In which each question mark has to be replaced by a letter. If the replace-
Ments are otherwise unrestricted, what variety (logarithmic) is there in the
Set of all possible such transformations ?
Ex. 8: (Continued.) If the closed transformation had n states what variety is
There?
Ex. 9: If the English vocabulary has variety of 10 bits per word, what is the stor-
Age capacity of 10 minutes, speech on a gramophone record, assuming the
Speech is at 120 words per minute?
Ex. 10: (Continued.) How does this compare with the capacity of a printed page
Of newspaper (approximately)?
Ex. 11: (Continued.) If a pamphlet takes 10 minutes to be read aloud, how does
Its variety compare with that of the gramophone record?
Ex. 12: What set is the previous Ex. referring to?
Ex. 13: Can a merely negative event— a light not being lit, a neuron not being
Excited, a telegram not arriving— be used as a contribution to variety ?
C ONS TR AI NT
A most important concept, with which we shall be much con-
Cerned later, is that of constraint. It is a relation between two sets,
And occurs when the variety that exists under one condition is less
Than the variety that exists under another. Thus, the variety in the
Human sexes is I bit; if a certain school takes only boys, the variety
In the sexes within the school is zero; so as 0 is less than 1, con-
Straint exists.
Another well-known example is given by the British traffic
Lights, which have three lamps and which go through the sequence
(where “+” means lit and “1” unlit):
Red:
Yellow:
Green:
(1) (2) (3) (4) (1) …
+ +00+ …
0+0+0 …
00+00 …
Four combinations are thus used. It will be noticed that Red is, at
Various times, both lit and unlit; so is Yellow; and so is Green. So
If the three lights could vary independently, eight combinations
Could appear. In fact, only four are used; so as four is less than
Eight, constraint is present.
A constraint may be slight or severe. Suppose, for instance,
That a squad of soldiers is to be drawn up in a single rank, and that
“independence” means that they may stand in any order they
Please. Various constraints might be placed on the order of stand-
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Ing, and these constraints may differ in their degree of restriction.
Thus, if the order were given that no man may stand next a man
Whose birthday falls on the same day, the constraint would be
Slight, for of all the possible arrangements few would be
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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
Q UA N TI TY O F V AR IE TY
Excluded. If, however, the order were given that no man was to
Stand at the left of a man who was taller than himself, the con-
Straint would be severe; for it would, in fact, allow only one order
Of standing (unless two men were of exactly the same height). The
Intensity of the constraint is thus shown by the reduction it causes
In the number of possible arrangements.
It seems that constraints cannot be classified in any simple
Way, for they include all cases in which a set, for any reason, is
Smaller than it might be. Here I can discuss only certain types of
Outstanding commonness and importance, leaving the reader to
Add further types if his special interests should lead him to them.
Constrain in vectors. Sometimes the elements of a set are
Vectors, and have components. Thus the traffic signal of S.7/8 was
A vector of three components, each of which could take two values.
In such cases a common and important constraint occurs if the
Actual number of vectors that occurs under defined conditions is
Fewer than the total number of vectors possible without conditions
(i.e. when each component takes its full range of values independ-
Ently of the values taken by the other components). Thus, in the
Case of the traffic lights, when Red and Yellow are both lit, only
Green unlit occurs, the vector with Green lit being absent.
It should be noticed that a set of vectors provides several varie-
Ties, which must be identified individually if confusion is not to
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Occur. Consider, for instance, the vector of S.3/5:
Age of car, Horse-power, Colour).
The first component will have some definite variety, and so will
The second component, and the third. The three varieties need not
Be equal. And the variety in the set of vectors will be different
Again.
The variety in the set of vectors has, however, one invariable
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