Abstract
Pawlak’s indiscernibility relation (which is an equivalence relation) represents a limit of our knowledge embedded in an information system. Covering approximation spaces generated by tolerance relations treat objects which are similar to a given object in the same way. Similarity based rough sets rely on the similarity of objects in general and preserve the benefit of pairwise disjoint system of base sets. By using correlation clustering not only a pairwise disjoint system of base sets can be generated but representative members of base sets can be defined. These representative members have an important logical usage. The author shows that there is a logical system relying on similarity base sets in which the truth values of first-order formulas can be counted in an effective simple way.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aszalós, L., Mihálydeák, T.: Rough clustering generated by correlation clustering. In: Ciucci, D., Inuiguchi, M., Yao, Y., Ślęzak, D., Wang, G. (eds.) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, pp. 315–324. Springer, Heidelberg (2013)
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(13), 89–113 (2004). https://doi.org/10.1023/B:MACH.0000033116.57574.95
Becker, H.: A survey of correlation clustering. In: Advanced Topics in Computational Learning Theory, pp. 1–10 (2005)
Ciucci, D., Mihálydeák, T., Csajbók, Z.E.: On definability and approximations in partial approximation spaces. In: Miao, D., Pedrycz, W., Ślȩzak, D., Peters, G., Hu, Q., Wang, R. (eds.) RSKT 2014. LNCS (LNAI), vol. 8818, pp. 15–26. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11740-9_2
Csajbók, Z., Mihálydeák, T.: A general set theoretic approximation framework. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) Advances on Computational Intelligence, pp. 604–612. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31709-5_61
Csajbók, Z.E., Mihálydeák, T.: From vagueness to rough sets in partial approximation spaces. In: Kryszkiewicz, M., Cornelis, C., Ciucci, D., Medina-Moreno, J., Motoda, H., Raś, Z.W. (eds.) Rough Sets and Intelligent Systems Paradigms, pp. 42–52. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08729-0_4
Golińska-Pilarek, J., Orłowska, E.: Logics of similarity and their dual tableaux a survey. In: Della Riccia, G., Dubois, D., Kruse, R., Lenz, H.J. (eds.) Preferences and Similarities, pp. 129–159. Springer, Vienna (2008). https://doi.org/10.1007/978-3-211-85432-7_5
Mani, A.: Choice inclusive general rough semantics. Inf. Sci. 181(6), 1097–1115 (2011)
Mihálydeák, T.: Partial first-order logic with approximative functors based on properties. In: Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J., Janicki, R., Hassanien, A.E., Yu, H. (eds.) Rough Sets and Knowledge Technology, pp. 514–523. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31900-6_63
Mihálydeák, T.: Aristotle’s Syllogisms in Logical Semantics Relying on Optimistic, Average and Pessimistic Membership Functions. In: Cornelis, C., Kryszkiewicz, M., Ślȩzak, D., Ruiz, E.M., Bello, R., Shang, L. (eds.) RSCTC 2014. LNCS (LNAI), vol. 8536, pp. 59–70. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08644-6_6
Mihálydeák, T.: First-order logic based on set approximation: a partial three-valued approach. In: 2014 IEEE 44th International Symposium on Multiple-Valued Logic, pp. 132–137, May 2014. https://doi.org/10.1109/ISMVL.2014.31
Nagy, D., Mihálydeák, T., Aszalós, L.: Similarity based rough sets. In: Polkowski, L., et al. (eds.) Rough Sets, pp. 94–107. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60840-2_7
Pawlak, Z.: Rough sets. Int. J. Parallel Programm. 11(5), 341–356 (1982)
Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning. Inf. Sci. 177(1), 41–73 (2007)
Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inf. Sci. 177(1), 3–27 (2007)
Pawlak, Z., et al.: Rough Sets: Theoretical Aspects of Reasoning About Data. System Theory, Knowledge Engineering and Problem Solving, vol. 9. Kluwer Academic Publishers, Dordrecht (1991)
Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27(2), 245–253 (1996)
Vakarelov, Dimiter: A modal characterization of indiscernibility and similarity relations in Pawlak’s information systems. In: Ślęzak, D., et al. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 12–22. Springer, Heidelberg (2005). https://doi.org/10.1007/11548669_2
Yao, J., Yao, Y., Ziarko, W.: Probabilistic rough sets: approximations, decision-makings, and applications. Int. J. Approx. Reason. 49(2), 253–254 (2008)
Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012). https://doi.org/10.1016/j.ins.2012.02.065. http://www.sciencedirect.com/science/article/pii/S0020025512001934
Acknowledgements
This work was supported by the construction EFOP–3.6.3–VEKOP–16–2017–00002. The project has been supported by the European Union, co-financed by the European Social Fund.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Mihálydeák, T. (2018). Logic on Similarity Based Rough Sets. In: Nguyen, H., Ha, QT., Li, T., Przybyła-Kasperek, M. (eds) Rough Sets. IJCRS 2018. Lecture Notes in Computer Science(), vol 11103. Springer, Cham. https://doi.org/10.1007/978-3-319-99368-3_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-99368-3_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99367-6
Online ISBN: 978-3-319-99368-3
eBook Packages: Computer ScienceComputer Science (R0)