Lattice Polynomial Functions Research Articles

Interpolation by lattice polynomial functions: A polynomial time algorithm

This paper deals with the problem of interpolating partial functions over finite distributive lattices by lattice polynomial functions. More precisely, this problem can be formulated as follows: Given a finite distributive lattice L and a partial function f from D⊆Ln to L, find all the lattice polynomial functions that interpolate f on D. If the set of lattice polynomials interpolating a function f is not empty, then it has a unique upper bound and a unique lower bound. This paper presents a new description of these bounds and proposes an algorithm for computing them that runs in polynomial time, thus improving existing methods. Furthermore, we present an empirical study on randomly generated datasets that illustrates our theoretical results.

K-maxitive Sugeno integrals as aggregation models for ordinal preferences

We consider an order variant of k-additivity, so-called k-maxitivity, and present an axiomatization of the class of k-maxitive Sugeno integrals over distributive lattices. To this goal, we characterize the class of lattice polynomial functions with degree at most k and show that k-maxitive Sugeno integrals coincide exactly with idempotent lattice polynomial functions whose degree is at most k. We also discuss the use of this parametrized notion in preference aggregation and learning. In particular, we address the question of determining optimal values of k through a case study on empirical data.

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.

Interpolation by polynomial functions of distributive lattices: a generalization of a theorem of R. L. Goodstein

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, respectively: given a function f : {0, 1} n → L , there exists a lattice polynomial function $${p: L^{n} \rightarrow L}$$ such that p| {0,1} = f 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 partial functions $${f : D \rightarrow L}$$ over a distributive lattice L, not necessarily bounded, and where $${{D}\, {\subseteq}\, {L}^{n}}$$ is allowed to range over n-dimensional rectangular boxes $${D = \{{a_{1}, b_{1}}\} {\times}. . . {\times} \{{a_{n}, b_{n}\}}}$$ with $${a_{i}, b_{i} \in L}$$ and $${a_{i} < b_{i}}$$ , and we determine the class of such partial functions that 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 that interpolate these partial functions, when such an interpolation is available.

Self-commuting lattice polynomial functions on chains

We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions on chains.

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions.

Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art

We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial functions, that is, functions built only on projections and minimum and maximum operations.

On the moments and distribution of discrete Choquet integrals from continuous distributions

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [J.-L. Marichal, I. Kojadinovic, Distribution functions of linear combinations of lattice polynomials from the uniform distribution, Statistics & Probability Letters 78 (2008) 985–991] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.

Cumulative distribution functions and moments of lattice polynomials

We give the cumulative distribution functions, the expected values, and the moments of lattice polynomials when regarded as real functions. Since lattice polynomial functions include order statistics, our results encompass the corresponding formulas for order statistics.