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Springer Handbook of Computational Intelligence pp 413–424Cite as

Generalized Rough Sets

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Abstract

This chapter reviews three formulations of rough set theory, i. e., element-based definition, granule-based definition, and subsystem-based definition. These formulations are adopted to generalize rough sets from three directions. The first direction is to use an arbitrary binary relation to generalize the equivalence relation in the element-based definition. The second is to use a covering to generalize the partition in the granule-based definition, and the third to use a subsystem to generalize the Boolean algebra in the subsystem-based definition. In addition, we provide some insights into the theoretical aspects of these generalizations, mainly with respect to relations with non-classical logic and topology theory.

Keywords

  • Rough Set (RS)
  • Arbitrary Binary Relation
  • Covering-based Rough Sets
  • Dual Approximation Operators
  • Pair Approximation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Abbreviations

DAL:

logic for data analysis

NIL:

nondeterministic information logic

References

  1. Z. Pawlak: Rough sets, Int. J. Parallel Program. 11(5), 341–356 (1982)

    MathSciNet  MATH  Google Scholar 

  2. Z. Pawlak: Rough Sets: Theoretical Aspects of Reasoning About Data (Kluwer, Dordrecht 1991)

    CrossRef  MATH  Google Scholar 

  3. Y.Y. Yao: Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason. 15(4), 291–317 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Y.Y. Yao, B.X. Yao: Covering based rough set approximations, Inf. Sci. 200, 91–107 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Y.Y. Yao: On generalizing rough set theory, Proc. Int. Conf. Rough Sets Fuzzy Sets Data Min. Granul. Comput. (Springer, Berlin Heidelberg 2003) pp. 44–51

    CrossRef  Google Scholar 

  6. Y.Y. Yao, T.Y. Lin: Generalization of rough sets using modal logic, Intell. Autom. Soft Comput. 2(2), 103–120 (1996)

    CrossRef  MathSciNet  Google Scholar 

  7. Y.Y. Yao: On generalizing Pawlak approximation operators, Proc. Int. Conf. Rough Sets Curr. Trends Comput. (Springer, Berlin Heidelberg 1998) pp. 298–307

    CrossRef  Google Scholar 

  8. Y.Y. Yao, T. Wang: On rough relations: An alternative formulation, Proc. Int. Conf. New Dir. Rough Sets Data Min. and Granul.-Soft Comput. (Springer, Berlin Heidelberg 1999) pp. 82–90

    Google Scholar 

  9. U. Wybraniec-Skardowska: On a generalization of approximation space, Bull. Pol. Acad. Sci. Math. 37(1-6), 51–62 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Z. Pawlak: Hard and soft sets, Rough Sets, Proc. Int. Workshop Rough Sets Knowl. Discov. (Springer, London 1994) pp. 130–135

    CrossRef  Google Scholar 

  11. Y.Y. Yao, X. Li, T.Y. Lin, Q. Liu: Representation and classification of rough set models, Proc. Int. Workshop Rough Sets Soft Comput. (SCS: Society for Computer Simulation, San Diego 1995) pp. 44–47

    Google Scholar 

  12. R. Slowinski, D. Vanderpooten: A generalized definition of rough approximations based on similarity, Knowl. Data Eng., IEEE Trans. 12(2), 331–336 (2000)

    Google Scholar 

  13. S. Greco, B. Matarazzo, R. Slowinski: Fuzzy similarity relation as a basis for rough approximations, Proc. Int. Conf. Rough Sets Curr. Trends Comput. (Springer, Berlin Heidelberg 1998) pp. 283–289

    CrossRef  Google Scholar 

  14. L.H. Guan, G.Y. Wang: Generalized approximations defined by non-equivalence relations, Inf. Sci. 193, 163–179 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Y.Y. Yao: Relational interpretations of neighborhood operators and rough set approximation operators, Inf. Sci. 111(1), 239–259 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. W.Z. Wu, W.X. Zhang: Neighborhood operator systems and approximations, Inf. Sci. 144(1), 201–217 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. H.M. Abu-Donia: Comparison between different kinds of approximations by using a family of binary relations, Knowl.-Based Syst. 21(8), 911–919 (2008)

    CrossRef  Google Scholar 

  18. Y.Y. Yao: Generalized rough set models. In: Rough Sets in Knowledge Discovery, ed. by L. Polkowski, A. Skowron (Physica, Heidelberg 1998) pp. 286–318

    Google Scholar 

  19. W. Zhu: Relationship between generalized rough sets based on binary relation and covering, Inf. Sci. 179(3), 210–225 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Y.Y. Yao, S.K.M. Wang, T.Y. Lin: A review of rough set models. In: Rough Sets and Data Mining: Analysis for Imprecise Data, ed. by L. Polkowski, A. Skowron (Kluwer, Boston 1997) pp. 47–75

