SYNTHESIS OF SWITCH I XG CIRCUITS 73 



Theorem 4. \(n) < 2"-' + 18 



d. Upper Limits for X(;0 witli Large //. 



Of course, it is not often necessary to synthesize a function of more than 

 say 10 variables, but it is of considerable theoretical interest to determine 

 as closely as possible the behavior of X(;0 for large n. 



n-2 



Fig. 14 — Disjunctive tree with {n — 2) bays. 



Fig. 15— Network giving all functions of {in + 1) variables constructed from one giving 

 all functions of m variables. 



We will tirst prove a theorem placing limits on the number of elements 

 required in a network analogous to Fig. 13 but generalized for m variables. 



Theorem 5. An N network realizing all 2'"' functions of m variables can 

 be constructed using not more than 2-f'" elements, i.e., not more than tico ele- 

 ments per function. Any network ivilh this property uses at least il — e) 

 elements per function for any e > with n sufficiently large. 



The first part will be proved by induction. We have seen it to be true 

 for m = 1,2. Suppose it is true for some m with the network .V of Fig. 15. 

 Any function of m + 1 variables can be written 



g= lX„,+i+/J[.YWi+/6l 



