Part 13

8. We make the latter statement despite the fact that we employ a statistical treatment of self-organization. We may predict the performance of, for example, the NPO by using a statistical description, but it does not necessarily follow that the NPO computes statistics.

9. McCulloch, W. S., and Pitts, W., “A Logical Calculus of the Ideas Imminent in Nervous Activity,” Bull-Math. Biophys =5=:115 (1943)

10. Newell, A., Shaw, J. C., and Simon, H. A., “Empirical Explorations of the Logic Theory Machine: A Case Study in Heuristic,” Proc. WJCC, pp. 218-230, 1957

10a. The spaces W, X, Y, and Z are stochastic spaces; that is, each space is defined as the ordered pair (X,p(X)) where p(X) = {p(x) ∋ x ∈ X}, p(x) ≥ 0, x ∈ X and ∫x p(x)dx = 1. Such spaces possess a metrizable topology.

11. We use the following convention for probability distributions: if the arguments of p( ) are different, they are different functions, thus: p(x) ≠ p(y) even if y = x.

12. One can prove the existence of a metric directly but in order to perform the metrization the space has to be decomposed first. But decomposing a space without having a metric calls for a neat trick, accomplished (as far as we know) only by the method used by the SOM.

12a. In this example we use a hemisphere; in general, it would be a spherical cap.

A Topological Foundation for Self-Organization

R. I. ŚCIBOR-MARCHOCKI

Northrop Nortronics Systems Support Department Anaheim, California

It is shown that by the use of Information Theory, any metrizable topology may be metrized as an orthogonal Euclidean space (with a random Gaussian probability distribution) times a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups. The necessary algorithm to accomplish this metrization from a statistical basis is presented. If such a basis is unavailable, a certain nilpotent projection operator has to be used instead, as is shown in detail in the companion paper. This operator possesses self-organizing features.

INTRODUCTION

In the companion article we will define a self-organizing system as one which, after observing the input and output of an unknown phenomenon (transfer relation), organizes itself into a simulation of the unknown phenomenon.

Kleyn, P. A., “Conceptual Design of Self-Organizing Machines,” Anaheim, California:Northrop Nortronics, NSS Report 2832, Nov. 14, 1963.

Within the mathematical model, the aforementioned phenomenon may be represented as a topological space thus omitting for the moment the (arbitrary) designation of input and output which, as will be shown, bears on the question of uniqueness. Hence, for the purpose of this paper, which emphasizes the mathematical foundation, an intelligent device is taken as one which carries out the task of studying a space and describing it.

In keeping with the policy that one should not ask someone (or something) else to do a task that he could not do himself (at least in principle), let us consider how we would approach such a problem.

In the first place, we have to select the space in which the problem is to be set. The most general space that we feel capable of tackling is a metrizable topology. On the other hand, anything less general would be unnecessarily restrictive. Thus, we choose a metrizable topological space.

As soon as we have made this choice, we regret it. In order to improve the situation somewhat, we show that there is no (additional) loss of generality in using an orthogonal Euclidean space times a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups.

This paper provides a survey of the problem and a method for solving it which is conceptually clear but not very practical. The companion paper provides a practical method for solving this problem by means of the successive use of a certain nilpotent projection operator.

Random cartesian product.

Kleyn, P. A., “Conceptual Design of Self-Organizing Machines,” Anaheim, California:Northrop Nortronics, NSS Report 2832, Nov. 14, 1963.

METRIZATION

We start with a metrizable topological space. There are many equivalent axiomatizations of a metrizable topology; e.g., see Kelley. Perhaps the easiest way to visualize a metrizable topology is to consider that one was given a metric space but that he lost his notes in which the exact form of the metric was written down. Thus one knows that he can do everything that he could in a metric space, if only he can figure out how.

The “figuring out how” is by no means trivial. Here, it will be assumed that a cumulative probability distribution has been obtained on the space by one of the standard methods; bird in cage, Munroe I, Munroe II, ordering (see Halmos or Kelley). This cumulative probability distribution is a function on X onto the interval [0,1] of real numbers. The inverse of this function, which exists by the Radon Nikodym theorem, provides a mapping from the real interval onto the non-trivial portion of X. This mapping induces all of the pleasant properties of the real numbers on the space X: topological, metric, and ordering.

Actually, it turns out that, especially if the dimensionality of the space is greater than one, the foregoing procedure not only provides one metrization, but many. Indeed, this lack of uniqueness is what makes the procedure exceedingly difficult. Only by imposing some additional conditions that result in the existence of a unique solution, does the problem become tractable.

We choose to impose the additional condition that the resulting metric space be a Euclidean geometry with a rectangular coordinate system.

Harman, W. W., “Principles of the Statistical Theory of Communication,” New York, New York:McGraw-Hill, 1963.

Munroe, M. E., “Introduction to Measure and Integration,” Cambridge, Mass.:Addison-Wesley, 1953.