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On the characterization of fuzzy rough sets based on a pair of implications

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Abstract

Ouyang et al. (Inf Sci 180:532–542, 2010) introduced the concept of (I, J)-fuzzy rough sets based on a pair of implications. Considering axiomatic characterization of approximation operators play a significant role in rough set theory, this paper devotes mainly to characterizing (I, J)-fuzzy rough sets based on a pair of implications from both constructive and axiomatic approaches. We firstly investigate the relationship between the lower and upper approximation operators based on a pair of ordinary implications and special fuzzy relations. And then (I, J)-fuzzy rough operators based on some special fuzzy relations are characterized by single axioms, which ensure the existence of polytypic fuzzy relations generating the same (I, J)-fuzzy rough approximation operators.

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References

  1. Baczýnski M (2004) Residual implications revisited. Notes on the Smets–Magrez theorem. Fuzzy Sets Syst 145:267–277

    Article  MathSciNet  Google Scholar 

  2. Baczýnski M, Jayaram B (2008) (S, N)- and R-implications: a state-of-art survey. Fuzzy Sets Syst 159:1836–1859

    Article  MathSciNet  Google Scholar 

  3. Boixader D, Jacas J, Recasens J (1999) Upper and lower approximations of fuzzy sets. Int J General Syst 29:555–568

    Article  MathSciNet  Google Scholar 

  4. Chen DG, He Q, Wang XZ (2010) FRSVMs: Fuzzy rough set based support vector machines. Fuzzy Sets Syst 161:596–607

    Article  MathSciNet  Google Scholar 

  5. Chen DG, Kwong S, He Q, Wang H (2012) Geometrical interpretation and applications of membership functions with fuzzy rough sets. Fuzzy Sets Syst 193:122–135

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Dai JH, Tian HW (2013) Fuzzy rough set model for set-valued data. Fuzzy Sets Syst 229:54–68

    Article  MathSciNet  Google Scholar 

  8. D’eer L, Verbiest N, Cornelis C, Godo L (2015) A comprehensive study of implicator-conjunctor-based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis. Fuzzy Sets Syst 275:1–38

    Article  MathSciNet  Google Scholar 

  9. Dubois D, Prade H (1990) Rough fuzzy set and fuzzy rough sets. Int J General Syst 17:191–209

    Article  Google Scholar 

  10. Fan BJ, Tsang ECC, Xu WH, Yu JH (2017) Double-quantitative rough fuzzy set based decisions: a logical operations method. Inf Sci 378:264–281

    Article  Google Scholar 

  11. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht, Boston, London

    Book  Google Scholar 

  12. Kryszkiewicz M (1998) Rough set approach to incomplete information systems. Inf Sci 112:39–49

    Article  MathSciNet  Google Scholar 

  13. Li TR, Luo C, Chen HM, Zhang JB (2015) PICKT: a solution for big data analysis. Lect Notes Comput Sci 9436:15C25

  14. Lin TY, Liu Q (1994) Rough approximate operators: axiomatic rough set theory. In: Ziarko W (ed) Rough sets fuzzy sets and knowledge discovery. Springer, Berlin, pp 256–260

    Chapter  Google Scholar 

  15. Liu G (2013) Using one axiom to characterize rough set and fuzzy rough set approximations. Inf Sci 223:285–296

    Article  MathSciNet  Google Scholar 

  16. Lu J, Li DY, Zhai YH et al (2016) A model for type-2 fuzzy rough sets. Inf Sci 328:359–377

    Article  Google Scholar 

  17. Ma ZM, Li JJ, Mi JS (2015) Some minimal axiom sets of rough sets. Inf Sci 312:40–54

    Article  MathSciNet  Google Scholar 

  18. Maji P, Pal SK (2011) Rough-fuzzy pattern recognition: applications in bioinformatics and medical imaging. Wiley

  19. Mas M, Monserrat M, Torrens J et al (2007) A survey on fuzzy implication functions. IEEE Trans Fuzzy Syst 15(6):1107–1121

    Article  Google Scholar 

  20. Mi JS, Leung Y, Zhao HY, Feng T (2008) Generalized fuzzy rough sets determined by a triangular norm. Inf Sci 178:3203–3213

