A Simplicial Model for $KB4_n$: Epistemic Logic with Agents that May Die

The standard semantics of multi-agent epistemic logic $S5$ is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that has been formalized as pure simplicial models in previous work from the authors. Here we extend the theory to encompass all simplicial models - including the ones that are not pure. The corresponding Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. This yields the epistemic logic $KB4$ which can reason about situations where some of the agents may die.