Formations, of transducer and inverter, are given, not in the
Abstract form of a table but by some mathematical function.
As a preliminary, consider building an inverter for the trans-
ducer with input a, variable n, and transformation n' = n + a. A
Suitable device for delay would be the transducer with parameter
n, variable p, and transformation p' = n. It is now easily verified
That, given the input a as shown, n (if started at 3) and p (if started
At 1) will change as:
A:
N:
P:
4 – 2 – 1
3 7 5
1 3 7
0
4
5
2 – 1 – 1
4 6 5
4 4 6
3
4
5
(a, A) SSS
(a, B) RRR
(a, C) (will not occur)
(a, D) Q Q Q
(b, A) Q Q Q
Etc.etc.
To which the input is the vector
State of delayer, state of coder).
148
It is now obvious that if the inverter, with a variable m, is to
Receive n and p as input, as vector (n,p), and give back a as output,
Then M, as transformation, must include such transitions as:
M: ↓
(7, 3) (5, 7) (4, 5) (4, 4)
4– 2 –10
149
…
…
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
T RA N SMISSI O N O F VA R IE TY
Examination of these in detail, to find how the transform follows
From the operand, shows that in all cases
m' = n – p
It is easily verified that the whole system will now emit the values
That the original input had two steps earlier.
(The reader might be tempted to say that as n' = n + a, therefore
a = n' – n, and the code is solved. This statement is true, but it
Does not meet our purpose, which is to build a machine (see pare.
Of S.8/7). It enables us to decode the message but it is not the
Specification of a machine. The building or specification requires
The complications of the previous paragraph, which finishes with
m' = n – p, a specification for a machine with input.)
The general rule is now clear. We start with the transducer’s
equation, n' = n + a, and solve it for the parameter: a = n' – n. The
delaying device has the transformation p' = n. The transformation
For the inverter is formed by the rules, applied to the equation
a = n' – n:
1: replace a by the new transducer’s symbol m';
|
|
|
2: replace n' by a parameter c;
Replace n by a parameter d.
Then, if this inverter is joined to the original transducer by putting
d = n, and to the delayer by c = p, it will have the required prop-
Erties.
If the original transducer has more than one variable, the proc-
Ess needs only suitable expansion. An example, without explana-
Tion, will be sufficient. Suppose the original transducer has
Parameters a1 and a2, variables x1 and x2, and transformation
x1' = 2x1 + a1x2
x2' = 2x2 + a1a2
Solving for the parameters gives
a1 = (x1' – 2x1)/x2
a2 = x2(x2' – 2x2)/(x1' – 2x1)
A delayer for x1 is p1' = x1, and one for x2 is p2' = x2. The equations
Of the inverter are formed from those for a1 and a2 by applying the
Rules:
1: replace each ai by a new symbol a1= m1', a2 = m2';
2: replace each xi 'by a parameter ci : x1' = c1, x2' = c2;
3: replace each xi by a parameter di: x1 = d1, x2 = d2;
150
There results the transducer
m1' = (c1 – 2d1)/d2
m2' = d2(c2 – 2d2)/(c1 – 2d1)
If now this transducer is joined to the original transducer by d1 =
x1, d2 = x2, and to the delayers by c1= p1, c2 = p2, then m1 and m2
Will give, respectively, the values that a1 and a2 had two steps ear-
Lier.
Ex. 1: Build an inverter for the transducer n' = an.
Ex. 2: Similarly for n' = n – 2a + 4.
Ex. 3: Similarly for x' = ax – by, y' = ax + by.
Ex. 4: Try to build an inverter for the transducer n' = n + a + b, why can it not
Be done ?
*Ex. 5: Build an inverter for the transducer
dx1/dt = a1x1x2 + a2
dx2/dt = (a1—1)x1 + a2x2.
Ex. 6: Why, in the section, does M have to transform (7,3) to 4, and not to — 2,
As the table a few lines higher might suggest ?
Size of the inverter. With the facts of the previous section, it
|
|
|
Is now possible to make some estimate of how much mechanism
Is necessary to invert the output of some given transducer. S.8/7
Makes clear that if the original transducer is not to lose distinc-
Tions it must have at least as many output values as the input has
Distinct values. Similarly the inverter must have at least as many,
But need not necessarily have more. The delayers will require lit-
Tle, for they are simple. It seems, therefore, that if the inverter is
Made of similar components to the original transducer then, what-
Ever the complexity or size of the original transducer, the inverter
Will have a complexity and size of the same order.
The importance of this observation is that one sometimes feels,
When thinking of the complexities in the cerebral cortex or in an
Ecological system, that any effect transmitted through the system
Must almost at once become so tangled as to be beyond all possi-
Ble unravelling. Evidently this is not so; the complications of cod-
Дата добавления: 2019-11-16; просмотров: 218; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!