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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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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...
The Computational Power of Continuous Time Neural Networks
 In Proc. SOFSEM'97, the 24th Seminar on Current Trends in Theory and Practice of Informatics, Lecture Notes in Computer Science
, 1995
"... We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11 ..."
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We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11], John Hopfield introduced a continuoustime version of the neural network model whose discretetime variant he had discussed in his seminal 1982 paper [10]. The 1984 paper also contains an electronic implementation scheme for the continuoustime networks, and an argument showing that for sufficiently largegain nonlinearities, these behave similarly to the discretetime ones, at least when used as associative memories. The power of Hopfield's discretetime networks as generalpurpose computational devices was analyzed in [17, 18]. In this paper we conduct a similar analysis for networks consisting of Hopfield's continuoustime units; however we are at this stage able to analyze only the gen...
An Overview Of The Computational Power Of Recurrent Neural Networks
 Proceedings of the 9th Finnish AI Conference STeP 2000{Millennium of AI, Espoo, Finland (Vol. 3: &quot;AI of Tomorrow&quot;: Symposium on Theory, Finnish AI Society
, 2000
"... INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. His ..."
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INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. Historically, the brain theory interest was primary [32], but with the advances in computer technology, the application potential of the statistical modeling techniques has shifted the balance. 1 The study of neural networks as general computational devices does not strictly follow this division of interests: rather, it provides a general framework outlining the limitations and possibilities aecting both research domains. The prime historic example here is obviously Minsky's and Papert's 1969 study of the computational limitations of singlelayer perceptrons [34], which was a major inuence in turning away interest from neural network learning to symbolic AI techniques for more
ContinuousTime Symmetric Hopfield Nets Are Computationally Universal
"... We establish a fundamental result in the theory of computation by continuoustime dynamical systems, by showing that systems corresponding to so called continuoustime symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained, Liapunovfunction ..."
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We establish a fundamental result in the theory of computation by continuoustime dynamical systems, by showing that systems corresponding to so called continuoustime symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained, Liapunovfunction controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discretetime binary neurons, with in general asymmetric coupling weights, can be simulated by a symmetric continuoustime Hopfield net containing only 18n+7 units employing the saturatedlinear activation function. Moreover, if the asymmetric network has maximum integer weight size w_max and converges in discrete time t*, then the corresponding Hopfield net can be designed to operate in continuous time &Theta;(t*/&epsilon;), for any &epsilon; > 0...
LETTER Communicated by Eduardo Sontag On the Effect of Analog Noise in DiscreteTime Analog Computations
"... We introduce a model for analog computation with discrete time in the presence of analog noise that is flexible enough to cover the most important concrete cases, such as noisy analog neural nets and networks of spiking neurons. This model subsumes the classical model for digital computation in th ..."
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We introduce a model for analog computation with discrete time in the presence of analog noise that is flexible enough to cover the most important concrete cases, such as noisy analog neural nets and networks of spiking neurons. This model subsumes the classical model for digital computation in the presence of noise. We show that the presence of arbitrarily small amounts of analog noise reduces the power of analog computational models to that of finite automata, and we also prove a new type of upper bound for the VCdimension of computational models with analog noise. 1