Topological structures in rough set theory: A survey

Year 2020, Volume: 49 Issue: 4, 1270 - 1294, 06.08.2020

Pankaj Kumar Singh Surabhi Tiwari

https://doi.org/10.15672/hujms.662711

Cited By: 15

Abstract

Rough set theory provides a mathematical tool to study vague, imprecise, inconsistent and uncertain knowledge. The topological notions are closely related to the notions and results of rough set theory, so the conjoint study of rough set theory and topology becomes essential. Researchers have widely discussed the topological aspects and their applications in rough set theory. This study highlights the interdependencies of topology and classical rough set theory and the significant work done in this area during the last twenty years.

Keywords

rough sets, topological spaces, dynamical systems

References

• [1] F.G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. 68, 17–25, 1999.
• [2] M. Banerjee and M.K. Chakraborty, Rough Consequence and Rough Algebra, in: Rough Sets, Fuzzy Sets and Knowledge Discovery, Proceedings of the International Workshop on Procedings of Rough Sets and Knowledge Discovery, (RSKD’93), Banff, Alberta, Canada, 1993, W.P. Ziarko, (Ed.), Springer Verlag, London, 196–207, 1994.
• [3] M. Barr and C. Wells, Toposes, Triples and Theories, Springer-Verlag, 1985.
• [4] W. Bartol, J. Miro, K. Pioro, and F. Rossello, On the coverings by tolerance classes, Inform. Sci. 166 (1-4), 193–211, 2004.
• [5] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ. 25, 1973.
• [6] R. Biswas, Rough metric spaces, Bull. Pour. Les. Sous. Ens. Flous. Appl.(France) 68, 21–32, 1996.
• [7] J. Dai, S. Gao, and G. Zheng, Generalized rough set models determined by multiple neighborhoods generated from a similarity relation, Soft Comput. 13, 2081–2094, 2013.
• [8] J.P. Doignon and J.C. Falmagne, Knowledge Spaces, Springer, Heidelberg, 1999.
• [9] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Springer Science and Business Media, 2013.
• [10] P. Eklund, M.A. Galán, and W. Gahler, Partially ordered monads for monadic topologies, Rough Sets and Kleene Algebras, Elect. Notes Theoret. Comp. Sci. 225, 67–81, 2009.
• [11] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer Science and Business Media, 2012.
• [12] F. Geerts and B. Kuijpers, Topological formulation of termination properties of iterates of functions, Inform. Process. Lett. 89 (1), 31–35, 2004.
• [13] M.A. Hajri, K. Belaid, and L.J. Belaid, Scattered spaces, compactification and an application to image classification problem, Tatra Mt. Math. Publ. 66 (1), 1–12, 2016.
• [14] P.C. Hammer, Extended topology: Structure of isotonic functions, J. Reine Angew. Math. 213, 174–186, 1964.
• [15] T. Herawan, Roughness of sets involving dependency of attributes in information systems, Int J. Softw. Eng. Appl. 9 (7), 111–126, 2015.
• [16] J.R. Isbell, Uniform Space, in: Math. Surveys Monogr No. 12, Amer. Math. Soc., 1964.
• [17] J. Järvinen, Approximations and rough sets based on tolerances, in: Lecture Notes in Comput. Sci. 2005, 182–189, Springer-Verlag, Heidelberg, 2001.
• [18] J. Järvinen, On the structure of rough approximations, Fund. Inform. 53, 135-153, 2002.
• [19] J. Järvinen, The ordered set of rough sets, in: Lecture Notes in Comput. Sci. 3066, 49–58, Springer-Verlag, Berlin, Heidelberg, 2004.
• [20] J. Järvinen, Lattice Theory for Rough Sets, in: Transactions on Rough Sets VI, LNCS 4374, 400–498, 2007.,
• [21] J. Järvinen, M. Kono, and J. Kortelainen, Modal-like operators in boolean algebras, Galois connections and fixed points, Fund. Inform. 76, 129–145, 2007.
• [22] J.L. Kelley, General Topology, Van Nostrand, 1955.
• [23] M. Khare and S. Tiwari, L-approach merotopies and their categorical perspective, Demonstr. Math. XLV (3), 2012.
• [24] M. Khare and S. Tiwari, Completion in a common supercategory of Met, UAP, wsAP and Near, Demonstr. Math. XLVI (1), 2013.
• [25] M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176, 589–600, 2006.
• [26] M. Kondo and W.A. Dudek, Topological structures of rough sets induced by equivalence relations, J. Advance Comput. Intell. Intell. Inform. 10 (5), 621–624, 2006.
