Hierarchies of Local Monotonicities and Lattice Derivatives for Boolean and Pseudo-Boolean Functions

Abstract : In this paper we report recent results in [1] concerning local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p -locally monotone if each of its partial derivatives keeps the same sign on tuples which differ on less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p -locally monotone functions 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 classication 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 variables for constants. This description is made explicit in the special case when p = 2 .
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Miguel Couceiro, Jean-Luc Marichal, Tamas Waldhauser. Hierarchies of Local Monotonicities and Lattice Derivatives for Boolean and Pseudo-Boolean Functions. ISMVL 2012 - 42nd IEEE International Symposium on Multiple-Valued Logic, May 2012, Victoria, BC, Canada. pp.262-267, ⟨10.1109/ISMVL.2012.10⟩. ⟨hal-01093657⟩

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