X on y was considered over a single step only, and this restriction
Is necessary in the basic theory. x was found to have no immediate
Effect on y; it may however happen that x has an immediate effect
On u and that u has an immediate effect on y, then x does have
Some effect on y, shown after a delay of one extra step. Such an
Effect, and those that work through even longer chains of vari-
Ables and with longer delay, will be referred to as ultimate
Effects. A diagram of ultimate effects can be constructed by
Drawing an arrow from A to B if and only if A has an ultimate
Effect on B. The two diagrams are simply related, for the diagram
57
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
Of immediate effects, if altered by the addition of another arrow
Wherever there are two joined head to tail, turning
u
x
y
x
u
→
y
What of y’s transitions? We can re-classify them, with x as
parameter, by representing, e.g. “00 → 01” as “when x = 0, y goes
From 0 to 1”. This gives the table
Y
0 1 2 ↓
01 0 1
X11 0 1
21 0 1
It shows at once that y’s transitions do not depend on the value
Of x. So x has no immediate effect on y.
Now classify x’s transitions similarly. We get:
Y
0 1 2 ↓
1 1 1
X0 0 2
1 2 1
What x will do (i.e. x’s transition) does depend on y’s value, so y
Has an immediate effect on x.
Thus, the diagram of immediate effects can be deduced from a
Statement of the primary transitions. It is, in fact,
y → x
And y has been proved to dominate x.
Ex.: A system has three variables— x, y, z— each of which can take only the val-
Ues 0 or 1. If the transformation is
000 001 010 011 100 101 110 111 ↓ 110 111 100 101 110 011 100 001
|
|
|
What is the diagram of immediate effects ? (Hint: First find how z’s transi-
Tions depend on the values of the others.)
And continuing this process until no further additions are possible,
Gives the diagram of ultimate effects.
If a variable or part has no ultimate effect on another, then the
Second is said to be independent of the first.
Both the diagrams, as later examples will show, have features
Corresponding to important and well-known features of the sys-
Tem they represent.
Ex. 1: Draw the diagrams of immediate effects of the following absolute sys-
Tems; and notice the peculiarity of each:
(i) x' = xy, y' = 2y.
(ii) x' = y, y' = z + 3, z' = x2.
(iii) u' = 2 + ux, v' = v – y, x' = u + x, y' = y + v2.
(iv) u' = 4u – 1, x' = ux, y' = xy + 1, z' = yz.
(v) u' = u + y, x' = 1 – y, y' = log y, z' = z + yz.
(vi) u' = sin 2u, x' = x2, y' = y + 1, z' = xy + u.
Ex. 2: If y' = 2uy – z, under what conditions does u have no immediate effect on
Y?
Ex. 3: Find examples of real machines whose parts are related as in the diagrams
Of immediate effects of Ex. 1.
Ex. 4: (Continued.) Similarly find examples in social and economic systems.
Ex. 5: Draw up a table to show all possible ways in which the kinematic graph
And the diagram of immediate effects are different.
0
1
2
In the discussion of the previous section, the system was
Given by algebraic representation; when described in this form,
The deduction of the diagram of immediate effects is easy. It
Should be noticed, however, that the diagram can also be deduced
Directly from the transformation, even when this is given simply
As a set of transitions.
Suppose, for instance that a system has two variables, x and y,
Each of which can take the values 0, 1 or 2, and that its (x,y)-states
Behave as follows (parentheses being omitted for brevity):
|
|
|
↓
00 01 02 10 11 12 20 21 22
01 00 11 11 00 21 11 20 11
58
Reducibility. In S.4/11 we noticed that a whole system may
Consist of two parts each of which has an immediate effect on the
Other:
P ← Q →
We also saw that the action may be only one way, in which case
One part dominates the other:
P → Q
59
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
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