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Reasoning about space: the modal way. (English) Zbl 1054.03015
This interesting paper reflects a current trend in modal logic, viz modelling and reasoning about space. The topological semantics of modal logic, interpreting $$\square$$ by the interior operator on topological spaces, is re-formulated with respect to topological models (i.e., topological spaces plus valuation). An appropriate notion of spatial bisimulation is introduced. And, in particular, a simplified proof of the McKinsey-Tarski Theorem on the completeness of the system S4 for the real line $$\mathbb{R}$$ (equipped with the usual topology) [cf. J. C. C. McKinsey and A. Tarski, Ann. Math. (2) 45, 141–191 (1944; Zbl 0060.06206)] is presented. The authors take up ideas from a paper by G. Mints [in: E. Orlowska (ed.), Logic at work: Essays dedicated to the memory of Helena Rasiowa. Heidelberg: Physica-Verlag, Stud. Fuzziness Soft Comput. 24, 79–88 (1999; Zbl 0923.03026)], where a short new proof of the completeness of S4 for the Cantor space was given.
Furthermore, the modal logic of the set of all serial subsets of $$\mathbb{R}$$, i.e., all finite unions of convex sets, is determined, and its one-variable fragment is characterized. The paper is closed with a broad discussion, among other things, on possible extensions concerning expressiveness.

##### MSC:
 03B45 Modal logic (including the logic of norms)
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