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34
Analog computers and recursive functions over the reals
 Journal of Complexity
, 2003
"... In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these mode ..."
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Cited by 40 (19 self)
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In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuoustime model of computation that solve these problems. We also show that this new model preserve all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions). We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals. 1
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 22 (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.
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 22 (14 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 21 (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
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 20 (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.
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 20 (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 Rsubrecursive functions
 THEORETICAL COMPUTER SCIENCE
, 2005
"... 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, linea ..."
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Cited by 16 (5 self)
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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 define 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.
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 13 (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
An analog characterization of elementarily computable functions over the real numbers
 31st Int. Col. on Automata, Languages and Programming (ICALP ’04
, 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 12 (2 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. 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. 1
The Complexity of Real Recursive Functions
 Unconventional Models of Computation (UMC'02), LNCS 2509
, 2002
"... We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. W ..."
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Cited by 11 (5 self)
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We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in recursion theory, and we study their complexity. In particular, we prove both upper and lower bounds for several classes of real recursive functions, which lie inside the primitive recursive functions and, therefore, can be characterized in terms of standard computational complexity.