A Computationally Universal Field Computer That is

A eld computer is a (spatial) continuum-limit neural net (MacLennan 1987). We investigate eld computers whose temporal dynamics is also continuum-limit, being governed by an integro-diierential equation. We prove that even when they are purely linear, such systems are computationally universal. The \trick" used to get such universal (and therefore in general nonlinear) behavior is quite similar to the way nonlinear macro-scopic physics arises from the purely linear microscopic physics of Schrr odinger's equation: one interprets the system in a non-linear way. In this paper, we show that simply using a unary code for the interpretation suuces. See Wolpert and MacLennan (1993) for full details and additional material. 1 A purely linear continuum-limit neural net The most natural way to extend conventional neural nets to the continuum limit in the set of neurons is to invoke the concept of a \\eld", that is a real-valued function on an n-dimensional Euclidean space (MacLen-nan 1987). In this extension the continuum of neurons is indexed by real vectors r 2 R n , and the activation of a neuron is represented by r = (r), the eld's value at that point. We represent the dynamics of such a eld with a function f : R ! (R n), where (R n), is some convenient space of allowed elds (e.g., L 2 (R n)). So = f(t 0) is interpreted as the state of a continuum-limit neural net at time t 0 ; r = f(t 0 ; r) is the activation value at the time t 0 of the neuron indexed by r. In this paper we concentrate on elds whose dynamics is purely linear. The dependence of f on t (i.e., the dynamics of the net) is determined by the continuum-limit version of what neural net dynamics would be if there were no sigmoidal non-linearity, i.e., by the continuum-limit version of multiplying by a weight matrix. More precisely, the dynamics is given by the (linear) integro-diierential evolution equation, @ t f(t; r) = Z dr 0 G(r; r 0)f(t; r 0); (1) which we abbreviate as _ f(t) = Gf(t): (2) The \weight matrix" of the net corresponds to the kernel (G 2 (R n R n)) of this evolution equation. In particular, a kernel that is non-zero for all values of its arguments corresponds to a neural net whose weight matrix is fully (recurrently) connected. The most natural way of assigning …