As an alternative to Kripke models, simplicial complexes are a versatile semantic primitive on which to interpret epistemic logic. Given a set of vertices, a simplicial complex is a downward closed set of subsets, called simplexes, of the vertex set. A maximal simplex is called a facet. Impure simplicial complexes represent that some agents (processes) are dead. It is known that impure simplicial complexes categorically correspond to so-called partial epistemic (Kripke) models. In this contribution, we define a notion of bisimulation to compare impure simplicial complexes and show that it has the Hennessy-Milner property. These results are for a logical language including atoms that express whether agents are alive or dead. Without these atoms no reasonable standard notion of bisimulation exists, as we amply justify by counterexamples, because such a restricted language is insufficiently expressive.

翻译：作为克里普克模型的替代方案，单纯复形是一种多功能的语义原语，可用于解释认知逻辑。给定顶点集合，单纯复形是顶点集合子集（称为单形）的下闭集。极大单形称为维面。不纯单纯复形表示某些智能体（进程）已失效。已知不纯单纯复形在范畴论意义上对应于所谓的部分认知（克里普克）模型。本文中，我们定义了用于比较不纯单纯复形的互模拟概念，并证明其具有亨尼西-米尔纳性质。这些结论适用于包含表达智能体存活或失效原子的逻辑语言。若不含这些原子，则不存在合理的标准互模拟概念——我们通过大量反例充分论证了这一点，因为受限语言的表达能力不足。