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Epistemic Logic is the modal logic of knowledge. It allows to reason about a finite set of agents who can know facts about the world, and about what the other agents know. The traditional Kripke-style semantics for epistemic logic is based on graphs whose vertices represent the possible worlds, and whose edges indicate the agents that cannot distinguish between two worlds. In this talk, I will present an alternative semantics for epistemic logic, based on combinatorial topology. The idea is to replace the Kripke graph by a simplicial complex, allowing for higher-dimensional connectivity between the possible worlds. In fact, every Kripke model can be turned into an equivalent simplicial model, thus uncovering its underlying geometric structure. Our notion of simplicial model is inspired from the “protocol complex” approach to distributed computing. I will show how our framework can be used to analyse distributed computing, where the agents are the processes, and the possible worlds are all the possible executions of the system. In order to prove impossibility results, one must find an epistemic logic formula representing the knowledge that the processes should acquire in order to solve a task; and argue that such knowledge cannot be achieved. This is joint work with Éric Goubault and Sergio Rajsbaum.
