1424 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Table III — Example of the Two Ways of Forming 

 A Character Having Two Dashes 



110 



6, 8, 10, 12, 14 character can be formed from theO, 2, 4, 6, and 8, 10, 12, 

 14 characters or the 0, 2, 8, 10, and 4, 6, 12, 14 characters or the 0, 4, 8, 

 12 and 2, 6, 10, 14 characters), four ways in which a character having 

 four dashes can be formed, etc. 



In general, any character can be formed by combining two characters 

 whose labels form an increasing sequence of decimal numbers when 

 placed together. It is possible to shorten the process of determining 

 prime implicants by not considering the combination of any characters 

 whose labels do not satisfy this requirement. For example, in Table 

 11(b) the possibility of combining the (0, 4) character with either the 

 (2, 6), (2, 10) or the (2, 18) character need not be considered. If the 

 process is so shortened, it is not sufficient to place check marks next to 

 the two characters from which a new character is formed; each member 

 of all pairs of characters which would produce the same new character 

 w^hen combined must also receive check marks. More simply, when a 

 new character is formed a check mark is placed next to all characters 

 whose labels contain only decimal numbers which occur in the label of 

 the new character. In Table II, when the (0, 2, 4, 6) character is formed 

 by combining the (0, 2) and (4, 6) characters, check marks must be 

 placed next to the (0, 4) and (2, 6) characters as well as the (0, 2) and 

 (4, 6) characters. If the process is not shortened as just described, the 

 fact that a character can be formed in several ways can serve as a check 

 on the accuracy of the process. 



It is possible to carry out the entire process of determining the prime 

 implicants solely in terms of the decimal labels without actually writing 

 the binary characters. If two binary characters can be combined as de- 

 scribed in this section, then the decimal label of one can be obtained 

 from the decimal label of the other character by adding some power of 

 two (corresponding to the position in which the two characters differ) 

 to each number in the character's label. For example, in Table lib the 

 label of the (4, G) (0 1 - 0) character can be obtained by adding 4 = (2^) 

 to the numbers of the label of the (0, 2) (0 - 0) character. By searching 

 for decimal labels which differ by a power of two, instead of binary char- 

 acters which differ in only one position, the prime implicants can be 



