SYNTHESIS or SWITCHING CIRCUITS 89 



I'or an inlinilc number of nt , 



fi(n) < (1 + e) V 

 n 



n+l 

 2~ 



The proof is direct and will be omitted. 



PART III: SPECIAL FUNCTIONS 

 8. Functional Relations 



We have seen that almost all functions require the order of 



2n+i 



elements per relay for their realization. Yet a little experience with the 

 circuits encountered in practice shows that this figure is much too large. 

 In a sender, for example, where many functions are realized, some of them 

 involving a large number of variables, the relays carry an average of perhaps 

 7 or 8 contacts. In fact, almost all relays encountered in practice have less 

 than 20 elements. What is the reason for this paradox? The answer, of 

 course, is that the functions encountered in practice are far from being a 

 random selection. Again we have an analogue with transcendental numbers 

 ^although almost all numbers are transcendental, the chance of first en- 

 countering a transcendental number on opening a mathematics book at 

 random is certainly much less than 1. The functions actually encountered 

 are simpler than the general run of Boolean functions for at least two major 

 reasons: 



(1) A circuit designer has considerable freedom in the choice of functions 

 to be realized in a given design problem, and can often choose fairly simple 

 ones. For example, in designing translation circuits for telephone work it is 

 common to use additive codes and also codes in which the same number of 

 relays are operated for each possible digit. The fundamental logical simplic- 

 ity of these codes reflects in a simplicity of the circuits necessary to handle 

 them. 



(2) Most of the things required of relay circuits are of a logically simple 

 nature. The most important aspect of this simplicity is that most circuits 

 can be broken down into a large number of small circuits. In place of 

 realizing a function of a large number of variables, we realize many functions, 

 each of a small number of variables, and then perhaps some function of these 

 functions. To get an idea of the effectiveness of this consider the following 

 example: Suppose we are to realize a function 



f{Xi , X2 , • ■ ■ , Xzn) 



