674 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



• • • , ocm)' For given values ro , ^o , • • • , Zo of the switching variables 

 ^1 , • • • , ^jt of the tree, there is only one closed path through the tree ; this 

 path ends at the output lead labelled groW + • • • + ^«ofe)- 



The hindrance from this point to the output lead of the right-hand net- 

 work is the hindrance of the network of Fig. 4. This hindrance is 

 just /(ro , ^0 , • • • , 2o , Xk+i , • • • , Xm) (note that the connections to the dis- 

 junctive tree do not cause any interconnections among the other leads of 

 the right hand network), which proves that the network has the required 

 hindrance. By proper choice of the number k of switches in the tree we 

 will obtain an economical design. 



Network to Produce All Functions 



To produce all of /(r, ^, • • • , z, Xk+i , • • 

 which produces every function of (xk+i , 



Xm) it suffices to build a circuit 

 • , Xm)' Let these variables be 



ei(Xi)+...+ei{Xk) 

 ei(x,)+...+e2(XK) 



TREE 



ep,(x,)+...+ep^(xk) 



Fig. 4 — Network for/(a;i 



f(i,...,i,Xk+i,---,XM) 



f{1,.--,2,Xk+i,---,XM) 

 NETWORK 



Xm)- 



relabelled yi , • • • , yL and have ranges pi , 

 Pi, " ' , pLhe called P. 



Theorem I. A network which produces every function of {yi , 

 be built with a number xj/l of contacts satisfying 



, pL . Let the largest of 

 , Jl) can 



(3) 



rpL < P2' 



The proof is by induction on L. Suppose that a network to produce every 

 function of (3/1 , • • • , yy_i) has been built with ^y_i contacts and try to 

 build one for every function oi {yi , • • • , yy). The number of functions which 

 the network must produce is 2^*"^' , for there are p\ " • pj different ways 

 of setting the switches and two choices (0 or 1) for the value of the function 

 for each state of the switches. Of these functions, the ^j_i network itself 

 provides 2^' '''"' functions with no additional contacts (these are the func- 

 tions independent of yy). Any one of the remaining (2 ''*'^' — 2^^'"^'^^) func- 



