Abstract
We present two different extensions of the spatial logic for closure spaces (SLCS), and its spatio-temporal variant (\(\tau \) SLCS), with spatial quantification operators. The first concerns the existential quantification on individual points of a space. The second concerns the quantification on sets of points. The latter amounts to a form of quantification over atomic propositions, thus without the full power of second order logic. The spatial quantification operators are useful for reasoning about the existence of particular spatial objects in a space, their spatial relation with respect to other spatial objects, and, in the spatio-temporal setting, to reason about the dynamic evolution of such spatial objects in time and space, including reasoning about newly introduced items. In this preliminary study we illustrate the expressiveness of the operators by means of several small, but representative, examples.
Research partially supported by the MIUR PRIN 2017FTXR7S IT-MaTTerS. The authors are listed in alphabetical order; they contributed to this work equally.
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Notes
- 1.
The disjoint union \((X_1,\mathcal {C}_1) + (X_2,\mathcal {C}_2)\) of closure spaces \((X_1,\mathcal {C}_1)\) and \((X_2,\mathcal {C}_2)\) is the closure space \((X,\mathcal {C})\) whose set of points X is the disjoint union \(X_1+X_2\triangleq \{(x,1) \,|\, x \in X_1\} \cup \{(x,2) \,|\, x \in X_2\}\) while, for \(A \subseteq X_1+X_2\) we define \(\mathcal {C}(A) \triangleq \{(x,1) \,|\, x \in A_1\} \cup \{(x,2) \,|\, x \in A_2\}\) with \(A_j \triangleq \{x \,|\, (x,j)\in A\}\) for \(j=1,2\).
- 2.
- 3.
- 4.
For the sake of notational simplicity, we refrain from giving an explicit syntactic characterisation of assertions here.
- 5.
In line with [19], we are not interested in the algorithm(s) used for deciding the winner of the game. We are only interested in providing a representation for the configurations of the game and investigating properties of any such configuration, as well as of the whole game, that can be expressed using the extensions of SLCS we discuss in the present paper.
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Bussi, L., Ciancia, V., Gadducci, F., Latella, D., Massink, M. (2022). On Binding in the Spatial Logics for Closure Spaces. In: Margaria, T., Steffen, B. (eds) Leveraging Applications of Formal Methods, Verification and Validation. Verification Principles. ISoLA 2022. Lecture Notes in Computer Science, vol 13701. Springer, Cham. https://doi.org/10.1007/978-3-031-19849-6_27
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