2 research outputs found
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-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
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts