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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 45 (21 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
Beyond Turing Machines
"... In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are pr ..."
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Cited by 39 (6 self)
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In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are presented. Other
Closedform Analytic Maps in One and Two Dimensions Can Simulate Turing Machines
, 1996
"... We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Int ..."
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Cited by 38 (4 self)
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We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduction Various authors have independently shown [9, 12, 4, 14, 1] that finitedimensional piecewiselinear maps and flows can simulate Turing machines. The construction is simple: associate the digits of the x and y coordinates of a point with the left and right halves of a Turing machine's tape. Then we can shift the tape head by halving or doubling x and y, and write on the tape by adding constants to them. Thus two dimensions suffice for a map, or three for a continuoustime flow. These systems can be thought of as billiards or optical ray tracing in three dimensions, recurrent neural networks, or hybrid systems. However, piecewiselinear functions are not very realistic from a physical p...
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.
A Survey of ContinuousTime Computation Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists o ..."
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Cited by 32 (5 self)
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Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists on the general theory of continuoustime models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
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.
Perturbed Turing Machines and Hybrid Systems
 In Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science. IEEE
, 2001
"... We investigate the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider in our study models for discrete and continuous time dynamical systems: Turing machines, Piecewise affine maps, Linear hybrid automata and Piecewise constant ..."
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Cited by 29 (1 self)
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We investigate the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider in our study models for discrete and continuous time dynamical systems: Turing machines, Piecewise affine maps, Linear hybrid automata and Piecewise constant derivative systems (a simple model of hybrid systems). We associate with each of these models a notion of perturbed dynamics by a small " (w.r.t. to a suitable metrics), and define the perturbed reachability relation as the intersection of all reachability relations obtained by "perturbations, for all possible values of ". We show that for the four kinds of models we consider, the perturbed reachability relation is corecursively enumerable, and that any cor.e. relation can be defined as the perturbed reachability relation of such models. A corollary of this result is that systems that are robust, i.e., their reachability relation is stable under infinitesimal perturbation, are decidable. 1
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
On the Computational Power of Dynamical Systems and Hybrid Systems
 Theoretical Computer Science
, 1996
"... We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as ..."
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Cited by 25 (5 self)
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We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as computational machines. We introduce a new discrete model of computation: the analog automaton model. We characterize the computational power of this model as P=poly in polynomial time and as unbounded in exponential time. We prove that many very simple dynamical systems from literature are able to simulate analog automata. From this results we deduce that many dynamical systems have intrinsically superTuring capabilities. 1 Introduction The computational power of abstract machines which compute over the reals in unbounded precision in constant time is still an open problem. We refer the reader to [18] for an upto date survey. Indeed, a basic model for their computations has been propose...