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The New Promise of Analog Computation
"... Abstract. We show that, using our more or less established framework of inductive definition of realvalued functions (work started by Cristopher Moore in [9]) together with ideas and concepts of standard computability we can prove theorems of Analysis. Then we will consider our ideas as a bridging ..."
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Abstract. We show that, using our more or less established framework of inductive definition of realvalued functions (work started by Cristopher Moore in [9]) together with ideas and concepts of standard computability we can prove theorems of Analysis. Then we will consider our ideas as a bridging tool between the standard Theory of Computability (and Complexity) on one side and Mathematical Analysis on the other, making real recursive functions a possible branch of Descriptive Set Theory. What follows is an Extended Abstract directed to a large audience of
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.
Author manuscript, published in "4th Conf. Computability in Europe (CiE~'08) (abstracts and extended abstracts of unpublished papers), Greece (2008)" Abstract geometrical computation
, 2010
"... beyond the ..."
A Survey on Continuous Time Computations (Chapter of the book “New Computational Paradigms”, To appear, Springer)
"... Abstract. We provide an overview of computation theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, sum ..."
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Abstract. We provide an overview of computation theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature. 1
ProjectTeam PROTHEO Constraints, Mechanized Deduction and Proofs of Software Properties
"... d ' ctivity ..."
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Chapter 4 On the Links Between Several Models
"... In this chapter, we present some of our results of comparisons between several continuous time models. We first focus on the General Purpose Analog Computer from Shannon [Shannon, 1941], and on polynomial Cauchy problems. Later on, we will focus on subclasses of Rrecursive functions. Rrecursive fu ..."
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In this chapter, we present some of our results of comparisons between several continuous time models. We first focus on the General Purpose Analog Computer from Shannon [Shannon, 1941], and on polynomial Cauchy problems. Later on, we will focus on subclasses of Rrecursive functions. Rrecursive functions were introduced by [Moore, 1998]. We relate them to computable functions in the sense of recursive analysis. All the results of this chapter have been obtained in collaborations. The results about the GPAC are the fruit of a collaboration with Manuel Campagnolo, Daniel Graça and Emmanuel Hainry (our PhD student). The results on Rrecursive functions also belong to the PhD thesis of Emmanuel Hainry. 4.1 A ChurchTuring Thesis for Analog Computations? According to ChurchTuring thesis, all sufficiently powerful “reasonable ” models of digital computations are computationally equivalent to Turing machines. No similar result is known when considering analog computations. Many analog models have been studied, including the BSS model [Blum et al., 1989], Moore’s Rrecursive functions [Moore, 1998], neural networks [Siegelmann, 1999], or computable analysis [PourEl and Richards, 1989], [Ko, 1991], [Weihrauch, 2000a], but none is able to affirm itself as “universal”. In part, this is due to the fact that few relations between them are known. Moreover some of the known results assert that these models are not equivalent, making
doi:10.1093/comjnl/bxs054 A Class of Contracting Stream Operators†
, 2011
"... for the semantics of analog networks operating on streams from topological algebras. Central to their model is a parametrized stream operator representing the network along with a theory that concerns the existence, uniqueness, continuity and computability of a fixed point of that stream operator. W ..."
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for the semantics of analog networks operating on streams from topological algebras. Central to their model is a parametrized stream operator representing the network along with a theory that concerns the existence, uniqueness, continuity and computability of a fixed point of that stream operator. We narrow the scope of this paper from general topological algebras to algebras of streams that assume values only from a Banach space. This restriction facilitates the definition of a fairly broad class of stream operators to which the theory described in the above two papers applies. As a demonstration in their original work, the authors provide two case studies: analog networks that model the behavior of simple massspringdamper systems. The case studies showcase the theory well, but they seem to require the imposition of somewhat peculiar conditions on the parameters (the masses, the spring constants and the damping coefficients). The extra conditions—while not catastrophic to the case studies—make them somewhat unsatisfying. We show here that while their original mass–spring–damper models do not fall within our new class, they can be trivially reconfigured into equivalent models that do. This modification obviates the extra conditions on the parameters.