SYNTHESIS OF SWITCHING CIRCUITS 



75 



corresponding to functions of less than m variables according to the equation 



g= (X-\- U){X' + U) = {X+ /lX=0)(X' + /2x=.l) 



if they are not already so connected. This means that all terminals of the 

 second class are then connected to a vanishingly small fraction of the total 

 terminals. We can then attribute two elements each to these terminals 

 and at least one and one-half each to the terminals of the third group. As 

 these two groups exhaust the terminals except for a fraction which -^ 

 as « ^ 20 , the theorem follows. 



If, in synthesizing a function of n variables, we break off the tree at the 

 (« — m)\\\ bay, the tree will contain 2"'""^^ — 2 elements, and we can find 

 an N network with not more than 2"'"- 2 elements exhibiting every function 

 of the remaining m variables. Hence 



\{n) < T 



-2+22' 



< 2"^'"'^^ + 2 2^" 



Fig. 17 — Possible situation in Fig. 16. 



for every integer m. We wish to find the integer M = M{n) minimizing 

 this upper bound. 



Considering m as a continuous variable and n fixed, the function 



/(m) = 2"-'"+'+ 2'"" -2 



clearly has just one minimum. This minimum must therefore lie between 

 m\ and Wi + 1, where 



/(wi) = /(wi + 1) 



I.e., 

 or 



^n ^ 2'"i+i(2''"'"'' _ 2''"') 



Now wi cannot be an integer since the right-hand side is a power of two 

 and the second term is less than half the first. It follows that to find the 

 inleger M making /(M) a minimum we must take for M the least integer 

 satisfying 



2" < 2^^'2''''" 



