This paper analyses models of a spatial logic with path operators based on the class of neighbourhood spaces, also called pretopological or closure spaces, a generalisation of topological spaces. For this purpose, we distinguish two dimensions: the type of spaces on which models are built, and the type of allowed paths. For the spaces, we investigate general neighbourhood spaces and the subclass of quasi-discrete spaces, which closely resemble graphs. For the paths, we analyse the cases of quasi-discrete paths, which consist of an enumeration of points, and topological paths, based on the unit interval. We show that the logic admits finite models over quasi-discrete spaces, both with quasi-discrete and topological paths. Finally, we prove that for general neighbourhood spaces, the logic does not have the finite model property, either for quasi-discrete or topological paths.
@InProceedings{linker_et_al:LIPIcs.MFCS.2021.72, author = {Linker, Sven and Papacchini, Fabio and Sevegnani, Michele}, title = {{Finite Models for a Spatial Logic with Discrete and Topological Path Operators}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {72:1--72:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.72}, URN = {urn:nbn:de:0030-drops-145120}, doi = {10.4230/LIPIcs.MFCS.2021.72}, annote = {Keywords: spatial logic, topology, finite models} }
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