Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 58

it functions so as to compute the desired logical function. This can be
accomplished by singling out input fibers of some of its elements and output
fibers of some other elements and then defining what original stimuli on
the former are to cause what ultimate responses of the latter.
Since the basic element of the networks to be discussed—the McCulloch
formal neuron—is capable of computing only some of all logical functions
which can be constructed from precisely two variables, and since an arbi-
trary set of propositions may contain temporal relationships, we require
three more steps to reach the generality claimed by the theorem. The first
step involves purely logical argument and shows (a) by using substitution
and the principle of induction the possibility of constructing n -variable
expressions from two-variable expressions, and (b) the possibility of
expressing uniquely any logical function of n -variables in a certain normal
form. The second step introduces an operator S that takes care of a single
synaptic delay and thus permits the representation of temporal relation-
ships, while the third step utilizes some formal properties of this operator
to obtain normal form expressions that are immediately translatable into
network language.
First step:
(a) Consider a logical function of the two variables A 1 and B 1

[ ]

A 11 , . (56)

Let B 1 be a logical function of the two variables A 2 and B 2 :

= [ ]

BAB

, ,

1 2 2

and, in general,

= [ ]

(57)

BAB
i

, .

i + 1 i + 1

Iterative substitution of (57) into (56) gives:

[ ]

AAA , , ,..., A nn , ,

1 2 3

which, by induction, holds for all.
(b) It can be shown (Hilbert and Ackermann, 1928) that any logical func-
tion of n arguments can be represented by a partial conjunction (·) of dis-
junctions (v) that contain each variable X i either affirmed ( x i = 1) or negated
( x i = 0). Let each disjunction be represented by D Z , where Z is the decimal
representation of the binary number

n

Â

0 Z = 2

i - 1

x 2

n

- 1 ,

(58)

i

1

and let K P represent the (partial) conjunction of those disjunctions present
( D Z = 1, otherwise D Z = 0) in the logical function, where P is the decimal
representation of the binary number