Abstract
Near sets were introduced by J.F. Peters in 2007 in the context and within the conceptual framework of rough sets, which were initiated by Z. Pawlak in the early 1980s. However, due to further evolution and development, near set theory has become an independent field of study. For this reason, nowadays, the relationships between near set theory and rough set theory are not easy to spot. In this short paper we would like to re-define near sets and to re-think their foundations and relationships to/bearing on rough sets. To this end we translate the basic concepts of near set theory into the framework of modal logic, which has already been successfully applied to rough sets. The concept of nearness of sets, however, was originally defined globally (that is, with respect to the whole underlying space), but modal logic is intrinsically local: the logical value of a formula is computed with respect to a single point and its neighbourhood. Our approach to near sets is local in the very same sense: we are concerned with nearness of sets seen from the perspective of a single point. Interestingly, this local perspective brings together rough set theory and near set theory, revealing their deep theoretical connections. Therefore, what we offer is a modal and algebraic “shared history” of the two theories at issue.
References
Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CSLI Publications, Stanford (1988)
Aczel, P., Mendler, N.: A final coalgebra theorem. In: Pitt, D.H., Rydeheard, D.E., Dybjer, P., Pitts, A.M., Poigné, A. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989). doi:10.1007/BFb0018361
Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley-Interscience Publication, London (1990)
Banerjee, M., Khan, M.A.: Propositional logics from rough set theory. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 1–25. Springer, Heidelberg (2007). doi:10.1007/978-3-540-71200-8_1
Bilkova, M., Palmigiano, A., Venema, Y.: Proof systems for the coalgebraic cover modality. Adv. Modal Log. 7, 1–23 (2008)
Bilkova, M., Palmigiano, A., Venema, Y.: Proof systems for Moss’ coalgebraic logic. Theor. Comput. Sci. 549, 36–60 (2014)
Čech, E.: Topological Spaces. Wiley, London (1966)
Efremovič, V.: The geometry of proximity I. Mat. Sb. 31, 189–200 (1951). (in Russian) MR 14, 1106
Herrlich, H.: A concept of nearness. Gen. Topol. Appl. 4, 191–212 (1974)
Jacobs, B.: Introduction to Coalgebra. Towards Mathematics of States and Observation. Cambridge University Press, Cambridge (2016)
Janin, D., Walukiewicz, I.: Automata for the modal \(\mu \)-calculus and related results. In: Wiedermann, J., Hájek, P. (eds.) MFCS 1995. LNCS, vol. 969, pp. 552–562. Springer, Heidelberg (1995). doi:10.1007/3-540-60246-1_160
Kupke, K., Kurz, A., Venema, Y.: Completeness for the coalgebraic cover modality. Log. Methods Comput. Sci. 8, 1–76 (2012)
Moss, L.: Coalgebraic logic. Ann. Pure Appl. Log. 96, 277–317 (1999)
Naimpally, S.: Proximity Approach to General Topology. Lakehead University, Thunder Bay (1974)
Naimpally, S., Peters, J.: Topology with Applications. Topological Spaces via Near and Far. World Scientific, Singapore (2013)
Pawlak, Z.: Classification of Objects by Means of Attributes. Institute of Computer Science, Polish Academy of Sciences PAS 429, Warsaw (1981)
Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)
Pawlak, Z.: Rough sets and decision tables. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 187–196. Springer, Heidelberg (1985). doi:10.1007/3-540-16066-3_18
Pawlak, Z.: Rough logic. Bull. Pol. Acad. Sci. (Tech. Sci.) 35(5–6), 253–258 (1987)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publisher, Berlin (1991)
Pawlak, Z.: Wiedza z Perspektywy Zbiorów Przybliżonych. Institute of Computer Science Report 23, Toronto (1992)
Peters, J.: Near sets. special theory about nearness of objects. Fundam. Inform. 75(1–4), 407–433 (2007)
Peters, J.: Near sets. General theory about nearness of objects. Appl. Math. Sci 1(53), 2609–2629 (2007)
Peters, J., Skowron, A., Stepaniuk, J.: Nearness in approximation spaces. In: Proceedings of Concurrency, Specification and Programming (CS&P 2006), Humboldt Universitat, pp. 435–445 (2006)
Peters, J., Naimpally, S.: Applications of near sets. Am. Math. Soc. Not. 59(4), 536–542 (2012). doi:10.1090/noti817.
Priest, G.: An Introduction to Non-classical Logic. Cambridge University Press, Cambridge (2001)
Riesz, F.: Stetigkeitsbegriff und abstrakte mengenlehre. IV Congresso Internazionale dei Matematici II, pp. 18–24 (1908)
Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logic. Intell. Autom. Soft Comput. 2(2), 103–120 (1996)
Acknowledgements
We would like to thank anonymous reviewers for careful reading the manuscript, their insightful comments, and corrections.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Wolski, M., Gomolińska, A. (2017). Rough and Near: Modal History of Two Theories. In: Polkowski, L., et al. Rough Sets. IJCRS 2017. Lecture Notes in Computer Science(), vol 10313. Springer, Cham. https://doi.org/10.1007/978-3-319-60837-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-60837-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-60836-5
Online ISBN: 978-3-319-60837-2
eBook Packages: Computer ScienceComputer Science (R0)