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A Uniform Framework for Rough Approximations Based on Generalized Quantifiers

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Transactions on Rough Sets XIX

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 8988))

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Abstract

The rough set theory provides an effective tool for decision analysis in the way of extracting decision rules from information systems. The rule induction process is based on the definitions of lower and upper approximations of the decision class. The condition attributes of the information system constitute an indiscernibility relation on the universe of objects. An object is in the lower approximation of the decision class if all objects indiscernible with it are in the decision class and it is in the upper approximation of the decision class if some objects indiscernible with it are in the decision class. Various generalizations of rough set theory have been proposed to enhance the capability of the theory. For example, variable precision rough set theory is used to improve the robustness of rough set analysis and fuzzy rough set approach is proposed to deal with vague information. In this paper, we present a uniform framework for different variants of rough set theory by using generalized quantifiers. In the framework, the lower and upper approximations of classical rough set theory are defined with universal and existential quantifiers respectively, whereas variable precision rough approximations correspond to probability quantifiers. Moreover, fuzzy rough set approximations can be defined by using different fuzzy quantifiers. We show that the framework can enhance the expressive power of the decision rules induced by rough set-based decision analysis.

A preliminary version of this paper was published in  [1].

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Notes

  1. 1.

    Also called knowledge representation systems, data tables, or attribute-value systems.

  2. 2.

    Recall that the \(\alpha \)-cut and the strict \(\alpha \)-cut of a membership function \(\nu :U\rightarrow [0,1]\) are defined as \(\{x\in U\mid \nu (x)\ge \alpha \}\) and \(\{x\in U\mid \nu (x)>\alpha \}\) respectively.

References

  1. Fan, T., Liau, C., Liu, D.: Variable precision fuzzy rough set based on relative cardinality. In: Proceedings of the Federated Conference on Computer Science and Information Systems (FedCSIS 2012), pp. 43–47 (2012)

    Google Scholar 

  2. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(15), 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kryszkiewicz, M.: Properties of incomplete information systems in the framework of rough sets. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, pp. 422–450. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  4. Lipski, W.: On databases with incomplete information. J. ACM 28(1), 41–70 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Myszkorowski, K.: Multiargument relationships in fuzzy databases with attributes represented by interval-valued possibility distributions. In: Kuznetsov, S.O., Slezak, D., Hepting, D.H., Mirkin, B.G. (eds.) RSFDGrC 2011. LNCS, vol. 6743, pp. 199–206. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Sakai, H., Nakata, M., Slezak, D.: A prototype system for rule generation in Lipski’s incomplete information databases. In: Kuznetsov, S.O., Slezak, D., Hepting, D.H., Mirkin, B.G. (eds.) RSFDGrC 2011. LNCS, vol. 6743, pp. 175–182. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  7. Yao, Y.Y., Liu, Q.: A generalized decision logic in interval-set-valued information tables. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS (LNAI), vol. 1711, pp. 285–293. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  8. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17(2–3), 191–209 (1990)

    Article  MATH  Google Scholar 

  9. Katzberg, J., Ziarko, W.: Variable precision extension of rough sets. Fundam. Inf. 27(2/3), 155–168 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46(1), 39–59 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cornelis, C., De Cock, M., Radzikowska, A.M.: Vaguely quantified rough sets. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 87–94. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  12. Cornelis, C., Verbiest, N., Jensen, R.: Ordered weighted average based fuzzy rough sets. In: Yu, J., Greco, S., Lingras, P., Wang, G., Skowron, A. (eds.) RSKT 2010. LNCS, vol. 6401, pp. 78–85. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  13. Mieszkowicz-Rolka, A., Rolka, L.: Fuzzy implication operators in variable precision fuzzy rough sets model. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 498–503. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  14. Mieszkowicz-Rolka, A., Rolka, L.: Remarks on approximation quality in variable precision fuzzy rough sets model. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 402–411. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  15. Mieszkowicz-Rolka, A., Rolka, L.: Variable precision fuzzy rough sets. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B., Swiniarski, R.W., Szczuka, M.S. (eds.) TRS I. LNCS, vol. 3100, pp. 144–160. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  16. Mieszkowicz-Rolka, A., Rolka, L.: An approach to parameterized approximation of crisp and fuzzy sets. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 127–136. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  17. Salido, J.F., Murakami, S.: Rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations. Fuzzy Sets Syst. 139(3), 635–660 (2003)

