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

Abstract

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 a function f : {0, 1} n → L, there exists a lattice polynomial function p : L n → L such that p| {0,1} n = 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 → L over a distributive lattice L, not necessarily bounded, and where D ⊆ L n is allowed to range over n-dimensional rectangular boxes D = {a 1 , b 1 } × • • • × {an, bn} with a i , b i ∈ L and a i < b i , 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.

Interpolation by lattice polynomial functions: A polynomial time algorithm

“…More precisely, we show that LPFs can help handling problems involving unknown values by enabling the use of distributive lattices as intermediary scales. In Section 4 we extend previous results [11,26,27] by showing that the upper and lower bounds of IP(f ) can be described by a polynomial number of constraints. In Section 5 we make use of the latter result to characterize the class of partial LPFs through the notion of semi-congruence.…”

Computing Version Spaces in the Qualitative Approach to Multicriteria Decision Aid

“…These results were then generalized in two different directions. In [21] an approach by "splines" was proposed, which enables elicitation of families of generalized Sugeno integrals from pieces of data where local and global evaluations may be imprecisely known, whereas in [5,11] lattice theoretic approaches were proposed not only to determine existence but also to provide explicit descriptions of all possible lattice polynomials interpolating a given data set S.…”

“…Unlike in the case of interpolation by real polynomial functions, solutions do not necessarily exist, and it is a nontrivial problem to determine the necessary Goodstein's theorem [15] provides a solution in the special case when the domain of g is the hypercube D = {0, 1} n , where 0 and 1 are the least and greatest elements of the bounded distributive lattice L: a function g : {0, 1} n → L can be interpolated by a polynomial function p : L n → L if and only if g is monotone, and in this case p is unique. This result was generalized in [11] by allowing L to be an arbitrary (possibly unbounded) distributive lattice and by considering functions…”

“…I ≤ c I ≤ c + I , then and only then there is a polynomial function p : L n → L such that p| D = g. For the special type of interpolation problem considered in [11], the condition for the existence of a solution was given by simple lattice inequalities, without referring to the Boolean algebra generated by the lattice. In the case when L is a finite chain such a condition was given in [23], where, rather than polynomial functions, the interpolating functions where assumed to be Sugeno integrals, i.e., idempotent polynomial functions (see [18,19]).…”

“…This situation of incomplete information pertains to preference learning, where the set of interpolating pseudo-polynomial functions constitutes its version space. This motivates the following extension of the interpolation problem (stated as Problem 5.1 in [11]):…”