13/18 AN INTRODUCTION TO CYBERNETICS 



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 

 variety.) 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 re- 

 striction is added "It must have no state of equilibrium"? 

 Ex. 2: (Continued.) When the restriction is "It must have three states of 



equilibrium"? 

 Ex. 3: In logarithmic measure, how much selection was exerted in Ex. 1 ? 



*Ex. 4: How much selection is exerted on an absolute system of n states, oi, 

 02, • • -, Cny with all transformations initially possible, if the restriction is 

 added "It must contain no state of equilibrium?" (Hint: To how many 

 states may oi 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.) 



13/18. 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 selector, 

 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 



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