Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 57

where y o denotes the original perturbation at f o = 0. For a chosen value of
m=10 this equation was numerically evaluated and served as a basis to
conxtruct fig. 18. The entries along the axis of symmetry indicate the focal
points of spines located at corresponding angles f o . The original perturba-
tion is assumed to be strong (y o ª 45 o ).
The sole purpose of this somewhat detailed account of a relatively
insignificant network was to suggest that organized behavior that seems to
be governed by a central control that operates according to an “action at a
distance” principle can very well arise from a localized point function that
permanently links elements in an infinitestimal neighborhood. Notice how
the behavior can be expressed in differential eqs. (52) or (54) which contain
local properties only. The whole system swings into action whenever the
“boundary value”—i.e., the stimulus—changes. The operational principle
here is “action by contagion”. We shall later discuss this principle in greater
detail in connection with interaction networks.
4.2. The McCulloch-Pitts Theorem
A network composed of McCulloch elements we shall call a “McCulloch
formal network”. The central issue of the McCulloch-Pitts (1943) theorem
is the synthesis of such networks, which compute any one of the 2 2 n logical
functions that can be defined by n propositions. In other words, any behav-
ior that can be defined at all logically, strictly, and unambiguously in a finite
number of words can be realized by such a formal network. Since in my
opinion this theorem not only is one of the most significant contributions
to the epistemology of the 20th century, but also gives important clues as
to the analysis of physiological neural nets, it is impossible in an article
about nerve nets not, at least, to touch upon the basic ideas and conse-
quences that are associated with this theorem. Its significance has best been
appraised in the words of the late John Von Neumann (1951):
“It has often been claimed that the activities and functions of the human
nervous system are so complicated that no ordinary mechanism could
possibly perform them. It has also been attempted to name specific func-
tions which by their nature exhibit this limitation. It has been attempted
to show that such specific functions, logically completely described, are
per se unable of mechanical neural realization. The McCulloch-Pitts result
puts an end to this. It proves that anything that can be exhaustively and
unambiguously described, anything that can be completely and unambigu-
ously put into words, is ipso facto realizable by a suitable finite neural
network”.
We shall give now a brief summary of the essential points of this theorem.
As already mentioned, the McCulloch-Pitts theorem shows that to any
logical function of an arbitrary number of propositions (variables) a
network composed of McCulloch elements can be synthesized that is equiv-
alent to any one of these logical functions. By “equivalence” is meant that