Abstract
We introduce the modal logic of planar polygonal subsets of the plane, with the modality interpreted as the Cantor-Bendixson derivative operator. We prove the finite model property of this logic and provide a finite axiomatization for it.
D. Gabelaia, M. Jibladze, E. Kuznetsov and L. Uridia—Supported by Shota Rustaveli National Science Foundation grant #DI-2016-25.
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Notes
- 1.
We introduce this terminology extending the terminology of [6] where reduction means taking a p-morphic image and subreduction means taking a p-morphic image of a subframe of the frame. Thus, up-reductions are special cases of subreduction, where the subframe under question is an up-set.
References
Aiello, M., van Benthem, J., Bezhanishvili, G.: Reasoning about space: the modal way. J. Log. Comput. 13, 889–920 (2003)
van Benthem, J., Bezhanishvili, G.: Modal logics of space. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4_5
Bezhanishvili, G., Esakia, L., Gabelaia, D.: Some results on modal axiomatization and definability for topological spaces. Stud. Logica 81, 325–355 (2005)
Bezhanishvili, N., Marra, V., McNeill, D., Pedrini, A.: Tarski’s theorem on intuitionistic logic, for polyhedra. Ann. Pure Appl. Logic 169, 373–391 (2017)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)
Engelking, R.: General Topology. Polish Scientific Publishers, Warszawa (1977)
Gabelaia, D., Gogoladze, K., Jibladze, M., Kuznetsov, E., Marx, M.: Modal logic of planar polygons. http://arxiv.org/abs/1807.02868
Kontchakov, R., Pratt-Hartmann, I., Zakharyaschev, M.: Interpreting topological logics over euclidean spaces. In: Proceedings of Twelfth International Conference on the Principles of Knowledge Representation and Reasoning, pp. 534–544. AAAI Press (2010)
Kontchakov, R., Pratt-Hartmann, I., Zakharyaschev, M.: Spatial reasoning with RCC8 and connectedness constraints in euclidean spaces. Artif. Intell. 217, 43–75 (2014)
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Gabelaia, D., Gogoladze, K., Jibladze, M., Kuznetsov, E., Uridia, L. (2019). An Axiomatization of the d-logic of Planar Polygons. In: Silva, A., Staton, S., Sutton, P., Umbach, C. (eds) Language, Logic, and Computation. TbiLLC 2018. Lecture Notes in Computer Science(), vol 11456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59565-7_8
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