MINIMIZATION OF BOOLEAN FUNCTIONS 1441 



The actual values (0 or 1) of these d-entries are chosen so as to simplify 

 the form of the transmission. This section will describe how to modify 

 the method for obtaining a minimum sum when the table of combina- 

 |{ tions contains rf-entries. 



The p-terms which correspond to rf-entries in the table of combinations 

 will be called d-terms. These d-terms should be included in the list of 

 p-terms which are used to form the prime implicants. See Table XIV. 

 However, in forming the prime implicant table, columns corresponding 

 to the d-terms should not be included. Table XlV(b). The d-terms are 

 used in forming the prime implicants in order to obtain prime impli- 

 cants containing the fewest possible literals. If columns corresponding 

 to the f/-terms were included in forming the prime implicant table this 

 would correspond to setting all the rf-entries in the table of combina- 

 tions equal to 1. This does not necessarily lead to the simplest minimum 

 sum. In the procedure just described, the rf-entries will automatically be 

 set equal to either or 1 so as to produce the simplest minimum sum. 

 For the transmission of Table XIV the 14 d-entry has been set eciual to 



I and the 9 c^-entry has been set equal to 0. 



II NON-CANONICAL SPECIFICATIONS 



A transmission is sometimes specified not by a table of combinations 

 or a canonical expansion, but as a sum of product terms (not necessarily 

 prime implicants). The method described in Section 3 is applicable to 

 such a transmission if the appropriate table of combinations (decimal 

 specification) is first obtained. However, it is possible to modify the 

 procedure to make use of the fact that the transmission is already partly 

 reduced. The first step is to express the transmission in a table of binary 

 characters such as Table XVa. Then each pair of characters is examined 

 to determine whether any different character could have been formed 

 from the characters used in forming the characters of the pair. For 

 example, in Table XV (a) a (1) (00 00 1) was used in forming the 

 (0, 1)(0000-) character and a (3) (000 1 1) was used in forming the 

 (3, 7)(0 - 1 1) character. These can be combined to form a new char- 

 acter (1, 3) (000- 1). The new characters formed by this process are 

 listed in another column such as Table XV (b). This process is continued 

 until no new characters are formed. 



In examining a pair of characters, it is sufficient to determine whether 

 there is only one position where one character has a one and the other 

 character has a zero. If this is true a new character is formed which has 

 a dash in this position and any other position in which both characters 

 have dashes, and has a zero (one) in any position in which either charac- 



