Abstract
Rough set theory was proposed by Pawlak to deal with the vagueness and granularity in information systems that are characterized by insufficient, inconsistent, and incomplete data. Its successful applications draw attentions from researchers in areas such as artificial intelligence, computational intelligence, data mining and machine learning. The classical rough set model is based on an equivalence relation on a set, but it is extended to generalized model based on binary relations and coverings. This paper reviews and summarizes the axiomatic systems for classical rough sets, generalized rough sets based on binary relations, and generalized rough sets based on coverings.
The first author is in part supported by the New Economy Research Fund of New Zealand and this work is also in part supported by two 973 projects (2004CB318103) and (2002CB312200) from the Ministry of Science and Technology of China.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
Pawlak, Z.: Rough sets. Internat. J. Comput. Inform. Sci. 11, 341–356 (1982)
Wang, F.Y.: Outline of a computational theory for linguistic dynamic systems: Toward computing with words. International Journal of Intelligent Control and Systems 2, 211–224 (1998)
Zadeh, L.: Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems 4, 103–111 (1996)
Lin, T.Y.: Granular computing - structures, representations, and applications. In: Wang, G., Liu, Q., Yao, Y., Skowron, A. (eds.) RSFDGrC 2003. LNCS (LNAI), vol. 2639, pp. 16–24. Springer, Heidelberg (2003)
Wang, F.Y.: On the abstraction of conventional dynamic systems: from numerical analysis to linguistic analysis. Information Sciences 171, 233–259 (2005)
Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1: Methodology and Applications, pp. 59–98 (1998)
Skowron, A., J.S.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)
Slowinski, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. On Knowledge and Data Engineering 12, 331–336 (2000)
Zakowski, W.: Approximations in the space (u, π). Demonstratio Mathematica 16, 761–769 (1983)
Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intentions in the rough set theory. Information Sciences 107, 149–167 (1998)
Zhu, F.: On covering generalized rough sets. Master’s thesis, The Universite of Arizona, Tucson, Arizona, USA (2002)
Zhu, W., Wang, F.Y.: Reduction and axiomization of covering generalized rough sets. Information Sciences 152, 217–230 (2003)
Ma, J.M., Zhang, W.X., Li, T.J.: A covering model of granular computing. In: Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, pp. 1625–1630 (2005)
Wu, W.Z., Zhang, W.X.: Constructive and axiomatic approaches of fuzzy approximation operator. Information Sciences 159, 233–254 (2004)
Dai, J., Chen, W., Pan, Y.: A minimal axiom group for rough set based on quasi-ordering. Journal of Zhejiang Unicersity Science 5, 810–815 (2004)
Yao, Y., Wang, F.Y., Wang, J.: ′′Rule + Exception′′ strategies for knowledge management and discovery. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 69–78. Springer, Heidelberg (2005)
Lin, T.Y., Liu, Q.: Rough approximate operators: axiomatic rough set theory. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 256–260. Springer, Heidelberg (1994)
Zhu, F., He, H.C.: The axiomization of the rough set. Chinese Journal of Computers 23, 330–333 (2000)
Sun, H., Liu, D.Y., Li, W.: The minimization of axiom groups of rough set. Chinese Journal of Computers 25, 202–209 (2002)
Thiele, H.: On axiomatic characterisations of crisp approximation operators. Information Sciences 129, 221–226 (2000)
Yao, Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning 15, 291–317 (1996)
Yao, Y.: Constructive and algebraic methods of theory of rough sets. Information Sciences 109, 21–47 (1998)
Cattaneo, G., Ciucci, D.: Algebraic structures for rough sets. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 208–252. Springer, Heidelberg (2004)
Yang, X.P., Li, T.J.: The minimization of axiom sets characterizing generalized approximation operators. Information Sciences 176, 887–899 (2006)
Zhu, W., Wang, F.Y.: Binary relation based rough set (manuscript, 2006)
Zhu, W.: Topological approaches to covering rough sets. Information Sciences (manuscript submitted, 2006)
Bryniaski, E.: A calculus of rough sets of the first order. Bull. Pol. Acad. Sci. 36, 71–77 (1989)
Pomykala, J.: Approximation, similarity and rough constructions, University of Amsterdam. ILLC Prepublication series, vol. CT-93-07 (1993)
Allam, A., Bakeir, M., Abo-Tabl, E.: New approach for basic rough set concepts. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 64–73. Springer, Heidelberg (2005)
Yao, Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 101, 239–259 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhu, W., Wang, F. (2006). Axiomatic Systems of Generalized Rough Sets. In: Wang, GY., Peters, J.F., Skowron, A., Yao, Y. (eds) Rough Sets and Knowledge Technology. RSKT 2006. Lecture Notes in Computer Science(), vol 4062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11795131_31
Download citation
DOI: https://doi.org/10.1007/11795131_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36297-5
Online ISBN: 978-3-540-36299-9
eBook Packages: Computer ScienceComputer Science (R0)