Connected Boolean Functions with a Locally Extremal Number of Prime Implicants

Abstract

The well-known lower bound for the maximum number of prime implicants of a Boolean function (the length of the reduced DNF) differs by $\Theta (\sqrt {n})$ times from the upper bound and is asymptotically attained at a symmetric belt function with belt width $n/3$. To study the properties of connected Boolean functions with many prime implicants, we introduce the notion of a locally extremal function in a certain neighborhood in terms of the number of prime implicants. Some estimates are obtained for the change in the number of prime implicants as the values of the belt function range over a $d$-neighborhood. We prove that the belt function for which the belt width and the number of the lower layer of unit vertices are asymptotically equal to $n/3$ is locally extremal in some neighborhood for $d \le \Theta (n)$ and not locally extremal if $d \ge 2^{\Theta (n)}$. A similar statement is true for the functions that have prime implicants of different ranks. The local extremality property is preserved after applying some transformation to the Boolean function that preserves the distance between the vertices of the unit cube.