672 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



Regarded as a hindrance function of the switching variable Xj , 



when Xj 9^ i 



Then by (1), 



e'iixj) = 



II when Xj = i 



e'iixj) = n ^h{xj). 



If the switch xj has pj positions, it takes pj — 1 contacts to build a circuit 

 with hindrance function ei{xj). 



Synthesis 



Suppose that M shafts, governed by switching variables Xi y - -- , Xm y 

 are given, together with a two-terminal hindrance function /(xi , • • • , Xm)- 

 The synthesis problem is to design a network with hindrance function 

 f{xi , • • • , Xm)) adding suitable rotors and contacts to form selector switches 

 from the given shafts. 



One solution can be found immediately: 



(i) As described above, express the hindrance function f{xi , • • • , Xm) 

 as a Boolean function ^(^1 , • • • , ^r) of Boolean variables ^1 , • • • , ^b 

 (which are the ei{xj) with new labels). Here R = pi + p2 -\r " ' -\- pM - 



(ii) Any of the well known methods of synthesizing relay networks can 

 be used to design a network operated by the Boolean variables ^1 , • • • , ^r 

 and with hindrance function ^fe , • • • , ^r). 



(iii) In the network found in (ii) replace each contact ^a by the appropri- 

 ate ei(xj) and each back contact ^a by the appropriate circuit ei{xj). 



The solution found in (iii) will ordinarily use up a number of contacts 

 which is unnecessarily large by many orders of magnitude. From Shannon's 

 theorems on relay networks we know that the probabiUty is high, that as 

 many as 



2" 



contacts will be needed in step (ii). The final circuit (iii) will have even more 

 contacts if some circuits ei{xj) are used. 



The synthesis process which follows replaces the exponent i? = X^^i pi 

 in the estimate of the number of contacts by the smaller number ^JLi 

 log pi . The reader may recognize the process as essentially the same as the 

 one given by Shannon for two-terminal relay networks. The network will 

 again take the form of a tree connected to a circuit which produces all 

 functions of the switching variables which govern it. 



