Hierarchies of Local Monotonicities and Lattice Derivatives for Boolean and Pseudo-Boolean Functions

In this paper we report recent results concerning local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if each of its partial derivatives keeps the same sign on tuples which differ on less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonic ties for pseudo-Boolean (Boolean) functions. Local monotonic ties are tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p-locally monotone functions 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 variables for constants. This description is made explicit in the special case when p=2.

I. Introduction

Throughout this paper, let and . We are interested in the so-called Boolean functions and pseudo-Boolean functions , where denotes the arity of . The pointwise ordering of functions is denoted by , i.e., means that for all . The negation of is defined by , where stands for addition modulo 2. For , we set , and .