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构组件组成，这些组件往往以组的形式组织。这些实体通过调整自身行为来追求个体或集体目标，从而相互交互。在此类系统中，实体的分布决定了空间——该空间可以是物理的，也可以是逻辑的。前者通过组件间的物理关系来定义，后者则依赖于逻辑关系（例如属于同一组）。在此背景下，空间属性的规约与验证在支持系统设计及预测其行为方面发挥着基础性作用。为此，研究者已提出多种工具和技术来规约和验证空间属性（主要被描述为图结构）。因此，这些方法通常使用空间关系模型来描述实体对之间的邻近关系。然而，当需要描述涉及实体组交互的空间属性时，基于图的模型无法处理涉及两个以上实体的关系。本文提出了一种基于单纯复形解释的空间逻辑。单纯复形是拓扑对象，能够高效表示曲面和体，并通过高阶边推广了图结构。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和逻辑公式的维度呈线性关系。本文通过经典互模拟和分支互模拟关系在单纯复形上的空间变体，研究了所提出逻辑的表达能力。