MINIMIZATION OF BOOLEAN FUNCTIONS 1437 



total number of selected rows (a check must be made that the two 

 selected rows being removed do not contain the only two selected-row 

 crosses in a column). In Table X no such interchange is possible. 



Next a check should be made as to whether two of the labeled non- 

 selected rows can be used to replace three selected rows, etc. In Table 

 X rows Y(a) and J(b) can replace rows E(a), F(b) and K or rows Y(a) 

 and P(d) can replace rows E(a), T(d) and K. The table which results 

 from replacing rows E, F and K by rows Y and J is given in Table XL 

 The number of selected rows is now 8 which is still greater than 7, the 

 minimum number possible. This table actually represents the minimum 

 sum for this transmission even though this cannot be proved rigorously 

 by the procedure being described. 



If it is assumed that a minimum sum can always be obtained by ex- 

 changing pairs of selected and nonselected rows until it finally becomes 

 possible to replace two or more selected row^s by a single selected row, 

 then it is possible to show directly that the Table XI does represent a 

 minimum sum. The only interchange possible in Table XI is that of 

 rows T and P. If this replacement is made then a table results in which 

 only rows J and F can be interchanged. Interchanging rows J and F 

 does not lead to the possibility of interchanging any new pairs of rows 

 so that this process cannot be carried any further. 



On the basis of experience with this method it seems that it is not 

 necessary to consider interchanges mvolving more than one non-selected 

 row. Such interchanges have only been necessary in order to obtain al- 

 ternate minimum sums; however, no proof for the fact that they are 

 never required in order to obtain a minimum sum has yet been dis- 

 covered. 



9 AN ALTERNATE EXACT PROCEDURE 



It is possible to represent the prime implicant table in an alternative 

 form such as that given in Table XII (b). From this form not only the 

 minimum sums but also all possible sum of products forms for the trans- 

 mission which correspond to consistent row sets can be obtained sys- 

 tematically. For concreteness, this representation will be explained in 

 terms of Table XII. Since column has crosses only in rows B and C, 

 any consistent row set must contain either row B or row C (or both). 

 Similarly, column 3 requires that any consistent row set must contain 

 either row D or row E (or both). When both columns and 3 are con- 

 sidered they require that any consistent row set must contain either 

 row B or row C (or both) and either row D or row E (or both). This 

 requirement can be expressed symbolically as (B -f C) (D -f E) where 



