1418 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Table I — Circuit Specifications 

 (a) Table of Combinations (b) p-terms 





 1 

 2 

 3 



4 

 5 

 6 



7 



(c) Canonical Expansion 



T = Xx'Xt'Xz + Xi'XtXz + Xi'XtXz + XxXi'Xz' + XiXt'Xz + XiXiXz 



p-term corresponding to a given row of a table of combinations is formed 

 by priming any variables which have a "zero" entry in that row of the 

 table and by leaving unprimed those variables which have "one" entries. 

 It is possible to write an algebraic expression for the over-all circuit 

 transmission directly from the table of combinations. This over-all 

 transmission, T", is the smn of the p-terms corresponding to those rows 

 of the table of combinations for which T is to have the value "one." 

 See Table 1(c). Any transmission which is a sum of p-terms is called a 

 canonical expansion. 



The decimal numbers in the first column of Table 1(a) are the decimal 

 equivalents of the binary numbers formed by the entries of the table 

 of combinations. A concise method for specifying a transmission function 

 is to list the decimal numbers of those rows of the table of combinations 

 for which the function is to have the value one. Thus the function of 

 Table I can be specified as ^(1, 2, 3, 4, 5, 6). 



One of the most basic problems of switching circuit theory is that of 

 writing a Boolean function in a simpler form than the canonical expan- 

 sion. It is frequently possible to realize savings in equipment by writing 

 a circuit transmission in simplified form. Methods for expressing a 

 Boolean function in the "simplest" sum of products form were published 

 by Karnaugh,^ Aiken, ^ and Quine.® These methods have the common 

 property that they all fail when the function to be simplified is reason- 

 ably complex. The following sections present a method for simplifying 

 functions which can be applied to more complex functions than previous 

 methods, is systematic, and can be easily programmed on a digital com- 

 puter. 



2 the minimum sum 



By use of the Boolean algebra theorem a;ia:;2 + a;/a;2 = Xo it is possible 

 to obtain from the canonical expansion other equivalent sum functions; 



