1432 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



A method for determining the group invariance for a specified trans^ 

 mission is presented in "Detection of Group Invariance or Total Sym; 

 metry of a Boolean Function."* 



8 AN APPROXIMATE SOLUTION FOR CYCLIC PRIME IMPLICANT TABLES 



It has not been possible to prove in general that the procedure pre 

 sented in this section will always result in a minimum sum. However, 

 this procedure should be useful when a reasonable approximation to a 

 minimum sum is sufficient, or when it is possible to devise a proof to! 

 show that the procedure does lead to a minimum sum for a specific trans- 

 mission (such proofs were discussed in Section 6). Since this procedure 

 is much simpler than enumeration, it should generally be tested beforef 

 resorting to enumeration. 



The first step of the procedure is to select from the prime implicant 

 table a set of rows such that (1) in each column of the table there is a 

 cross from at least one of the selected rows and (2) none of the selected 

 rows can be discarded without destroying property (1). Any set of rows 

 having these properties will be called a consistent row set. Each consistent 

 row set corresponds to a sum of products expression from which no 

 product term can be eliminated directly by any of the theorems of 

 Boolean Algebra. In particular, the consistent row sets having the fewest 

 members correspond to minimum sums. The first step of the procedure 

 to be described here is to select a consistent row-set. This is done by 

 choosing one of the columns, counting the total number of crosses in each 

 row which has a cross in this column, and then selecting the row with 

 the most crosses. If there is more than one such row, the topmost row is 

 arbitrarily selected. The selected row is marked with a check. In Table 

 IX, column 30 was chosen and then row A was selected since rows A and 

 Z each have a cross in column 30, but row A has 4 crosses while row Z 

 has only 2 crosses. The selected row and each column in which it has a 

 cross is then lined out. The process just described is repeated by selecting 

 another column (which is not lined out). Crosses in lined-out columns 

 are not counted in determining the total number of crosses in a row. The 

 procedure is repeated until all columns are lined out. 



The table is now rearranged so that all of the selected rows are at the 

 top, and a line is drawn to separate the selected rows from the rest. 

 Table X results from always choosing the rightmost column in Table 

 IX. If any column contains only one cross from a selected row, the single 

 selected-row cross is circled. Any selected row which does not have any 



See page 1445 of this issue. 



