A Simplicial Model for $KB4_n$: Epistemic Logic with Agents that May Die
This work addresses a foundational issue in multi-agent epistemic logic for distributed systems, but it is incremental as it builds on prior simplicial models.
The paper tackles the problem of extending epistemic logic to handle agents that may disappear, by developing a simplicial model for KB4_n that accounts for non-reflexive accessibility relations. The result is a formal framework that applies to distributed computing scenarios with process crashes, though no concrete numerical results are provided.
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 we formalized as pure simplicial models in a previous work (Information and Computation, 2021). Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4 . Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash.