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BELL SYSTEM TECHNICAL JOURNAL 



Theorem 3: Any function of four variables can be realized with not more 

 than 14 elements. 

 c. Functions of More Than Four Variables: 



Any function of five variables may be written 

 f(Xx, ••• , X,) = [Xs + fiiXr, ■■■ , X,)]-[Xi + /2(Xi, ••• , X,)] 

 and since, as we have just shown, the two functions of four variables can 

 be realized with 14 elements each, f{Xi , • • ■ .Y5) can be realized with 30 



x'»- 



-00- 



- X' 



yV 



X' Y»- 



X'+Y'»- 



X'Y'»- 



X +Y'»- 

 X' + Y»- 



x' 



xyV 



X Y'+ X'Y 



H::>k:;n- 



X Y + X' Y'l 



<^:} 



Fig. 13 — Network giving all functions of two variables. 



Now consider a function /(Xi , X2, ■■■ , Xn) of n variables. For 

 5 < « < 13 we get the best limit by expanding about all but two variables. 



/(Xi , X2 , • • • , Xn) = [Xi + X2 + • • • + X„_2 + Fi(X„_l , X„)] 



[X( + X; + • • • + xLo + F.(X„_x , \\)] (4) 



The F's are all functions of the variables .Y„_i , A'„ and may be obtained 



from the general X network of Fig. 13, in which every function of two 



variables appears. This network contains 20 elements which are grouped 



into five transfer elements for one variable and five for the other.* The 



M network for (4), shown in Fig. 14, requires in general 2" ' - 2 elements. 



Thus we have: 



* Several other networks with ihe same property as Fig. 13 have been found, but they 

 all require 20 elements. 



