MINIMIZATION OF BOOLEAN FUNCTIONS 



1439 



Table XIII — Determination of the Minimum Sums for the 



Prime Implicant Table of Table VII by Means of 



THE Boolean Representation 



(a) Boolean representation of the Prime Implicant Table of Table VI 

 (A+B) (A+C) (B+D) (C+E) (D+F) (G) (E+F+H) (G+I) (H+J) (1+ J) 



(b) The expression of (a) after multiplying out. (The terms in italic 



correspond to minimum sums) 



ADEJG + ACDFJG + ACDHJG + ADEHIG + ACDHIG + ABEFJG 



+ ABEFHIG + BCDEJG + BCDHJG + BCDHIG + BCFJG + BCFHIG 



-G- 



A 



(c) Tree circuit equivalent of (b) 



J 



B E---F- 



---D 



■--H- 

 --H- 



--I 



--J 



--I 



--J 



-E--- 



-B--- 



-C 



D- 



-_F 



5 

 6 

 5 

 5 

 5 

 5 

 4 V 



5 

 5 

 5 

 5 



4 V 



k is the number of crosses in the row.* To select the minimum sums, the 

 sum of the weights of the rows should be calculated for each row set 

 containing the fewest rows. The row sets having the smallest total weight 

 correspond to minimum sums. If, instead of the minimum sum, the form 

 leading to the two-stage diode-logic circuit requiring fewest diodes is 

 desired, a slightly different procedure is appropriate. To each row set 

 is assigned a total weight equal to the sum of the weights of the rows 

 plus the number of rows in the set. The desired form then corresponds to 

 the row set having the smallest total weight. 



The procedure for an arbitrary table is analogous. A more compli- 

 cated example is given in Table XIII. In this example the additional 



* n-log2 k is the number of literals in the prime implicant coriesponding to 

 a row containing k crosses. 



