Paper:

# Rough Set Approximation as Formal Concept

## Nozomi Ytow^{*}, David R. Morse^{**}, and David McL. Roberts^{***}

^{*}Graduate School of Life and Environmental Sciences, University of Tsukuba

^{**}Faculty of Mathematics and Computing, The Open University

^{***}The Natural History Museum

Formal Concept Analysis (FCA) defines a formal concept as a pair of sets: objects and attributes, called extent and intent respectively. A rough set, on the other hand, approximates a concept using sets of objects only (in terms of FCA). We show that 1) a formal concept can be composed using a set of objects and its complement, 2) such object-based formal concepts are isomorphic to formal concepts based on objects and attributes, 3) upper and lower approximations of rough sets give generalization of formal concept, and 4) the pair of positive and negative sets (sensu rough set theory) are isomorphic to complemental formal concepts when the equivalence of the rough set gives positive and negative sets unique to each of the formal concepts. Implications of this are discussed.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.10, No.5, pp. 606-611, 2006.

- [1] S. P. Demri and E. S. Orłowska, “Incomplete information: Structure, inference, complexity,” Springer-Verlag,

ISBN 3-540-41904-7,

2002. - [2] B. Ganter and R. Wille, “Formal concept analysis. Mathematical foundations,” Springer-Verlag,

ISBN 3-540-62771-5,

1999. - [3] R. Kent, “Rough concept analysis: a synthesis of rough sets and formal concept analysis,” Fundamenta Informaticae, 27, pp. 169-181, 1996.
- [4] D. R. Morse, N. Ytow, D. M. Roberts, and A. Sato, “Comparison of multiple taxonomic hierarchies using TaxoNote,” In VIS2003 Poster Compendium IEEE Conference on Visualization, pp. 126-127, 2003.
- [5] Z. Pawlak, “Rough sets: Theoretical aspects of Reasoning about data,” Kluwer Academic Publishers,

ISBN 0-7923-1472-7,

1991. - [6] L. Polkowski, “Rough sets: Mathematical foundations,” Advances in soft computing, Physica-Verlag,

ISBN 3-7908-1510-1,

2002. - [7] J.-J. Qi, L. Wei, and Z.-Z. Li, “A Partitional View of Concept Lattice,” In D. Slezak, G. Wang, M. Szczuka, I. Düntsch, and Y. Yao (eds.), Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31-September 3, 2005, Proceedings, Part I, volume 3641 of Lecture Notes in Computer Science, ISBN: 3-540-28653-5, pp. 74-83, 2005.
- [8] J. Saquer and J. S. Deogun, “Formal rough concept analysis,” In A. S. Ning Zhong and S. Ohsuga (eds.), New Directions in Rough Sets, Data Mining, and Granular-Soft Computing, 7th International Work-shop, RSFDGrC ’99, Lecture Notes in Computer Science 1711, Springer, pp. 91-99, 1999.
- [9] J. Saquer and J. S. Deogun, “Concept approximations based on rough sets and similarity measures,” International Journal of Applied Mathematics and Computer Science, 11, pp. 655-674, 2001.
- [10] M.-W. Shao and W.-X. Zhang, “Approximation in Formal Concept Analysis,” In D. Slezak, G. Wang, M. Szczuka, I. Düntsch, and Y. Yao (eds.), Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005,Regina, Canada, August 31-September 3, 2005, Proceedings, Part I, volume 3641 of Lecture Notes in Computer Science, Springer, ISBN:3-540-28653-5, pp. 43-53, 2005.
- [11] P. Wasilewski, “Concept Lattices vs. Approximation Spaces,” In D. Slezak, G. Wang, M. Szczuka, I. Düntsch, and Y. Yao (eds.), Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31-September 3, 2005, Proceedings, Part I, volume 3641 of Lecture Notes in Computer Science, Springer,

ISBN: 3-540-28653-5,

pp. 114-123, 2005. - [12] K. Wolff, “A conceptual view of knowledge bases in rough set theory,” In Rough Sets and Current Trends in Computing, Second International Conference, RSCTC 2000, volume 2005 of Lecture Notes in Computer Science, Springer, pp. 220-228, 2001.
- [13] Y. Y. Yao, “A Comparative Study of Formal Concept Analysis and Rough Set Theory in Data Analysis,” In H. J. K. S. Tsumoto, R. Slowinski, and J. W. Grzymala-Busse (eds.), Rough Sets and Current Trends in Computing, 4th International Conference (RSCTC 2004) Proceedings, volume 3066 of Lecture Notes in Computer Science, Springer, pp. 59-68, 2004.
- [14] Y. Yao, “Concept Lattices in Rough Set Theory,” In W. P. S. Dick, L. Kurgan, and M. Reformat (eds.), Proceedings of 2004 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 2004), IEEE Catalog Number: 04TH8736, pp. 796-801, 2004.
- [15] Y. Yao and Y. H. Chen, “Rough Set Approximations in Formal Concept Analysis,” In W. P. S. Dick, L. Kurgan, and M. Reformat (eds.), Proceedings of 2004 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 2004), IEEE Catalog Number: 04TH8736, pp. 73-78, 2004.
- [16] N. Ytow, D. R. Morse, and D. M. Roberts, “Nomencurator: a nomenclatural history model to handle multiple taxonomic views,” Biological Journal of the Linnean Society, 73, pp. 81-98, 2001.

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