Logo PTI Logo FedCSIS

Communication Papers of the 17th Conference on Computer Science and Intelligence Systems

Annals of Computer Science and Information Systems, Volume 32

Algebraic structures gained from rough approximation in incomplete information systems

,

DOI: http://dx.doi.org/10.15439/2022F90

Citation: Communication Papers of the 17th Conference on Computer Science and Intelligence Systems, M. Ganzha, L. Maciaszek, M. Paprzycki, D. Ślęzak (eds). ACSIS, Vol. 32, pages 7377 ()

Full text

Abstract. We give an algebraic approach for defining rough sets on incomplete information systems. The constructed approximation sets are based on objects. Given several attributes, the value of each attribute can be known or unknown for each object. In the current paper, we introduce four different approaches, a real value, a binary, a ternary and a likelihood approach. Furthermore, we define operations on the elements of the introduced approximation sets. For all three cases we can show that the achieved structure is a quasi-Brouwer-Zadeh distributive lattice with the defined operations. We also show that the introduced lower and upper approximations build up commutative monoids with the introduced operations.

References

  1. Z. Pawlak, “Rough sets,” International Journal of Parallel Programming, vol. 11, no. 5, pp. 341–356, 1982.
  2. Z. Pawlak and A. Skowron, “Rough sets and boolean reasoning,” Information sciences, vol. 177, no. 1, pp. 41–73, 2007.
  3. Z. Pawlak et al., “Rough sets: Theoretical aspects of reasoning about data,” Systern Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, 199l, vol. 9, 1991.
  4. A. Skowron and J. Stepaniuk, “Tolerance approximation spaces,” Fundamenta Informaticae, vol. 27, no. 2, pp. 245–253, 1996.
  5. Y. Yao and B. Yao, “Covering based rough set approximations,” Information Sciences, vol. 200, pp. 91 – 107, 2012.
  6. Z. Pawlak and A. Skowron, “Rudiments of rough sets,” Information sciences, vol. 177, no. 1, pp. 3–27, 2007.
  7. J. Yao, Y. Yao, and W. Ziarko, “Probabilistic rough sets: Approximations, decision-makings, and applications,” International Journal of Approximate Reasoning, vol. 49, no. 2, pp. 253–254, 2008.
  8. Z. Csajbók and T. Mihálydeák, “A general set theoretic approximation framework,” in Advances on Computational Intelligence (S. Greco, B. Bouchon-Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo, and R. R. Yager, eds.), (Berlin, Heidelberg), pp. 604–612, Springer Berlin Heidelberg, 2012.
  9. D. Nagy, T. Mihálydeák, and L. Aszalós, “Similarity based rough sets,” in Rough Sets (L. Polkowski, Y. Yao, P. Artiemjew, D. Ciucci, D. Liu, D. Ślęzak, and B. Zielosko, eds.), (Cham), pp. 94–107, Springer International Publishing, 2017.
  10. G. Cattaneo and D. Ciucci, “Algebraic structures for rough sets,” in Transactions on Rough sets II, pp. 208–252, Springer, 2004.
  11. G. Liu and W. Zhu, “The algebraic structures of generalized rough set theory,” Information Sciences, vol. 178, no. 21, pp. 4105–4113, 2008.
  12. A. Mani, G. Cattaneo, and I. Düntsch, Algebraic methods in general rough sets. Springer, 2018.
  13. D. Ciucci, T. Mihálydeák, and Z. E. Csajbók, “On definability and approximations in partial approximation spaces,” in International Conference on Rough Sets and Knowledge Technology, pp. 15–26, Springer, 2014.
  14. L. Polkowski, Rough sets. Springer, 2002.
  15. T. Murai, S. Miyamoto, M. Inuiguchi, Y. Kudo, and S. Akama, “Fuzzy multisets in granular hierarchical structures generated from free monoids,” Journal of Advanced Computational Intelligence and Intelligent Informatics, vol. 19, no. 1, pp. 43–50, 2015.
  16. B. Praba, V. Chandrasekaran, and A. Manimaran, “A commutative regular monoid on rough sets,” Italian Journal of Pure and Applied Mathematics, vol. 31, pp. 307–318, 2013.