Tions of this chapter will have made clear that a theory of such
Unanalysed states can be rigorous.
Nevertheless, systems often have states whose specification
Demands (for whatever reason) further analysis. Thus suppose a
News item over the radio were to give us the “state”, at a certain
Hour, of a Marathon race now being run; it would proceed to give,
For each runner, his position on the road at that hour. These posi-
Tions, as a set, specify the “state” of the race. So the “state” of the
Race as a whole is given by the various states (positions) of the
Various runners, taken simultaneously. Such “compound” states
Are extremely common, and the rest of the book will be much con-
30
Is merely a vector written vertically.
Two vectors are considered equal only if each component of
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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
TH E D ET ERM IN A TE MA C HI N E
The one is equal to the corresponding component of the other.
Thus if there is a vector (w,x,y,z), in which each component is
Some number, and if two particular vectors are (4,3,8,2) and
(4,3,8,1), then these two particular vectors are unequal; for, in the
Fourth component, 2 is not equal to 1. (If they have different com-
Ponents, e.g. (4,3,8,2) and (H,T), then they are simply not compa-
Rable.)
When such a vector is transformed, the operation is in no way
Different from any other transformation, provided we remember
That the operand is the vector as a whole, not the individual com-
Ponents (though how they are to change is, of course, an essential
Part of the transformation’s definition). Suppose, for instance, the
“system” consists of two coins, each of which may show either
Head or Tail. The system has four states, which are
H,H) (H,T) (T,H) and (T,T).
Suppose now my small niece does not like seeing two heads up,
But always alters that to (T,H), and has various other preferences.
It might be found that she always acted as the transformation
N: ↓
(H,H) (H,T) (T,H) (T,T)
(T,H) (T,T) (T,H) (H,H)
Ex. 4: (Continued.) Express the transformation as a kinematic graph.
Ex. 5: The first operand, x, is the vector (0,1,1); the operator F is defined thus:
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I) the left-hand number of the transform is the same as the middle number
Of the operand;
Ii) the middle number of the transform is the same as the right-hand number
Of the operand;
Iii) the right-hand number of the transform is the sum of the operand’s mid-
Dle and right-hand numbers.
Thus, F(x) is (1,1,2), and F2(x) is (1,2,3). Find F3(x), F4(x), F5(x). (Hint:
Compare Ex. 2/14/9.)
As a transformation on four elements, N differs in no way from
Those considered in the earlier sections.
There is no reason why a transformation on a set of vectors
Should not be wholly arbitrary, but often in natural science the
Transformation has some simplicity. Often the components
Change in some way that is describable by a more or less simple
Rule. Thus if M were:
(H,T) (T,H) (T,T)M: ↓ (H,H) (T,T) (H,H) (H,T)(T,H)
It could be described by saying that the first component always
Changes while the second always remains unchanged.
Finally, nothing said so far excludes the possibility that some or
all of the components may themselves be vectors! (E.g. S.6/3.)
But we shall avoid such complications if possible.
Ex. 1: Using ABC as first operand, find the transformation generated by repeated
Application of the operator “move the left-hand letter to the right” (e.g. ABC
→ BCA).
Ex. 2: (Continued.) Express the transformation as a kinematic graph.
Ex. 3: Using (1, – 1) as first operand, find the other elements generated by
Repeated application of the operator “interchange the two numbers and then
Multiply the new left-hand number by minus one”.
Notation. The last exercise will have shown the clumsiness of
Trying to persist in verbal descriptions. The transformation F is in
Fact made up of three sub-transformations that are applied simul-
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Taneously, i.e. always in step. Thus one sub-transformation acts on
the left-hand number, changing it successively through 0 → 1 →
1 → 2 → 3 → 5, etc. If we call the three components a, b, and c,
Then F, operating on the vector (a, b, c), is equivalent to the simul-
Taneous action of the three sub-transformations, each acting on
One component only:
a' = b
F: b' = c
c' = b + c
Thus, a' = b says that the new value of a, the left-hand number in
The transform, is the same as the middle number in the operand;
And so on. Let us try some illustrations of this new method; no
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