Mining Formal Concepts Using Implications Between Items

  • Aimene BelfodilEmail author
  • Adnene BelfodilEmail author
  • Mehdi Kaytoue
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)


Formal Concept Analysis (FCA) provides a mathematical tool to analyze and discover concepts in Boolean datasets (i.e. Formal contexts). It does also provide a tool to analyze complex attributes by transforming them into Boolean ones (i.e. items) thanks to conceptual scaling. For instance, a numerical attribute whose values are \(\{1,2,3\}\) can be transformed to the set of items \(\{\le 1, \le 2, \le 3, \ge 3, \ge 2, \ge 1\}\) thanks to interordinal scaling. Such transformations allow us to use standard algorithms like Close-by-One (CbO) to look for concepts in complex datasets by leveraging a closure operator. However, these standard algorithms do not use the relationships between attributes to enumerate the concepts as for example the fact that \(\le 1\) implies \(\le 2\) and so on. For such, they can perform additional closure computations which substantially degrade their performance. We propose in this paper a generic algorithm, named CbOI for Close-by-One using Implications, to enumerate concepts in a formal context using the inherent implications between items provided as an input. We show that using the implications between items can reduce significantly the number of closure computations and hence the time effort spent to enumerate the whole set of concepts.



This work has been partially supported by the project ContentCheck ANR-15-CE23-0025 funded by the French National Research Agency, the ANRt French program and the APRC Conf Pap-CNRS project. The authors would like to thank the reviewers for their valuable remarks. They also warmly thank Anes Bendimerad for interesting discussions.


  1. 1.
    Aho, A.V., Garey, M.R., Ullman, J.D.: The transitive reduction of a directed graph. SIAM J. Comput. 1(2), 131–137 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Belfodil, A., Cazalens, S., Lamarre, P., Plantevit, M.: Flash points: discovering exceptional pairwise behaviors in vote or rating data. In: Ceci, M., Hollmén, J., Todorovski, L., Vens, C., Džeroski, S. (eds.) ECML PKDD 2017. LNCS (LNAI), vol. 10535, pp. 442–458. Springer, Cham (2017). Scholar
  3. 3.
    Belfodil, A., Kuznetsov, S.O., Kaytoue, M.: Pattern setups and their completions. In: CLA, pp. 243–253 (2018)Google Scholar
  4. 4.
    Boley, M., Horváth, T., Poigné, A., Wrobel, S.: Listing closed sets of strongly accessible set systems with applications to data mining. Theor. Comput. Sci. 411(3), 691–700 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bordat, J.P.: Calcul pratique du treillis de galois d’une correspondance. Mathématiques et Sciences humaines 96, 31–47 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cellier, P., Ferré, S., Ridoux, O., Ducassé, M.: An algorithm to find frequent concepts of a formal context with taxonomy. In: CLA, pp. 226–231 (2006)zbMATHGoogle Scholar
  7. 7.
    Dietrich, B.L.: Matroids and antimatroids-a survey. Discrete Math. 78(3), 223–237 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999). Scholar
  9. 9.
    Ganter, B.: Two basic algorithms in concept analysis. Technical report, Technische Hoschule Darmstadt (1984)Google Scholar
  10. 10.
    Ganter, B., Kuznetsov, S.O.: Pattern structures and their projections. In: Delugach, H.S., Stumme, G. (eds.) ICCS-ConceptStruct 2001. LNCS (LNAI), vol. 2120, pp. 129–142. Springer, Heidelberg (2001). Scholar
  11. 11.
    Ganter, B., Wille, R.: Conceptual scaling. In: Roberts, F. (ed.) Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, pp. 139–167. Springer, New York (1989). Scholar
  12. 12.
    Gély, A.: A generic algorithm for generating closed sets of a binary relation. In: Ganter, B., Godin, R. (eds.) ICFCA 2005. LNCS (LNAI), vol. 3403, pp. 223–234. Springer, Heidelberg (2005). Scholar
  13. 13.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kaytoue, M., Kuznetsov, S.O., Napoli, A.: Revisiting numerical pattern mining with formal concept analysis. In: IJCAI, pp. 1342–1347 (2011)Google Scholar
  15. 15.
    Korte, B., Lovász, L.: Mathematical structures underlying greedy algorithms. In: Gécseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 205–209. Springer, Heidelberg (1981). Scholar
  16. 16.
    Krajca, P., Outrata, J., Vychodil, V.: Advances in algorithms based on CbO. In: CLA, pp. 325–337 (2010)Google Scholar
  17. 17.
    Kuznetsov, S.O.: A fast algorithm for computing all intersections of objects in a finite semi-lattice. Nauchno-Tekhnicheskaya Informatsiya ser. 2(1), 17–20 (1993)Google Scholar
  18. 18.
    Kuznetsov, S.O.: Learning of simple conceptual graphs from positive and negative examples. In: Żytkow, J.M., Rauch, J. (eds.) PKDD 1999. LNCS (LNAI), vol. 1704, pp. 384–391. Springer, Heidelberg (1999). Scholar
  19. 19.
    Kuznetsov, S.O.: Pattern structures for analyzing complex data. In: Sakai, H., Chakraborty, M.K., Hassanien, A.E., Ślęzak, D., Zhu, W. (eds.) RSFDGrC 2009. LNCS (LNAI), vol. 5908, pp. 33–44. Springer, Heidelberg (2009). Scholar
  20. 20.
    Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303. ACM (2014)Google Scholar
  21. 21.
    Lumpe, L., Schmidt, S.E.: Pattern structures and their morphisms. In: CLA, vol. 1466, pp. 171–179 (2015)Google Scholar
  22. 22.
    Roman, S.: Lattices and Ordered Sets. Springer, New York (2008). Scholar
  23. 23.
    Soulet, A., Rioult, F.: Efficiently depth-first minimal pattern mining. In: Tseng, V.S., Ho, T.B., Zhou, Z.-H., Chen, A.L.P., Kao, H.-Y. (eds.) PAKDD 2014. LNCS (LNAI), vol. 8443, pp. 28–39. Springer, Cham (2014). Scholar
  24. 24.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, vol. 83, pp. 445–470. Springer, Dordrecht (1982). Scholar

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Authors and Affiliations

  1. 1.Univ Lyon, INSA Lyon, CNRS, LIRIS UMR 5205LyonFrance
  2. 2.Mobile Devices IngénierieVillejuifFrance
  3. 3.InfologicBourg-Lès-ValenceFrance

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