MINIMIZATION OF BOOLEAN FUNCTIONS 



1429 



Table VII — Determination of Basis Rows for a 

 Cyclic Prime Implicant Table 



'a) Selection of Single Asterisk Rows 

 4 16 12 24 19 28 27 29 31 



A. 

 B 



C 

 D 

 E 

 F 

 G 

 H 



(c) Selection of Row 1 as a Trial Basis 

 Row (Column 0) 



X X 

 X X 

 X X 



X X 



X X 

 X X 

 X X 



X I X 



X X 



X X 



4 16 12 24 19 28 27 29 31 



A 



B 



C 



D 



E 



F 



G 



H 



I 



J 



** 

 ** 



(b) Selection of Double Asterisk Rows 

 4 16 12 24 19 28 27 29 31 



A 



B 



C 



D 



E 



F 



G 



H 



I 



J 



(d) Selection of Row 2 as a Trial Basis 

 Row (Column 0) 



4 16 12 24 19 28 27 29 31 



A 



B 



C 



D 



E 



F 



G 



H 



I 



J 





* 



to be cyclic. If any column has crosses in only two rows, at least one of 

 these rows must be included in any set of basis rows. Therefore, the 

 basis rows for a cyclic table can be discovered by first determining 

 whether any column contains only two crosses, and if such a column 

 exists, by then selecting as a trial basis row one of the rows in which the 

 crosses of this column occur. If no column contains only two crosses, 

 then a column which contains three crosses is selected, etc. All columns 

 in which the trial basis row has crosses are lined out and the process of 

 lining out rows which are covered by other rows and selecting each row 

 which contains the only cross of some column is carried out as described 

 above. Either all columns will be lined out or another cyclic table will 

 result. Whenever a cyclic table occurs, another trial row must be se- 

 lected. Eventually all columns will be lined out. However, there is no 

 guarantee that the selected rows are actually basis rows. The possibility 

 exists that a different choice of trial rows would have resulted in fewer 

 selected rows. In general, it is necessary to carry out the procedure of 

 selecting rows several times, choosing different trial rows each time, so 



