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On Bisimilarity for Polyhedral Models and SLCS

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Formal Techniques for Distributed Objects, Components, and Systems (FORTE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13910))

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Abstract

The notion of bisimilarity plays an important role in concurrency theory. It provides formal support to the idea of processes having “equivalent behaviour” and is a powerful tool for model reduction. Furthermore, bisimilarity typically coincides with logical equivalence of an appropriate modal logic enabling model checking to be applied on reduced models. Recently, notions of bisimilarity have been proposed also for models of space, including those based on polyhedra. The latter are central in many domains of application that exploit mesh processing and typically consist of millions of cells, the basic components of face-poset models, discrete representations of polyhedral models. This paper builds on the polyhedral semantics of the Spatial Logic for Closure Spaces (SLCS) for which the geometric spatial model checker PolyLogicA has been developed, that is based on face-poset models. We propose a novel notion of spatial bisimilarity for face-poset models, called ±-bisimilarity. We show that it coincides with logical equivalence induced by SLCS on such models. The latter corresponds to logical equivalence with respect to SLCS on polyhedra which, in turn, coincides with simplicial bisimilarity, a notion of bisimilarity for continuous spaces.

Research partially supported by MUR projects PRIN 2017FTXR7S, “IT-MaTTerS”, PRIN 2020TL3X8X “T-LADIES”, bilateral project between CNR (Italy) and SRNSFG (Georgia) “Model Checking for Polyhedral Logic” (#CNR-22-010), and European Union - Next Generation EU - Italian MUR project PNRR PRI ECS00000017 PRR.AP008.003 “THE - Tuscany Health Ecosystem”. The authors are listed in alphabetical order, as they equally contributed to the work presented in this paper.

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Notes

  1. 1.

    Available from the VoxLogicA repository at https://github.com/vincenzoml/VoxLogicA.

  2. 2.

    \(\mathbf {v_0},\ldots ,\mathbf {v_d}\) are affinely independent if \(\mathbf {v_1} - \mathbf {v_0}, \ldots ,\mathbf {v_d} - \mathbf {v_0}\) are linearly independent. In particular, this condition implies that \(d \le m\).

  3. 3.

    Note that the colours of the classes have only an illustrative purpose; in particular they have nothing to do with the colours expressing the evaluation function of atomic proposition letters.

  4. 4.

    Recall that partial orders are transitive and reflexive.

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Acknowledgements

We thank Nick Bezhanishvili, Gianluca Grilletti and Jan Friso Groote for interesting discussions concerning various aspects of polyhedral model-checking, bisimulations and model reduction techniques.

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Authors and Affiliations

  1. Consiglio Nazionale delle Ricerche - ISTI, Pisa, Italy

    Vincenzo Ciancia, Diego Latella & Mieke Massink

  2. TSU Razmadze Mathematical Institute, Tbilisi, Georgia

    David Gabelaia

  3. Eindhoven University of Technology, Eindhoven, The Netherlands

    Erik P. de Vink

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  1. Vincenzo Ciancia
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  2. David Gabelaia
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  3. Diego Latella
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  4. Mieke Massink
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  5. Erik P. de Vink
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Correspondence to Vincenzo Ciancia .

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Editors and Affiliations

  1. University of Twente, Enschede, The Netherlands

    Marieke Huisman

  2. NOVA School of Science and Technology and NOVA LINCS, Caparica, Portugal

    António Ravara

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Ciancia, V., Gabelaia, D., Latella, D., Massink, M., de Vink, E.P. (2023). On Bisimilarity for Polyhedral Models and SLCS. In: Huisman, M., Ravara, A. (eds) Formal Techniques for Distributed Objects, Components, and Systems. FORTE 2023. Lecture Notes in Computer Science, vol 13910. Springer, Cham. https://doi.org/10.1007/978-3-031-35355-0_9

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