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23
Polynomial differential equations compute all real computable functions on computable compact intervals
, 2007
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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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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 general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
 Theory and Applications of Models of Computation, Third International Conference, TAMC 2006
, 2006
"... Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, ..."
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Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models. 1
The Elementary Computable Functions over the Real Numbers: Applying Two New Techniques. Archives for Mathematical Logic
, 2008
"... 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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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 C2 real elementary computable functions; our result does not require the restriction to C2 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. 1
Distributed Learning of Wardrop Equilibria
 IN "7TH INTERNATIONAL CONFERENCE ON UNCONVENTIONAL COMPUTATION  UC 2008) LECTURE NOTES IN COMPUTER SCIENCE, AUTRICHE VIENNE
"... We consider the problem of learning equilibria in a well known game theoretic traffic model due to Wardrop. We consider a distributed learning algorithm that we prove to converge to equilibria. The proof of convergence is based on a differential equation governing the global macroscopic evolution of ..."
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We consider the problem of learning equilibria in a well known game theoretic traffic model due to Wardrop. We consider a distributed learning algorithm that we prove to converge to equilibria. The proof of convergence is based on a differential equation governing the global macroscopic evolution of the system, inferred from the local microscopic evolutions of agents. We prove that the differential equation converges with the help of Lyapunov techniques.
Using approximation to relate computational classes over the reals
 MCU 2007, Lecture Notes in Computer Science 4664 (2007
"... Abstract. We use our method of approximation to relate various classes of computable functions over the reals. In particular, we compare Computable Analysis to the two analog models, the General Purpose Analog Computer and Real Recursive Functions. There are a number of existing results in the lite ..."
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Abstract. We use our method of approximation to relate various classes of computable functions over the reals. In particular, we compare Computable Analysis to the two analog models, the General Purpose Analog Computer and Real Recursive Functions. There are a number of existing results in the literature showing that the different models correspond exactly. We show how these exact correspondences can be broken down into a two step process of approximation and completion. We show that the method of approximation has further application in relating classes of functions, exploiting the transitive nature of the approximation relation. This work builds on our earlier work with our method of approximation, giving more evidence of the breadth of its applicability. 1
Differential Recursion
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2009
"... We present a redevelopment of the theory of realvalued recursive functions that was introduced by C. Moore in 1996 by analogy with the standard formulation of the integervalued recursive functions. While his work opened a new line of research on analog computation, the original paper contained som ..."
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We present a redevelopment of the theory of realvalued recursive functions that was introduced by C. Moore in 1996 by analogy with the standard formulation of the integervalued recursive functions. While his work opened a new line of research on analog computation, the original paper contained some technical inaccuracies. We discuss possible attempts to remove the ambiguity in the behaviour of the operators on partial functions, with a focus on his “primitive recursive” functions generated by the differential recursion operator that solves initial value problems. Under a reasonable reformulation, the functions in this class are shown to be analytic and computable in a strong sense in Computable Analysis. Despite this wellbehavedness, the class turns out to be too big to have the originally purported relation to differentially algebraic functions, and hence to C. E. Shannon’s model of analog computation.
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
Implicit complexity in recursive analysis
 TENTH INTERNATIONAL WORKSHOP ON LOGIC AND COMPUTATIONAL COMPLEXITY LCC'09
, 2009
"... Recursive analysis is a model of analog computation which is based on type 2 Turing machines. Various classes of functions computable in recursive analysis have recently been characterized in a machine independent and algebraical context. In particular nice connections between the class of computabl ..."
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Recursive analysis is a model of analog computation which is based on type 2 Turing machines. Various classes of functions computable in recursive analysis have recently been characterized in a machine independent and algebraical context. In particular nice connections between the class of computable functions (and some of its sub and supclasses) over the reals and algebraically defined (sub and sup) classes of Rrecursive functions à la Moore have been obtained. We provide in this paper a framework that allows to dive into complexity for functions over the reals. It indeed relates classical computability and complexity classes with the corresponding classes in recursive analysis. This framework opens the field of implicit complexity of functions over the reals. While our setting provides a new reading of some of the existing characterizations, it also provides new results: inspired by Bellantoni and Cook’s characterization of polynomial time computable functions, we provide the first algebraic characterization of polynomial time computable functions over the reals.
Algebraic Characterizations of ComplexityTheoretic Classes of Real Functions, Laboratoire d’informatique de l’école polytechnique
 Alexandria University  Alexandria University
, 2009
"... Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the senseof recursive analysis can bedefined (or characterized) in an algebraic machine independent way, without resorting ..."
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Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the senseof recursive analysis can bedefined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub and supclasses) over the reals and algebraically defined (sub and sup) classes of Rrecursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomialtime computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level. 1