80 BELL SYSTEM TECHNICAL JOURNAL 



It may be of interest to show that for large K the number of networks is 

 greater than 



This may be done by an inversion of the above argument. Let/(/\') be the 



number of networks with A' branches. Now, since there are 2" functions of 



2"+- 

 n variables and each can be realized with (1 + e) elements {n sufficiently 



// 



large), 



./Yd + e) Y V2;o"^'>'^''"^'"^ >r 



for n large. But assuming /(A') < (6A)^ reverses the inequality, as 

 is readily verified. Also, for an infinite sequence of A, 



/(A) > (6A)^''^ 



Since there is no obvious reason why /(A) should be connected with powers 

 of 2 it seems likely that this is true for all large A'. 



We may summarize what we have proved concerning the behavior of 



2n+l 



\{n) for large n as follows. \{n) varies somewhat as — ; if we let 



n 



ryn + l 



\{n) = An — 

 n 



then, for large n, An lies between 5 — e and (2 + e), while, for an infinite 

 sequence ofw, 5— e<^„<l+ e. 



We have proved, incidentally, that the new design method cannot, in a 

 sense, be improved very much. W'ith series-parallel circuits the best known 

 limit* for X{n) is 



X(;0 < 3.2»-i + 2 



2" 

 and almost all functions require (1 — e) ,- - elements.' We have lowered 



logo n 



n 

 the order of infinity, dividing by at least . and possibly by //. The 



best that can be done now is to divide by a constant factor < 4, and for 

 some n, < 2. The possibility of a design method which does this seems, 

 however, quite unlikely. Of course, these remarks apply only to a perfectly 

 general design method, i.e., one applicable to any function. Many special 

 classes of functions can be realized by special methods with a great saving. 



* Mr. J. Riordan hcas pointed out an error in my reasoning in (6) leading to the statement 

 that this limit is actually reached by the function A'l ® A'^ . . . ® A'„, and has shown that 

 this function and its negative can be realized with about n- elements. The error occurs 

 in Part IV after equation 19 and lies in the assumption that the factorization given is 

 the best. 



