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22
Polynomial differential equations compute all real computable functions on computable compact intervals
, 2007
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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
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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Cited by 17 (6 self)
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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 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 10 (2 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
, 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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Cited by 9 (4 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 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
How to compare the power of computational models
 In Computability in Europe 2005: New Computational Paradigms
, 2005
"... Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physica ..."
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Cited by 3 (3 self)
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Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.” 1
A network model of analogue computation over metric algebras
 Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science
, 2005
"... Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the ..."
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Cited by 3 (1 self)
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Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C[T,A] p ×A r →C[T,A] q (p, q> 0,r ≥ 0), where C[T,A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custommade concrete computation theory over C[T,A] and show that if the modules are concretely computable then so is the function Φ. 1
What lies beyond the mountains? Computational systems beyond the Turing limit
 BULLETIN OF THE EUROPEAN ASSOCIATION FOR THEORETICAL COMPUTER SCIENCE
, 2005
"... Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without ..."
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Cited by 2 (0 self)
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Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without a program. When we observe natural phenomena and endow them with computational significance, it is not the algorithm we are observing but the process. Some objects near us may be performing hypercomputation: we observe them, but we will never be able to simulate their behaviour on a computer. What is then the profit of such a theory of computation to Science?