Skip to main content
Springer Nature Link
Log in

On Generalized Rough Fuzzy Approximation Operators

  • Conference paper
Transactions on Rough Sets V

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 4100))

Abstract

This paper presents a general framework for the study of rough fuzzy sets in which fuzzy sets are approximated in a crisp approximation space. By the constructive approach, a pair of lower and upper generalized rough fuzzy approximation operators is first defined. The rough fuzzy approximation operators are represented by a class of generalized crisp approximation operators. Properties of rough fuzzy approximation operators are then discussed. The relationships between crisp relations and rough fuzzy approximation operators are further established. By the axiomatic approach, various classes of rough fuzzy approximation operators are characterized by different sets of axioms. The axiom sets of rough fuzzy approximation operators guarantee the existence of certain types of crisp relations producing the same operators. The relationship between a fuzzy topological space and rough fuzzy approximation operators is further established. The connections between rough fuzzy sets and Dempster-Shafer theory of evidence are also examined. Finally multi-step rough fuzzy approximations within the framework of neighborhood systems are analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bodjanova, S.: Approximation of a fuzzy concepts in decision making. Fuzzy Sets and Systems 85, 23–29 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boixader, D., Jacas, J., Recasens, J.: Upper and lower approximations of fuzzy sets. International Journal of General Systems 29(4), 555–568 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chakrabarty, K., Biswas, R., Nanda, S.: Fuzziness in rough sets. Fuzzy Sets and Systems 110, 247–251 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chuchro, M.: A certain conception of rough sets in topological Boolean algebras. Bulletin of the Section of Logic 22(1), 9–12 (1993)

    MATH  Google Scholar 

  5. Chuchro, M.: On rough sets in topological Boolean algebras. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 157–160. Springer, Berlin (1994)

    Google Scholar 

  6. Comer, S.: An algebraic approach to the approximation of information. Fundamenta Informaticae 14, 492–502 (1991)

    MATH  MathSciNet  Google Scholar 

  7. Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38, 325–339 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Denoeux, T.: Modeling vague beliefs using fuzzy-valued belief structures. Fuzzy Sets and Systems 116, 167–199 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dubois, D., Prade, H.: Twofold fuzzy sets and rough sets—some issues in knowledge representation. Fuzzy Sets and Systems 23, 3–18 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17, 191–208 (1990)

    Article  MATH  Google Scholar 

  11. Dubois, D., Prade, H.: Fundamentals of Fuzzy Sets. Kluwer Academic Publishers, Boston (2000)

    MATH  Google Scholar 

  12. Inuiguchi, M., Hirano, S., Tsumoto, S. (eds.): Rough Set Theory and Granular Computing. Springer, Berlin (2003)

    MATH  Google Scholar 

  13. Kortelainen, J.: On relationship between modified sets, topological space and rough sets. Fuzzy Sets and Systems 61, 91–95 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kuncheva, L.I.: Fuzzy rough sets: application to feature selection. Fuzzy Sets and Systems 51, 147–153 (1992)

    Article  MathSciNet  Google Scholar 

  15. Lin, T.Y.: Neighborhood systems—application to qualitative fuzzy and rough sets. In: Wang, P.P. (ed.) Advances in Machine Intelligence and Soft-Computing. Department of Electrical Engineering, pp. 132–155. Duke University, Durham (1997)

    Google Scholar 

  16. Lin, T.Y., Liu, Q.: Rough approximate operators: axiomatic rough set theory. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 256–260. Springer, Berlin (1994)

    Google Scholar 

  17. Lin, T.Y., Yao, Y.Y., Zadeh, L.A. (eds.): Data Mining, Rough Sets and Granular Computing. Physica-Verlag, Heidelberg (2002)

    MATH  Google Scholar 

  18. Lowen, R.: Fuzzy topological spaces and fuzzy compactness. Journal of Mathematical Analysis and Applications 56, 621–633 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mi, J.-S., Leung, Y., Wu, W.-Z.: An uncertainty measure in partition-based fuzzy rough sets. International Journal of General Systems 34(1), 77–90 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mi, J.-S., Zhang, W.-X.: An axiomatic characterization of a fuzzy generalization of rough sets. Information Sciences 160, 235–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough sets. Fuzzy Sets and Systems 100, 327–342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nakamura, A., Gao, J.M.: On a KTB-modal fuzzy logic. Fuzzy Sets and Systems 45, 327–334 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nanda, S., Majumda, S.: Fuzzy rough sets. Fuzzy Sets and Systems 45, 157–160 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pal, S.K.: Roughness of a fuzzy set. Information Sciences 93, 235–246 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pawlak, Z.: Information systems, theoretical foundations. Information Systems 6, 205–218 (1981)

