Bilities open. The change to transformation U
A b c d
U: ↓ c * b *
Represents a partial selection. U also represents a set of transfor-
Mations, though a smaller set. So does V:
V: ↓
a
B c d
↓ b a b c
a
b
c
d
And
↓ c b c a ?
a
b
c
d
Ex. 5: How much design goes to the production of a penny stamp, (i) as con-
Sisting of 15,000 half-tone dots each of which may be at any one of 10
Intensities? (ii) as the final form selected by Her Majesty from three sub-
Mitted forms? Explain the lack of agreement.
Ex. 6: How much variety must be supplied to reduce to one the possible
Machines on a given n states? (Hint: Ex. 7/7/8.)
Ex. 7: (Continued.) Similarly when the machine’s states number n and the
Input’s states (after design) number i.
b or c * * *
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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
RE GU LA TI N G TH E V ER Y LA R GE SY STE M
Which excludes all single-valued transformations that include the
transitions a → a or a → d. A machine can thus be selected in
Stages, and the stages may be defined in various ways.
What is fundamental quantitatively is that the overall selection
Achieved cannot be more than the sum (if measured logarithmically)
Of the separate selections. (Selection is measured by the fall in vari-
Ety.) Thus if a pack of cards is taken, and a 2-bit selection is made
And then a 3-bit, a unique card cannot be indicated unless a further
Selection of at least 0.7 bits is made, for log2 52 is 5.7. The limitation
Is absolute, and has nothing to do (if a machine is selected) with the
Type of machine or with the mode of selection used.
Ex. 1: How many possibilities are removed when, to the closed, single-valued
Transformation on a, b and c with all 27 forms initially possible, the restric-
Tion is added “It must have no state of equilibrium” ?
Ex. 2: (Continued.) When the restriction is “It must have three states of equilib-
Rium” ?
Ex. 3: In logarithmic measure, how much selection was exerted in Ex. I ?
*Ex. 4: How much selection is exerted on an absolute system of n states, a1, a2,
…, an, with all transformations initially possible, if the restriction is added
|
|
|
It must contain no state of equilibrium?” (Hint: To how many states may
Al now transform, instead of to the n previously?) (Cf. Ex. 1.)
*Ex. 5: (Continued.) To what does this quantity tend as n tends to infinity?
(Hint: Calculate it for n = 10, 100, 1000.) (This estimation can be applied
To the machine of S.12/15.)
*Ex. 6: If, as described in this section, the cards of a shuffled pack are searched
Without further shuffling) one by one in succession for a particular card,
How much information is gained, on the average, as the first, second, third,
Etc., cards are examined? (Systematic searching.)
*Ex. 7: (Continued.) How much if, after each failure, the wrong card is
Replaced and the pack shuffled before the next card is drawn? (Random
Searching.)
Supplementation of selection. The fact that selection can
Often be achieved by stages carries with it the implication that the
Whole selection can often be carried out by more than one selec-
Tor, so that the action of one selector can be supplemented by the
Action of others.
An example would occur if a husband, selecting a new car from
the available models, first decided that it must cost less than £1000,
And then allowed his wife to make the remainder of the selection. It
Would occur again if the wife, having reduced the number to two
Models, appealed to the spin of a coin to make the final decision.
Examples are ubiquitous. (Those that follow show supplementa-
Tion by random factors, as we shall be interested in them in the next
Chapter.) At Bridge, the state of the game at the moment when the
258
First card is led has been selected partly by the bids of the players and
Partly by chance— by the outcome of the statistically standardised
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