    CrossRef  Google Scholar 

  21. Y.Y. Yao: Constructive and algebraic methods of the theory of rough sets, J. Inf. Sci. 109(1), 21–47 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. J.T. Yao, Y.Y. Yao: Induction of classification rules by granular computing, Proc. Int. Conf. Rough Sets Curr. Trends Comput. (Springer, Berlin Heidelberg 2002) pp. 331–338

    CrossRef  Google Scholar 

  23. J.T. Yao, A.V. Vasilakos, W. Pedrycz: Granular computing: Perspectives and challenges, IEEE Trans. Cybern. 43(6), 1977–1989 (2013)

    CrossRef  Google Scholar 

  24. W. Zakowski: Approximations in the space (u, Π), Demonstr. Math. 16(40), 761–769 (1983)

    MathSciNet  MATH  Google Scholar 

  25. J.A. Pomykala: Approximation operations in approximation space, Bull. Pol. Acad. Sci. Math. 35, 653–662 (1987)

    MathSciNet  MATH  Google Scholar 

  26. J.A. Pomykała: On definability in the nondeterministic information system, Bull. Pol. Acad. Sci.: Math. 36(3/4), 193–210 (1988)

    MathSciNet  MATH  Google Scholar 

  27. I. Couso, D. Dubois: Rough sets, coverings and incomplete information, Fundam. Inf. 108(3), 223–247 (2011)

    MathSciNet  MATH  Google Scholar 

  28. M. Restrepo, C. Cornelis, J. Gómez: Duality, conjugacy and adjointness of approximation operators in covering-based rough sets, Int. J. Approx. Reason. 55(1), 469–485 (2014)

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. J. Wang, D. Dai, Z. Zhou: Fuzzy covering generalized rough sets, J. Zhoukou Teach. Coll. 21(2), 20–22 (2004), in Chinese

    Google Scholar 

  30. Z. Xu, Q. Wang: On the properties of covering rough sets model, J. Henan Norm. Univ. 33(1), 130–132 (2005), in Chinese

    MathSciNet  MATH  Google Scholar 

  31. W.H. Xu, W.X. Zhang: Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets Syst. 158(22), 2443–2455 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. W. Zhu, F.Y. Wang: Some results on covering generalized rough sets, Pattern Recogn. Artif. Intell. 15(1), 6–13 (2002)

    Google Scholar 

  33. W. Zhu, F.Y. Wang: Reduction and axiomization of covering generalized rough sets, Inf. Sci. 152, 217–230 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. W. Zhu: Properties of the second type of covering-based rough sets, Proc. Int. Web Intell. Intell. Agent Technol. (IEEE, Piscataway 2006) pp. 494–497

    Google Scholar 

  35. W. Zhu, F.Y. Wang: A new type of covering rough set, Proc. Int. Conf. Intell. Syst. (IEEE, Piscataway 2006) pp. 444–449

    Google Scholar 

  36. W. Zhu, F.Y. Wang: On three types of covering-based rough sets, IEEE Trans. Knowl. Data Eng. 19(8), 1131–1144 (2007)

    CrossRef  Google Scholar 

  37. G.L. Liu: The relationship among different covering approximations, Inf. Sci. 250, 178–183 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Y.-L. Zhang, M.-K. Luo: Relationships between covering-based rough sets and relation-based rough sets, Inf. Sci. 225, 55–72 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. P. Samanta, M.K. Chakraborty: Generalized rough sets and implication lattices, Trans. Rough Sets 14, 183–201 (2011)

    CrossRef  MATH  Google Scholar 

  40. J.A. Pomykala: Approximation, Similarity and Rough Construction, ILLC Prepublication Series for Computation and Complexity Theory, Vol. 93 (Univ. Amsterdam, Amsterdam 1993)

    Google Scholar 

  41. T.-J. Li: Rough approximation operators in covering approximation spaces, RSCTC2006 Proc. (Springer, Berlin Heidelberg 2006) pp. 174–182

    Google Scholar 

  42. D. Slezak, P. Wasilewski: Granular sets – Foundations and case study of tolerance spaces, RSFDGrC2007 Proc. (Springer, Berlin Heidelberg 2007) pp. 435–442

    Google Scholar 

  43. G. Cattaneo, D. Ciucci: Lattices with interior and closure operators and abstract approximation spaces, Trans. Rough Sets 10, 67–116 (2009)

    MATH  Google Scholar 

  44. Z. Bonikowski: A certain conception of the calculus of rough sets, Notre Dame J. Formal Log. 33(3), 412–421 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. K.Y. Qin, Y. Gao, Z. Pei: On covering rough sets, RSKT2007 Proc. (Springer, Berlin Heidelberg 2007) pp. 34–41

    Google Scholar 

  46. W. Zhu: Topological approaches to covering rough sets, Inf. Sci. 177(6), 1499–1508 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. Y.Y. Yao, Y.H. Chen: Subsystem based generalizations of rough set approximations, Proc. Int. Conf. Found. Intell. Syst. (Springer, Berlin Heidelberg 2005) pp. 210–218

