Equivocation. A suitable measure for the degree of corrup-
Tion has not, so far as I am aware, been developed for use in the
Basic cases. In the case of the channel that transmits incessantly,
However, Shannon has developed the appropriate measure.
It is assumed first that both the original signals and the received
Signals form Markov chains of the type defined in S.9/4. The data
Of the messages can then be presented in a form which shows the
Frequencies (or probabilities) with which all the possible combi-
Nations of the vector (symbol sent, symbol received) occur. Thus,
To use an example of Shannon’s suppose 0’s and 1’s are being
Sent, and that the probabilities (here relative frequencies) of the
Symbols being received are:
Symbol sent0011
Symbol received 0101
Probability0.495 0.005 0.005 0.495
Of every thousand symbols sent, ten arrive in the wrong form, an
Error of one per cent.
At first sight this “one per cent wrong” might seem the natural
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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
I N CESSA N T TR AN SMI SSIO N
Ex. 4: Find – p log2 p when p is 0.00025. (Hint: Write p as 2.5 × 10– 4 and use (i)).
Ex. 5: During a blood count, lymphocytes and monocytes are being examined
Under the microscope and discriminated by the haematologist. If he mistakes
One in every hundred lymphocytes for a monocyte, and one in every two
Hundred monocytes for a lymphocyte, and if these cells occur in the blood in
The ratio of 19 lymphocytes to 1 monocyte, what is his equivocation? (Hint:
Use the results of the previous two exercises.)
Error-free transmission. We now come to Shannon’s funda-
Mental theorem on the transmission of information in the presence
Of noise (i.e. when other, irrelevant, inputs are active). It might be
Thought that when messages are sent through a channel that sub-
Jects each message to a definite chance of being altered at random,
Then the possibility of receiving a message that is correct with cer-
Tainty would be impossible. Shannon however has shown conclu-
Sively that this view, however plausible, is mistaken. Reliable
Messages can be transmitted over an unreliable channel. The
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Reader who finds this incredible must go to Shannon’s book for
The proof; here I state only the result.
Let the information to be transmitted be of quantity H, and sup-
Pose the equivocation to be E, so that information of amount H– E
Is received. (It is assumed, as in all Shannon’s book, that the trans-
Mission is incessant.) What the theorem says is that if the channel
Capacity be increased by an amount not less than E— by the provi-
Sion perhaps of another channel in parallel— then it is possible so
To encode the messages that the fraction of errors still persisting
May be brought as near zero as one pleases. (The price of a very
Small fraction of errors is delay in the transmission, for enough
Message-symbols must accumulate to make the average of the
Accumulated material approach the value of the average over all
Time.)
Conversely, with less delay, one can still make the errors as few
As one pleases by increasing the channel capacity beyond the min-
Imal quantity E.
The importance of this theorem can hardly be overestimated in
Its contribution to our understanding of how an intricately con-
Nected system such as the cerebral cortex can conduct messages
Without each message gradually becoming so corrupted by error
And interference as to be useless. What the theorem says is that if
Plenty of channel capacity is available then the errors may be
Kept down to any level desired. Now in the brain, and especially
In the cortex there is little restriction in channel capacity, for
More can usually be obtained simply by the taking of more fibres,
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Whether by growth in embryogeny or by some functional tak-
Ing-over in learning.
The full impact of this theorem on neuropsychology has yet to
Be felt. Its power lies not so much in its ability to solve the prob-
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Lem “How does the brain overcome the ever-increasing corruption
Of its internal messages?” as in its showing that the problem
Hardly arises, or that it is a minor, rather than a major, one.
The theorem illustrates another way in which cybernetics can
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