Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 29

“Equivalent” here means that a connecting pathway between a receptor in
A and an effector in B should again be represented by a connection, and
the same should hold for no connections.
The answer to this question is in the affirmative; the resulting action
matrix C m is the matrix product of A m and B m :

c = a ¥ b , (3)

ij ij ij

m m m

where according to the rules of matrix multiplication the elements c ij are
defined by

m

 1

c = a b .

(4)

ij ik kj

k =

Fig. 5b shows the transformation of the two cascaded nets into the single
action net. Clearly, this process can be repeated over and over again, and
we have

k

) = '

( Cas AAAA ... A A . (5)

1234 k mi

m

1

Here we have one indication of the difficulty of establishing uniquely the
receptor field of a particular element, because an observer who is aware of
the presence of cascades would maintain that elements e 1 and e 3 in the
second layer of fig. 5a constitute the receptor field of element e 2 in layer
III, while an observer who is unaware of the intermediate layer (fig. 5b) will
argue that elements e 1 , e 2 and e 3 in the first layer define the receptor field
of this element.
In passing, it may be pointed out that matrix multiplication preserves the
multiplicity of pathways as seen in fig. 5, where in the cascaded system
element e 2 , bottom row, can be reached from e 3 , top row, via e 1 as well as
via e 3 , middle row. All other connections are single-valued.
As a final example, we will apply an interesting result in matrix algebra
to cascades of action networks. It can be shown that a square matrix whose
rows are all alike

aa
ij

= , (6)

kj

and each row of which adds up to unity

m

Â

a ij = 1

(7)

j = 1

generates the same matrix, when multiplied by itself:

a ¥ a = a , (8)

ij ij ij

m m m

or

A mm

2 =

. (9)