Prime and non-prime implicants in the minimization of multiple-valued logic functions

Minimal sum-of-products expressions for multiple-valued logic functions for realization by programmable logic arrays are investigated. The 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 realize all functions optimally has been answered in the affirmative. The same question is considered for higher radix functions. When the combining operation is MAX, prime implicants are sufficient. However, it is shown that this is not the case with SUM. It is also shown that all functions cannot be optimally realized by successively selecting implicants that are prime with respect to the intermediate functions. In fact, the number of implicants in a solution using prime implicants successively can be significantly larger than the number of implicants in a minimal solution