Locally monotone Boolean and pseudo-Boolean functions
Waldhauser, Tamás; Marichal, Jean-Luc; Couceiro, Miguel (2012), Locally monotone Boolean and pseudo-Boolean functions, Discrete Applied Mathematics, 160, 12, p. 1651-1660. http://dx.doi.org/10.1016/j.dam.2012.03.006
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/1107.1161
Journal nameDiscrete Applied Mathematics
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Abstract (EN)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.
Subjects / KeywordsBoolean function; pseudo-Boolean function; local monotonicity; discrete partial derivative; join and meet derivatives
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