Hierarchies of local monotonicities and lattice derivatives for Boolean and pseudo-Boolean functions
English
Couceiro, Miguel[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Marichal, Jean-Luc[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Waldhauser, Tamás[University of Szeged, Szeged, Hungary > Bolyai Institute]
2012
42nd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2012, Victoria, BC, Canada, May 14-16, 2012
Miller, D. Michael
Gaudet, Vincent C.
IEEE
262-267
Yes
No
International
978-1-4673-0908-0
IEEE 42nd Int. Symposium on Multiple-Valued Logic (ISMVL-2012)
from 14-05-2012 to 16-05-2012
Victoria, BC
Canada
[en] 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 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.
[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
