Locally monotone Boolean and pseudo-Boolean functions
We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p p -locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p 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 p -locally monotone functions are shown to have p 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 p -locally monotone functions, as well as of functions having p 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 p = 2 .
local monotonicity pseudo-boolean function boolean function discrete partial derivative join and meet derivatives