6 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
Decision-making with Sugeno integrals: Bridging the gap between multicriteria evaluation and decision under uncertainty
International audienceThis paper clarifies the connection between multiple criteria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical formulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent preference relations between uncertain acts or between multifactorial alternatives where attributes share the same totally ordered domain. This paper proposes a generalized form of the Sugeno integral that can cope with attributes which have distinct domains via the use of qualitative utility functions. It is shown that in the case of decision under uncertainty, this model corresponds to state-dependent preferences on act consequences. Axiomatizations of the corresponding preference functionals are proposed in the cases where uncertainty is represented by possibility measures, by necessity measures, and by general order-preserving set-functions, respectively. This is achieved by weakening previously proposed axiom systems for Sugeno integrals
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
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts
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
Axiomatizations and factorizations of sugeno utility functions
In this paper we consider a multicriteria aggregation model
where local utility functions of different sorts are aggregated
using Sugeno integrals, and which we refer to as Sugeno utility
functions. We propose a general approach to study such functions
via the notion of pseudo-Sugeno integral (or, equivalently,
pseudo-polynomial function), which naturally generalizes that of
Sugeno integral, and provide several axiomatizations for this
class of functions. Moreover, we address and solve the problem
of factorizing a Sugeno utility function as a composition
q(phi(1)(x(1)), ... ,phi(n)(x(n))) of a Sugeno integral q with
local utility functions phi(i), if such a factorization exists