Abstract
It is well known that rough set-based approximations of concepts and possible world semantics of modal logics are closely related. In this chapter, we review the relationships between two types of possible world semantic models, i.e., Kripke model and measure-based model, and two variation of rough sets, i.e., Pawlak’s rough set and variable precision rough set.
Dedicated to Jair Minoro Abe for his 60th birthday
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References
Balbiani, P., Iliev, P., Vakarelov, D.: A modal logic for pawlaks approximation spaces with rough cardinality \(n\). Fundamenta Informaticae, 83, 451E464 (2008)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press (1980)
Greco, S., Matarazzo, B., Słowiński, R.: Rough set theory for multicriteria decision analysis. Eur. J. Oper. Res. 129, 1–47 (2002)
Järvinen, J., Kondo, M., Kortelainen, J.: Logics from Galois connections. Int. J. Approximate Reasoning 49, 595 E606 (2008)
Kondo, M.: On the structure of generalized rough sets. Inf. Sci. 176, 589–E00 (2006)
Kripke, S.A.: Semantical analysis of modal logic I. Normal modal propositional calculi. Zeitschr. 1. math. Logik und Otundlagen d. Math. 9, 67–96 (1963)
Kudo, Y., Murai, T.: Approximation of concepts and reasoning based on rough sets. J. Japn. Soc. Artif. Intell. 22(5), 597–604 (2007) (in Japanese)
Kudo, Y., Murai, T., Akama, S.: A granularity-based framework of deduction, induction, and abduction. Int. J. Approximate Reasoning 50, 1215–1226 (2009)
Liau, C.J.: An overview of rough set semantics for modal and quantifier logics. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 8(1), 93–118 (2000)
Liau, C.J.: Modal reasoning and rough set theory. In: Artificial Intelligence: Methodology, Systems, and Applications, LNCS, vol. 1480, pp. 317–30. Springer (2006)
Murai, T., Miyakoshi, M., Shimbo, M.: Measure-Based Semantics for Modal Logic. In: Fuzzy Logic : State of the Art, pp. 395–405. Kluwer (1993)
Murai, T., Miyakoshi, M., Shimbo, M.: A logical foundation of graded modal operators defined by fuzzy measures. In: Proceedings of 4th FUZZ-IEEE, pp. 151–156 (1995)
Orłowska, E. (ed.): Incomplete Information: Rough Set Analysis. Physica-Verlag, Springer (1998)
Pawlak, Z.: Rough Sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer (1991)
Pawlak, Z., Skowron, A.: Rough membership functions: a tool for reasoning with uncertainty. In: Algebraic Methods in Logic and In Computer Science, vol. 28. Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa (1993)
Polkowski, L.: Rough sets: Mathematical Foundations. Advances in Soft Computing. Physica-Verlag (2002)
Ślȩzak, D., Ziarko, W.: The investigation of the Bayesian rough set model. Int. J. Approximate Reasoning 40, 81–91 (2005)
Thiele, H.: Generalizing the explicit concept of rough set on the basis of modal logic. In: Reusch, B., et al. (eds.) Computational Intelligence in Theory and Practice. Springer, Berlin (2001)
Yao, Y.Y.: Generalized rough set models. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, pp. 286–318. Physica-Verlag, Heidelberg (1998)
Yao, Y.Y., Li, X.: Comparison of rough-set and interval-set models for uncertain reasoning. Fundamenta Informaticae 27(2–3), 289–298 (1996)
Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logics. Intell. Autom. Soft Comput. 2(2), 103–120 (1996)
Yao, Y.Y., Wang, S.K.M., Lin, T.Y.: A review of rough set models. In: Rough Sets and Data Mining, pp. 47–75. Kluwer (1997)
Zakowski, W.: Approximations in the space \((u,\pi )\). Demonstratio Mathematica 16, 761–769 (1983)
Zhu, W.: Generalized rough sets based on relations. Inf. Sci. 177, 4997E5011 (2007)
Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46, 39–59 (1993)
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Kudo, Y., Murai, T., Akama, S. (2016). A Review on Rough Sets and Possible World Semantics for Modal Logics. In: Akama, S. (eds) Towards Paraconsistent Engineering. Intelligent Systems Reference Library, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-40418-9_8
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