Which we shall use as the standard form. (In this chapter we shall
Continue to discuss only transformations that are closed and sin-
Gle-valued.)
A transformation corresponds to a machine with a characteris-
Tic way of behaving (S.3/1); so the set of three— R 1, R2, and R3—
If embodied in the same physical body, would have to correspond
To a machine with three ways of behaving. Can a machine have
Three ways of behaving?
It can, for the conditions under which it works can be altered.
Many a machine has a switch or lever on it that can be set at any
One of three positions, and the setting determines which of three
Ways of behaving will occur. Thus, if a, etc., specify the machine’s
States, and R1 corresponds to the switch being in position 1, and R2
Corresponds to the switch being in position 2, then the change of
R’s subscript from 1 to 2 corresponds precisely with the change of
The switch from position 1 to position 2; and it corresponds to the
Machine’s change from one way of behaving to another.
It will be seen that the word “change” if applied to such a
Machine can refer to two very different things. There is the change
From state to state, from a to b say, which is the machine’s behav-
Iour, and which occurs under its own internal drive, and there is
The change from transformation to transformation, from R1 to R2
Say, which is a change of its way of behaving, and which occurs
At the whim of the experimenter or some other outside factor. The
Distinction is fundamental and must on no account be slighted.
R’s subscript, or any similar symbol whose value determines
Which transformation shall be applied to the basic states will be
Called a parameter. If numerical, it must be carefully distin-
Guished from any numbers that may be used to specify the oper-
Ands as vectors.
R1
R2
R3
43
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 MA C HI N E WI TH I N PUT
A real machine whose behaviour can be represented by such a
Set of closed single-valued transformations will be called a trans-
Ducer or a machine with input (according to the convenience of
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The context). The set of transformations is its canonical represen-
Tation. The parameter, as something that can vary, is its input.
Ex. 1: If S is
a b ↓ b a ,
How many other closed and single-valued transformations can be formed on
The same two operands?
Ex. 2: Draw the three kinematic graphs of the transformations R1, R2, and R3
Above. Does change of parameter-value change the graph?
Ex. 3: With R (above) at R1, the representative point is started at c and allowed
To move two steps (to R12(c)); then, with the representative point at this new
State, the transformation is changed to R2, and the point allowed to move two
More steps. Where is it now?
Ex. 4: Find a sequence of R’s that will take the representative point (i) from d to
A, (ii) from c to a.
Ex. 5: What change in the transformation corresponds to a machine having one
Of its variables fixed? What transformation would be obtained if the system
x' = – x + 2y
y' = x – y
Were to have its variable x fixed at the value 4?
Ex. 6: Form a table of transformations affected by a parameter, to show that a
Parameter, though present, may in fact have no actual effect.
Quantities that appear on the right, but not on the left, must be
Parameters. The examples below will clarify the facts.
Ex. 1: What are the three transformations obtained by giving parameter a the val-
ues – 1, 0, or +1 in Ta :
Ta : g' = (1 – a)g + (a – 1)h
h' = 2g + 2ah
Ex. 2: What are the two transformations given when the parameter a takes the
Value 0 or 1 in S?:
h' = (1 – α )j + log (1 + α + sin α h)
S: j' = (1 + sin α j) e( α – 1)h
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Ex. 3: The transducer n' = n + a2, in which a and n can take only positive integral
values, is started at n = 10. (i) At what value should a be kept if, in spite of
Repeated transformations, n is to remain at 10? (ii) At what value should a be
kept if n is to advance in steps of 4 at a time (i.e. 10, 14 18, …)? (iii) What
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