Computing with continuous-time Liapunov systems

• Authors: Jirí Šíma;Pekka Orponen
• Venue: STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
• Year: 2001

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. More precisely, we prove that any function computed by a discrete-time asymmetric recurrent network of n threshold gates can also be computed by a continuous-time symmetrically-coupled Hopfield system of dimension 18n+7. Moreover, if the threshold logic network has maximum weight w_{\max} and converges in discrete time t^*, then the corresponding Hopfield system can be designed to operate in continuous time &THgr;(t^*/&egr;), for any value 0w_{\max}2^{3n}\leq\&egr; 2^{1/&egr;}.The result appears at first sight counterintuitive, because the dynamics of any symmetric Hopfield system is constrained by a Liapunov, or energy function defined on its state space. In particular, such a system always converges from any initial state towards some stable equilibrium state, and hence cannot exhibit nondamping oscillations, i.e. strictly speaking cannot simulate even a single alternating bit. However, we show that if one only considers terminating computations, then the Liapunov constraint can be overcome, and one can in fact embed arbitrarily complicated computations in the dynamics of Liapunov systems with only a modest cost in the system's dimensionality.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.