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Elementarily computable functions over the real numbers and subrecursive functions
 Theoretical Computer Science, 348(23):130 – 147, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALPA
, 2004
"... Abstract We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composit ..."
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Abstract We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. This paper improves several previous partial characterizations and has a dual interest: Concerning recursive analysis, our results provide machineindependent characterizations of natural classes of computable functions over the real numbers, allowing to dene these classes without usual considerations on higherorder (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers and provide new insights for understanding the relations between several analog computational models. 1
The Nature of the Extended Analog Computer
"... During the past decade, researchers have asked fundamental questions about the nature of Rubel’s extended analog computer, the EAC. The questions have made it clear that the design, implementation, and applications of the EAC are based on a paradigm unfamiliar to most users of conventional digital c ..."
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During the past decade, researchers have asked fundamental questions about the nature of Rubel’s extended analog computer, the EAC. The questions have made it clear that the design, implementation, and applications of the EAC are based on a paradigm unfamiliar to most users of conventional digital computers. The basic difference is that the EAC’s components visibly implement only a few explicit functions. The rest are implicit, being properties of nature that are described mathematically. A new approach to bridge the “paradigm gap ” is needed. This paper introduces the!digraph, a directed graph that can be labeled to show how both unconventional and conventional computers relate nature, mathematics and computer architecture. The!digraph defines a semantic hierarchy that bridges the paradigms of analogy and algorithm. It shows how many applications for the EAC are developed by choosing the semantics for an analogy, rather than programming an algorithm. Finally, concise case studies show how society is in the early stages of adopting the EAC. These applications suggest the future for unconventional computers. PACS codes and keywords:
Abstract How much can analog and hybrid systems be proved
"... Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring.
Abstract Relativistic computers and the Turing barrier q
"... We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lup ..."
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We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic 6 (3) (2000) 265–283] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing nonTuring computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability. Ó 2005 Elsevier Inc. All rights reserved. Of all the entities I have encountered in my life in physics, none approaches the black hole in fascination. And none, I think, is a more important constituent of this universe we call home. The black hole epitomizes the revolution wrought by general relativity. It pushes to an extreme—and therefore tests to the limit—the features of general relativity (the dynamics of curved spacetime) that set it apart from special relativity (the physics of static, ‘‘flat’ ’ spacetime) and the earlier mechanics of Newton. Spacetime curvature. Geometry as part of physics. Gravitational radiation. All of these things become, with black holes, not tiny corrections to older physics, but the essence of newer physics.
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
Philosophy and Complexity: Essays on Epistemology, Evolution, and
, 2006
"... The aim of this paper is to address the question: Can an artificial neural network (ANN) model be used as a possible characterization of the power of the human mind? We will discuss what might be the relationship between such a model and its natural counterpart. A possible characterization of the di ..."
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The aim of this paper is to address the question: Can an artificial neural network (ANN) model be used as a possible characterization of the power of the human mind? We will discuss what might be the relationship between such a model and its natural counterpart. A possible characterization of the different power capabilities of the mind is suggested in terms of the information contained (in its computational complexity) or achievable by it. Such characterization takes advantage of recent results based on natural neural networks (NNN) and the computational power of arbitrary artificial neural networks (ANN). The possible acceptance of neural networks as the model of the human mind’s operation makes the aforementioned quite relevant. 1.