Document Type : Research Paper


1 Department of Computer Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

2 Department of Computer Science, Yazd University, Yazd, Iran

3 Department of Mathematics, Yazd University, Yazd, Iran


In this paper, we study the concept of S-approximation spaces in fuzzy set theory and investigate its properties. Along introducing three pairs of lower and upper approximation operators for fuzzy S-approximation spaces, their properties under different assumptions, e.g. monotonicity and weak complement compatibility are studied. By employing two thresholds for minimum acceptance accuracy and maximum rejection error, these spaces are interpreted in three-way decision systems by defining the corresponding positive, negative and boundary regions.


[1] M. Alamuri, B. R. Surampudi and A. Negi, A survey of distance/similarity measures for
categorical data, In 2014 International Joint Conference on Neural Networks, IJCNN (2014),
[2] N. Azam and J. T. Yao, Analyzing uncertainties of probabilistic rough set regions with game-
theoretic rough sets, International Journal of Approximate Reasoning, 55(1) (2014), 142-155,
[3] C. Cornelis, M. De Cock and A. M. Radzikowska, Fuzzy rough sets: from theory into practice,
Handbook of Granular computing, (2008), 533-552.
[4] B. Davvaz, A short note on algebraic T-rough sets, Information Sciences, 178 (2008), 3247-
[5] B. Davvaz, Approximations in n-ary algebraic systems, Soft Computing, 12(4) (2008), 409-
[6] B. Davvaz, Approximations in a semigroup by using a neighbourhood system, International
Journal of Computer Mathematics, 88(4) (2011), 709-713.
[7] B. Davvaz and M. Mahdavipour, Rough approximations in a general approximation space
and their fundamental properties, International Journal of General Systems, 37(3) (2008),
[8] A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals
of Mathematical Statistics, 38(2) (1967), 325{339.
[9] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of
General Systems, 17(2-3) (1990), 191{209.
[10] A. Gomolinska, Approximation spaces based on relations of similarity and dissimilarity of
objects, Fundamenta Informaticae, 79(3) (2007), 319{333.
[11] A. Gomolinska, On certain rough inclusion functions, Transactions on Rough Sets IX,
Springer Berlin Heidelberg, (2008), 35{55.
[12] A. Gomolinska, Rough approximation based on weak q-RIFs, Transactions on Rough Sets X,
Springer Berlin Heidelberg, (2009), 117{135.
[13] A. Gomolinska and M.Wolski, Rough inclusion functions and similarity indices, Fundamenta
Informaticae, 133(2) (2014), 149{163.
[14] S. Greco, B. Matarazzo and R. Slowinski, Rough approximation of a preference relation by
dominance relations, European Journal of Operational Research, 117(1) (1999), 63{83.
[15] J. W. Grzymala Busse, Knowledge acquisition under uncertainty { a rough set approach,
Journal of Intelligent and Robotic Systems, 1(1) (1988), 3{16.
[16] J. W. Grzymala Busse, Rough-set and Dempster-Shafer approaches to knowledge acquisition
under uncertainty { a comparison, Technical report, University of Kansas, 1987.
[17] M. R. Hooshmandasl, A. Shakiba, A. K. Goharshady and A. Karimi, S-approximation: A
new approach to algebraic approximation, Journal of Discrete Mathematics, 2014 (2014),
[18] M. J. Lesot, M. Rifqi and H. Benhadda, Similarity measures for binary and numerical data:
a survey, International Journal of Knowledge Engineering and Soft Data Paradigms, 1(1)
(2009), 63{84.
[19] T. J. Li, Rough approximation operators on two universes of discourse and their fuzzy ex-
tensions, Fuzzy Sets and Systems, 159(22) (2008), 3033{3050.
