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Reduction and Introducers in d-contexts

  • Alexandre Bazin
  • Giacomo KahnEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)

Abstract

Concept lattices are well-known conceptual structures that organise interesting patterns – the concepts – extracted from data. In some applications, the size of the lattice can be a problem, as it is often too large to be efficiently computed and too complex to be browsed. In others, redundant information produces noise that makes understanding the data difficult. In classical FCA, those two problems can be attenuated by, respectively, computing a substructure of the lattice – such as the AOC-poset – and reducing the context. These solutions have not been studied in d-dimensional contexts for \(d > 3\). In this paper, we generalise the notions of AOC-poset and reduction to d-lattices, the structures that are obtained from multidimensional data in the same way that concept lattices are obtained from binary relations.

References

  1. 1.
    Rudolph, S., Sacarea, C., Troanca, D.: Reduction in triadic data sets. In: Proceedings of the 4th International Workshop “What Can FCA do for Artificial Intelligence?” FCA4AI 2015, Co-located with the International Joint Conference on Artificial Intelligence (IJCAI 2015), Buenos Aires, Argentina, 25 July 2015, pp. 55–62 (2015)Google Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, vol. 25. American Mathematical Society, New York (1940)zbMATHGoogle Scholar
  3. 3.
    Barbut, M., Monjardet, B.: Ordre et classification. Algebre et Combinatoire, Volumes 1 and 2 (1970)Google Scholar
  4. 4.
    Ganter, B., Wille, R.: Formal Concept Analysis - Mathematical Foundations. Springer, Heidelberg (1999).  https://doi.org/10.1007/978-3-642-59830-2CrossRefzbMATHGoogle Scholar
  5. 5.
    Lehmann, F., Wille, R.: A triadic approach to formal concept analysis. In: Ellis, G., Levinson, R., Rich, W., Sowa, J.F. (eds.) ICCS-ConceptStruct 1995. LNCS, vol. 954, pp. 32–43. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60161-9_27CrossRefGoogle Scholar
  6. 6.
    Wille, R.: The basic theorem of triadic concept analysis. Order 12(2), 149–158 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Voutsadakis, G.: Polyadic concept analysis. Order 19(3), 295–304 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Krantz, D.H., Suppes, P., Luce, R.D.: Foundations of Measurement. Academic Press, New York (1971)zbMATHGoogle Scholar
  9. 9.
    Wille, U.: Geometric representation of ordinal contexts. Ph.D. thesis, University Gießen (1995)Google Scholar
  10. 10.
    Godin, R., Mili, H.: Building and maintaining analysis-level class hierarchies using Galois lattices. In: 8th Conference on Object-Oriented Programming Systems, Languages, and Applications (OOPSLA), pp. 394–410 (1993)Google Scholar
  11. 11.
    Biedermann, K.: Powerset trilattices. In: Mugnier, M.-L., Chein, M. (eds.) ICCS-ConceptStruct 1998. LNCS, vol. 1453, pp. 209–221. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0054916CrossRefGoogle Scholar
  12. 12.
    Voutsadakis, G.: Dedekind-Macneille completion of n-ordered sets. Order 24(1), 15–29 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Lorraine, CNRS, Inria, LORIANancyFrance
  2. 2.Université d’Orléans, INSA Centre Val de Loire, LIFOOrléansFrance

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