MINIMIZATION OF BOOLEAN FUNCTIONS 1419 



that is, other sum functions which correspond to the same table of com- 

 binations. These functions are still siuiis of products of variables but 

 not all of the variables appear in each term. For example, the transmis- 

 sion of Table 1, T = xix-Zx^ + X1X2X3 + XiXiX-s + aia-2'a;3' -f a-i.t;2'a;3 + 

 .Tia;2.r3' = (xiXz'Xi + X1X2X5) -f (aTi'a;2a;3' + a-ia,-2i-3') + (a;i.r2'ar3' + .t-ia-2'a;3) = 

 (.ri'a:2'.i'3 + X1X2X3) + (.<■/.^•2.^■3' + .r/.r2a;3) + (.-»ia;2'a;3' + a;i.T2a:;3') can be 

 written as either T = .r/.rs + .r2.r3' + xix-/ or T = x^x^ -f a;i'x2 + XiX^' . 

 A literal is defined as a variable with or without the associated prime 

 {xi , x-i are literals) . The sum functions which have the fewest terms of 

 all equivalent sum functions will be called minimum sums unless these 

 functions having fewest terms do not all involve the same number of 

 literals. In such cases, only those functions which involve the fewest 

 literals will be called minimum sums. For example, the function 



T = E(7, 9, 10, 12, 13, 14, 15) 

 can be written as either 



T = XiXoXi' + .r;i.r2.ri + .r4.r2'.ri + .r4.r3.r1' 



or as 



T = Xi^XiXx -f .r3a-2.ri + Xi^ci'xx + Xi\,x% 



Only the second expression is a minimum sum since it involves 11 literals 

 while the first expression involves 12 literals. 



The minimum sum defined here is not necessarily the expression con- 

 taining the fewest total literals, or the expression leading to the most 

 economical two-stage diode logic circuit,^ even though these three ex- 

 pressions are identical for many transmissions. The definition adopted 

 here lends itself well to computation and results in a form which is useful 

 in the design of contact networks. A method is presented in Section 9 

 for obtaining directly the expressions corresponding to the optimum 

 two-stage diode logic circuit or the e.xpressions containing fewest literals. 



In principle it is possible to obtain a minimum sum for any given 

 transmission by enumerating all possible eciuivalent sum functions then 

 selecting those functions which have the fewest terms, and finally select- 

 ing from these the functions which contain fewest literals. Since the 

 number of equivalent sum functions may be c^uite large, this procedure 

 is not generally practical. The following sections present a practical 

 method for obtaining a ixiinimum sum without resorting to an enumera- 

 tion of all eciuivalent sum functions. 



3 PRIME IMPLICANTS 



When the theorem XxXt + Xxx^ = x\ is used to replace by a single 

 term, two p-terms, which correspond to rows i and j of a table of combi- 



