In some one of the states S1, S2, . . ., Sn, which will be assumed
Here to be finite in number. It has one or more parameters that can
Take, at each moment, some one of a set of values P1, P2, . . ., Pk.
Each of these values will define a transformation of the S’s. We
Now find that such a system can accept a message, can code it, and
Can emit the coded form. By “message” I shall mean simply some
Succession of states that is, by the coupling between two systems,
At once the output of one system and the input of the other. Often
The state will be a vector. I shall omit consideration of any “mean-
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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
T RA N SMISSI O N O F VA R IE TY
Ing” to be attached to the message and shall consider simply what
Will happen in these determinate systems.
For simplicity in the example, suppose that M can take any one
Of four states: A, B, C, and D; that the parameters provide three
States Q, R, and S. These suppositions can be shown in tabular
Form, which shows the essentials of the “transducer” (as in S.4/l):
↓
Ex. 6: Pass the message “314159 . . .” (the digits of π) through the transducer n'
= n + a— 5, starting the transducer at n = 10.
Ex. 7: If a and b are parameters, so that the vector (a,b) defines a parameter state,
And if the transducer has states defined by the vector (x,y) and transformation
x' = ax + by
y' = x + (a – b)y,
A
B
C D
Complete the trajectory in the table:
a
b
x
y
1
– 1
2
1
– 20
11
12
4 – 11
– 1
0
?
?
2
1
?
?
5
– 2
?
?
– 2
0
?
?
Q C C A B
R A C B B
S B D C D
Given its initial state and the sequence of values given to the
Parameter, its output can be found without difficulty, as in S.4/1.
Thus, suppose it starts at B and that the input is at R; it will change
To C. If the input goes next to Q, it will go from C to A. The results
So far can be shown in tabular form:
Input-state:
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Transducer-state:
R
B
Q
C
A
*Ex. 8: A transducer, with parameter u, has the transformation dx/dt = – (u + 4)x;
it is given, from initial state x = 1, the input u = cos t; find the values of x as
Output.
*Ex. 9: If a is input to the transducer
dx/dt = y
dy/dt = – x – 2y + a,
With diagram of immediate effects
a → y → x, ←
what is the output from x if it is started at (0,0) with input a = sin t? (Hint:
Use the Laplace transform.)
*Ex. 10: If a is input and the transducer is
dx/dt = k(a – x)
What characterises x’s behaviour as k is made positive and numerically larger
And larger?
It can now easily be verified that if the initial state is B and the
Input follows the sequence R Q R S S Q R R Q S R, the output
Will follow the sequence B C A A B D B C B C C B.
There is thus no difficulty, given the transducer, its initial state,
And the input sequence, in deducing its trajectory. Though the
Example may seem unnatural with its arbitrary jumps, it is in fact
Quite representative, and requires only more states, and perhaps
The passage to the limit of continuity to become a completely nat-
Ural representation. In the form given, however, various quantita-
Tive properties are made obvious and easily calculable, where in
The continuous form the difficult technique of measure theory
Must be used.
Ex. 1: Pass the same message (R Q R S S Q R R Q S R) through the same trans-
Ducer, this time starting at A.
Ex. 2: Pass the message “R1, R2, R3, R1, R2, R3” through the transducer of S.4/1,
Starting it at a.
Ex. 3: (Continued.) Encode the same message through the same transducer, start-
Ing it at b.
Ex. 4: (Continued.) Does a transducer’s output depend, for given input, on its ini-
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Tial state?
Ex. 5: If the transducer is n' = n – a, where a is a parameter, what will its trajec-
tory be if it starts at n = 10 and receives the input sequence 2, 1, – 3, – 1, 2, 1?
I NVER TI NG A CO DED M ES S AGE
In S.8/4 it was emphasised that, for a code to be useful as a
Message-carrier, the possibility of its inversion must exist. Let us
Attempt to apply this test to the transducer of S.8/5, regarding it as
A coder.
There are two transformations used, and they must be kept care-
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