9/16 AN INTRODUCTION TO CYBERNETICS 



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 redundancy 

 if regarded as the generators of a different set of vectors. Suppose 

 that the lights are clock-operated, rather than traffic-operated, so 

 that they go through the regular cycle of states (as numbered above) 



3 4 1 '' 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 4"; 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 earher steps. (Compare 

 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 trafllic lights provides two grossly 

 different sets of vectors illustrates yet again that great care is 

 necessary 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 considered ; 

 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 some 



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