Treating the changes that each state and parameter undergo indi-
Vidually, shows the relations that are involved in “coupling” with
Perfect clarity and generality. Various modifications can be devel-
Oped without any loss of this clarity.
Thus suppose the machines are specified, as is common, in
Terms of vectors with numerical components; then the rule for
Coupling remains unaltered: each machine must have one or more
Parameters, and the coupling is done by specifying what function
These parameters are to be of the other machine’s variables. Thus
The machines M and N
a' = a2 + pb c' = rsc + ud2
M: b' = – qaN: d' = 2tue
e' = uce
Might be joined by the transformations U and V:
=a + b
r =a – b
p = 2cs
U: q = de2V: t = – a u = b2
U is a shorthand way of writing a whole set of transitions from a
Value of (c,d,e) to a value of (p,q), e.g.
(0,0,0) (0,0,1) (1,3,5) (2,2,4)U: ↓ (0,0) (0,0) (2,75) (4,32)
Similarly for V, a transformation from (a,b) to (r,s,t,u), which
includes, e.g. (5,7) → (12, – 2, – 5, 49) (and compare P of S.6/9).
The result of the coupling is the five-variable system with rep-
Resentation:
a' = a2 + 2bc
b' = – ade2
c' = (a2 – b2)c + b2d2
d' – 2ab2e
e' = b2ce
(Illustrations of the same process with differential equations have
Been given in Design for a Brain, S.21/6.)
Ex. 1.: Which are the parameters in M? Which in N?
Ex. 2.: Join M and N by W and X, and find what state (1, 0, 0, 1, 0), a value of (a,
B, c, d, e), will change to:
r = a
p = ds = ab
W:
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X: t = a q = cu = a
Ex. 4/7/4 has already shown that parts can, in general, be
Coupled in different ways to form a whole. The defining of the
Component parts does not determine the way of coupling.
From this follows an important corollary. That a whole machine
Should be built of parts of given behaviour is not sufficient to
Determine its behaviour as a whole: only when the details of cou-
Pling are added does the whole’s behaviour become determinate.
FE EDB AC K
In S.4/7, P and R were joined so that P affected R while R
Had no effect on P. P is said to dominate R, and (to anticipate S.4/
We may represent the relation between the parts by
P → R
(The arrow cannot be confused with that used to represent a tran-
Sition (S.2/2), for the latter always relates two states, whereas the
Arrow above relates two parts. In the diagrams to come, parts will
Always be shown boxed.)
Cybernetics is, however, specially interested in the case of S.4/8
Where each affects the other, a relation that may be represented by
P ← R →
When this circularity of action exists between the parts of a
Dynamic system, feedback may be said to be present.
The definition of feedback just given is that most in accord with
The spirit of this book, which is concerned essentially with princi-
Ples.
Other definitions, however, are possible, and there has been
Some dispute as to the best; so a few words in explanation may be
Useful. There are two main points of view that have to be consid-
Ered.
On the one side stand those who are following the path taken by
This book— those whose aim is to get an understanding of the prin-
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Ciples behind the multitudinous special mechanisms that exhibit
Them To such workers, “feedback” exists between two parts when
Each affects the other, as for instance, in
x' = 2xy
y' = x – y2
For y’s value affects how x will change and so does x’s value affect
Y. By contrast, feedback would not be said to be present in
x' = 2x
y' = x – y2
53
52
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E MA C HI N E WI TH I N PUT
For x’s change does not now depend on y’s value; x dominates y,
And the action is one way only.
On the other side stand the practical experimenters and con-
Structors, who want to use the word to refer, when some forward
Effect from P to R can be taken for granted, to the deliberate con-
Duction of some effect back from R to P by some connexion that
I; physically or materially evident. They object to the mathemati-
Cian’s definition, pointing out that this would force them to say
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