General interpolation by polynomial functions of distributive lattices
Waldhauser, Tamás; Rico, Agnés; Prade, Henri; Dubois, Didier; Couceiro, Miguel (2012), General interpolation by polynomial functions of distributive lattices, in Greco, Salvatore; Bouchon-Meunier, Bernadette; Coletti, Giulianella; Fedrizzi, Mario; Matarazzo, Benedetto; Yager, Ronald R., Advances in Computational Intelligence. 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Catania, Italy, July 9-13, 2012, Proceedings, Part III, Springer : Berlin, p. 347-355. http://dx.doi.org/10.1007/978-3-642-31718-7_36
Type
Communication / ConférenceDate
2012Conference title
14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU2012)Conference date
2012-07Conference city
CataneConference country
ItalieBook title
Advances in Computational Intelligence. 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Catania, Italy, July 9-13, 2012, Proceedings, Part IIIBook author
Greco, Salvatore; Bouchon-Meunier, Bernadette; Coletti, Giulianella; Fedrizzi, Mario; Matarazzo, Benedetto; Yager, Ronald R.Publisher
Springer
Series title
Communications in Computer and Information ScienceSeries number
vol 299Published in
Berlin
ISBN
978-3-642-31717-0
Number of pages
630Pages
347-355
Publication identifier
Metadata
Show full item recordAbstract (EN)
For a distributive lattice L, we consider the problem of interpolating functions f : D → L defined on a finite set D ⊆ L n , by means of lattice polynomial functions of L. Two instances of this problem have already been solved. In the case when L is a distributive lattice with least and greatest elements 0 and 1, Goodstein proved that a function f : {0,1} n → L can be interpolated by a lattice polynomial function p : L n → L if and only if f is monotone; in this case, the interpolating polynomial p was shown to be unique. The interpolation problem was also considered in the more general setting where L is a distributive lattice, not necessarily bounded, and where D ⊆ L n is allowed to range over cuboids D=a1,b1×⋯×an,bn with a i ,b i ∈ L and a i < b i . In this case, the class of such partial functions that can be interpolated by lattice polynomial functions was completely described. In this paper, we extend these results by completely characterizing the class of lattice functions that can be interpolated by polynomial functions on arbitrary finite subsets D ⊆ L n . As in the latter setting, interpolating polynomials are not necessarily unique. We provide explicit descriptions of all possible lattice polynomial functions that interpolate these lattice functions, when such an interpolation is available.Subjects / Keywords
Distributive lattice; polynomial function; interpolationRelated items
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