The Reduction of Redundancy in Solving Prime Implicant Tables

Abstract

This paper is primarily concerned with finding, in the most efficient possible way, the set of all solutions to a cyclic prime implicant table. (A solution is a set of rows such that every column contains at least one marked entry in a row belonging to the set and such that no row can be deleted from the set without destroying this property.) Extensive use is made of the relationship between this and the problem of efficiently reducing a Boolean frontal function from the form of a product of sums of single literals to a sum of products. The transformation methods commonly in use today have the disadvantage that they tend to introduce duplicate and redundant products. (A redundant product includes at least one row which can be removed, and the remaining rows will still constitute a solution.) Several methods which appreciably reduce the number of such redundancies are presented. One of these methods (called row branching), applies repeatedly the algebraic transformation f = af(α = 1) + f(μ= 0) where μ is the Boolean variable corresponding to a row of the table, and f is the function corresponding to the given table. The mechanism by which a redundant solution is generated is described. The paper includes other useful transformation procedures such as, for example, a tabular method in which redundancies are avoided systematically at each step.