Topological Spatial Model Checking is a recent paradigm that combines Model Checking with the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. In particular, the spatial model checker VoxLogicA, that uses an extended version of SLCS, has been used successfully in the domain of medical imaging. However, SLCS is not restricted to discrete space. Following a recently developed geometric semantics of Modal Logic, we show that it is possible to assign an interpretation to SLCS in continuous space, admitting a model checking procedure, by resorting to models based on polyhedra. In medical imaging such representations of space are increasingly relevant, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We demonstrate feasibility of our approach via a new tool, PolyLogicA, aimed at efficient verification of SLCS formulas on polyhedra, while inheriting some well-established optimization techniques already adopted in VoxLogicA. Finally, we cater for a geometric definition of bisimilarity, proving that it characterises logical equivalence.
翻译:顶端空间模型检查是将模型检查与对摩达尔逻辑的地形解释相结合的近期范例。封闭空间空间空间空间逻辑(SLCS)空间逻辑(SLCS)扩展了模型逻辑(modal Lologic)和可达性连接(可达性连接),而这些功能又可以用来表达有趣的空间属性,例如“接近”或“被环绕 ” 。 SLCS 是一个坚实的逻辑框架的核心,用于解释离散空间的推理,例如图表和数字图像,被解释为准离散封闭空间。特别是,在医学成像领域成功地使用了扩大版的SLCS的空间模型检查器VoxLogicA, 空间逻辑性特征应用了扩大版的SLCS,但SL的模型逻辑性应用了多种可视性技术,我们通过一个不断更新的SLFIL方法, 展示了在持续空间上对SL的解析法的诠释。我们最后通过一个基于多元性模型的模型的模型检查程序,这种空间的描述越来越具有相关性,因为最近开发了3D扫描和可预见性技术,利用了内流层的SLABILSBIL方法,我们通过一个对SBIL的精确化工具的精确化方法来验证。