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Hypercomputation: computing more than the Turing machine
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
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Cited by 31 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the negative results of the 20 th Century. I then explain how the ChurchTuring Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability. Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were previously ignored because of long held misconceptions about the limits of computation.
Beyond The Universal Turing Machine
, 1998
"... We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a phi ..."
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Cited by 31 (1 self)
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We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Super TuringMachines
"... to a practical application. A dozen years later the first storedprogram electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to Compu ..."
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Cited by 5 (1 self)
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to a practical application. A dozen years later the first storedprogram electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to Computability? Many textbooks on the fundamentals of computer science offer examples of informationprocessing tasks that are, it is claimed, absolutely uncomputable, in the sense that no machine can be specified to carry out these tasks. For example, it is said that no machine can repond to any given (finite) string of binary digits in accordance with the following rules: 3 (1) Answer '1' if the string is a program that will cause a universal Turing machine on whose tape it is inscribed to execute only a finite number of operations (such programs are called 'terminating'). (2) Answer '0' if the string is not a terminating program; i.e. if the st
Computational Power of Infinite Quantum Parallelism
 pp.2057–2071 in International Journal of Theoretical Physics vol.44:11
, 2005
"... Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of class ..."
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Cited by 2 (1 self)
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Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.
The Invalidity of Diagonalization Proof for Halting Problem
"... We examine the converse of ChurchTuring thesis and establish the existence of uncountable number of accelerated Turing machines. This leads to the case that these machines are una#ected by Godel's incompleteness theorem. ..."
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We examine the converse of ChurchTuring thesis and establish the existence of uncountable number of accelerated Turing machines. This leads to the case that these machines are una#ected by Godel's incompleteness theorem.
Forthcoming in Minds and Machines, 2011. On the Possibilities of Hypercomputing Supertasks 1
, 2010
"... This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses prop ..."
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This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zenomachines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zenomachines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though noneffective methods or special rules for semidecidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the ChurchTuring thesis in its traditional interpretation. 1
Thèse de Church. Autres Modèles de Calculs
, 2009
"... Un des résultats fondamentaux les plus inattendus du vingtième siècle est le théorème d’incomplétude de Gödel, qui affirme qu’aucun système de preuve ne peut capturer pleinement le raisonnement mathématique: toute théorie suffisante pour capturer les raisonnements arithmétiques est nécessairement in ..."
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Un des résultats fondamentaux les plus inattendus du vingtième siècle est le théorème d’incomplétude de Gödel, qui affirme qu’aucun système de preuve ne peut capturer pleinement le raisonnement mathématique: toute théorie suffisante pour capturer les raisonnements arithmétiques est nécessairement incomplète, c’estàdire telle qu’il existe des énoncés qui ne sont pas démontrables et dont la négation n’est pas non plus démontrable. En particulier, on peut exprimer la cohérence d’une théorie mathématique par un énoncé, qui ne peut être démontré, ou infirmé. Les arguments de Kurt Gödel dans l’article original [1] sont en fait très intimement basés sur une notion (informelle) de déduction algorithmique. Alan Turing, travaillant sur le problème de la décision de Hilbert (Entscheidungsproblem, formulé ainsi par Turing: “peuton décider mécaniquement si un énoncé est démontrable ou non”) proposa dans l’article [52] son célèbre modèle de machine, capable de capturer la déduction dans les systèmes formels, et en particulier la notion de déduction utilisée par Gödel dans sa preuve.