Finite Models for a Spatial Logic with Discrete and Topological Path Operators

Linker, Sven and Papacchini, Fabio and Sevegnani, Michele (2021) Finite Models for a Spatial Logic with Discrete and Topological Path Operators. In: 46th International Symposium on Mathematical Foundations of Computer Science :. Leibniz International Proceedings in Informatics, LIPIcs, 202 . Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 72:1-72:16. ISBN 9783959772013

Abstract

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.