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Three Paths to Effectiveness
"... For Yuri, profound thinker, esteemed expositor, and treasured friend. Abstract. Over the past two decades, Gurevich and his colleagues have developed axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in a new framework of abstract state ..."
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For Yuri, profound thinker, esteemed expositor, and treasured friend. Abstract. Over the past two decades, Gurevich and his colleagues have developed axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in a new framework of abstract state machines. Recently, this approach was extended to suggest axiomatic foundations for the notion of effective computation over arbitrary countable domains. This was accomplished in three different ways, leading to three, seemingly disparate, notions of effectiveness. We show that, though having taken different routes, they all actually lead to precisely the same concept. With this concept of effectiveness, we establish that there is – up to isomorphism – exactly one maximal effective model across all countable domains.
Around the physical ChurchTuring thesis: Cellular automata, formal languages, and the principles of quantum theory
 In Proc. 6th International Conference on Language and Automata Theory and Applications (LATA 2012, A Coruña
, 2012
"... Abstract. The physical ChurchTuring thesis explains the Galileo thesis, but also suggests an evolution of the language used to describe nature. It can be proved from more basic principle of physics, but it also questions these principles, putting the emphasis on the principle of a bounded density ..."
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Abstract. The physical ChurchTuring thesis explains the Galileo thesis, but also suggests an evolution of the language used to describe nature. It can be proved from more basic principle of physics, but it also questions these principles, putting the emphasis on the principle of a bounded density of information. This principle itself questions our formulation of quantum theory, in particular the choice of a field for the scalars and the origin of the infinite dimension of the vector spaces used as state spaces1. 1 The ChurchTuring Thesis and Its Various Forms 1.1 Why a Thesis? It is a quite common situation in mathematics, that a notion, first understood intuitively, receives a formal definition at some point. For instance, the notion of a real number has been understood intuitively in geometry, for instance as the length of a segment, before it has been formally defined in the 19th century, by Cauchy and Dedekind. Another example is the notion of an algorithm, that has been understood intuitively for long, before a formal definition of the notion of a
Honest Universality
, 2012
"... We extend the notion of universality of a function, due to Turing, to arbitrary (countable) effective domains, taking care to disallow any cheating on the part of the representations used. universal function, representation, encoding, effectiveness, comKeywords: putability 1 ..."
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We extend the notion of universality of a function, due to Turing, to arbitrary (countable) effective domains, taking care to disallow any cheating on the part of the representations used. universal function, representation, encoding, effectiveness, comKeywords: putability 1
Towards an Axiomatization of Simple Analog Algorithms
"... Abstract. We propose a formalization of analog algorithms, extending the framework of abstract state machines to continuoustime models of computation. The states of ‘continuous ’ machinery... form a continuous manifold, and the behaviour of the machine is described by a curve on this manifold. All ..."
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Abstract. We propose a formalization of analog algorithms, extending the framework of abstract state machines to continuoustime models of computation. The states of ‘continuous ’ machinery... form a continuous manifold, and the behaviour of the machine is described by a curve on this manifold. All machinery can be regarded as continuous, but when it is possible to regard it as discrete it is usually best to do so. The property of being ‘discrete ’ is only an advantage for the theoretical investigator, and serves no evolutionary purpose, so we could not expect Nature to assist us by producing truly ‘discrete ’ brains. Alan M. Turing, Intelligent Machinery, 1948 1
Abstract State Machines: A Generic Model of Computation
, 2010
"... Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new framework of abstract state machines. This approach has been extended to suggest a formalization of ..."
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Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new framework of abstract state machines. This approach has been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein.
Intrinsic computability over algebraic structures
"... Abstract. The notion of computable function can be extended from the natural numbers to other domains, using an indexing. Usually, the choice of the indexing affects the set of computable functions and there is no intrinsic notion of computability over these domains. In this paper, we show that, in ..."
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Abstract. The notion of computable function can be extended from the natural numbers to other domains, using an indexing. Usually, the choice of the indexing affects the set of computable functions and there is no intrinsic notion of computability over these domains. In this paper, we show that, in contrast, when we extend the notion of computable function to algebraic structures, we obtain an intrinsic notion in many cases. We give examples of such structures having an intrinsic notion of computability and characterize them as finitely generated structures. 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.
Effectiveness
, 2011
"... We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. ..."
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We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis.
RAM is as Good as it Gets
"... Abstract. We prove that any algorithm, running on any effective operational model can be simulated by a randomaccess machine (RAM) with only constant overhead of time, when memory access is charged per bit. The first rule of any technology used in a business is that automation applied to an efficie ..."
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Abstract. We prove that any algorithm, running on any effective operational model can be simulated by a randomaccess machine (RAM) with only constant overhead of time, when memory access is charged per bit. The first rule of any technology used in a business is that automation applied to an efficient operation will magnify the efficiency. The second is that automation applied to an inefficient operation will magnify the inefficiency. —William Henry Gates III 1