Rough Sets in Incomplete Information Systems with Order Relations Under Lipski’s Approach

  • Michinori Nakata
  • Hiroshi Sakai
  • Keitarou Hara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10313)


Rough sets and rule induction based on them are described in incomplete information tables where attribute values are ordered. We apply possible world semantics to an incomplete information table, as Lipski did in incomplete databases. The set of possible tables on a set of attributes is derived from the original incomplete information table. Rough sets, a pair of lower and upper approximations, are obtained from every possible table. An object is certainly included in an approximation when it is in the approximation in all possible tables, while an object is possibly included in an approximation when it is in the approximation in some possible tables. From this, we obtain certain and possible approximations. The actual approximation is greater than the certain one and less than the possible one. Finally, we obtain the approximation in the form of interval sets. There exists a gap between rough sets and rule induction from them. To bridge rough sets and rule induction, we give expressions that correspond to certain and possible approximations. The expressions consist of a pair of an object and a rule that the object supports. Consequently, four types of rule supports: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent supports, are obtained from the expressions. The formulae can be applied to the case where not only attributes used to approximate but also attributes approximated have a value with incomplete information. The results give a correctness criterion of rough sets and rule induction based on them in incomplete ordered information systems, as the results of Lipski’s work are so in incomplete databases.


Rough sets Rule induction Incomplete information systems Ordered domains Possible world semantics 



The authors wish to thank the anonymous reviewers for their valuable comments.