    Article  MATH  Google Scholar 

  26. Pawlak, Z.: Rough sets. International Journal of Computer and Information Science 11, 341–356 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Boston (1991)

    MATH  Google Scholar 

  28. Pei, D.W., Xu, Z.-B.: Rough set models on two universes. International Journal of General Systems 33, 569–581 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Peters, J.F., Pawlak, Z., Skowron, A.: A rough set approach to measuring information granules. In: Proceedings of COMPSAC 2002, pp. 1135–1139 (2002)

    Google Scholar 

  30. Peters, J.F., Skowron, A., Synak, P., Ramanna, S.: Rough sets and information granulation. In: De Baets, B., Kaynak, O., Bilgiç, T. (eds.) IFSA 2003. LNCS (LNAI), vol. 2715, pp. 370–377. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  31. Pomykala, J.A.: Approximation operations in approximation space. Bulletin of the Polish Academy of Sciences: Mathematics 35, 653–662 (1987)

    MATH  MathSciNet  Google Scholar 

  32. Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets and Systems 126, 137–155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  34. Skowron, A.: The relationship between the rough set theory and evidence theory. Bulletin of Polish Academy of Science: Mathematics 37, 87–90 (1989)

    Google Scholar 

  35. Skowron, A.: The rough sets theory and evidence theory. Fundamenta Informatica 13, 162–245 (1990)

    MathSciNet  Google Scholar 

  36. Skowron, A., Grzymala-Busse, J.: From rough set theory to evidence theory. In: Yager, R., et al. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 193–236. Wiley, New York (1994)

    Google Scholar 

  37. Skowron, A., Stepaniuk, J.: Information granules: towards foundations of granular computing. International Journal of Intelligent Systems 16, 57–85 (2001)

    Article  MATH  Google Scholar 

  38. Skowron, A., Świniarski, R.W., Synak, P.: Approximation spaces and information granulation. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III. LNCS, vol. 3400, pp. 175–189. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  39. Slowinski, R., Vanderpooten, D.: Similarity relation as a basis for rough approximations. In: Wang, P.P. (ed.) Advances in Machine Intelligence and Soft-Computing. Department of Electrical Engineering, pp. 17–33. Duke University, Durham (1997)

    Google Scholar 

  40. Slowinski, R., Vanderpooten, D.: A Generalized definition of rough approximations based on similarity. IEEE Transactions on Knowledge and Data Engineering 12(2), 331–336 (2000)

    Article  Google Scholar 

  41. Thiele, H.: On axiomatic characterisations of crisp approximation operators. Information Sciences 129, 221–226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Thiele, H.: On axiomatic characterizations of fuzzy approximation operators: I. The fuzzy rough set based case. In: Ziarko, W.P., Yao, Y. (eds.) RSCTC 2000. LNCS, vol. 2005, pp. 239–247. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  43. Thiele, H.: On axiomatic characterisation of fuzzy approximation operators II, the rough fuzzy set based case. In: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, pp. 330–335 (2001)

    Google Scholar 

  44. Thiele, H.: On axiomatic characterization of fuzzy approximation operators III, the fuzzy diamond and fuzzy box cases. In: The 10th IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1148–1151 (2001)

    Google Scholar 

  45. Vakarelov, D.: A modal logic for similarity relations in Pawlak knowledge representation systems. Fundamenta Informaticae 15, 61–79 (1991)

    MATH  MathSciNet  Google Scholar 

  46. Wasilewska, A.: Conditional knowledge representation systems—model for an implementation. Bulletin of the Polish Academy of Sciences: Mathematics 37, 63–69 (1990)

