Equivalent Representations of Set Functions
Mathematics of Operations Research
Disjunctive and conjunctive normal forms of pseudo-Boolean functions
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions
Mathematics of Operations Research
Meet and Join Derivatives and Their Use in Switching Theory
IEEE Transactions on Computers
Pseudo-polynomial functions over finite distributive lattices
ECSQARU'11 Proceedings of the 11th European conference on Symbolic and quantitative approaches to reasoning with uncertainty
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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.