Paper:

# A Theoretical Formulation of Object-Oriented Rough Set Models

## Yasuo Kudo^{*} and Tetsuya Murai^{**}

^{*}Department of Computer Science and Systems Engineering, Muroran Institute of Technology, 27-1 Mizumoto, Muroran 050-8585, Japan

^{**}Graduate School of Information Science and Technology, Hokkaido University, Kita 14, Nishi 9, Kita-Ku, Sapporo 060-0814, Japan

We introduce object-oriented paradigm into rough set theory. First, we provide concepts of class, object, and name, respectively. Class structures represent abstract data forms, and abstract structural hierarchy based on is-a relationship and has-a relationship. Object structures illustrate many kinds of objects and actual dependence among objects by is-a relationship and has-a relationship. Name structures provide concrete design of objects, and connect class structures and object structures consistently. Next, combining class, name and object structures, we propose object-oriented information systems, which include “traditional” information systems as special cases. Moreover, we introduce indiscernibility relations on the set of objects, lower and upper approximations, and object-oriented rough sets in the object-oriented information systems.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.10, No.5, pp. 612-620, 2006.

- [1] Z. Pawlak, “Rough Sets,” International Journal of Computer and Information Science, Vol.11, pp. 341-356, 1982.
- [2] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publisher, 1991.
- [3] T. A. Budd, “An Introduction of Object-Oriented Programming, 2
^{nd}Edition,” Addison Wesley Longman, 1997. - [4] A. C. Kay, “The Early History of Smalltalk,” Proceedings of the 2
^{nd}SCMSIGPLAN History of Programming languages Conference, ACM SIGPLAN Notice 28, Vol.3, pp. 69-75, 1993. - [5] R. Socher-Ambrosius and P. Johann, “Deduction Systems,” Springer, 1996.
- [6] K. Kaneiwa, “Order-Sorted Logic Programming with Predicate Hierarchy,” Artificial Intelligence, Vol.158, pp. 155-188, 2004.
- [7] S. Popkorn, “First Steps in Modal Logic,” Cambridge University Press, 1994.
- [8] http://www.uml.org/
- [9] http://www.omg.org/
- [10] R. Wille, “Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts,” Ordered Sets, I. Rival (Ed.), pp. 445-470, Reidel, 1982.
- [11] K. E. Wolff, “First Course in Formal Concept Analysis –How to Understand Line Diagrams–,” Advances in Statistical Software, F. Faulbaum (Ed.), Vol.4, pp. 429-438, VCH Publishers, 1994.
- [12] Y. Y. Yao, “Concept Lattices in Rough Set Theory,” Proceedings of 2004 Annual Meeting of the North American Fuzzy Information Processing Society, S. Dick, L. Kurgan, W. Pedrycz, and M. Reformat (Eds.), pp. 796-801, IEEE, 2004.
- [13] M. W. Shao and W. X. Zhang, “Approximation in Formal Concept Analysis,” Proceedings of RSFDGrC 2005, LNAI 3641, pp. 43-53, Springer, 2005.
- [14] P. Wasilewski, “Concept Lattices vs Approximation Spaces,” Proceedings of RSFDGrC 2005, LNAI 3641, pp. 114-123, Springer, 2005.
- [15] W. X. Zhang, L. Wei, and J. J. Qi, “Attribute Reduction in Concept Lattice Based on Discernibility Matrix,” Proceedings of RSFDGrC 2005, LNAI 3641, pp. 157-165, Springer, 2005.
- [16] T. Murai, M. Nakata, and Y. Sato, “A Note on Filtration and Granular Reasoning,” New Frontiers in Artificial Intelligence, T. Terano et al. (Eds.), LNAI 2253, pp. 385-389, Springer, 2001.
- [17] T. Murai, G. Resconi, M. Nakata, and Y. Sato, “Operations of Zooming In & Out on Possible Worlds for Semantic Fields,” Knowledge-Based Intelligent Information Engineering Systems and Allied Technologies, E. Damiani et al. (Eds.), pp. 1083-1087, IOS Press, 2002.
- [18] T. Murai, G. Resconi, M. Nakata, and Y. Sato, “Granular Reasoning Using Zooming In & Out: Part 2. Aristotle’s Categorical Syllogism,” Proceedings of International Workshop on Rough Sets in Knowledge Discovery and Soft Computing, Electronical Notices in Theoretical Computer Science, Vol.82, Elsevier, 2003.
- [19] T. Murai, G. Resconi, M. Nakata, and Y. Sato, “Granular Reasoning Using Zooming In & Out: Part 1. Propositional Reasoning,” Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, G. Wang et al. (Eds.), LNAI 2639, pp. 421-424, Springer, 2003.
- [20] T. Murai, M. Sanada, Y. Kudo, and M. Kudo, “A Note on Ziarko’s Variable Precision Rough Set Model and Nonmonotonic Reasoning,” Rough Sets & Current Trends in Computing, S. Tsumoto et al. (Eds.), LNCS 3066, pp. 421-424, Springer, 2004.
- [21] T. Murai and Y. Kudo, “Granularity and Ordinary Reasoning,” Proceedings of the 19th Annual Conference of the Japanese Society for Artificial Intelligence, 2005 (in Japanese).