On the Determination of the ps Maximal Implicants of a Switching Function

``ps maximal implicant'' of a function T is every function F having the following properties: 1) F implies TX; 2) the cost of the minimal ``product of sums of literals'' expression of F (psMF) is inferior to the cost of the minimal ``product of sums of literals'' of T (psMT); 3) no function V, implying T and implied by F, has a ps expression whose cost is inferior to the cost of the psMF. ps maximal implicants have, for the third-order minimization, the same importance that conventional prime implicants have for the second-order minimization. In this paper a method for determining the ps maximal implicants of a function is described. The procedure consists in the determination of two sets of functions, called ``first-stage'' and ``second-stage functions,'' and in the search for ps maximal implicants among the complements of those functions. While the determination of the first-stage functions offers no difficulty, the problem of finding the second-stage functions is an intricate one. However, it can be solved by a topological approach. It is important to note that the knowledge of the second-stage functions is not necessary for the determination of the minimal third-order expressions of the functions considered in the examples, but we could not prove this to be a general property.