    Article  MathSciNet  Google Scholar 

  21. Mi JS, Zhang W (2004) An axiomatic characterization of a fuzzy generalization of rough sets. Inf Sci 160:235–249

    Article  MathSciNet  Google Scholar 

  22. Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342

    Article  MathSciNet  Google Scholar 

  23. Ouyang Y, Wang ZD, Zhang HP (2010) On fuzzy rough sets based on tolerance relations. Inf Sci 180:532–542

    Article  MathSciNet  Google Scholar 

  24. Pawlak Z (1982) Rough set. Int J Comp Inf Sci 11:341–356

    Article  Google Scholar 

  25. Pawlak Z (1991) Rough set: theoretical aspects of reasoning about data. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  26. Qin KY, Yang JL, Pei Z (2008) Generalized rough sets based on transitive and reflexive relations. Inf Sci 178:4138–4141

    Article  Google Scholar 

  27. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155

    Article  MathSciNet  Google Scholar 

  28. She YH, Wang GJ (2009) An axiomatic approach of fuzzy rough sets based on residuated lattices. Comp Math Appl 58:189–201

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  30. Song XX, Wang X, Zhang WX (2013) Independence of axiom sets characterizing formal concepts. Int J Mach Learn Cybern 4:459–468

    Article  Google Scholar 

  31. Wang CY, Hu BQ (2013) Fuzzy rough sets based on generalized residuated lattice. Inf Sci 248:31–49

    Article  MathSciNet  Google Scholar 

  32. Wu WZ, Leung Y, Mi JS (2005) On characterizations of (\(I, T\))-fuzzy rough approximation operators. Fuzzy Sets Syst 15:76–102

    Article  MathSciNet  Google Scholar 

  33. Wu WZ, Leung Y, Shao MW (2013) Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int J Approx Reason 54:1388–1409

    Article  MathSciNet  Google Scholar 

  34. Wu WZ, Li TJ, Gu SM (2015) Using one axiom to characterize fuzzy rough approximation operators determined by a fuzzy implication operator. Fund Inf 142:87–104

    MathSciNet  MATH  Google Scholar 

  35. Wu WZ, Xu YH, Shao MW, Wang GY (2016) Axiomatic characterizations of (\(S, T\))-fuzzy rough approximation operators. Inf Sci 334–335:17–43

    MATH  Google Scholar 

  36. Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254

    Article  MathSciNet  Google Scholar 

  37. Xia BH, Ge Y, Wang JF, Li DY, Liao YL, Zheng XY (2014) A method for extracting rules from spatial data based on rough fuzzy sets. Knowl Based Syst 57:28C40

  38. Xu WH, Guo Y (2016) Generalized multigranulation double-quantitative decision-theoretic rough set. Knowl Based Syst 105:190–205

    Article  Google Scholar 

  39. Xu WH, Li TW (2016) Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans Cybern 46:366–379

    Article  Google Scholar 

  40. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 101:239–259

    Article  MathSciNet  Google Scholar 

  41. Yao YY (1998) Constructive and algebraic methods of theory of rough sets. Inf Sci 109:21–47

    Article  MathSciNet  Google Scholar 

  42. Yao YY (1998) A comparative study of fuzzy sets and rough sets. Inf Sci 109:227–242

    Article  MathSciNet  Google Scholar 

  43. Zadeh LA (1971) Similarity relation and fuzzy orderings. Inf Sci 3:177–200

    Article  MathSciNet  Google Scholar 

  44. Zakowski W (1982) On a concept of rough sets. Demonstratio Mathematica XV:1129–1133

  45. Zhang XX, Chen DG, Tsang ECC (2017) Generalized dominance rough set models for the dominance intuitionistic fuzzy information systems. Inf Sci 378:1–25

    Article  MathSciNet  Google Scholar 

  46. Zhang YL, Li JJ, Wu WZ (2010) On axiomatic characterizations of three pairs of covering based approximation operators. Inf Sci 180:274–287

    Article  MathSciNet  Google Scholar 

  47. Zhang XH, Zhou B, Li P (2012) A general frame for intuitionistic fuzzy rough sets. Inf Sci 216:34–49

    Article  MathSciNet  Google Scholar 

  48. Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees and the Editor-in-Chief for their valuable comments. This work was funded by the National Natural Science Foundation of China (Grant Nos. 61673352, 41631179, 61573321).

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Authors and Affiliations

  1. School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, 316022, Zhejiang, China

    Dechao Li & Weizhi Wu

  2. Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhoushan, 316022, Zhejiang, China

    Dechao Li & Weizhi Wu

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  1. Dechao Li
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  2. Weizhi Wu
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Correspondence to Dechao Li.

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Li, D., Wu, W. On the characterization of fuzzy rough sets based on a pair of implications. Int. J. Mach. Learn. & Cyber. 9, 2081–2092 (2018). https://doi.org/10.1007/s13042-017-0690-8

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  • DOI: https://doi.org/10.1007/s13042-017-0690-8

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