On the Computational Power of Discrete Hopfield Nets

Abstract

We prove that polynomial size discrete synchronous Hoppeld networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly. 1 Background Recurrent, or cyclic, neural networks are an intriguing model of massively parallel computation. In the recent surge of research in neural computation, such networks have been considered mostly from the point of view of two types of applications: pattern classiication and associative memory (e. recurrent networks are capable also of more general types of computation, and issues of what exactly such networks can compute, and how they should be programmed, are becoming increasingly topical as work on hardware implementations progresses. (See, for instance, the several designs for analog, digital, hybrid, and optoelectronic implementations of recurrent networks contained in the compendium 30].) In this paper we address the task of comparing the computational power of the most basic recurrent network model, the discrete Hoppeld net 17], to more traditional models of computation. More precisely, our main interest is in the problem of Boolean function computation by undirected, or symmetric networks of weighted threshold logic units; but to get there, we rst have to consider directed, or asymmetric nets. In our model of network computation, the input to a net is initially loaded onto a set of designated input units; then the states of the units in the network are updated repeatedly, according to their local update rules until the network (possibly) converges to some stable global state, at which point the output is read from a set of designated output units. We only consider nite networks of units with binary states. Since our networks in general contain cycles, their behavior may depend signii-cantly on the update order of the units. (Note that we only deal with discrete-time networks here.) The updates may be synchronous, in which case all the units are updated simultaneously in parallel, or asynchronous, in which case the units are selected for updating one at a time in some order. We concentrate here on synchronous networks; not only because they are easier to deal with, but also because we do not

DOI: 10.1007/3-540-56939-1_74

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Cite this paper

@inproceedings{Orponen1993OnTC, title={On the Computational Power of Discrete Hopfield Nets}, author={Pekka Orponen}, booktitle={ICALP}, year={1993} }