In the companion article[8] 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.
[8] 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[9]
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[10] provides a practical method for solving this problem by means
of the successive use of a certain nilpotent projection operator.
[9] Random cartesian product.
[10] Kleyn, P. A., “Conceptual Design of Self-Organizing Machines,”
Anaheim, California:Northrop Nortronics, NSS Report 2832, Nov. 14, 1963.