Continuous-time symmetric Hopfield nets are computationally universal

  • Authors:

  • Jiří Šíma;Pekka Orponen

  • Affiliations:

  • Institute of Computer Science, Academy of Sciences of the Czech Republic, P.O. Box 5 182 07 Prague 8, Czech Republic;Laboratory for Theoretical Computer Sciecne, Helsinki University of Technology, P.O. Box 5400, FIN-02015 HUT, Finland

  • Venue:
  • Neural Computation
  • Year:
  • 2003

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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 Lyapunov-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 Wmax 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 such that Wmax212n ≤ ε21/ε. In terms of standard discrete computation models, our result implies that any polynomially space-bounded Turing machine can be simulated by a family of polynomial-size continuous-time symmetric Hopfield nets.