Sur le cardinal maximum de la base complete d'une fonction booleenne, en fonction du nombre de conjonctions de l'une de ses formes normales.

On k-Term DNF with the Largest Number of Prime Implicants

“…After some preliminaries in Section 2, the results of [4,14,15,17] are presented in Section 3. The proof of Theorem 1 is given in Section 4.…”

“…In this section we describe the results of [4,14,15,17] on prime implicants of k-term DNF. We give a complete presentation in order to make the paper self-contained, to clarify what are the consequences of the separate assumptions of being an implicant, a prime implicant, resp.…”

“…For a recent application in computational learning theory, see Hellerstein and Raghavan [9]. It was shown by Laborde [14], Levin [15] and McMullen and Shearer [17] that the bound is sharp, i.e., there are k-term DNF with 2 k − 1 prime implicants (Chandra and Markowsky gave an example with more than 2 k/2 prime implicants). In view of these results, we call a DNF maximal if it has k terms and 2 k − 1 prime implicants for some k.…”

“…In this paper we complete the results of [4,14,15,17] by determining all the maximal disjunctive normal forms. In order to formulate the description, let us introduce the following definition.…”

“…In view of this correspondence, with some abuse of terminology, we can talk about a DNF being a NUD, rather than being equivalent to a NUD. The maximal DNF of [14,17] (resp., [15]) corresponds to a tree which is a single path (resp., a complete binary tree), without any z literals. A NUD generalizes these examples by allowing for an arbitrary tree and for the additional z literals.…”