Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes.

翻译：集体自适应系统通常由许多异质组件组成，这些组件常以群体形式组织。这些实体通过调整自身行为以实现个体或集体目标而相互交互。在这些系统中，实体的分布决定了空间属性，该空间可以是物理空间或逻辑空间。前者由组件间的物理关系定义，后者则依赖于逻辑关系（例如属于同一群体）。在此背景下，空间属性的规范与验证在支持系统设计及行为预测中起到关键作用。因此，研究者提出了多种描述空间属性的工具和技术，这些方法通常以图结构为基础。现有方法一般通过二元空间关系描述实体间的邻近性，但基于图模型的方法无法处理涉及三个及以上实体间的关系——这种关系在需要描述群体交互的空间特征时至关重要。本文提出一种基于单纯复形解释的空间逻辑。单纯复形是一种拓扑结构，能高效表达表面和体积，并通过高阶边推广图模型。我们讨论了如何通过正确完备的模型检测算法验证逻辑公式的可满足性，该算法的时间复杂度与单纯复形维度和逻辑公式的规模呈线性关系。通过定义在单纯复形上的经典互模拟和分支互模拟关系的空间变体，本文研究了所提出逻辑的表达能力。