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BELL SYSTEM TECHNICAL JOURNAL 



again be as in Fig. 8, and by the same type of reasoning it can be seen that 

 X(4) < 16. 



Using a shghtly more complicated method, however, it is possible to 

 reduce this limit. Let the function be expanded in the following way: 



f{W, X, Y, Z) = [W-\-X+ Fi(F, Z)HW -\-X' + ^2(7, Z)] 



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



We may use a network of the type of Fig. 7 for M. The V functions are 

 now functions of two variables Y and Z and may be any of the 16 functions: 







B{ 



We have divided the functions into five groups, A, B,C, D and E for later 

 reference. We are going to show that any function of four variables can 



Fig. 10 — Simplifying network. 



be realized with not more than 14 elements. This means that we must 

 construct a network N using not more than eight elements (since there are 

 six in the M network) for any selection of four functions from those listed 

 above. To prove this, a number of special cases must be considered and 

 dealt with separately: 



(1) If all four functions are from the groups, A, B, C, and D, X will 

 certainly not contain more than eight elements, since eight letters at most 

 can appear in the four functions. 



(2) We assume now that just one of the functions is from group E; 

 without loss of generality we may take it to be YZ' + Y'Z, for it is the other, 

 replacing Y by Y' transforms it into this. If one or more of the remaining 

 functions are from groups A ov B the situation is satisfactory, for this func- 

 tion need require no elements. Obviously and 1 require no elements and 

 F, Y', Z or Z' may be "tapped oflf" from the circuit for YZ' + Y'Z by writing 

 it as (F + Z)(Y' + Z'). For example, Y' may be obtained with the circuit 

 of Fig. 10. This leaves four elements, certainly a sufficient number for 

 any two functions from A, B,C, or D. 



