We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. Our logical semantics has a satisfaction relation defined simultaneously with a definability relation. The latter restricts which formulas are allowed to have a truth value: dead processes cannot know or be ignorant of any proposition, and live processes cannot know or be ignorant of propositions involving processes they know to be dead. The logic satisfies some but not all axioms and rules of the modal logic S5. Impure simplicial complexes correspond to Kripke models where each agent's accessibility relation is an equivalence relation on a subset of the domain only, and otherwise empty, and where each propositional variable is known by an agent. We also propose a notion of bisimulation for impure simplexes and show bisimulation correspondence on certain finitary simplexes. % Dynamic aspects of our semantics, such as how to formalize possibly incomplete tasks and algorithms in distributed computing, is left for future research.

中文翻译：

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被通缉的死或活：不纯的简单复合体的认识逻辑

我们提出了不纯单纯复形的知识逻辑。不纯的简单复合体表示不确定性下的分布式系统，在这些不确定性上，哪些进程仍处于活动状态（活动），哪些进程已失败或崩溃（死亡）。我们的工作概括了Goubault等人在所有过程都存在的情况下的纯单纯形复杂物的知识逻辑。我们的逻辑语义具有与定义关系同时定义的满足关系。后者限制了允许哪些公式具有真实值：死进程不能知道或不知道任何命题，而活动进程不能知道或不知道涉及他们知道已经死的过程的命题。该逻辑满足模态逻辑S5的一些但不是全部公理和规则。不纯朴素的复合体与Kripke模型相对应，其中每个代理的可访问性关系仅是域子集上的等价关系，否则为空，并且每个命题变量都是代理已知的。我们还提出了不纯单纯形的双仿真概念，并在某些最终单纯形上显示了双仿真对应关系。我们语义的动态方面，例如如何形式化分布式计算中可能不完整的任务和算法，还有待进一步研究。