On the Minimization of Boolean Functions for Additive Complexity Measures

Abstract

The problem of minimizing Boolean functions for additive complexity measures in a geometric interpretation, as covering a subset of vertices in the unit cube by faces, is a special type of a combinatorial statement of the weighted problem of a minimal covering of a set. Its specificity is determined by the family of covering subsets, the faces of the unit cube, that are contained in the set of the unit vertices of the function, as well as by the complexity measure of the faces, which determines the weight of the faces when calculating the complexity of the covering. To measure the complexity, we need nonnegativity, monotonicity in the inclusion of faces, and equality for isomorphic faces. For additive complexity measures, we introduce a classification in accordance with the order of the growth of the complexity of the faces depending on the dimension of the cube and study the characteristics of the complexity of the minimization of almost all Boolean functions.