Abstract

We establish a fundamental result in the theory of computation by continuous-time dynamical systems, by showing that systems corresponding to so called continuous-time symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained, Liapunov-function 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 discrete-time binary neurons, with in general asymmetric coupling weights, can be simulated by a symmetric continuous-time Hopfield net containing only 18n+7 units employing the saturated-linear 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 Θ(t*/ε), for any ε > 0...

Citations