A Computational Universal Field Computer That is Purely Linear

As defined in MacLennan (1987), a {\it field computer} is a (spatial) continuum-limit neural net. This paper investigates field computers whose dynamics is also contiuum-limit, being governed by a purely linear integro-differential equation. Such systems are motivated both as as a means of studing neural nets and as a model for cognitive processing. As this paper proves, such systems are computationally universal. The ``trick'' used to get such universal nonlinear behavior from a purely linear system is quite similar to the way nonlinear macroscopic physics arises from the purely linear microsopic physics of Schr\"odinger's equation. More precisely, the ``trick'' involves two parts. First, the kind of field computer studied in this paper is a continuum-limit threshold neural net. That is, the meaning of the system's output is determined by which neurons have an activation exceeding a threshold (which in ths paper is taken to be 0), rather than by the actual activation values of the neurons. Second, the occurence of output is determined in the same thresholding fashion; output is available only when certain {\it output-flagging} neurons exceed the threshold, rather than after a certain fixed number of iterations of the system. In addition to proving and discussing their computational universality, this paper cursorily investigates the dynamics of these systems.