[Peano Axioms]] briefly, gently introduce readers to formal logic.
* [[Taylor Booth]], ''Sequential Machines and Automata Theory'', Wiley, New York, 1967. Cf Chapter 9, Turing Machines. Difficult book, meant for electrical engineers and technical specialists. Discusses recursion, partial-recursion with reference to Turing Machines, halting problem. Has a [[Turing Machine]] model in it. References at end of Chapter 9 catch most of the older books (i.e. 1952 until 1967 including authors Martin Davis, F. C. Hennie, H. Hermes, S. C. Kleene, M. Minsky, T. Rado) and various technical papers. See note under Busy-Beaver Programs.
* [[Busy Beaver]] Programs are described in Scientific American, August 1984, also March 1985 p. 23. A reference in Booth attributes them to Rado, T.(1962), On non-computable functions, Bell Systems Tech. J. 41. Booth also defines Rado's Busy Beaver Problem in problems 3, 4, 5, 6 of Chapter 9, p. 396.
* [[David Bolter]], ''Turing’s Man: Western Culture in the Computer Age'', The University of North Carolina Press, Chapel Hill, 1984. For the general reader. May be dated. Has yet another (very simple) Turing Machine model in it.
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