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 Booleanfunction (the length of the reduced DNF) differs by Θ(√n) times from the upper bound and is asymptoticallyattained at a symmetric belt function with belt width n/3. To study the properties of connected Booleanfunctions with many prime implicants, we introduce the notion of a locally extremal function ina certain neighborhood in terms of the number of prime implicants. Some estimates are obtainedfor 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 forwhich 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 ≤ Θ(n) and not locally extremal if d ≥ 2Θ(n). A similar statement is true for the functions thathave prime implicants of different ranks. The local extremality property is preserved afterapplying some transformation to the Boolean function that preserves the distance between thevertices of the unit cube. © 2021, Pleiades Publishing, Ltd.