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11
Iteration, Inequalities, and Differentiability in Analog Computers
, 1999
"... Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPACcomputable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G s ..."
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Cited by 36 (16 self)
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Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPACcomputable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+k , then any function of x computable by a Turing machine in T (x) time is also.
An analog characterization of the Grzegorczyk hierarchy
 Journal of Complexity
, 2002
"... We study a restricted version of Shannon's General . . . ..."
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Cited by 36 (16 self)
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We study a restricted version of Shannon's General . . .
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 23 (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.
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
Upper and Lower Bounds on ContinuousTime Computation
"... We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive fun ..."
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Cited by 10 (2 self)
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We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.
An analog characterization of the subrecursive functions
 PROC. 4TH CONFERENCE ON REAL NUMBERS AND COMPUTERS
, 2000
"... We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a dif ..."
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Cited by 7 (1 self)
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We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions. Furthermore, we show that if the machine has access to an oracle which computes a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n − 3 nonlinear differential equations of a certain kind. Therefore, we claim that there is a close connection between analog complexity classes, and the dynamical systems that compute them, and classical sets of subrecursive functions.
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?
Physics and Computation: Essay on the unity of science through computation
"... S. Barry Cooper and Piergiorgio Odifreddi have written the most interesting articles in our times on the philosophy of computing. In this paper I will try to reconcile their own views with established science and scientific criticism. I will take [3] as the main reference containing pointers to many ..."
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S. Barry Cooper and Piergiorgio Odifreddi have written the most interesting articles in our times on the philosophy of computing. In this paper I will try to reconcile their own views with established science and scientific criticism. I will take [3] as the main reference containing pointers to many ideas exposed in previous work of the same authors. Computability Theory has been considered a corpse for mathematicians who did forget the old debate about whether computability theory has useful consequences for mathematics other than those whose statements depend on recursion theoretic terminology. In this context, hypercomputation is a forbidden word because it is not implementable, as foundational criticism says, although mathematicians do not mind to explore Turing degrees such as K KK··· We explore the origins of this criticism and misinterpretations of concepts such as superTuring computational power. To make the discussion opened for all generations of mathematicians and physicists we developed our argumentation in a basis of a language of the late sixties and early seventies, decade of decline of the most enthusiastic Debates in Science: in Mathematics, in Physics, in Cosmology, etc., the times of Radio Programs by Sir Fred Hoyle, the times of the phone calls at the middle of the night between Sir Roger Penrose and Stephen Hawking, the times of the solution of Hilbert’s Tenth
Analog Computers and the Iteration Functional
, 1998
"... In this paper we extend the class of differentially algebraic functions computed by Shannon's General Purpose Analog Computer (GPAC). We relax PourEl's definition of GPAC to obtain new operators and we use recursion theory on the reals to define a new class of analog computable functions. ..."
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In this paper we extend the class of differentially algebraic functions computed by Shannon's General Purpose Analog Computer (GPAC). We relax PourEl's definition of GPAC to obtain new operators and we use recursion theory on the reals to define a new class of analog computable functions. We show that a function F (t; x) which simulates t timesteps of a Turing machine on input x, and more generally a functional that allows us to define the t'th iterate of a definable function, are definable in this system. Therefore, the analog model we propose is closed under (discrete) iteration.