7 research outputs found
Pseudo-polynomial functions over finite distributive lattices
In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for
arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as
f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice
polynomial function over Y, and each uk is a map from Xk to Y. The resulting
functions are referred to as pseudo-polynomial functions. We present an
axiomatization for this class of pseudo-polynomial functions which differs from
the previous ones both in flavour and nature, and develop general tools which
are then used to obtain all possible such factorizations of a given
pseudo-polynomial function.Comment: 16 pages, 2 figure
A generalization of Goodstein's theorem: interpolation by polynomial functions of distributive lattices
We consider the problem of interpolating functions partially defined over a
distributive lattice, by means of lattice polynomial functions. Goodstein's
theorem solves a particular instance of this interpolation problem on a
distributive lattice L with least and greatest elements 0 and 1, resp.: Given
an n-ary partial function f over L, defined on all 0-1 tuples, f can be
extended to a lattice polynomial function p over L if and only if f is
monotone; in this case, the interpolating polynomial p is unique. We extend
Goodstein's theorem to a wider class of n-ary partial functions f over a
distributive lattice L, not necessarily bounded, where the domain of f is a
cuboid of the form D={a1,b1}x...x{an,bn} with ai<bi, and determine the class of
such partial functions which can be interpolated by lattice polynomial
functions. In this wider setting, interpolating polynomials are not necessarily
unique; we provide explicit descriptions of all possible lattice polynomial
functions which interpolate these partial functions, when such an interpolation
is available.Comment: 12 page
Apprentissage d’intégrales de Sugeno à partir de données inconsistantes
National audienceThe basic setting of this article is multicriteria decision making and preference aggregation. The problem treated is that of learning a Sugeno integral from inconsistent data, where values are elements of a totally ordered set. This is a difficult optimization problem : indeed, a Sugeno integral is determined by 2^n values, with n being the number pf parameters. In this article we propose two learning methods : the first one is an application of simulated annealing, and the second is a new algorithm which relies on the selection of a consistant subset of data and for which the value of n doesn't affect the running time significantly.En prenant pour cadre de référence l'aidè a la décision multi-critères et l'agrégation de préférences, cet article traite de l'apprentissage de l'intégrale de Sugenò a partir de données inconsistantes, et dont les valeurs appartiennent à un ensemble totalement ordonné. Il s'agit d'un problème d'optimisation difficile, puisqu'une intégrale de Sugeno est définie d'après 2^n valeurs, où n est le nombre de paramètres. Dans cet article nous considérons deux méthodes : la premìère est une application du recuit simulé, et la seconde est un nouvel algorithme reposant sur la séléction préalable d'un sous-ensemble de données consistantes, dont le temps d'exécution est peu sensible à la valeur de n
Locally monotone Boolean and pseudo-Boolean functions
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
Pseudo-polynomial functions over finite distributive lattices
International audienceIn this paper we extend the authors’ previous works by considering a multi-attribute aggregation model based on a composition of a polynomial function over a finite distributive lattice with local utility functions; these are referred to as pseudo-polynomial functions. We present an axiomatization for this class of pseudo-polynomial functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-polynomial function