• [27] E.F. Lashin, A.M. Kozae, A.A. Abo Khadra, and T. Medhat, Rough set theory for topological spaces, Internat. J. Approx. Reason. 40, 35–43, 2005.
• [28] T.J. Li and Y.L. Jing, Rough set model on granular structures and rule induction, Intern. J. Database Theory Appl. 4 (1), 7–18, 2011.
• [29] Z. Li, T. Xie, and Q. Li, Topological structure of generalized rough sets, Comput. Math. Appl. 63, 1066–1071, 2012.
• [30] T.Y. Lin, Q. Liu, and Y.Y. Yao, Logic systems for approximate reasoning: via rough sets and topology, Methodologies for Intelligent Systems. Berlin, Germany: Springer- Verlag, 1994.
• [31] W.J. Liu, Topological space properties of rough sets, in: Proceedings of the Third International Conferences on Machine Learning and Cybernetics, 2353–2355, Shanghai, 2004.
• [32] E.G. Manes, Algebraic Theories, Springer, 1976.
• [33] B.P. Mathew and S.J. John, On rough topological spaces, Intern. J. Math. Archive 3 (9), 3413–3421, 2012.
• [34] T. Medhat, Topological spaces and covering rough sets, Advances Inform. Sci. Service Sci. 3 (8), 283–289, 2011.
• [35] K. Monks, A category of topological spaces encoding a cyclic set theoretic dynamics, in: Proceedings of the International Conference on the Collatz Problem and Related Topics, 2000.
• [36] S.A. Naimpally and B.D. Warrack, Proximity Spaces, Cambridge Tract in Math. Math. Phys, Cambridge University Press 59, 1970.
• [37] P. Pagliani and M. Chakraborty, A Geometry of Approximation, Springer, Dordrecht, 2008.
• [38] Z. Pawlak, Information systems-theoretical foundations, Inf. Syst 6 (3), 205–218, 1981.
• [39] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5), 341–356, 1982.
• [40] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers Boston, 1991.
• [41] Z. Pawlak and W. Marek, Rough sets and information systems, ICS. PAS. Reports 441, 481–485, 1981.
• [42] Z. Pawlak and A. Skowron, Rudiment of rough sets, Inform. Sci. 177, 3–27, 2007.
• [43] J.F. Peters, Near sets. Special theory about nearness of objects, Fund. Inform. 75(3-4), 407–433, 2007.
• [44] J.F. Peters, A. Skowron, and J. Stepaniuk, Nearness of objects: Extension of approximation space model, Fund. Inform. 79 (3-4), 497–512, 2007.
• [45] J.F. Peters, A. Skowron, and J. Stepaniuk , Nearness of visual objects. Application of rough sets in proximity spaces, Fund. Inform. 128, 159–176, 2013.
• [46] L. Polkowski, Rough Sets: Mathematical Foundations, Heidelberg New York, Physica- Verl., 2002.
• [47] L. Polkowski and A. Skowron, Rough sets in Knowledge Discovery 1: Methodology and Applications, Stud. Fuzziness Soft Comput. 18, Physica-Verlag, Heidelberg, Germany, 1998.
• [48] L. Polkowski and A. Skowron, Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems, Stud. Fuzziness Soft Comput. 19, Physica-Verlag, Heidelberg, Germany , 1998.
• [49] A.N. Prior, Tense-logic and the continuity of time, Studia Logica 13 (1), 133–148, 1962.
• [50] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151, 601–613, 2005.
• [51] K. Qin, J. Yang, and Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inform. Sci. 178, 4138–4141, 2008.
• [52] S. Ramanna, A.H. Meghdadi, and J.F. Peters, Nature-inspired framework for measuring visual image resemblance: A near rough set approach, Theoret. Comput. Sci. 412, 5926–5938, 2011.
• [53] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, Stud. Logic Found. Math. 78, North-Holland Publishing Company, Amsterdam, London - PWN, Warsaw, 1974.
• [54] M. Restrepo, C. Cornelis, and J. Gomez, Characterization of neighborhood operators for covering based rough sets, using duality and adjoining, in: Fourth International Workshop Proceedings Eureka-2013, 98–104, 2013.
• [55] A.S. Salama, Some Topological properties of rough sets with tools for data mining, Int. J comp. sci. Issues 8 (2), 588–595, 2011.
• [56] A. Skowron, On the topology in information systems, Bull. Pol. Acad. Sci. Math 36, 477–480, 1988.
• [57] M.H. Stone, Topological representation of distributive lattices and Brouwerian logics, J. Symb. Log. 3 (2), 90–91, 1938.
• [58] A. Tarski, Grundzüge der Syntemenkalküls I, Fund. Math. 25 (1), 503–526, 1935.
