SYNTIJE^IS OF SWITCIJING CIRCUITS 69 



It seems probable that, in general, the function 



Ai e A'2 e ■ • • e a'„ 



requires 4(;/ — 1) elements, but no proof has been found. Proving that a 

 certain function cannot be realized with a small number of elements is 

 somewhat like proving a number transcendental; we will show later that 

 almost all* functions require a large number of elements, but it is difBcult 

 to show that a particular one does. 



-•b 



• 



Fig. 8 — Network giving all functions of one variable. 



Y 



Fig. 9 — Disjunctive tree with three bays, 

 b. Functions of Four \'ariables: 



In synthesizing functions of four variables by the same method, two 

 courses are open. First, we may expand the function as follows: 



f{\\\ X, Y, Z) = [W + X + 1' + V^{Z)\-{W + A' + I" + V.{Z)]. 

 [W + X' -V Y + Vz{Z)\-[W + X' + I" + V,{Z)\. 

 [W + X^ Y + V,{Z)\-{W' + X + y + ^5(Z)]. 



[w + X' + r + v,{z)\-[w' + r + y' + v,{z)]. 



By this expansion we would let Ui = W + A^ + Y, U2 = IF + A' + I ', • • • , 

 f/g = II" 4- A"' + I" and construct the M network in Fig. 9. X would 



* We use the expression "almost all" in the arithmetic sense: e.g., a property is true 

 of almost all functions of n variables if the fraction of all functions of n variables for which 

 it is not true — > as « — > « . 



