On the minimal third order expression of a Boolean function

The determination of the minimal third-order expression of a Boolean function (sum of products of sums or product of sums of products of the variables) is pratically an un solved problem. In this paper a procedure is set forth for the determination of the mini mal sum of products of sums; this procedure, if applied also to the complementary of the function assigned, makes it possible to determine, by duality, the minimal product of sums of products, and consequently leads in the end to the determination of the minimal third-order expression. The procedure is based on the definition of ps maximal implicant. The importance of the definition lies in the fact that the addends of the minimal sum of products of sums are the minimal products of sums of some ps maximal implicants. Consequently the determination of the minimal sum of products of sums of an assigned function T is carried out in three successive stages: I) determination of the ps maximal implicants of T; II) elimination of some of the ps maximal implicants by comparison with sums of prime implicants including the former; III) selection of the uneliminable ps maximal implicants, so that the addends of the minimal sum of products of sums can be found out.