Locally monotone Boolean and pseudo-Boolean functions

We propose local versions of monotonicity for Boolean and pseudoBoolean 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 ppermutable 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.