[20] P. Lingras and R. Jensen, Survey of rough and fuzzy hybridization, IEEE International Fuzzy
Systems Conference, (2007), 1{6.
[21] C. Liu, D. Miao and N. Zhang, Graded rough set model based on two universes and its
properties, Knowledge-Based Systems, 33(0) (2012), 65 { 72.
[22] G. Liu, Rough set theory based on two universal sets and its applications, Knowledge-Based
Systems, 23(2) (2010), 110{1150.
[23] N. N. Morsi and M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems,
100(1) (1998), 327{342.
[24] A. Nakamura, Fuzzy rough sets, Note on Multiple-valued Logic in Japan, 9(8) (1988), 1{8.
[25] S. Nanda and S. Majumdar, Fuzzy rough sets, Fuzzy Sets and Systems, 45(2) (1992), 157{
[26] S. K. Pal, L. Polkowski and A. Skowron, Rough-Neural Computing: Techniques for Comput-
ing with Words, Arti cial intelligence, Springer Berlin Heidelberg, 2004.
[27] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic
Publishers, 1991.
[28] Z. Pawlak and A. Skowron, Rough membership functions, Advances in the Dempster-Shafer
Theory of Evidence, John Wiley & Sons, (1994), 251{271.
[29] Z. Pawlak, S. K. M. Wong, and W. Ziarko, Rough sets: Probabilistic versus deterministic
approach, International Journal of Man-Machine Studies, 29 (1988), 81{95.
[30] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11
(1982), 341{356.
[31] Z. Pawlak, Vagueness and uncertainty: A rough set perspective, Computational Intelligence,
11(2) (1995), 227{232.
[32] Z. Pawlak, Rough set theory and its applications to data analysis, Cybernetics & Systems,
29(7) (1998), 661{688.
[33] Z. Pei and Z. B. Xu, Rough set models on two universes, International Journal of General
Systems, 33(5) (2004), 569{581.
[34] L. Polkowski, Rough sets: Mathematical foundations, Physica Verlag, 15 (2002).
[35] L. Polkowski and A. Skowron, Rough mereology: A new paradigm for approximate reasoning,
International Journal of Approximate Reasoning, 15(4) (1996), 333{365.
[36] L. Polkowski and A. Skowron, Rough mereology, Methodologies for Intelligent Systems,
Springer Berlin Heidelberg, (1994), 85{94.
[37] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and
Systems, 126(2) (2002), 137{155.
[38] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and
Systems, 126(2) (2002), 137{155.
[39] G. Shafer, A mathematical theory of evidence, Princeton university press Princeton, 1 (1976).
[40] A. Shakiba and M. R. Hooshmandasl, S-approximation spaces: A three-way decision ap-
proach, Fundamenta Informaticae, 139(3) (2015), 307{328.
[41] A. Shakiba and M. R. Hooshmandasl, Neighborhood system S-approximation spaces and
applications, Knowledge and Information Systems, (2015), 1{46.
[42] Y. Shen and F. Wang, Variable precision rough set model over two universes and its proper-
ties, Soft Computing, 15(3) (2011),557{567.
[43] A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae,
27(2) (1996), 245{253.
[44] R. Slowinski and D. Vanderpooten, A generalized de nition of rough approximations based on
similarity, IEEE Transactions on Knowledge and Data Engineering, 12(2) (2000), 331{336.
[45] B. Sun, Z. Gong and D. Chen, Fuzzy rough set theory for the interval-valued fuzzy information
systems, Information Sciences, 178(13) (2008), 2794{2815.
[46] B. Sun and W. Ma, Fuzzy rough set model on two di erent universes and its application,
Applied Mathematical Modelling, 35(4) (2011), 1798{1809.
[47] S. K. M. Wong, L. S. Wang, and Y. Y. Yao, Interval structure: A framework for representing
uncertain information, Proceedings of the Eighth International Conference on Uncertainty
in Arti cial Intelligence, UAI'92 (1992), 336{343.
