Minimization of Boolean Functions* 



E. J. McCLUSKEY, Jr. 



(Manuscript received June 26, 1956) 



A systematic procedure is presented for writing a Boolean function as 

 a minimum sum of products. This procedure is a simplification and exten- 

 sion of the method presented hy W. V. Quine. Specific attention is given to 

 terms which can be included in the function solely for the designer's con- 

 venience. 



1 INTRODUCTION 



In designing switching circuits such as digital computers, telephone 

 central offices, and digital machine tool controls, it is common practice 

 to make use of Boolean algebra notation.^- 2.3.4 'pj-^g performance of a 

 single-output circuit is specified by means of a Boolean function of the 

 input variables. This function, which is called the circuit transmission, 

 is equal to 1 when an output is present and equals when there is no 

 output. A convenient means of specifying a transmission is a table of 

 combinations such as that given in Table I. This table lists, in the column 

 under T, the output condition for each combination of input conditions. 

 If there are some combinations of input conditions for which the output 

 is not specified (perhaps because these combinations can never occur), 

 d-entries are placed in the T-column of the corresponding rows of the 

 table of combinations. The actual values (0 or 1) assigned to these rows 

 are usually chosen so as to simplify the circuit which is designed to 

 satisfy the requirements specified in the table of combinations. 



For each row of the table of combinations a transmission can be written 

 which equals "one" only when the variables have the values listed in 

 that row of the table. These transmissions will be called elementary 

 product terms (or more simply, p-terms) since any transmission can 

 always be written as a sum of these p-terms. Table I (b) lists the p-terms 

 for Table 1(a). Note that every variable appears in each p-term. The 



* This paper is derived from a thesis submitted to the Massachusetts Institute 

 of Technology in partial fulfillment of the requirements for the degree of Doctor 

 of Science on April 30, 1956. 



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