Soft Computing

, Volume 17, Issue 7, pp 1241–1252 | Cite as

Multi-granulation rough sets based on tolerance relations

  • Weihua XuEmail author
  • Qiaorong Wang
  • Xiantao Zhang


The original rough set model is primarily concerned with the approximations of sets described by a single equivalence relation on the universe. Some further investigations generalize the classical rough set model to rough set model based on a tolerance relation. From the granular computing point of view, the classical rough set theory is based on a single granulation. For some complicated issues, the classical rough set model was extended to multi-granulation rough set model (MGRS). This paper extends the single-granulation tolerance rough set model (SGTRS) to two types of multi-granulation tolerance rough set models (MGTRS). Some important properties of the two types of MGTRS are investigated. From the properties, it can be found that rough set model based on a single tolerance relation is a special instance of MGTRS. Moreover, the relationship and difference among SGTRS, the first type of MGTRS and the second type of MGTRS are discussed. Furthermore, several important measures are presented in two types of MGTRS, such as rough measure and quality of approximation. Several examples are considered to illustrate the two types of multi-granulation tolerance rough set models. The results from this research are both theoretically and practically meaningful for data reduction.


Rough set Multi-granulation Tolerance relation Upper approximation Lower approximation 



This paper is supported by National Natural Science Foundation of China (No.61105041,71071124 and 11001227), Natural Science Foundation Project of CQ CSTC (No.cstc2011jjA40037), and Science and Technology Program of Board of Education of Chongqing (KJ120805).


  1. Dübois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J General Syst 17:191–209CrossRefGoogle Scholar
  2. Jarinen J (2005) Approximations and rough sets based on tolerances. Springer Berlin/Heidelberg, pp 182–189Google Scholar
  3. Kim D (2001) Data classification based on tolerant rough set. Pattern Recogn Lett 34:1613–1624zbMATHCrossRefGoogle Scholar
  4. Liang JY, Qian YH (2006) Axiomatic approach of knowledge granulation in information systems. Lect Notes Artif Intell 4304:1074–1078MathSciNetGoogle Scholar
  5. Ma JM, Zhang WX, Leung Y, Song XX (2007) Granular computing and dual Galois connection. Inf Sci 177:5365–5377MathSciNetzbMATHCrossRefGoogle Scholar
  6. Ouyang Y, Wang ZD, Zhang HP (2010) On fuzzy rough sets based on tolerance relations. Inf Sci 180:532–542MathSciNetzbMATHCrossRefGoogle Scholar
  7. Pomykala JA (2002) Rough sets and current trends in computing: about tolerance and similarity relations in information systems, vol 2475. Springer Berlin/Heidelberg, pp 175–182Google Scholar
  8. Pomykala JA (1988) On definability in the nondeterministic information system. Bull Polish Acad Sci Math 36:193–210MathSciNetzbMATHGoogle Scholar
  9. Pei DW (2005) A generalized model of fuzzy rough sets. Int J General Syst 34:603–613zbMATHCrossRefGoogle Scholar
  10. Qian YH, Liang JY, Dang CY (2009) Knowledge structure, knowledge granulation and knowledge distance in a knowledge base. Int J Approx Reason 50:174–188MathSciNetzbMATHCrossRefGoogle Scholar
  11. Qian YH, Liang JY, Yao YY, Dang CH (2010a) MGRS: a multi-granulation rough set. Inf Sci 180:949–970MathSciNetzbMATHCrossRefGoogle Scholar
  12. Qian YH, Liang JY, Pedrycz W, Dang CY (2010b) Positive approximation: an accelerator for attribute reduction in rough set theory. Artif Intell 174:597618MathSciNetCrossRefGoogle Scholar
  13. Qian YH, Liang JY, Wei W (2010c) Pessimistic rough decision. In: Second international workshop on rough sets theory 440449Google Scholar
  14. Skowron A, Stepaniuk J (1996) Tolerance approximation space. Fundam Inf 27:245–253MathSciNetzbMATHGoogle Scholar
  15. Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12:331–336CrossRefGoogle Scholar
  16. Xu BZ, Hu XG, Wang H (2004) A general rough set model based on tolerance. International Academic Publishers, World Publishing Corporation, San Fransisco, pp 770–774Google Scholar
  17. Xu WH, Wang QR, Zhang XT (2011a) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Fuzzy Syst 13:246–259MathSciNetGoogle Scholar
  18. Xu WH, Wang QR, Zhang XT (2011b) Multi-granulation fuzzy rough set model on tolerance relations. In: Fourth international workshop on advanced computational intelligence, Wuhan, Hubei, China, October 19(21):359–366Google Scholar
  19. Xu WH, Sun WX, Zhang XY, Zhang WX (2012a) Multiple granulation rough set approach to ordered information systems. Int J General Syst 41:475–501MathSciNetCrossRefGoogle Scholar
  20. Xu WH, Zhang XT, Wang QR (2012b) A generalized multi-granulation rough set approach. Lect Notes Bioinformatics 1:681–689Google Scholar
  21. Yao YY, Lin TY (1996) Generalization of rough sets using modal logic. Intell Automat Soft Comput 2:103–120Google Scholar
  22. Yao YY (2003) On generalizing rough set theory, rough sets, fuzzy sets, data mining, and granular computing. In: Proceedings of the 9th international conference (RSFDGrC 2003), LNCS(LNAI) 2639:44–51Google Scholar
  23. Yao YY (2004) A comparative study of formal concept analysis and rough set theory in data analysis. In: Proceedings of RSCTC’ 04 LNCS (LNAI 3066), pp 59–68Google Scholar
  24. Yao YY (2000) Granular computing basis issues and possible solutions. In: Proceedings of the fifth international conference on computing and information, pp 186–189Google Scholar
  25. Yao YY (2005) Perspectives of granular computing. In: Proceedings of 2005 IEEE international conference on granular computing, pp 85–90Google Scholar
  26. Zakowski W (1983) Approximations in the space (U,π). Demonstr Math XVI:761–769Google Scholar
  27. Zheng Z, Hu H, Shi ZZ (2005) Rough sets, fuzzy sets, data mining, and granular computing: tolerance relation based granular space, vol 3641. Springer, Berlin/Heidelberg, pp 682–691Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsChongqing University of TechnologyChongqingPeople’s Republic of China

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