Fully distinct. The first is that which corresponds to U of S.8/4,
And whose operands are the individual messages; the second is
That of the transducer. Suppose the transducer of S.8/5 is to be
Given a “message” that consists of two letters, each of which may
Be one of Q, R, S. Nine messages are possible:
QQ, QR, QS, RQ, RR, RS, SQ, SR, SS
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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
And these correspond to M1, M2, . . ., M9 of U. Suppose the trans-
Ducer is always started at A; it is easy to verify that the corre-
Sponding nine outputs will be (if we ignore the initial and
Invariable A):
CA, CB, CC, AC, AA, AB, BC, BC, BB.
These are the C1, C2, . . ., C9 of U. Now the coding performed by
The transducer is not one-one, and there has been some loss of
Variety, for there are now only eight distinguishable elements, BC
Being duplicated. This transducer therefore fails to provide the
Possibility for complete and exact decoding; for if BC arrives,
There is no way of telling whether the original message was SQ or
SR.
In this connexion it must be appreciated that an inability to
Decode may be due to one of two very different reasons. It may be
Due simply to the fact that the decoder, which exists, is not at hand.
This occurs when a military message finds a signaller without the
Code-book, or when a listener has a gramophone record (as a
Coded form of the voice) but no gramophone to play it on. Quite
Different is the inability when it is due to the fact that two distinct
Messages may result in the same output, as when the output BC
Comes from the transducer above. All that it indicates is that the
Original message might have been SQ or SR, and the decoder that
Might distinguish between them does not exist.
It is easy to see that if, in each column of the table, every state
Had been different then every transition would have indicated a
Unique value of the parameter; so we would thus have been able
To decode any sequence of states emitted by the transducer. The
Converse is also true; for if we can decode any sequence of states,
Each transition must determine a unique value of the parameter,
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And thus the states in a column must be all different. We have thus
Identified the characteristic in the transducer that corresponds to
Its being a perfect coder.
Ex. 1: In a certain transducer, which has 100 states, the parameters can take 108
Combinations of values; can its output always be decoded? (Hint: Try simple
Examples in which the number of transformations exceeds that of the states.)
Ex. 2: (To emphasise the distinction between the two transformations.) If a trans-
Ducer’s input has 5 states, its output 7, and the message consists of some
Sequence of 12, (i) how many operands has the transducer’s transformation,
And (ii) how many has the coding transformation U?
Ex. 3: If a machine is continuous, what does “observing a transition” correspond
To in terms of actual instrumentation ?
*Ex. 4: If the transducer has the transformation dx/dt = ax, where a is the input,
Can its output always be decoded ? (Hint: Solve for a.)
Designing an inverter. The previous section showed that pro-
Vided the transducer did not lose distinctions in transmission from
Input to output, the coded message given as output could always
Be decoded. In this section we shall show that the same process
Can be done automatically, i.e. given a machine that does not lose
Distinctions, it is always possible to build another machine that,
Receiving the first’s output as input, will emit the original message
As its own output.
We are now adopting a rather different point of view from that
Of the previous section. There we were interested in the possibility
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