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.
References
Baldan, P., Corradini, A., König, B., Lluch Lafuente, A.: A temporal graph logic for verification of graph transformation systems. In: Fiadeiro, J.L., Schobbens, P.-Y. (eds.) WADT 2006. LNCS, vol. 4409, pp. 1–20. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71998-4_1
Bednarczyk, B., Demri, S.: Why propositional quantification makes modal logics on trees robustly hard? In: LICS 2019, pp. 1–13. IEEE (2019)
Belmonte, G., Broccia, G., Ciancia, V., Latella, D., Massink, M.: Feasibility of spatial model checking for nevus segmentation. In: Bliudze, S., Gnesi, S., Plat, N., Semini, L. (eds.) FormaliSE@ICSE 2021, pp. 1–12. IEEE (2021)
Belmonte, G., Ciancia, V., Latella, D., Massink, M.: Innovating medical image analysis via spatial logics. In: ter Beek, M.H., Fantechi, A., Semini, L. (eds.) From Software Engineering to Formal Methods and Tools, and Back. LNCS, vol. 11865, pp. 85–109. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30985-5_7
Belmonte, G., Ciancia, V., Latella, D., Massink, M.: VoxLogicA: a spatial model checker for declarative image analysis. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 281–298. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_16
van Benthem, J., Bezhanishvili, G.: Modal logics of space. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 217–298. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4_5
Bezhanishvili, N., Ciancia, V., Gabelaia, D., Grilletti, G., Latella, D., Massink, M.: Geometric model checking of continuous space. CoRR arXiv:abs/2105.06194 (2021)
Bull, R.A.: On modal logic with propositional quantifiers. J. Symb. Log. 34(2), 257–263 (1969)
Buonamici, F.B., Belmonte, G., Ciancia, V., Latella, D., Massink, M.: Spatial logics and model checking for medical imaging. Int. J. Softw. Tools Technol. Transf. 22(2), 195–217 (2020)
Bussi, L., Ciancia, V., Gadducci, F.: Towards a spatial model checker on GPU. In: Peters, K., Willemse, T.A.C. (eds.) FORTE 2021. LNCS, vol. 12719, pp. 188–196. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-78089-0_12
Bussi, L., Ciancia, V., Gadducci, F., Latella, D., Massink, M.: Towards model checking video streams using VoxLogicA on GPU’s. In: DataMod 2021. LNCS, Springer, Cham (2022, to appear)
Ciancia, V., Gilmore, S., Grilletti, G., Latella, D., Loreti, M., Massink, M.: Spatio-temporal model checking of vehicular movement in public transport systems. Int. J. Softw. Tools Technol. Transf. 20(3), 289–311 (2018)
Ciancia, V., Grilletti, G., Latella, D., Loreti, M., Massink, M.: An experimental spatio-temporal model checker. In: Bianculli, D., Calinescu, R., Rumpe, B. (eds.) SEFM 2015. LNCS, vol. 9509, pp. 297–311. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-49224-6_24
Ciancia, V., Latella, D., Loreti, M., Massink, M.: Specifying and verifying properties of space. In: Diaz, J., Lanese, I., Sangiorgi, D. (eds.) TCS 2014. LNCS, vol. 8705, pp. 222–235. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44602-7_18
Ciancia, V., Latella, D., Loreti, M., Massink, M.: Model checking spatial logics for closure spaces. J. Log. Methods Comput. Sci. 12(4) (2016)
Ciancia, V., Latella, D., Massink, M., Paškauskas, R., Vandin, A.: A tool-chain for statistical spatio-temporal model checking of bike sharing systems. In: Margaria, T., Steffen, B. (eds.) ISoLA 2016. LNCS, vol. 9952, pp. 657–673. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-47166-2_46
Ciancia, V., Latella, D., Massink, M., de Vink, E.P.: On bisimilarities for closure spaces - preliminary version. CoRR arXiv:abs/2105.06690 (2021)
Demri, S., Lazić, R.: LTL with the freeze quantifier and register automata. ACM Trans. Comput. Log. 10(3) (2009)
Gadducci, F., Lluch-Lafuente, A., Vandin, A.: Counterpart semantics for a second-order \(\mu \)-calculus. Fund. Inform. 118(1–2), 177–205 (2012)
Galton, A.: A generalized topological view of motion in discrete space. Theoret. Comput. Sci. 305(1–3), 111–134 (2003)
Grosu, R., Smolka, S.A., Corradini, F., Wasilewska, A., Entcheva, E., Bartocci, E.: Learning and detecting emergent behavior in networks of cardiac myocytes. Commun. ACM 52(3), 97–105 (2009)
Haghighi, I., Jones, A., Kong, Z., Bartocci, E., Grosu, R., Belta, C.: Spatel: a novel spatial-temporal logic and its applications to networked systems. In: Girard, A., Sankaranarayanan, S. (eds.) HSCC 2015, pp. 189–198. ACM (2015)
Holliday, W.H.: A note on algebraic semantics for S5 with propositional quantifiers. Notre Dame J. Formal Log. 60(2), 321–332 (2017)
Kripke, S.A.: A completeness theorem in modal logic. J. Symb. Log. 24(1), 1–14 (1959)
Kröger, F., Merz, S.: First-order linear temporal logic. In: Kröger, F., Merz, S. (eds.) Temporal Logic and State Systems, pp. 153–179. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68635-4_5
Kurtonina, N., de Rijke, M.: Bisimulations for temporal logic. J. Logic Lang. Inform. 6(4), 403–425 (1997)
Loreti, M., Quadrini, M.: A spatial logic for a simplicial complex model. CoRR arXiv:abs/2105.08708 (2021)
Nenzi, L., Bortolussi, L., Ciancia, V., Loreti, M., Massink, M.: Qualitative and quantitative monitoring of spatio-temporal properties with SSTL. J. Log. Methods Comput. Sci. 14(4) (2018)
Patthak, A., Bhattacharya, I., Dasgupta, A., Dasgupta, P., Chakrabarti, P.: Quantified computation tree logic. Inf. Process. Lett. 82(3), 123–129 (2002)
Rensink, A.: Model checking quantified computation tree logic. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 110–125. Springer, Heidelberg (2006). https://doi.org/10.1007/11817949_8
Smyth, M.B., Webster, J.: Discrete spatial models. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 713–798. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4_12
Stirling, C.: Modal and temporal logics. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, pp. 477–563. Oxford University Press, Oxford (1993)
Čech, E.: Topological spaces. In: Pták, V. (ed.) Topological Spaces, chap. III, pp. 233–394. Publishing House of the Czechoslovak Academy of Sciences/Interscience Publishers, Wiley, Prague/London-New York-Sydney (1966). Revised edition by Z. Frolíc, M. Katětov. Scientific editor, V. Pták. Editor of the English translation, C.O. Junge. MR0211373
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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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