Journal of Applied Mathematics

On the Calculation of Formal Concept Stability

Hui-lai Zhi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The idea of stability has been used in many applications. However, computing stability is still a challenge and the best algorithms known so far have algorithmic complexity quadratic to the size of the lattice. To improve the effectiveness, a critical term is introduced in this paper, that is, minimal generator, which serves as the minimal set that makes a concept stable when deleting some objects from the extent. Moreover, by irreducible elements, minimal generator is derived. Finally, based on inclusion-exclusion principle and minimal generator, formulas for the calculation of concept stability are proposed.

Article information

J. Appl. Math., Volume 2014 (2014), Article ID 917639, 6 pages.

First available in Project Euclid: 2 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Zhi, Hui-lai. On the Calculation of Formal Concept Stability. J. Appl. Math. 2014 (2014), Article ID 917639, 6 pages. doi:10.1155/2014/917639.

Export citation


  • S. O. Kuznetsov, “Stability as an estimate of the degree of substantiation of hypotheses derived on the basis of operational similarity,” Nauchno-Tekhnicheskaya Informatsiya Ser.2 (Automat. Document. Math. Linguist), no. 12, pp. 21–29, 1990.
  • S. O. Kuznetsov, “On stability of a formal concept,” Annals of Mathematics and Artificial Intelligence, vol. 49, no. 1–4, pp. 101–115, 2007.
  • S. O. Kuznetsov, S. Obiedkov, and C. Roth, “Reducing the representation complexity of lattice-based taxonomies,” in Proceedings of the 15th International Conference on Conceptual Structures (ICCS '07), U. Priss, S. Polovina, and R. Hill, Eds., vol. 4604 of Lecture Notes in Artificial Intelligence, pp. 241–254, Springer, 2007.
  • C. Roth, S. Obiedkov, and D. G. Kourie, “On succinct representation of knowledge community taxonomies with formal concept analysis,” International Journal of Foundations of Computer Science, vol. 19, no. 2, pp. 383–404, 2008.
  • C. Roth, S. Obiedkov, and D. Kourie, “Towards concise representation for taxonomies of epistemic communities,” in Proceedings of the 4th International Conference on Concept Lattices and Their Applications, pp. 240–255, Tunis, Tunisia, 2006.
  • B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, Pa, USA, 1982.
  • R. Carnap, The Logical Foundations of Probability, University of Chicago Press, Chicago, Ill, USA, 1962.
  • M. A. Babin and S. O. Kuznetsov, “Approximating concept stability,” in Formal Concept Analysis: Proceedings of the 10th International Conference, ICFCA 2012, Leuven, Belgium, May 7–10, 2012, F. Domenach, D. Ignatov, and J. Poelmans, Eds., vol. 7278 of Lecture Notes in Artificial Intelligence, pp. 7–15, 2012.
  • V. K. Finn, “On machine-oriented formalization of plausible reasoning in the style of F. Bacon-J. S. Mill,” Semiotika i Informatika, no. 20, pp. 35–101, 1983.
  • S. O. Kuznetsov, “Mathematical aspects of concept analysis,” Journal of Mathematical Sciences, vol. 80, no. 2, pp. 1654–1698, 1996.
  • B. Ganter and S. O. Kuznetsov, “Formalizing hypotheses with concepts,” in Proceedings of the 8th International Conference on Conceptual Structures (ICCS '00), vol. 1867 of Lecture Notes in Computer Science, pp. 342–356, 2000.
  • R. Wille, “Concept lattices and conceptual knowledge systems,” Computers and Mathematics with Applications, vol. 23, no. 6–9, pp. 493–515, 1992.
  • R. Wille, “Restructuring lattice theory: an approach based on hierarchies of concepts,” in Ordered Sets, vol. 83 of NATO Advanced Study Institutes Series, pp. 445–470, Springer, Berlin, Germany, 1982.
  • B. Ganter and S. O. Kuznetsov, “Hypotheses and version Spaces,” in Proceedings of the 10th International Conference on Conceptual Structures (ICCS '03), A. de Moor, W. Lex, and B. Ganter, Eds., vol. 2746 of Lecture Notes in Artificial Intelligence, pp. 83–95, Dresden, Germany, 2003.
  • W. Hong, J. Mao, J. Yu, and J. Song, “The complete definitions of attributes and abstract description of attribute features of the formal context,” ICIC Express Letters, vol. 7, no. 3, pp. 997–1003, 2013.
  • B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundation, Springer, Berlin, Germany, 1999.
  • S. O. Kuznetsov and S. A. Obiedkov, “Comparing performance of algorithms for generating concept lattices,” Journal of Experimental and Theoretical Artificial Intelligence, vol. 14, no. 2-3, pp. 189–216, 2002. \endinput