8/8 AN INTRODUCTION TO CYBERNETICS 



Examination of these in detail, to find how the transform follows 

 from the operand, shows that in all cases 



m' = n — p 



It is easily verified that the whole system will now emit the values that 

 the original input had two steps earlier, 



(The reader might be tempted to say that as n' = n + a, therefore 

 a = n' — n, and the code is solved. This statement is true, but it 

 does not meet our purpose, which is to build a machine (see para. 

 2 of S.8/7). It enables us to decode the message but it is not the 

 specification of a machine. The building or specification requires 

 the complications of the previous paragraph, which finishes with 

 /;/ = «—/?, a specification for a machine with input.) 



The general rule is now clear. We start with the transducer's 

 equation, n' = n -{■ a, and solve it for the parameter: a = n' — n. 

 The delaying device has the transformation p' = n. The trans- 

 formation for the inverter is formed by the rules, applied to the 

 equation a = n' — n: 



1 : replace a by the new transducer's symbol m'; 

 2: replace n' by a parameter c; 

 3 : replace « by a parameter d. 



Then, if this inverter is joined to the original transducer by putting 

 d = n, and to the delayer by c = p, it will have the required proper- 

 ties. 



If the original transducer has more than one variable, the process 

 needs only suitable expansion. An example, without explanation, 

 will be sufficient. Suppose the original transducer has parameters 

 ai and 02, variables x^ and X2, and transformation 



Xi = 2^1 + aiX2 

 X2 = 2^2 + aia2 



Solving for the parameters gives 



«i = C^i' - 2xi)lx2 



ai = •V2(-^'2' - 2x2)/Cvi' - 2xi) 



A delayer for x^ is p^' = Xi, and one for X2 is p2 = X2. The 

 equations of the inverter are formed from those for Cj and ^2 by 

 applying the rules: 



1 : replace each a^ by a new symbol: a^ = nii, Ui = ^2', 

 2: replace each x/ by a parameter c,-: x^ = Ci, x{ = C2; 

 3: replace each x^ by a parameter d^'. Xi = d^, X2 = ^2^ 



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