Abstract
Granular computing is an emerging computing paradigm of information processing. Rough set theory is a kind of models of granular computing and is used to deal with the vagueness and granularity in information systems. Covering-based rough set theory is one of the most important extensions of the classical Pawlak rough set theory. In the covering-based rough set theory, the covering lower and upper approximation operators are two basic concepts. The axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators, so the rough set axiomatic system is the foundation of the covering-based rough set theory. In this paper, we present a new covering upper approximation operator and explore the basic properties of the covering upper approximation operator to find an axiomatic set for characterizing the covering upper approximation operator. We also discuss the relationships between the covering upper approximation operator and three other covering upper approximation operators, compare the covering upper approximation operator with the other covering upper approximation operators by defining precision degrees and rough degrees, and present the necessary and sufficient conditions for the covering upper approximation operators to be identical. The results not only are beneficial to enrich the kinds of the covering rough sets, but also have theoretical and actual significance to covering rough sets.
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References
Bonikowski Z (1994) Algebraic structures of rough sets. In: Ziarko W (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, Berlin, pp 243–247
Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inf Sci 107:149–167
Bryniaski E (1989) A calculus of rough sets of the first order. Bull Pol Acad Sci 36(16):71–77
Cattaneo G (1998) Abstract approximation spaces for rough theories, rough sets in knowledge discovery 1: methodology and applications. Springer, Berlin, pp 59–98
Chen DG, Wang CZ, Hu QH (2007) A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf Sci 177:3500–3518
Chen DG, Li WL, Zhang X, Kwong S (2014) Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets. Int J Approx Reason 55:908–923
Chen DG, Zhang XX, Li WL (2015) On measurements of covering rough sets based on granules and evidence theory. Inf Sci 317:329–348
Chen JK, Lin YJ, Lin GP, Li JJ, Zhang YL (2017) Attribute reduction of covering decision systems by hypergraph model. Knowl Based Syst 118:93–104
Deng T, Chen Y (2006) On reduction of morphological covering rough sets. In: The third international conference on fuzzy systems and knowledge discovery (FSKD 2006). Lecture Notes in Artificial Intelligence, vol 4223, pp 266–275
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17(2–3):191–209
Feng T, Mi J S, Wu W Z (2006) Covering-based generalized rough fuzzy sets. In: The first international conference on rough sets and knowledge technology (RSKT 2006). Lecture Notes in Computer Science, vol 4062, pp 208–215
Mi JS, Zhang WX (2004) An axiomatic characterization of a fuzzy generalization of rough sets. Inf Sci 160:235–249
Mi JS, Leung Y, Zhao HY, Feng T (2008) Generalized fuzzy rough sets determined by a triangular norm. Inf Sci 178:3203–3213
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356
Pomykala JA (1987) Approximation operations in approximation space. Bull Pol Acad Sci 35(9–10):653–662
Qin KY, Pei Z (2005) On the topological properties of fuzzy rough sets. Fuzzy Sets Syst 151(3):601–613
Qin K, Gao Y, Pei Z (2007) On covering rough sets. In: The second international conference on rough sets and knowledge technology (RSKT 2007). Lecture notes in computer science, vol 4481, pp 34–41
Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155
Restrepo M, Cornelis C, Gómez J (2014) Partial order relation for approximation operators in covering based rough sets. Inf Sci 284:44–59
Skowron A, Stepaniuk J (1996) Tolerance approximation spaces. Fundam Inf 27:245–253
Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12(2):331–336
Tan AH, Li JJ, Lin YJ, Lin GP (2015) Matrix-based set approximations and reductions in covering decision information systems. Int J Approx Reason 59:68–80
Thiele H (2000) On axiomatic characterization of fuzzy approximation operators I, the fuzzy rough set based case. In: The second international conference on rough sets and current trends in computing (RSCTC 2000). Lecture notes in computer science, vol 2005, pp 239–247
Thiele H (2001) On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy set based case. In: Proceedings of 31st IEEE international symposium on multiple-valued logic. Warsaw, Poland, pp 330–335
Tsang E, Cheng D, Lee J, Yeung D (2004) On the upper approximations of covering generalized rough sets. In: Proceedings of the 3rd international conference machine learning and cybernetics, vol 7, pp 4200–4203
Wang J, Dai D, Zhou Z (2004) Fuzzy covering generalized rough sets. J Zhoukou Teach Coll 21(2):20–22
Wang CZ, He Q, Chen DG, Hu HQ (2014) A novel method for attribute reduction of covering decision systems. Inf Sci 254:181–196
Wang CZ, Shao MW, Sun BQ, Hu QH (2015) An improved attribute reduction scheme with covering based rough sets. Appl Soft Comput 26:235–243
Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159(3–4):233–254
Wu WZ, Mi JS, Zhang WX (2003) Generalized fuzzy rough sets. Inf Sci 151:263–282
Wu WZ, Leung Y, Mi JS (2005) On characterizations of (I, T)-fuzzy rough approximation operators. Fuzzy Sets Syst 154(1):76–102
Xu Z, Wang Q (2005) On the properties of covering rough sets model. J Henan Normal Univ (Nat Sci) 33(1):130–132 (in Chinese)
Xu WH, Zhang WX (2007) Measuring roughness of generalized rough sets induced by a covering. Fuzzy Sets Syst 158:2443–2455
Yao YY (1998) Constructive and algebraic methods of theory of rough sets. Inf Sci 109:21–47
Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111(1–4):239–259
Yao YY, Yao BX (2012) Covering based rough set approximations. Inf Sci 200:91–107
Yun ZQ, Ge X, Bai XL (2011) Axiomatization and conditions for neighborhoods in a covering to form a partition. Inf Sci 181:1735–1740
Żakowski W (1983) Approximations in the space \((U,\Pi )\). Demonstr Math 16:761–769
Zhang YL, Luo MK (2011) On minimization of axiom sets characterizing covering-based approximation operators. Inf Sci 181:3032–3042
Zhang YL, Li JJ, Wu WZ (2010) On axiomatic characterizations of three pairs of covering-based approximation operators. Inf Sci 180:174–187
Zhao ZG (2016) On some types of covering rough sets from topological points of view. Int J Approx Reason 68:1–14
Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508
Zhu W, Wang FY (2003) Reduction and axiomization of covering generalized rough sets. Inf Sci 152:217–230
Zhu W, Wang FY (2007) On three types of covering-based rough sets. IEEE Trans Knowl Data Eng 19(8):1131–1144
Acknowledgements
This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11701258, 11526109, 61603173), Natural Science Foundation of Fujian (Nos. 2016J01671, 2016J01315, 2017J01771, 2017J01507), and the outstanding youth foundation of the Education Department of Fujiang Province.
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Zhang, YL., Li, CQ. & Li, J. On characterizations of a pair of covering-based approximation operators. Soft Comput 23, 3965–3972 (2019). https://doi.org/10.1007/s00500-018-3321-8
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DOI: https://doi.org/10.1007/s00500-018-3321-8