90 BELL SYSTEM TECHNICAL JOURNAL 



of 2n variables. The best limit we can put on the total number of elements 



necessary is about . However, if we know that / is a function of two 



In 



functions /i and/2 , each involving only n of the variables, i.e. if 



/ = g{h,h) 



f\ = f\ {Xl , X2 , • • • , Xn) 



fi = j2\Xn-^\ , -X^n+2 , ' " ' , -^2n) 



then we can realize / with about 



4 • — 



n 



elements, a much lower order of infinity than — — . If g is one of the simpler 



functions of two variables ; for example if g(/i , /2) = /i + /2 , or in any case 

 at the cost of two additional relays, we can do still better and realize /with 



about 2 elements. In general, the more we can decompose a synthesis 



n 



problem into a combination of simple problems, the simpler the final circuits. 

 The significant point here is that, due to the fact that / satisfies a certain 

 functional relation 



/=g(A,/2), 



we can find a simple circuit for it compared to the average function of the 

 same number of variables. 



This type of functional relation may be called functional separability. It 

 is often easily detected in the circuit requirements and can always be used 

 to reduce the limits on the number of elements required. We will now show 

 that most functions are not functionally separable. 



Theorem 12: The fraction of all functions of n variables that can he written 

 in the form 



f = g(h(X, ■ ■ • X.), X.+ 1 ,-■■ ,Xn) 



where 1 < s < n — 1 approaches zero as n approaches 00 . 



We can select the 5 variables to appear in // in ( j ways; the function h 



then has 2^' possibilities and g has 2"" ' possibilities, since it has n— s-\- \ 

 arguments. The total number of functionally separable functions is there- 

 fore dominated by 



