Abstract

We propose a unifying approach to the analysis of computational aspects of symmetric Hopfield nets which is based on the concept of "energy source". Within this framework we present different results concerning the computational power of various Hopfield model classes. It is shown that polynomial-time computations by nondeterministic Turing machines can be reduced to the process of minimizing the energy in Hopfield nets (the MIN ENERGY problem). Furthermore, external and internal sources of energy are distinguished. The external sources include e.g. energizing inputs from so-called Hopfield languages, and also certain external oscillators that prove finite analog Hopfield nets to be computationally Turing universal. On the other hand, the internal source of energy can be implemented by a symmetric clock subnetwork producing an exponential number of oscillations which are used to energize the simulation of convergent asymmetric networks by Hopfield nets. This shows that infinite families of polynomial-size Hopfield nets compute the complexity class PSPACE/poly. A special attention is paid to generalizing these results for analog states and continuous time to point out alternative sources of efficient computation. 1

Citations