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.