Would be 8 states. In fact, only 4 combinations are used, so the set
Shows constraint.
Now reconsider these facts after recognising that a variety of
Four signals is necessary:
I) Stop
Ii) Prepare to go
(iii) Go
(iv) Prepare to stop.
If we have components that can each take two values, + or – , we can
Ask how many components will be necessary to give this variety.
The answer is obviously two; and by a suitable re-coding, such as
+ + = Stop
+ – = Prepare to go
– – = Go
– + = Prepare to stop
The same variety can be achieved with a vector of only two com-
Ponents. The fact that the number of components can be reduced
(from three to two) without loss of variety can be expressed by
181
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
I N CESSA N T TR AN SMI SSIO N
Saying that the first set of vectors shows redundancy, here of one
Lamp.
The constraint could clearly be taken advantage of. Thus, if
Electric lights were very expensive, the cost of the signals, when
Re-coded to the new form, would be reduced to two-thirds.
Exactly the same lights may also show quite a different redun-
Dancy if regarded as the generators of a different set of vectors.
Suppose that the lights are clock-operated, rather than traf-
Fic-operated, so that they go through the regular cycle of states (as
Numbered above)
...3, 4, 1, 2, 3, 4, 1, 2, 3, ...
The sequence that it will produce (regarded as a vector, S.9/9)
Can only be one of the four vectors:
(i) (1, 2, 3, 4, 1, 2, …)
(ii) (2, 3, 4, 1, 2, 3, …)
(iii) (3, 4, 1, 2, 3, 4, …)
(iv) (4, 1, 2, 3, 4, 1, …)
Were there independence at each step, as one might get from a
Four-sided die, and n components, the variety would be 4n; in fact
It is only 4. To make the matter quite clear, notice that the same
Variety could be obtained by vectors with only one component:
I) (1)
Ii) (2)
|
|
|
Iii) (3)
Iv) (4)
All the components after the first being omitted; so all the later
Components are redundant.
Thus a sequence can show redundancy if at each step the next
Value has not complete independence of the earlier steps. (Com-
Pare S.9/10.) If the sequence is a Markov chain, redundancy will
Be shown by its entropy having a value less than the maximum.
The fact that the one set of traffic lights provides two grossly
Different sets of vectors illustrates yet again that great care is nec-
Essary when applying these concepts to some object, for the object
Often provides a great richness of sets for discussion. Thus the
Question “Do traffic lights show redundancy?” is not admissible;
For it fails to indicate which of the sets of vectors is being consid-
Ered; and the answer may vary grossly from set to set.
This injunction is particularly necessary in a book addressed to
Workers in biological subjects, for here the sets of vectors are
Often definable only with some difficulty, helped out perhaps with
182
Some arbitrariness. (Compare S.6114.) There is therefore every
Temptation to let one’s grasp of the set under discussion be intui-
Tive and vague rather than explicit and exact. The reader may
Often find that some intractable contradiction between two argu-
Ments will be resolved if a more accurate definition of the set
Under discussion is achieved; for often the contradiction is due to
The fact that the two arguments are really referring to two distinct
Sets, both closely associated with the same object or organism.
Ex. 1: In a Table for the identification of bacteria by their power to ferment sug-
Ars, 62 species are noted as producing “acid”, “acid and gas”, or “nothing”
From each of 14 sugars. Each species thus corresponds to a vector of 14 com-
|
|
|
Ponents, each of which can take one of three values. Is the set redundant ?
To how many components might the vector be reduced ?
Ex. 2: If a Markov chain has no redundancy, how may its matrix be recognised
At a glance?
It is now possible to state what is perhaps the most funda-
Mental of the theorems introduced by Shannon. Let us suppose
That we want to transmit a message with H bits per step, as we
Might want to report on the movements of a single insect in the
Pool. H is here 0 84 bits per step (S.9/12), or, as the telegraphist
would say, per symbol, thinking of such a series as … P W B W
B B B W P P P W B W P W …. Suppose, for definiteness, that 20
Seconds elapse between step and step. Since the time-rate of these
Events is now given, H can also be stated as 2.53 bits per minute.
Дата добавления: 2019-11-16; просмотров: 204; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!