Locally monotone Boolean and pseudo-Boolean functions

TL;DR

This paper introduces a hierarchy of local monotonicity for Boolean and pseudo-Boolean functions, linking it to lattice derivatives and providing classifications based on forbidden sections.

Contribution

It defines p-local monotonicity, relates it to lattice derivatives, and classifies functions using forbidden sections, advancing understanding of monotonicity in Boolean functions.

Findings

p-local monotonicity forms a hierarchy of monotonicity.

p-locally monotone functions have p-permutable lattice derivatives.

Classification of functions based on forbidden sections, especially for p=2.

Abstract

We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are shown to be tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p-locally monotone functions are shown to have p-permutable lattice derivatives and, in the case of symmetric functions, these two notions coincide. We provide further results relating these two notions, and present a classification of p-locally monotone functions, as well as of functions having p-permutable derivatives, in terms of certain forbidden "sections", i.e., functions which can be obtained by substituting constants for variables. This description is made explicit in the special case when p=2.