Hierarchies of local monotonicities and lattice derivatives for Boolean and pseudo-Boolean functions
Waldhauser, Tamás; Marichal, Jean-Luc; Couceiro, Miguel (2012), Hierarchies of local monotonicities and lattice derivatives for Boolean and pseudo-Boolean functions, ISMVL '12 Proceedings of the 2012 IEEE 42nd International Symposium on Multiple-Valued Logic, IEEE : Washington, p. 262-267. http://dx.doi.org/10.1109/ISMVL.2012.10
TypeCommunication / Conférence
Conference titleIEEE 42nd International Symposium on Multiple-Valued Logic (ISMVL2012)
Book titleISMVL '12 Proceedings of the 2012 IEEE 42nd International Symposium on Multiple-Valued Logic
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Abstract (EN)In this paper we report recent results 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 monotonic ties for pseudo-Boolean (Boolean) functions. Local monotonic ties 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 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 variables for constants. This description is made explicit in the special case when p=2.
Subjects / Keywordspseudo-Boolean function; Boolean function; local monotonicity; discrete partial derivative; join and meet derivatives
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