MINIMIZATION OF BOOLEAN FUNCTIONS 1431 



CYCLIC PRIME IMPLICANT TABLES AND GROUP INVARIANCE 



It is not always necessary to resort to enumeration in order to deter- 



ne all minimum sums for a cyclic prime implicant table. Often 

 here is a simple relation among the various minimum sums for a trans- 

 nission so that they can all be determined directly from any single 

 ninimum sum by simple interchanges of variables. The process of select- 

 ng basis rows for a cyclic table can be shortened by detecting before- 

 aand that the minimum sums are so related. 



An example of a transmission for which this is true is given in Table 

 VIII. If the variables a'l and x-2 are interchanged, one of the minimum 

 sums is changed into the other. In the prime implicant table the inter- 

 change of Xi and Xz leads to the interchange of columns 1 and 2, 5 and 6, 

 9 and 10, 13 and 14, and rows A and B, C and D, E and F, G and H. 

 The transmission itself remains the same after the interchange. 



In determining the basis rows for the prime imphcant table, Table 

 VIII (d), either row G or row H can be chosen as a trial basis row. If row 

 G is selected the i-set of basis rows will result and if row H is selected 

 the ii-set of basis rows will result. It is unnecessary to carry out the 

 procedure of determining both sets of basis rows. Once the i-set of basis 

 rows is known, the ii-set can be determined directly by interchanging 

 the Xi and X2 variables in the i-set. Thus no enumeration is necessary in 

 order to determine all minimum sums. 



In general, the procedure for a complex prime implicant table is to 

 determine whether there are any pairs of variables which can be inter- 

 changed without effecting the transmission. If such pairs of variables 

 exist, the corresponding interchanges of pairs of rows are determined. 

 A trial basis row is then selected from a pair of rows which contain the 

 only two crosses of a column and which are interchanged when the varia- 

 bles are permuted. After the set of basis rows has been determined, the 

 other set of basis rows can be obtained by replacing each basis row by 

 the row with which it is interchanged w^hen variables are permuted. If 

 any step of this procedure is not possible, it is necessary to resort to 

 enumeration. 



In the preceding discussion only simple interchanges of variables have 

 been mentioned. Actually all possible permutations of the contact varia- 

 bles should be considered. It is also possible that priming variables or 

 both priming and permuting them will leave the transmission unchanged. 

 For example, ii T = Xi Xs Xo Xi + x/ Xs x-/ xi , priming all the variables 

 leaves the function unchanged. Also, priming Xi and x^ and then inter- 

 changing X4 and x^ does not change the transmission. The general name 

 for this property is group invariance. This was discussed by Shannon.^ 



