SYNTIIESrS OF SWITCHING CIRCUITS 77 



(r) There is an injhiile sequence of nifor which 



\(n,) < — (1 + e) e > 0. 



n 



These results can be proved rigorously without much difTiculty. 



e. A Lower Limit for X(;0 with Large //. 



Up to now most of our work has been toward the determination of upper 

 limits for X{n). We have seen that for all n 



X{n) < B- . 

 n 



2" 

 We now ask whether this function 5 — is anywhere near the true value 



n 



of X(»), or may \{n) be perhaps dominated by a smaller order of infinity, 

 e.g., n^. It was thought for a time, in fact, that \{n) might be limited by 

 n~ for all //, arguing from the first few values: 1, 4, 8, 14. We will show that 



2» 

 this is far from the truth, for actually — is the correct order of magni- 



tude of \{n): 



A-< X(n) < B- 

 n n 



for all n. A closely associated question to which a partial answer will be 

 given is the following: Suppose we define the "complexity" of a given func- 

 tion / of )i variables as the ratio of the number of elements in the most 

 economical realization of/ to X(77). Then any function has a complexity 

 lying between and L Are most functions simple or complex? 



Theorem 7: For all sufficiently large n, all functions of n variables excepting 



a fraction 8 require at least (1 — e) — elements, where e and 8 are arbitrarily 



small positive numbers. Hence for large n 



\{n) > (1 - e) - 

 n 



and almost all functions have a complexity > |(1 — e). For a certain sequence 



Hi almost all functions have a complexity > ^(1 — e). 



The proof of this theorem is rather interesting, for it is a pure existence 



proof. We do not show that any particular function or set of functions 



2" 

 requires (1 — e) — elements, but rather that it is impossible for all functions 



