Prime and Non-Prime Implicants in the Minimization of Multiple-Valued Logic Functions

Abstract

We investigate minimal sum-of-products expressions for multiple-valued logic functions for realization by programmable logic arrays. Our focus is on expressions where product terms consist of the MIN of interval literals on input variables and are combined using one of two operations SUM or MAX. In binary logic, the question of whether or not prime implicants are sufficient to optimally realize all functions has been answered in the affiimative. We consider the same question for higher radix functions. When the combining operation is MAX, prime implicants are sufficient. However, we show that this is not the case with SUM. There is also the question of whether all functions can be optimally realized by successively selecting implicants that are prime with respect to the intermediate functions. We show that this is not true either. In fact, the number of implicants in a solution using prime implicants successively can be sigmficantly larger than the number of implicants in a minimal solution.