    CrossRef  Google Scholar 

  48. A. Wiweger: On topological rough sets, Bull. Pol. Acad. Sci. Math. 37, 89–93 (1989)

    MathSciNet  MATH  Google Scholar 

  49. J. Járvinen: On the structure of rough approximations, Fundam. Inf. 53(2), 135–153 (2002)

    MathSciNet  MATH  Google Scholar 

  50. G. Cattaneo: Abstract approximation spaces for rough theories, Rough Sets Knowl. Discov. 1, 59–98 (1998)

    MathSciNet  MATH  Google Scholar 

  51. M. Gehrke, E. Walker: On the structure of rough sets, Bull. Pol. Acad. Sci. Math. 40, 235–245 (1992)

    MathSciNet  MATH  Google Scholar 

  52. B.F. Chellas: Modal Logic: An Introduction (Cambridge Univ. Press, Cambridge 1988)

    MATH  Google Scholar 

  53. E. Orlowska, Z. Pawlak: Representation of nondeterministic information, Theor. Comput. Sci. 29, 27–39 (1984)

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. D. Vakarelov: A model logic for similarity relations in Pawlak knowledge representation systems, Fundam. Inf. 15(1), 61–79 (1991)

    MathSciNet  MATH  Google Scholar 

  55. D. Vakarelov: Modal logics for knowledge representation systems, Theor. Comput. Sci. 90(2), 433–456 (1991)

    MathSciNet  MATH  Google Scholar 

  56. P. Balbiani, D. Vakarelov: A modal logic for indiscernibility and complementarity in information systems, Fundam. Inf. 50(3/4), 243–263 (2002)

    MathSciNet  MATH  Google Scholar 

  57. F. del Cerro, L.E. Orlowska: DAL – A logic for data analysis, Theor. Comput. Sci. 36, 251–264 (1985)

    CrossRef  MathSciNet  MATH  Google Scholar 

  58. M. Banerjee, K. Chakraborty: Algebras from rough sets. In: Rough-Neural Computing: Techniques for Computing with Words, ed. by S.K. Pal, A. Skowron, L. Polkowski (Springer, Berlin Heidelberg 2004) pp. 157–188

    CrossRef  Google Scholar 

  59. M. Banerjee, M.A. Khan: Propositional logics from rough set theory, Trans. Rough Sets 6, 1–25 (2007)

    MathSciNet  MATH  Google Scholar 

  60. D. Ciucci, D. Dubois: Truth-functionality, rough sets and three-valued logics, Proc. ISMVL (IEEE, Piscataway 2010) pp. 98–103

    Google Scholar 

  61. Z. Bonikowski, E. Bryniarski, U. Wybraniec-Skardowska: Extensions and intentions in the rough set theory, Inf. Sci. 107(1–4), 149–167 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. G. Cattaneo, D. Ciucci: On the lattice structure of preclusive rough sets, IEEE Int. Conf. Fuzzy Syst., Piscataway (2004)

    Google Scholar 

  63. B. Konikowska: Three-valued logic for reasoning about covering-based rough sets. In: Rough Sets and Intelligent Systems – Professor Z. Pawlak in Memoriam, Intelligent Systems Reference Library, Vol. 42, ed. by A. Skowron, Z. Suraj (Springer, Berlin Heidelberg 2013) pp. 439–461

    CrossRef  Google Scholar 

  64. S. Greco, B. Matarazzo, R. Słowiński: Algebra and topology for dominance-based rough set approach. In: Advances in Intelligent Information Systems, Studies in Computational Intelligence, Vol. 265, ed. by Z.W. Ras, L.-S. Tsay (Springer, Berlin Heidelberg 2010) pp. 43–78

    CrossRef  Google Scholar 

  65. B.A. Davey, H.A. Priestley: Introduction to Lattices and Order (Cambridge Univ. Press, Cambridge 1990)

    MATH  Google Scholar 

  66. N. Caspard, B. Monjardet: The lattices of closure systems, closure operators, and implicational systems on a finite set: A survey, Discret. Appl. Math. 127(2), 241–269 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dep. Computer Science, University of Regina, 3737 Wascana Parkway, S4S 0A2, Regina, Saskatchewan, Canada

    JingTao Yao & Yan Zhang

  2. Dep. Informatics, Systems and Communications, University of Milano-Bicocca, viale Sarca 336/14, 20126, Milano, Italy

    Davide Ciucci

Authors
  1. JingTao Yao
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  2. Davide Ciucci
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  3. Yan Zhang
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Correspondence to JingTao Yao .

Editor information

Editors and Affiliations

  1. Systems Research Inst., Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

  2. Dep. Electrical and Computer Engineering, University of Alberta, 116 Street 9107, T6J 2V4, Edmonton, Alberta, Canada

    Witold Pedrycz

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Yao, J., Ciucci, D., Zhang, Y. (2015). Generalized Rough Sets. In: Kacprzyk, J., Pedrycz, W. (eds) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43505-2_25

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  • DOI: https://doi.org/10.1007/978-3-662-43505-2_25

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