• [59] M.L. Thivagar, C. Richard, and N.R. Paul, Mathematical innovations of a modern topology in medical events, Int. J. inform. Sci. 2 (4), 33–36, 2012.
• [60] N.D. Thuan, Covering rough sets from a topological point of view, Int. J. Comp. Theory Eng. 1 (5), 606–609, 2009.
• [61] S. Tiwari, Ultrafilter completeness in ε-approach nearness Spaces, Math. Comput. Sci. 7, 107–111, 2013.
• [62] S. Tiwari and P.K. Singh, An approach of proximity in rough set theory, Fund. Inform. 166 (3), 251–271, 2019.
• [63] S. Tiwari and P.K. Singh, Čech rough proximity spaces, Mat. Vesnik, accepted, 2019.
• [64] M. Vlach, Algebraic and topological aspects of rough set theory, in: IEEE Fourth International Workshop on Computational Intelligence and Application 2008 (1), 23–30, 2008.
• [65] S. Wang, P. Zhu, and W. Zhu, Structures of covering-based rough sets, Int. J. Math. Computer Sci. 6 (3), 147–150, 2010.
• [66] A. Wasilewska, Topological Rough Algebras, Rough sets and data mining. Springer, Boston MA, 411–425, 1997.
• [67] A. Wiweger, On topological rough sets, Bull. Pol. Acad. Sci. Math. 37, 51–62, 1988.
• [68] M. Wolski, Formal concept analysis and rough set theory from the perspective of finite topological approximations, in: Transaction on Rough Sets III, LNCS 3400, 230–243, 2005.
• [69] M. Wolski, Rough sets in terms of Discrete Dynamical system, International Conference on rough Sets and Current Trend in Computing 2010, 237–246, 2010.
• [70] M.Wolski, Granular computing: Topological and categorical aspects of near and rough set approaches to granulation of knowledge, in: Transaction on Rough Sets XVI, LNCS 7736, 34–52, 2013.
• [71] W.Z. Wu, Study on relationship between fuzzy rough approximation operators and fuzzy topological spaces, International Conference on Fuzzy Systems and Knowledge Discovery, LNCS 3613, 167–174, 2005.
• [72] W.Z. Wu and J.S. Mi, Some mathematical structures of generalized rough sets in infinite universes of discourse, in: Transactions on Rough Sets XIII, LNCS 6499, 175–206, 2011.
• [73] Q.E. Wu, T. Wang, Y.X. Huang, and J.S. Li, Topology theory on rough sets, IEEE Transactions on Systems, man and Cybernetics, Part B (Cybernetics) 38 (1), 68–77, 2008.
• [74] T. Yang, Q. Li, and B. Zhou, On the topological structures of granular reducts with covering rough sets, IEEE International Conference on Granular Computing, 601–604, 2012.
• [75] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci. 109 (1-4), 21–47, 1998.
• [76] Y.Y. Yao, Relational interpretations of neighborhood operator and rough set approximation operators, Inform. Sci. 111 (1-4), 239–259, 1998.
• [77] Y.Y. Yao and T.Y. Lin, Generalization of rough sets using modal logic, Intell. Autom. Soft Co. 2 (2), 103–120, 1996.
• [78] Y.Y. Yao and B. Yao, Covering based rough sets approximations, Inform. Sci. 200, 91–107, 2012.
• [79] Y.Y. Yao and N. Zhong, Potential applications of granular computing in knowledge discovery and data mining, in: Proceedings of World Multiconference on Systemics, Cybernetics and Informatics 5, 573–580, 1999.
• [80] W. Zakowski, Approximations in the space (U, Π), Demonstr. Math. 16 (3), 761–769, 1983.
• [81] Y.L. Zhang, J. Li, and C. Li, Topological structure of relational-based generalized rough sets, Fund. Inform. 147 (4), 477–491, 2016.
• [82] N. Zhong, J.Z. Dong, and S. Ohsuga, Using rough sets with heuristics to feature selection, J. Intell. Inf. Syst. 16 (3), 199–214, 2001.
• [83] N. Zhong, Y. Yao, and M. Ohshima, Peculiarity oriented multidatabase mining, IEEE T. Knowl. Data Eng. 15 (4), 952–960, 2003.
• [84] F. Zhu, On Covering Generalized Rough Sets, Master’s thesis, The University of Arizona, Tucson, AZ, USA, May 2002.
• [85] W. Zhu, Topological approaches to covering rough sets, Inform. Sci. 177 (6), 1499– 1508, 2007.
• [86] W. Zhu and F.Y. Wang, Reduction and axiomization of covering generalized rough sets, Inform. Sci. 152, 217–230, 2003.
• [87] W. Zhu and F.Y. Wang, Covering based granular computing for analysis of conflict, Lecture Notes in Comput. Sci 3975, 566–571, 2006.
• [88] W. Zhu and F.Y. Wang, Topological properties in covering-based rough sets, Fourth International Conference on Fuzzy Systems and Knowledge Discovery 1, 289–293, 2007.