  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley Publishing Company, Boston (1995)zbMATHGoogle Scholar
  2. 2.
    Bosc, P., Duval, L., Pivert, O.: An initial approach to the evaluation of possibilistic queries addressed to possibilistic databases. Fuzzy Sets Syst. 140, 151–166 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z., Shi, P., Liu, P., Pei, Z.: Criteria reduction of set-valued ordered decision system based on approximation quality. Int. J. Innov. Comput. Inf. Control 9(6), 2393–2404 (2013)Google Scholar
  4. 4.
    Du, W.S., Hu, B.Q.: Dominance-based rough set approach to incomplete ordered information systems. Inf. Sci. 346–347, 106–129 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grahne, G.: The Problem of Incomplete Information in Relational Databases. LNCS, vol. 554. Springer, Heidelberg (1991). doi: 10.1007/3-540-54919-6 zbMATHGoogle Scholar
  6. 6.
    Greco, S., Matarazzo, B., Słowinski, R.: Handling missing values in rough set analysis of multi-attribute and multi-criteria decision problems. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS, vol. 1711, pp. 146–157. Springer, Heidelberg (1999). doi: 10.1007/978-3-540-48061-7_19 CrossRefGoogle Scholar
  7. 7.
    Greco, S., Matarazzo, B., Slowinski, R.: Rough sets theory for multicriteria decision analysis. Eur. J. Oper. Res. 129, 1–47 (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hu, M., Yao, Y.: Definability in incomplete information tables. In: Flores, V., Gomide, F., Janusz, A., Meneses, C., Miao, D., Peters, G., Ślęzak, D., Wang, G., Weber, R., Yao, Y. (eds.) IJCRS 2016. LNCS, vol. 9920, pp. 177–186. Springer, Cham (2016). doi: 10.1007/978-3-319-47160-0_16 CrossRefGoogle Scholar
  9. 9.
    Imielinski, T.: Incomplete information in logical databases. Data Eng. 12, 93–104 (1989)Google Scholar
  10. 10.
    Imielinski, T., Lipski, W.: Incomplete information in relational databases. J. ACM 31, 761–791 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kryszkiewicz, M.: Rules in incomplete information systems. Inf. Sci. 113, 271–292 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lipski, W.: On semantics issues connected with incomplete information databases. ACM Trans. Database Syst. 4, 262–296 (1979)CrossRefGoogle Scholar
  13. 13.
    Lipski, W.: On databases with incomplete information. J. ACM 28, 41–70 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Luo, C., Li, T., Chen, H., Liu, D.: Incremental approaches for updating approximations in set-valued ordered information systems. Knowl.-Based Syst. 50, 218–233 (2013)CrossRefGoogle Scholar
  15. 15.
    Luo, G., Yang, X.: Limited dominance-based rough set model and knowledge reductions in incomplete decision system. J. Inf. Sci. Eng. 26, 2199–2211 (2010)MathSciNetGoogle Scholar
  16. 16.
    Nakata, M., Sakai, H.: Applying rough sets to information tables containing missing values. In: Proceedings of 39th International Symposium on Multiple-Valued Logic, pp. 286–291. IEEE Computer Society Press (2009)Google Scholar
  17. 17.
    Nakata, M., Sakai, H.: Twofold rough approximations under incomplete information. Int. J. Gen. Syst. 42, 546–571 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Paredaens, J., De Bra, P., Gyssens, M., Van Gucht, D.: The Structure of the Relational Database Model. Springer, Heidelberg (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Parsons, S.: Current approaches to handling imperfect information in data and knowledge bases. IEEE Trans. Knowl. Data Eng. 8, 353–372 (1996)CrossRefGoogle Scholar
  20. 20.
    Parsons, S.: Addendum to current approaches to handling imperfect information in data and knowledge bases. IEEE Trans. Knowl. Data Eng. 10, 862 (1998)Google Scholar
  21. 21.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Qi, Y., Sun, H., Yang, X., Song, Y., Sun, Q.: Approches to approximate distribution reduct in incomplete ordered decision system. J. Inf. Comput. Sci. 3(3), 189–198 (2008)Google Scholar
  23. 23.
    Qian, Y.H., Liang, J.Y., Song, P., Dang, C.Y.: On dominance relations in disjunctive set-valued ordered information systems. Int. J. Inf. Technol. Decis. Mak. 9(1), 9–33 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Sakai, H., Okuma, A.: Basic algorithms and tools for rough non-deterministic information analysis. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 209–231. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-27794-1_10 CrossRefGoogle Scholar
  25. 25.
    Sakai, H., Ishibashi, R., Koba, K., Nakata, M.: Rules and apriori algorithm in non-deterministic information systems. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 328–350. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-89876-4_18 CrossRefGoogle Scholar
  26. 26.
    Sakai, H., Liu, C., Zhu, X., Nakata, M.: On NIS-apriori based data mining in SQL. In: Flores, V., et al. (eds.) IJCRS 2016. LNCS, vol. 9920, pp. 514–524. Springer, Cham (2016). doi: 10.1007/978-3-319-47160-0_47 CrossRefGoogle Scholar
  27. 27.
    Sakai, H., Wu, M., Nakata, M.: Apriori-based rule generation in incomplete information databases and non-deterministic information systems. Fundam. Inform. 130(3), 343–376 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Shao, M., Zhang, W.: Dominance relation and rules in an incomplete ordered information system. Int. J. Intell. Syst. 20, 13–27 (2005)CrossRefzbMATHGoogle Scholar
  29. 29.
    Stefanowski, J., Tsoukiàs, A.: On the extension of rough sets under incomplete information. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS, vol. 1711, pp. 73–81. Springer, Heidelberg (1999). doi: 10.1007/978-3-540-48061-7_11 CrossRefGoogle Scholar
  30. 30.
    Wang, H., Guan, Y., Huang, J., Shen, J.: Decision rules acquisition for inconsistent disjunctive set-valued ordered decision information systems. Math. Prob. Eng. 2015, Article ID 936340, 8 p. (2015)Google Scholar
  31. 31.
    Wei, L., Tang, Z., Wang, R., Yang, X.: Extensions of dominance-based rough set approach in incomplete information system. Autom. Control Comput. Sci. 42(5), 255–263 (2008)CrossRefGoogle Scholar
  32. 32.
    Yang, X., Dou, H.: Valued dominance-based rough set approach to incomplete information system. In: Gavrilova, M.L., Tan, C.J.K. (eds.) Transactions on Computational Science XIII. LNCS, vol. 6750, pp. 92–107. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22619-9_5 CrossRefGoogle Scholar
  33. 33.
    Yang, X., Yang, J., Wu, C., Yu, D.: Dominance-based rough set approach and knowledge reductions in incomplete ordered information system. Inf. Sci. 178, 1219–1234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zimányi, E., Pirotte, A.: Imperfect information in relational databases. In: Motro, A., Smets, P. (eds.) Uncertainty Management in Information Systems: From Needs to Solutions, pp. 35–87. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Michinori Nakata
    • 1
  • Hiroshi Sakai
    • 2
  • Keitarou Hara
    • 3
  1. 1.Faculty of Management and Information ScienceJosai International UniversityToganeJapan
  2. 2.Faculty of Engineering, Department of Mathematics and Computer Aided SciencesKyushu Institute of TechnologyTobataJapan
  3. 3.Department of InformaticsTokyo University of Information SciencesWakaba-kuJapan

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