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 pp-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than pp 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, pp-locally monotone functions are shown to have pp-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 pp-locally monotone functions, as well as of functions having pp-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=2p=2.