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Computability with Polynomial Differential Equations
, 2007
"... In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time. ..."
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Cited by 31 (20 self)
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In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.
Robust simulations of Turing machines with analytic maps and flows
 CiE 2005: New Computational Paradigms, LNCS 3526
, 2005
"... Abstract. In this paper, we show that closedform analytic maps and flows can simulate Turing machines in an errorrobust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time. 1 ..."
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Cited by 25 (7 self)
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Abstract. In this paper, we show that closedform analytic maps and flows can simulate Turing machines in an errorrobust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time. 1
Some recent developments on Shannon’s general purpose analog computer
 Mathematical Logic Quarterly
"... This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be ..."
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Cited by 22 (7 self)
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This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann’s Zeta function. 1
Real recursive functions and their hierarchy
, 2004
"... ... onsidered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1–14). However, one of the operators ..."
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Cited by 22 (2 self)
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... onsidered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1–14). However, one of the operators introduced in the seminal paper by Moore (1996), the minimalization operator, has not been considered: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper, we show that a most natural operator captured from analysis—the operator of taking a limit—can be used properly to enhance the theory of recursion over the reals, providing good solutions to puzzling problems raised by the original model.
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 14 (3 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
Computational bounds on polynomial differential equations
, 2008
"... In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simul ..."
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Cited by 5 (3 self)
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In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuoustime dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initialvalue problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.
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.
Computability on Reals, Infinite Limits and Differential Equations
"... We study a countable class of realvalued functions inductively defined from a basic set of trivial functions by composition, solving firstorder differential equations and the taking of infinite limits. This class is the analytical counterpart of Kleene’s partial recursive functions. By counting th ..."
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We study a countable class of realvalued functions inductively defined from a basic set of trivial functions by composition, solving firstorder differential equations and the taking of infinite limits. This class is the analytical counterpart of Kleene’s partial recursive functions. By counting the number of nested limits required to define a function, this class can be stratified by a potentially infinite hierarchy — a hierarchy of infinite limits. In the first meaningful level of the hierarchy we have the extensions of classical primitive recursive functions. In the next level we find partial recursive functions, and in the following level we find the solution to the halting problem. We use methods from numerical analysis to show that the hierarchy does not collapse, concluding that the taking
A foundation for real recursive function theory
"... The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpos ..."
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The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpose Analog Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of REC(R) where seen to represent different classes of recursive functions, e.g., recursive, primitive recursive and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies. The class of real recursive functions was then stratified in a natural way, and REC(R) and the analytic hierarchy were recently recognized as two faces of the same mathematical concept. In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognize both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added.