8/13 AN INTRODUCTION TO CYBERNETICS 



no further increase at the second step, even though T still has some 

 variety. 



It will be noticed how important in the argument are the pairings 

 between the states of T and the states of U, i.e. which value of T 

 and which of U occur in the same machine. Evidently merely 

 knowing the quantities of variety in T and in U (over the set of 

 replicates) is not sufficient for the prediction of how they will 

 change. 



8/13. Transmission through a channel. We can now consider how 

 variety, or information, is transmitted through a small intermediate 

 transducer — a "channel" — where "small" refers to its number of 

 possible states. Suppose that two large transducers Q and S are 

 connected by a small transducer R, so that Q dominates R, and R 

 dominates S. 



Q 



R 



As usual, let there be a great number of replicates of the whole 

 triple system. Let i?'s number of possible states be r. Put log2r 

 equal to p. Assume that, at the initial state, the Q's have a variety 

 much larger than r states, and that the 7?'s and 5"s, for simplicity, 

 have none. (Had they some variety, S.8/11 shows that the new 

 variety, gained from Q, would merely add, logarithmically, to what 

 they possess already.) 



Application of S. 8/11 to i? and S shows that, at the first step, 5"s 

 variety will not increase at all. So if the three initial varieties, 

 measured logarithmically, were respectively A'^, and 0, then after 

 the first step they may be as large as jV, p, and 0, but cannot be larger. 



At the next step, R cannot gain further in variety (by S.8/12), but 

 S can gain in variety from R (as is easily verified by considering 

 an actual example such as Ex. 2). So after the second step the 

 varieties may be as large as N, p and p. Similarly, after the third 

 step they may be as large as A'^, p and 2p\ and so on. 5"s variety 

 can thus increase with time as fast as the terms of the series, 0, p, 

 2p, 3p, . . ., but not faster. The rule is now obvious: a transducer 

 that cannot take more than r states cannot transmit variety at more 

 than Iog2T bits per step. This is what is meant, essentially, by diff"erent 

 transducers having different "capacities" for transmission. 



Conversely, as 5"s variety mounts step by step we can see that 

 the amount of variety that a transducer (such as R) can transmit is 

 proportional to the product of its capacity, in bits, and the number of 

 steps taken. From this comes the important corollary, which will 



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