The Lower Approximation Number in Covering-Based Rough Set

  • Hui LiuEmail author
  • William ZhuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)


Covering-based rough set has attracted much research interest with significant achievements. However, there are few analysis that have been conducted to quantify covering-based rough set. The approximation number is viewed as a quantitative tool for analyzing the covering-based rough set. In this paper, we focus on the lower approximation number. Firstly, we investigate some key properties of the lower approximation number. Secondly, we establish a lattice and two semilattice structures in covering-based rough set with the lower approximation number. Finally, based on the lower approximation number, a pair of matroid approximation operators is constructed. Moreover, we investigate the relationship between the pair of matroid approximation operators and a pair of lattice approximation operators.


Covering Rough set The lower approximation number Granular computing 



This work is in part supported by The National Nature Science Foundation of Chi- na under Grant Nos. 61170128, 61379049 and 61379089, the Key Project of Education Department of Fujian Province under Grant No. JA13192, the Project of Education De- partment of Fujian Province under Grant No. JA14194, the Zhangzhou Municipal Nat- ural Science Foundation under Grant No. ZZ2013J03, and the Science and Technology Key Project of Fujian Province, China Grant No. 2012H0043.


  1. 1.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf Sci. 11, 341–356 (1982)CrossRefGoogle Scholar
  2. 2.
    Pawlak, Z.: Rough classification. Int. J. Man-Mach. Stud. 20, 469–483 (1984)CrossRefGoogle Scholar
  3. 3.
    Yao, Y., Chen, Y.: Rough set approximations in formal concept analysis. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 285–305. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  4. 4.
    Drwal, G.: Rough, and fuzzy-rough classification methods implemented in RClass system. In: Ziarko, W.P., Yao, Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, pp. 152–159. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  5. 5.
    Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co-entropies of coverings with application to incomplete information systems. Fundamenta Informaticae 75, 77–105 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, D., Wang, C., Hu, Q.: A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf. Sci. 177, 3500–3518 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, F., Yin, Y.: Approaches to knowledge reduction of covering decision systems based on information theory. Inf. Sci. 179, 1694–1704 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhu, F., He, H.: Logical properties of rough sets. In: The Fourth International Conference on High Performance Computing in the Asia-Pacific Region, pp. 670–671. IEEE Press (2000)Google Scholar
  9. 9.
    Zhu, F., He, H.: The axiomization of the rough set. Chin. J. Comput. 23, 330–333 (2000)Google Scholar
  10. 10.
    Zhu, W.: Relationship among basic concepts in covering-based rough sets. Inf. Sci. 179, 2478–2486 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ma, L.: On some types of neighborhood-related covering rough sets. Int. J. Approx. Reasoning 53, 901–911 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intentions in the rough set theory. Inf. Sci. 107, 149–167 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bartol, W., Miró, J., Pióro, K., Rosselló, F.: On the coverings by tolerance classes. Inf. Sci. 166, 193–211 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, G., Sai, Y.: A comparison of two types of rough sets induced by coverings. Int. J. Approx. Reasoning 50, 521–528 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhu, W., Wang, F.: The fourth type of covering-based rough sets. Inf. Sci. 201, 80–92 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, G., Sai, Y.: Invertible approximation operators of generalized rough sets and fuzzy rough sets. Inf. Sci. 180, 2221–2229 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, S., Zhu, W., Zhu, Q., Min, F.: Covering base. J. Inf. Comput. Sci. 9, 1343–1355 (2012)Google Scholar
  19. 19.
    Wang, S., Zhu, Q., Zhu, W., Min, F.: Matroidal structure of rough sets and its characterization to attribute reduction. Knowl.-Based Syst. 36, 155–161 (2012)CrossRefGoogle Scholar
  20. 20.
    Min, F., Zhu, W.: Attribute reduction of data with error ranges and test costs. Inf. Sci. 211, 48–67 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qian, Y., Liang, J., Li, D., Wang, F., Ma, N.: Approximation reduction in inconsistent incomplete decision tables. Knowl.-Based Syst. 23, 427–433 (2010)CrossRefGoogle Scholar
  22. 22.
    Wang, S., Zhu, W.: Matroidal structure of covering-based rough sets through the upper approximation number. Int. J. Granular Comput., Rough Sets Intel. Syst. 2, 141–148 (2011)CrossRefGoogle Scholar
  23. 23.
    Birhoff, G.: Lattice Theory. American Mathematical Society, Rhode Island (1995)Google Scholar
  24. 24.
    Zhang, W., Yao, Y., Liang, Y.: Rough set and concept lattice. Xi’an Jiaotong University Press (2006)Google Scholar

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Authors and Affiliations

  1. 1.Lab of Granular ComputingMinnan Normal UniversityZhangzhouChina

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