syxruEsis or switciiinc circiits 79 



plied by the number of assignments of the variables to the branches, i.e., 

 it is less than 



(l-.)(2«/n) 



H = (InY'-''''""'' h{\ - e) ^ 



Hence 



log2 // = (1 - e) - log In + (1 - e) - log (1 - e) - • 6 

 // n 11 



= (1 — e) 2" + terms dominated by this term for large n. 



By choosing n so large that - 2" dominates the other terms of log H we 

 arrive at the inequality 



log2 H <{\- ei) 2" 



But there are 5 = 2 functions of n variables and 



22" 



— » as n — » 06 . 



Hence ahnost all functions require more than (1 — €1)2" eleriientS. 



Now, since for all n> N there is at least one function requiring mofe than 



1 2" 



(say) - — elements and since \{n) > for « > 0, we can say that for all n, 



2 n 



2* 



X(w) > A~ 



n 



for some constant A > 0, for we need only choose A to be the minimum 

 number in the finite set: 



1 Ml) X(2) X(3) XC^) 



2 ' 2* ' 2^ ' 2' ' " " ' 2^ 



I 2 3 iV 



2" 

 Thus X(w) IS of the order of magnitude of — . The other parts of Theorem 



8 follow easily from what we have already shown. 



The writer is of the opinion that almost all functions have a complexity 

 nearly 1, i.e., > 1 — e. This could be shown at least for an infinite sequence 

 Hi if the Lemma could be improved to show that the number of networks is 

 less than {6K)^'^ for large K. Although several methods have been used 

 in counting the networks with K branches they all give the result (6K)'^. 



