SVNTHKSIS OF SWITCH fNC, CIRCUITS 65 



must operate very quickly, wliile .V2 and .V3 have no essential time 



limitations but are ordinary U-type relays, and .V4 is a muiticontact 



relay on which many contacts are available, we would probably try 



to design a circuit for/(.Vi , .Y2 , A'3 , .Y4) in such a way as, hrst of all, 



to minimize the loading on .Yi , next to equalize the loading on Y2 



and A'3 keeping it at the same time as low as possible, and finally 



not to load A'l any more than necessary. Problems of this sort may 



be called problems in spring-load dislribulioii. 



Although all equivalent circuits representing a given function / which 



contain only series and j)arallcl connections can be found with the aid of 



lioolean Algebra, the most economical circuit in any of the above senses will 



often not be of this type. The problem of synthesizing non-series-parallel 



circuits is exceedingly ditlicult. It is even more dilBcult to show that a 



circuit found in some way is the most economical one to realize a given 



function. The difficulty springs from the large number of essentially 



different networks available and more particularly from the lack of a 



simple mathematical idiom for representing these circuits. 



We will describe a new design method whereby any function /(-Yi , A'2, • • • , 

 X„) may be realized, and frequently with a considerable saving of elements 

 over other methods, particularly when the number of variables n is large. 

 The circuits obtained by this method will not, in general, be of the series- 

 parallel type, and, in fact, they will usually not even be planar. This 

 method is of interest theoretically as well as for practical design purposes, 

 for it allows us to set new upper limits for certain numerical functions asso- 

 ciated with relay circuits. Let us make the following definitions: 



\(n) is defined as the least number such that any function of n variables 

 can be realized with not more than \{n) elements.* Thus, any function of 

 n variables can be realized with X(//) elements and at least one function with 

 no less. • 



IJL(n) is defined as the least number such that given any function/ of n 

 variables, there is a two-terminal network having the hindrance / and using 

 not more than n(n) elements on the most heavily loaded relay. 



The first part of this paper deals with the general design method and the 

 behaviour of X(;/). The second part is concerned with the possibility of 

 various types of spring load distribution, and in the third part we will study 

 certain classes of functions that are especially easy to synthesize, and give 

 some miscellaneous theorems on switching networks and functions. 



2. Fundamental Design Theorem 

 The method of design referred to above is based on a simple theorem deal- 

 ing with the interconnection of two switching networks. We shall first 



* An element means a make or break contact on one relay. .\ transfer element means 

 a make-and-break with a common spring, and contains two elements. 



