678 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



However, one can show that all the requirements (7) hold if and only if the 



N(N - 1){N - 2) 

 requirements 



''Bia + Baj = implies Bij = 0" 



hold. 



N(N — 1) 

 Conversely, any set of hindrance functions which satisfy (7) 



determine a realizable iV-terminal network. One way of synthesizing the 

 network is just to connect, between each pair i,j of terminals, a two-terminal 

 network with hindrance function Bij . It follows from theorem II that 



Theorem III. Any N -terminal switching function of switches with P or 

 fewer positions can be synthesized with no more than 



^(^-^K^ + 2-)f^ 



2 1og^ 



contacts when H > 4 bits. 



The network can also be synthesized using N trees, each of which pro- 

 duces all of the possible functions Crixi) -{■ es{x2) + • • • + e^ixM)- Each 

 terminal is connected to the input lead of one of the trees; and the output 

 leads, to which the terminals are connected in any given state (xi , • • • , Xm), 

 are interconnected in the way one wants the terminals to be interconnected 

 in that state. The number of contacts used in this type of synthesis is less 

 than 



The synthesis using two-terminal networks ordinarily requires fewer con- 

 tacts than the one using trees as long as 



H-2\ogH>^{N-l){P-\-i) 



An example illustrating the design of a typical three-terminal network is 

 given in the appendix. 



Number of Functions 



Every A^-terminal switching function determines a realizable matrix of 

 hindrance functions Bij{xi , • • • , Xm)- It is important to know the number 

 of different switching functions oi {xi , • • • , Xm)- 



A state of the N terminals is determined by specifying the groups of 

 terminals which are connected together. The number (t)(N) of such states 

 is the number of ways that N different objects can be distributed into 

 1, 2, • • • , or .V parcels when the parcels are indistinguishable from one 

 another and no parcel is left empty. 