    MathSciNet  Google Scholar 

  47. Wiweger, R.: On topological rough sets. Bulletin of Polish Academy of Sciences: Mathematics 37, 89–93 (1989)

    MATH  MathSciNet  Google Scholar 

  48. Wu, W.-Z.: A study on relationship between fuzzy rough approximation operators and fuzzy topological spaces. In: Wang, L., Jin, Y. (eds.) FSKD 2005. LNCS (LNAI), vol. 3613, pp. 167–174. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  49. Wu, W.-Z.: Upper and lower probabilities of fuzzy events induced by a fuzzy set-valued mapping. In: Ślęzak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 345–353. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  50. Wu, W.-Z., Leung, Y., Mi, J.-S.: On characterizations of (\({\cal I, T}\))-fuzzy rough approximation operators. Fuzzy Sets and Systems 154(1), 76–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  51. Wu, W.-Z., Leung, Y., Zhang, W.-X.: Connections between rough set theory and Dempster-Shafer theory of evidence. International Journal of General Systems 31(4), 405–430 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  52. Wu, W.-Z., Mi, J.-S., Zhang, W.-X.: Generalized fuzzy rough sets. Information Sciences 151, 263–282 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  53. Wu, W.-Z., Zhang, W.-X.: Neighborhood operator systems and approximations. Information Sciences 144, 201–217 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  54. Wu., W.-Z., Zhang, W.-X.: Constructive and axiomatic approaches of fuzzy approximation operators. Information Sciences 159, 233–254 (2004)

    Article  MathSciNet  Google Scholar 

  55. Wu., W.-Z., Zhang, W.-X., Li, H.-Z.: Knowledge acquisition in incomplete fuzzy information systems via rough set approach. Expert Systems 20(5), 280–286 (2003)

    Article  Google Scholar 

  56. Wybraniec-Skardowska, U.: On a generalization of approximation space. Bulletin of the Polish Academy of Sciences: Mathematics 37, 51–61 (1989)

    MATH  MathSciNet  Google Scholar 

  57. Yao, Y.Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning 15, 291–317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  58. Yao, Y.Y.: Combination of rough and fuzzy sets based on α-level sets. In: Lin, T.Y., Cercone, N. (eds.) Rough Sets and Data Mining: Analysis for Imprecise Data, pp. 301–321. Kluwer Academic Publishers, Boston (1997)

    Google Scholar 

  59. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Journal of Information Sciences 109, 21–47 (1998)

    Article  MATH  Google Scholar 

  60. Yao, Y.Y.: Generalized rough set model. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1. Methodology and Applications, pp. 286–318. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  61. Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111, 239–259 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  62. Yao, Y.Y.: Information granulation and rough set approximation. International Journal of Intelligent Systems 16, 87–104 (2001)

    Article  MATH  Google Scholar 

  63. Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing, an International Journal 2, 103–120 (1996)

    Google Scholar 

  64. Yao, Y.Y., Lingras, P.J.: Interpretations of belief functions in the theory of rough sets. Information Sciences 104, 81–106 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  65. Yen, J.: Generalizing the Dempster-Shafer theory to fuzzy sets. IEEE Transaction on Systems, Man, and Cybernetics 20(3), 559–570 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  66. Zadeh, L.A.: Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications 23, 421–427 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  67. Zadeh, L.A.: Fuzzy sets and information granularity. In: Gupta, M.M., Ragade, R.K., Yager, R.R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North-Holland, Amsterdam (1979)

    Google Scholar 

  68. Zadeh, L.A.: Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems 19, 111–127 (1997)

    Article  MathSciNet  Google Scholar 

  69. Zakowski, W.: On a concept of rough sets. Demonstratio Mathematica 15, 1129–1133 (1982)

    MATH  MathSciNet  Google Scholar 

  70. Zhang, M., Wu, W.-Z.: Knowledge reductions in information systems with fuzzy decisions. Journal of Engineering Mathematics 20(2), 53–58 (2003)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information College, Zhejiang Ocean University, Zhoushan, Zhejiang, 316004, P.R. China

    Wei-Zhi Wu

  2. Department of Geography and Resource Management, Center for Environmental Policy and Resource Management, and Institute of Space and Earth Information Science, The Chinese University of Hong Kong, Hong Kong

    Yee Leung

  3. Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaan’xi, 710049, P.R. China

    Wen-Xiu Zhang

Authors
  1. Wei-Zhi Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yee Leung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wen-Xiu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Manitoba, R3T 5V6, Winnipeg, Manitoba, Canada

    James F. Peters

  2. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Skowron

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Wu, WZ., Leung, Y., Zhang, WX. (2006). On Generalized Rough Fuzzy Approximation Operators. In: Peters, J.F., Skowron, A. (eds) Transactions on Rough Sets V. Lecture Notes in Computer Science, vol 4100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847465_13

Download citation

  • DOI: https://doi.org/10.1007/11847465_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39382-5

  • Online ISBN: 978-3-540-39383-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics