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Lindig’s Algorithm for Concept Lattices over Graded Attributes

  • Radim Belohlavek
  • Bernard De Baets
  • Jan Outrata
  • Vilem Vychodil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4617)

Abstract

Formal concept analysis (FCA) is a method of exploratory data analysis. The data is in the form of a table describing relationship between objects (rows) and attributes (columns), where table entries are grades representing degrees to which objects have attributes. The main output of FCA is a hierarchical structure (so-called concept lattice) of conceptual clusters (so-called formal concepts) present in the data. This paper focuses on algorithmic aspects of FCA of data with graded attributes. Namely, we focus on the problem of generating efficiently all clusters present in the data together with their subconcept-superconcept hierarchy. We present theoretical foundations, the algorithm, analysis of its efficiency, and comparison with other algorithms.

Keywords

Fuzzy Logic Formal Concept Concept Lattice Formal Concept Analysis Fuzzy Concept 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer, Academic/Plenum Publishers, New York (2002)zbMATHGoogle Scholar
  2. 2.
    Belohlavek, R.: Fuzzy Galois connections. Math. Logic Quarterly 45(4), 497–504 (1999)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Belohlavek, R.: Algorithms for fuzzy concept lattices. In: Proc. Fourth Int. Conf. on Recent Advances in Soft Computing. Nottingham, United Kingdom, pp. 200–205 (December 12–13, 2002)Google Scholar
  4. 4.
    Belohlavek, R.: Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic 128(1-3), 277–298 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Belohlavek, R., Vychodil, V.: Reducing the size of fuzzy concept lattices by fuzzy closure operators. In: Proc. SCIS & ISIS 2006, pp. 309–314. Tokyo Institute of Technology, Japan, (September 20–24, 2006), ISSN 1880–3741Google Scholar
  6. 6.
    Carpineto, C., Romano, G.: Concept Data Analysis. Theory and Applications. J. Wiley, Chichester (2004)zbMATHGoogle Scholar
  7. 7.
    Ganter, B.: Two basic algorithms in concept analysis. FB4-Preprint No. 831, TH Darmstadt (1984)Google Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal concept analysis. Mathematical Foundations. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  9. 9.
    Gerla, G.: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht (2001)zbMATHGoogle Scholar
  10. 10.
    Goguen, J.A.: The logic of inexact concepts. Synthese 18, 325–373 (1968-1969)Google Scholar
  11. 11.
    Gratzer, G.A.: General Lattice Theory, 2nd edn. Birkhauser (1998)Google Scholar
  12. 12.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)zbMATHGoogle Scholar
  13. 13.
    Hájek, P.: On very true. Fuzzy Sets and Systems 124, 329–333 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Höhle, U.: On the fundamentals of fuzzy set theory. J. Math. Anal. Appl. 201, 786–826 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)zbMATHGoogle Scholar
  16. 16.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice-Hall, Englewood Cliffs (1995)zbMATHGoogle Scholar
  17. 17.
    Kuznetsov, S.O., Obiedkov, S.A.: Comparing performance of algorithms for generating concept lattices. J. Exp. Theor. Artif. Intelligence 14(2/3), 189–216 (2002)zbMATHCrossRefGoogle Scholar
  18. 18.
    Lindig, C.: Fast concept analysis. In: Ganter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS, vol. 1867, pp. 152–161. Springer, Heidelberg (2000)Google Scholar
  19. 19.
    Pollandt, S.: Fuzzy Begriffe. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  20. 20.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht, Boston (1982)Google Scholar
  21. 21.
    Zaki, M.: Mining non-redundant association rules. Data Mining and Knowledge Discovery 9, 223–248 (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Radim Belohlavek
    • 1
    • 3
  • Bernard De Baets
    • 2
  • Jan Outrata
    • 3
  • Vilem Vychodil
    • 3
  1. 1.Dept. Systems Science and Industrial Engineering, T. J. Watson School of Engineering and Applied Science, Binghamton University–SUNY, PO Box 6000, Binghamton, NY 13902–6000USA
  2. 2.Dept. Appl. Math., Biometrics, and Process Control, Ghent University, Coupure links 653, B-9000 GentBelgium
  3. 3.Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 OlomoucCzech Republic

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