A^-Terminal Switching Circuits 



By E. N. GILBERT 



(Manuscript Received Feb. 14, 1951) 



The circuits considered have N accessible terminals and are operated by 

 gangs of selector switches. Synthesis of any iV-terminal switching function is 

 accomplished. The synthesis method is proved to be economical in the sense 

 that the switching functions which can be synthesized by any other method 

 using much fewer contacts comprise a vanishingly small fraction of the total 

 of all possible switching functions. 



Introduction 



In a recent issue of The Bell System Technical JoumaP, C. E. Shannon 

 discussed the synthesis of two-terminal relay contact networks. Some of 

 his results will be generalized in this paper to A^-terminal networks which 

 use selector switches with any number of positions instead of the two of a 

 relay. The kind of circuit which will be considered may be visualized as a 

 black box with N accessible terminals and with M shafts extending from it. 

 Each shaft operates a selector switch (which will usually consist of several 

 simple selector switches ganged together) inside the box. The rotors and con- 

 tacts inside the box are connected electrically to one another and to the N 

 terminals so that each way of setting the M shafts determines a pattern of 

 interconnection of the N terminals. 



We do not permit the black box to contain relay magnets or other devices 

 which would allow the circuit to operate sequentially. Because of this re- 

 striction our results apply only to the simplest kind of switching circuit in 

 which the state of the N terminals depends only on the present state of the 

 M shafts, and not on the past history of the shafts. We may then use the 

 term N -terminal switching function to mean a rule which assigns to each way 

 of setting the M shafts a state of the terminals. We are concerned with the 

 problem of synthesis: given an TV-terminal switching function/, to find a 

 switching circuit for which the states of the shafts and terminals correspond 

 in the way indicated by /. 



Let />i , • • • , />Af be the numbers of positions which the M shafts can 

 assume. Then there are Pi " • pM different states of the shafts and the shafts 

 have a memory^ 



» C. E. Shannon, B.S.TJ., 28, pp. 59-98 (1949). 

 « C. E. Shannon, B.S.TJ., 29, pp. 343-349 (1950). 



668 



