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11
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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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 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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Cited by 7 (1 self)
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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
, 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.
Distributed Learning of Wardrop Equilibria
 in "7th International Conference on Unconventional Computation  UC 2008) Lecture Notes in Computer Science, Autriche Vienne
, 30
"... Abstract. 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 ev ..."
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Cited by 2 (0 self)
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Abstract. 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. 1
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
Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of
"... 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.
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.
Algebraic Characterization of Complexity Classes of Computable Real Functions
"... Abstract. Several algebraic machineindependant characterizations of computable functions over the reals have been obtained recently. 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 ..."
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Abstract. Several algebraic machineindependant characterizations of computable functions over the reals have been obtained recently. 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. We provide in this paper a framework that allows to relate these classes to classical computability and complexity classes over the integers. While our setting provides a new reading of some of the existing characterizations, this also provides new results: in particular, we provide an algebraic characterization of polynomial time computable real functions. 1