[48] W. Z.Wu, J. Sheng Mi and W. Xiu Zhang, Generalized fuzzy rough sets, Information Sciences,
151 (2003), 263{282.
[49] W. Xu, W. Sun, Y. Liu and W. Zhang, Fuzzy rough set models over two universes, Interna-
tional Journal of Machine Learning and Cybernetics, 4(6) (2013), 631{645.
[50] S. Yamak, O. Kazanci and B. Davvaz, Generalized lower and upper approximations in a ring,
Information Sciences, 180(9) (2010), 1759{1768.
[51] R. Yan, J. Zheng, J. Liu and C. Qin, Rough set over dual-universes in fuzzy approximation
space, Iranian Journal of Fuzzy Systems, 9(3) (2012), 79{91
[52] Q. Yang and X. Wu, 10 challenging problems in data mining research, International Journal
of Information Technology & Decision Making, 5(4) (2006), 597{604.
[53] Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, 109(14)
(1998), 227{242.
[54] Y. Y. Yao, Probabilistic approaches to rough sets, Expert Systems, 20(5) (2003), 287{297.
[55] Y. Y. Yao and P. J. Lingras, Interpretations of belief functions in the theory of rough sets,
Information Sciences, 104(1-2) (1998), 81{106.
[56] Y. Y. Yao, Probabilistic rough set approximations, International Journal of Approximate
Reasoning, 49(2) (2008), 255{271.
[57] Y. Y. Yao, Three-way decision: an interpretation of rules in rough set theory, International
Conference on Rough Sets and Knowledge Technology, Springer Berlin Heidelberg, (2009),
[58] Y. Y. Yao, An outline of a theory of three-way decisions, Rough Sets and Current Trends in
Computing, Springer Berlin Heidelberg, (2012), 1{17.
[59] Y. Y. Yao and X. Deng, Quantitative rough sets based on subsethood measures, Information
Sciences, 267 (2014), 306{322.
[60] Y. Y. Yao, Combination of rough and fuzzy sets based on -level sets, Rough sets and Data
Mining, Springer, (1997), 301{321.
[61] Y. Y. Yao, Generalized rough set models, Rough Sets in Knowledge Discovery 1: Methodology
and Approximations, Studies in Fuzziness and Soft Computing, Physica Verlag Heidelberg,
1 (1998), 286{318.
[62] Y. Y. Yao and T. Y. Lin, Generalization of rough sets using modal logic, Intelligent Automa-
tion and Soft Computing, 2(2) (1996), 103{120.
[63] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.
[64] L. A. Zadeh, The role of fuzzy logic in the management of uncertainty in expert systems,
Fuzzy Sets and Systems, 11(1) (1983), 197{198.
[65] L. A. Zadeh, A simple view of the Dempster-Shafer theory of evidence and its implication
for the rule of combination, AI magazine, 7(2) (1986), 85{90.
[66] H. Y. Zhang, W. X. Zhang and W. Z. Wu, On characterization of generalized interval-
valued fuzzy rough sets on two universes of discourse, International Journal of Approximate
Reasoning, 51(1) (2009), 56{70.
[67] W. X. Zhang and Y. Leung, Theory of including degrees and its applications to uncertainty
inferences, Fuzzy Systems Symposium, Soft Computing in Intelligent Systems and Informa-
tion Processing., Proceedings of the 1996 Asian, (1996), 496{501.
[68] W. Zhu and F. Y.Wang, A new type of covering rough set, 3rd International IEEE Conference
on Intelligent Systems, (2006), 444{449.
[69] W. Zhu and F. Y. Wang, On three types of covering-based rough sets, IEEE Transactions on
Knowledge and Data Engineering, 19(8) (2007), 1131{1144.
[70] W. Zhu and F. Y. Wang, The fourth type of covering-based rough sets, Information Sciences,
201 (2012), 80{92.
[71] W. Ziarko, Variable precision rough set model, Journal of Computer and System Sciences,
46(1) (1993), 39{59.
[72] W. Ziarko, Probabilistic decision tables in the variable precision rough set model, Computa-
tional Intelligence, 17(3) (2001), 593{603.