A Map Technique for Identifying Variables of Symmetry

Recognition of symmetry in circuit design often can drastically reduce the problem of finding the least expensive circuit configuration. Multi-output circuits frequently have a symmetric circuit as a common portion so no single error will result in a wrong output. As a consequence, numerous papers and chapters of books have presented recognition of symmetry in a switching function. 1-34 However, none of these articles has presented a technique that is simple to apply and has natural extensions to accommodate both completely and incompletely specified functions that are almost symmetrizable.* This paper presents such a technique. Caldwell1,2 has demonstrated a technique using Karnaugh maps for recognizing symmetrizable functions (SF's) of three or four variables, and has also demonstrated a procedure for extending this to functions of more variables. The extension requires the use of a large number of maps and the use of an expansion theorem a multiplicity of times. The Caldwell technique requires mapping all possible submaps in four of the variables. * See Section II for definitions. 801