Optimizations in computing the Duquenne–Guigues basis of implications

  • Konstantin Bazhanov
  • Sergei ObiedkovEmail author


In this paper, we consider algorithms involved in the computation of the Duquenne–Guigues basis of implications. The most widely used algorithm for constructing the basis is Ganter’s Next Closure, designed for generating closed sets of an arbitrary closure system. We show that, for the purpose of generating the basis, the algorithm can be optimized. We compare the performance of the original algorithm and its optimized version in a series of experiments using artificially generated and real-life datasets. An important computationally expensive subroutine of the algorithm generates the closure of an attribute set with respect to a set of implications. We compare the performance of three algorithms for this task on their own, as well as in conjunction with each of the two algorithms for generating the basis. We also discuss other approaches to constructing the Duquenne–Guigues basis.


Implications Horn formulae Duquenne–Guigues basis Formal Concept Analysis Algorithms LinClosure 

Mathematics Subject Classifications (2010)

06A15 68T99 68W05 68W40 


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  1. 1.
    Angluin, D.: Queries and concept learning. Mach. Learn. 2, 319–342 (1988)Google Scholar
  2. 2.
    Angluin, D., Frazier, M., Pitt, L.: Learning conjunctions of Horn clauses. Mach. Learn. 9, 147–164 (1992)zbMATHGoogle Scholar
  3. 3.
    Arias, M., Balcázar, J.L.: Construction and learnability of canonical Horn formulas. Mach. Learn. 85(3), 273–297 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Armstrong, W.W.: Dependency structures of data base relationships. In: Proc. IFIP Congress, pp. 580–583 (1974)Google Scholar
  5. 5.
    Baader, F., Ganter, B., Sertkaya, B., Sattler, U.: Description logic knowledge bases using formal concept analysis. In: Veloso, M.M. (ed.) Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI’07), pp. 230–235 (2007)Google Scholar
  6. 6.
    Babin, M.A., Kuznetsov, S.O.: Recognizing pseudo-intents is coNP-complete. In: Kryszkiewicz, M., Obiedkov, S. (eds.) Proceedings of the 7th International Conference on Concept Lattices and Their Applications, pp. 294–301. University of Sevilla, Spain (2010)Google Scholar
  7. 7.
    Beeri, C., Bernstein, P.: Computational problems related to the design of normal form relational schemas. ACM Trans. Database Syst. 4(1), 30–59 (1979)CrossRefGoogle Scholar
  8. 8.
    Bertet, K., Monjardet, B.: The multiple facets of the canonical direct unit implicational basis. Theor. Comput. Sci. 411(22–24), 2155–2166 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Blake, C., Merz, C.: UCI repository of machine learning databases (1998).
  10. 10.
    Day, A.: The lattice theory of functional dependencies and normal decompositions. Int. J. Algebra Comput. 2, 409–431 (1992)CrossRefzbMATHGoogle Scholar
  11. 11.
    Demming, R., Duffy, D.: Introduction to the Boost C++ Libraries. Datasim Education Bv (2010). See Accessed 3 May 2013
  12. 12.
    Distel, F., Sertkaya, B.: On the complexity of enumerating pseudo-intents. Discrete Appl. Math. 159, 450–466 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ganter, B.: Two basic algorithms in concept analysis. Preprint 831, Technische Hochschule Darmstadt, Germany (1984)Google Scholar
  14. 14.
    Ganter, B.: Attribute exploration with background knowledge. Theor. Comput. Sci. 217(2), 215–233 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Guigues, J.L., Duquenne, V.: Familles minimales d’implications informatives resultant d’un tableau de donnees binaires. Math. Sci. Hum. 95(1), 5–18 (1986)MathSciNetGoogle Scholar
  17. 17.
    Kautz, H., Kearns, M., Selman, B.: Horn approximations of empirical data. Artif. Intell. 74(1), 129–145 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Khardon, R.: Translating between Horn representations and their characteristic models. J. Artif. Intell. Res. (JAIR) 3, 349–372 (1995)zbMATHGoogle Scholar
  19. 19.
    Klimushkin, M., Obiedkov, S., Roth, C.: Approaches to the selection of relevant concepts in the case of noisy data. In: Kwuida, L., Sertkaya, B. (eds.) Formal Concept Analysis, Lecture Notes in Computer Science, vol. 5986, pp. 255–266. Springer, Berlin/Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. J. Exp. Theor. Artif. Intell. 14(2/3), 189–216 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kuznetsov, S.O., Obiedkov, S.: Some decision and counting problems of the Duquenne–Guigues basis of implications. Discrete Appl. Math. 156(11), 1994–2003 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Maier, D.: The theory of relational databases. Computer software engineering series. Computer Science Press, Rockville (1983)Google Scholar
  23. 23.
    Mannila, H., Räihä, K.J.: The design of relational databases. Addison-Wesley Longman Publishing Co., Inc., Boston, MA (1992)zbMATHGoogle Scholar
  24. 24.
    Obiedkov, S., Duquenne, V.: Attribute-incremental construction of the canonical implication basis. Ann. Math. Artif. Intell. 49(1–4), 77–99 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Obiedkov, S., Kourie, D., Eloff, J.: Building access control models with attribute exploration. Comput. Secur. 28(1–2), 2–7 (2009)CrossRefGoogle Scholar
  26. 26.
    Reeg, S., Wei, W.: Properties of Finite Lattices. Diplomarbeit, TH Darmstadt (1990)Google Scholar
  27. 27.
    Roth, C., Obiedkov, S., Kourie, D.G.: On succinct representation of knowledge community taxonomies with formal concept analysis. Int. J. Found. Comput. Sci. 19(2), 383–404 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Ryssel, U., Distel, F., Borchmann, D.: Fast computation of proper premises. In: Int. Conf. on Concept Lattices and Their Applications, pp. 101–113. INRIA Nancy–Grand Est and LORIA, France (2011)Google Scholar
  29. 29.
    Taouil, R., Bastide, Y.: Computing proper implications. In: Proc. ICCS-2001 International Workshop on Concept Lattices-Based Theory, Methods and Tools for Knowledge Discovery in Databases, pp. 290–303 (2001)Google Scholar
  30. 30.
    Valtchev, P., Duquenne, V.: On the merge of factor canonical bases. In: Medina, R., Obiedkov, S. (eds.) ICFCA, Lecture Notes in Computer Science, vol. 4933, pp. 182–198. Springer, New York (2008)Google Scholar
  31. 31.
    Wild, M.: Computations with finite closure systems and implications. In: Computing and Combinatorics, pp. 111–120 (1995)Google Scholar
  32. 32.
    Yevtushenko, S.A.: System of data analysis “Concept Explorer” (in Russian). In: Proceedings of the 7th National Conference on Artificial Intelligence KII-2000, pp. 127–134. Russia (2000). Accessed 3 May 2013

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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