    Article  MATH  Google Scholar 

  18. Zadeh, L.: A computational approach to fuzzy quantifiers in natural languages. Comput. Math. Appl. 9(1), 149–184 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Keenan, D., Westerståhl, D.: Generalized quantifiers in linguistics and logic. In: Benthem, J.V., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 837–893. Elsevier, Beijing (2011)

    Google Scholar 

  20. Lindström, P.: First order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)

    MathSciNet  MATH  Google Scholar 

  21. Mostowski, A.: On a generalization of quantifiers. Fundam. Math. 44(1), 12–36 (1957)

    MathSciNet  MATH  Google Scholar 

  22. Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Clarendon Press, Wotton-under-Edge, UK (2006)

    Google Scholar 

  23. Westerståhl, D.: Quantifiers in natural language: a survey of some recent work. In: Krynicki, M., Mostowski, M., Szczerba, L. (eds.) Quantifiers: Logic, Models and Computation, vol. 1, pp. 359–408. Kluwer Academic Publishers, Dordrecht (1995)

    Chapter  Google Scholar 

  24. Westerståhl, D.: Generalized quantifiers. In: Zalta, E. (ed.) The Stanford Encyclopedia of Philosophy (2011)

    Google Scholar 

  25. Glöckner, I.: Evaluation of quantified propositions in generalized models of fuzzy quantification. Int. J. Approx. Reason. 37(2), 93–126 (2004)

    Article  MATH  Google Scholar 

  26. Thiele, H.: On T-quantifiers and S-quantifiers. In: Proceedings of the 24th IEEE International Symposium on Multiple-Valued Logic, pp. 264–269 (1994)

    Google Scholar 

  27. Delgado, M., Sánchez, D., Martín-Bautista, M., Vila, M.: A probabilistic definition of a nonconvex fuzzy cardinality. Fuzzy Sets Syst. 126(2), 177–190 (2002)

    Article  MATH  Google Scholar 

  28. Pawlak, Z., Skowron, A.: Rough membership functions. In: Yager, R., Fedrizzi, M., Kacprzyk, J. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 251–271. John Wiley & Sons, New York (1994)

    Google Scholar 

  29. Pawlak, Z.: Rough Sets-Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  30. Slezak, D.: Rough sets and bayes factor. In: Peters, J.F., Skowron, A. (eds.) TRS III. LNCS, vol. 3400, pp. 202–229. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  31. Slezak, D., Ziarko, W.: The investigation of the Bayesian rough set model. Int. J. Approx. Reason. 40(1–2), 81–91 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. de Luca, A., Termini, S.: A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Inf. Control 20(4), 301–312 (1972)

    Article  MATH  Google Scholar 

  33. Zadeh, L.: The concept of a linguistic variable and its applications in approximate reasoning. Inf. Sci. 8, 199–251 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zadeh, L.: A theory of approximate reasoning. In: Hayes, J., Mitchie, D., Mikulich, L. (eds.) Machine Intelligence, vol. 9, pp. 149–194 (1979)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Penghu University of Science and Technology, Penghu, 880, Taiwan

    Tuan-Fang Fan

  2. Institute of Information Science, Academia Sinica, Taipei, 115, Taiwan

    Churn-Jung Liau

  3. Institute of Information Management, National Chiao-Tung University, Hsinchu, 300, Taiwan

    Duen-Ren Liu

Authors
  1. Tuan-Fang Fan
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  2. Churn-Jung Liau
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  3. Duen-Ren Liu
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Correspondence to Churn-Jung Liau .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada

    James F. Peters

  2. University of Warsaw, Warsaw, Poland

    Andrzej Skowron

  3. University of Warsaw, Warsaw, Poland

    Dominik Ślȩzak

  4. University of Warsaw, Warsaw, Poland

    Hung Son Nguyen

  5. University of Rzeszów, Rzeszów, Poland

    Jan G. Bazan

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Fan, TF., Liau, CJ., Liu, DR. (2015). A Uniform Framework for Rough Approximations Based on Generalized Quantifiers. In: Peters, J., Skowron, A., Ślȩzak, D., Nguyen, H., Bazan, J. (eds) Transactions on Rough Sets XIX. Lecture Notes in Computer Science(), vol 8988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47815-8_1

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  • DOI: https://doi.org/10.1007/978-3-662-47815-8_1

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