Figure 10—Some possible networks of NPO’s
CONCLUSION
The definition of self-organizing behavior suitably represented has permitted the use of Information Theoretic techniques to synthesize a (mathematical) mechanism for a self-organizing machine. Physical mechanization in the form of an NPO has been accomplished and has introduced the experimental phase of the program. From among the many items deserving of further study we may mention: more economical physical mechanization through introduction of modern technology; identification of networks of NPO’s with their group theoretic descriptions; analysis of the dimensionality of tasks which a SOM might be called on to simulate, and prototype SOM applications to related tasks. It is hoped that progress along these lines can be reported in the future.
REFERENCES
| 1. | Ścibor-Marchocki, Romuald I., |
| “A Topological Foundation for Self-Organization,” | |
| Anaheim, California:Northrop Nortronics, NSS Report 2828, | |
| November 14, 1963 | |
| 2. | It is true that our definition is very similar to that proposed by Hawkins (reference 5). Compare for example his definition of learning machines (page 31 of reference 5). But the subsequent developments reviewed therein are different from the one we have followed. |
| 3. | Ashby, W. R., |
| “The Set Theory of Mechanism and Homeostasis,” | |
| Technical Report 7, University of Illinois, | |
| September 1962 | |
| 4. | Ashby, W. R., |
| “Systems and Information,” | |
| Transactions PTGME MIL-7:94-97 | |
| (April-July, 1963) | |
| 5. | Hawkins, J. K., |
| “Self-Organizing Systems—A Review and Commentary,” | |
| Proc. IRE. 49:31-48 (January 1961) | |
| 6. | Mesarovic, M. D., |
| “On Self Organizational Systems,” | |
| Spartan Books, pp. 9-36, 1962 | |
| 7. | Braverman, D., |
| “Learning Filters for Optimum Pattern Recognition,” | |
| PGIT IT-8:280-285 (July 1962) | |
| 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.