Its end. If the transformation is discrete, the following method
Gives its length rigorously: Let the sequence of states go on till
Repetition becomes evident, thus
A B C D C D C D C D C … or H E F G G G G G G G …
Then, coming in from the right, make the mark “1” as soon as the
Sequence departs from the cycle, thus
A B1 C D C D C D C D C … or H E F1 G G G G G G G …
Next add the mark “2”, to the right of 1, to include one complete
Cycle, thus
A B1 C D2 C D C D C D C … or H E F1 G2 G G G G G G G…
Then the transient is defined as the sequence of states from the
Initial state to the mark 2: A B C D, or H E F G.
Rigorous form can now be given to the intuitive impression that
Complex systems can produce, in constant conditions, more com-
Plex forms of behaviour than can the simple. By drawing an arbi-
Trary kinematic graph on N states it is easy to satisfy oneself that
If a closed single-valued transformation with N operands is
Applied repeatedly, then the length of transient cannot exceed N
States.
Ex. 1: What property must the graph have if the onset of a recurrence is to be
Postponed as long as possible?
Ex. 2: What is the transient of the system of Ex. 3/6/6, started from the state
(8,5)?
So that they become “coupled” as two automobiles may be locked
Together after an accident. This form, however, is of little interest
To us, for the automobiles are too much changed by the process. U
Flat we want is a way of coupling that does no violence to each
Machine’s inner working, so that after the coupling each machine
Is ,till the same machine that it was before.
For this or this to be so, the coupling must be arranged so that, in
Principlen each machine affects the other only by affecting its con-
Ditions, i. e. by, affecting its input. Thus, if the machines are to
Retain their individual natures after being coupled to form a whole,
The coupling must be between the (given) inputs and outputs, other
Parts being left alone no matter how readily accessible they may be.
Now trace the operation in detail. Suppose a machine (trans-
Ducer) P is to be joined to another, R. For simplicity assume that
P is going to affect R, without R affecting P, as when a micro-
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Phone is joined to an amplifier, or a motor nerve grows down to
Supply an embryonic muscle. We must couple P’s output to R’s
Input. Evidently R’s behaviour, or more precisely the transforma-
Tion that describes R’s changes of state, will depend on, and
Change with, the state of P. It follows that R must have parame-
Ters, for input, and the values of these parameters must be at each
Moment some function of the state of P. Suppose for definiteness
That the machine or transducer R has the three transformations
Shown in S 4/1, i.e.
↓ a b c d
C d d b
B a d c
D c d b
And that P has the transformation, on the three states i, j, k:
R1
R2
R3
i j kP: ↓ k i i
P and R are now to be joined by our specifying what value R’s
Parameter, call it x, is to take when P has any one of its states.
Suppose we decide on the relation Z (a transformation, single-val-
Ued but not closed):
state of P:i j kZ: value of α : ↓ 2 3 2
(The relation between P and α has been made somewhat irregular
To emphasise that the details are quite arbitrary and are completely
49
C O U PLI N G S YS T EM S
A fundamental property of machines is that they can be cou-
Pled. Two or more whole machines can be coupled to form one
Machine; and any one machine can be regarded as formed by the
Coupling of its parts, which can themselves be thought of as small,
Sub-, machines. The coupling is of profound importance in sci-
Ence, for when the experimenter runs an experiment he is cou-
Pling himself temporarily to the system that he is studying. To
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What does this process, the joining of machine to machine or of
Part to part, correspond in the symbolic form of transformations?
Of what does the operation of “coupling” consist?
Before proceeding to the answer we must notice that there is
More than one answer. One way is to force them roughly together,
48
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
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