Now the total combinations possible are eight; Pavlov presented
Only four. It was an essential part of the experiment that the full
Set should not be given, for otherwise there would be nothing par-
Ticular for the animal to learn. Constraint was an essential feature
Of the experiment.
The same principle can be seen more simply in learning by
Association. Suppose one wanted the subject, given a letter, to
Reply with a number according to the rule
A given : reply with 2
B,,: ,,,, 5
C,,: ,,,, 3
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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
The subject might then be given a sequence such as A2, B5, C3,
B5, C3, A2, A2, C3, and so on.
Now this sequence, as a sequence of vectors with two compo-
Nents, shows constraint; and if learning is to occur the constraint
Is necessary; for without constraint A would be followed equally
By 2, 3 or 5; and the subject would be unable to form any specific
Associations. Thus learning is possible only to the extent that the
Sequence shows constraint.
The same is true of learning a maze. For this to occur the maze
Must retain the same pattern from day to day during the period of
Learning. Were the maze to show no constraint, the animal would
Be unable to develop the particular (and appropriate) way of
Behaving. Thus, learning is worth while only when the environ-
Ment shows constraint. (The subject is taken up again in S.13/7.)
V AR I ETY IN MA C H IN E S
We can now turn to considering how variety is affected by
A machine’s activities, making our way towards an understanding
Of what happens to information when it is handled by a machine.
First, let us notice a fundamental peculiarity of the single-valued
Transformation in its relation to variety.
Consider for instance the single-valued transformation
Z: ↓
A B C
B C C
It is not necessary that the transformation should be closed.
Thus if the same set of ten letters is transformed by Y:
A B CY: ↓ p q p
Giving q q p p p p p p q p, the variety falls. It is easy to see that
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Only when the transformation is one-one (over the letters that
Actually occur in the set to be transformed) is the set’s variety
Unchanged; and this is a very special type of transformation.
Ex. 1: Write the letters A to Z in a row; under it, letter by letter, write the first 26
Letters of some well known phrase. The transition from upper row to lower
Now defines a single-valued transformation (u). Write your name in full, find
The variety among its letters, transform by u (i.e. “code” it) and find the vari-
Ety in the new set of letters How has the variety changed ? Apply u repeat-
Edly; draw a graph of how the variety changes step by step.
Ex. 2: In a certain genus of parasite, each species feeds off only one species of
Host. If the varieties (in our sense) of parasites’ species and hosts’ species are
Unequal, which is the larger?
Ex. 3: “A multiplicity of genotypes may show the same phenotypic feature.” If
The change from each genotype to its corresponding phenotype is a transfor-
Mation V, what change in variety does V cause ?
Ex. 4: When a tea-taster tastes a cup of tea, he can be regarded as responsible for
A transformation Y converting “sample of leaf” as operand to “opinion” as
Transform. If the taster is perfect, Y will be one-one. How would he be
Described if Y were many-one ?
Ex. 5: When read to the nearest degree on each of seven occasions, the temper-
Atures of the room and of a thermostatically-controlled water-bath were
Found to be
Room: 65, 62, 68, 63, 62, 59, 61.
Water-bath: 97, 97, 98, 97, 97, 97, 97.
How much variety is shown (i) by the room’s temperatures, (ii) by those of
The bath ? What would have been said had the variety in (i) exceeded that of
Ii)?
* Ex. 6: If the transformation has been formed by letting each state go to one state
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Selected at random from all the states (independently and with equal proba-
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