MINIMIZATION OF BOOLEAN FUNCTIONS 1425 



determined as described above without ever actually writing the binary 

 characters. 



4 PRIME IMPLICANT TABLES 



The minimum sum is formed by picking the fewest prime imphcants 

 whose sum will equal one for all rows of the table of combinations for 

 which the transmission is to equal one. In terms of the characters used 

 in Section 3 this means that each number in the decimal specification 

 of the function must appear in the label of at least one character which 

 corresponds to a ms-term (term of the minimum sum). 



The ms-terms are selected from the prime implicants by means of a 

 prime implicant table,* Table IV. Each column of the prime implicant 

 table corresponds to a row of the table of combinations for which the 

 transmission is to have the value one. The decimal number at the top of 

 each column specifies the corresponding row of the table of combinations. 

 Thus the numbers which appear at the tops of the columns are the same 

 as those which specify the transmission. Each row of the prime implicant 

 table represents a prime implicant. If a prime implicant equals "one" for 

 a given row of the table of combinations, a cross is placed at the inter- 

 section of the corresponding row and column of the prime implicant 

 table. All other positions are left blank. The table can be written directly 

 from the characters obtained in Section 3 by identifying each row of the 

 table with a character and then placing a cross in each column whose 

 number appears in the label of the character. 



It is convenient to arrange the rows in the order of the number of 

 crosses they contain, with those rows containing the most crosses at the 

 top of the table. Also, horizontal lines should be drawn partitioning the 

 table into groups of rows which contain the same number of crosses, 

 Table IV. If, in selecting the rows which are to correspond to ms-terms, 

 a choice between two equally appropriate rows is required, the row hav- 

 ing more crosses should be selected. The row with more crosses has 

 fewer literals in the corresponding prime implicant. This choice is more 

 obvious when the table is partitioned as suggested above. 



A minimum sum is determined from the prime implicant table by 

 selecting the fewest rows such that each column has a cross in at least 

 one selected row. The selected rows are called basis roivs, and the prime 

 implicants corresponding to the basis rows are the ms-terms. If any 

 column has only one entry, the row in which this entry occurs must be a 

 basis row. Therefore the fir.st step in selecting the basis rows is to place 



* This table was first discussed by Quine."' However, no sj'stematic procedure 

 for obtaining a minimum sum from the prime implicant table was presented. 



