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Recursive analysis characterized as a class of real recursive functions
 Fundamenta Informaticae
, 2006
"... Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real r ..."
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Cited by 18 (8 self)
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Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.
A survey on continuous time computations
 New Computational Paradigms
"... Abstract. We provide an overview of 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 resu ..."
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Cited by 11 (2 self)
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Abstract. We provide an overview of 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
The elementary computable functions over the real numbers: Applying two new techniques
 ARCHIVES FOR MATHEMATICAL LOGIC
, 2007
"... The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). ..."
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Cited by 6 (3 self)
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The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C 2 real elementary computable functions; our result does not require the restriction to C 2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort.
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... 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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Cited by 5 (1 self)
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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. 1
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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Cited by 1 (1 self)
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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
Characterizing Computable Analysis with Differential Equations
, 2008
"... The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to ..."
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The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to characterize the twice continuously differentiable functions of Computable Analysis, restricted to certain compact domains. In a similar model, Shannon’s General Purpose Analog Computer, Bournez et. al. 2007 [3] also characterize the functions of Computable Analysis. We combine the results of [5] and Graça et. al. [13], to show that a different function algebra also yields Computable Analysis. We believe that our function algebra is an improvement due to its simple definition and because the operations in our algebra are less obviously designed to mimic the operations in the usual definition of the recursive functions using the primitive recursion and minimization operators. 1
COntinuous tiMe comPUTations. Computations on the Reals.
, 2007
"... We propose to contribute to understand computation theories for continuous time systems. This is motivated by • understanding algorithmic complexity of automatic verification procedures for continuous and hybrid systems; • understanding some new models of computations. New models of computations und ..."
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We propose to contribute to understand computation theories for continuous time systems. This is motivated by • understanding algorithmic complexity of automatic verification procedures for continuous and hybrid systems; • understanding some new models of computations. New models of computations under study include analog electronics models, and some recent sensor and telecommunication networks models. Hybrid systems include all systems that mix continuous dynamics with discrete transitions. We propose to do so to develop the model of Rrecursive functions introduced by Moore in [49], using the recent framework of [24]. We expect by the end of this project to • Develop significantly computation theory for continuous time systems to noisy and robust systems. Expected implications are contributions to understand a famous conjecture in verification about decidability and termination of verification procedures for hybrid systems, and hence possibly new verification tools. • Revisit computations on the reals, to avoid references to Turing machines. Expected implications are lower and upper bounds on the algorithmic complexity of natural problems in verification and control, motivated by automatic verification procedures for continuous and hybrid systems. • Understand deeply some new computational models. Expected implications are better understanding of some recent models of sensor and telecommunication networks, that could be used to better program them. • Contribute to understand better the computational properties of models of natural inspiration, and in particular contribute to understand whether edgeofchaos regimes may provide an appropriate setting for computational processes.
IOS Press Recursive Analysis Characterized as a Class of Real Recursive Functions ∗
"... Abstract. Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class ..."
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Abstract. Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.
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 ..."
Abstract
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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.
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