Semi-simplicial Set Models for Distributed Knowledge

Abstract

In recent years, a new class of models for multi-agent epistemic logic has emerged, based on simplicial complexes. Since then, many variants of these simplicial models have been investigated, giving rise to different logics and axiomatizations. In this paper, we present a further generalization, where a group of agents may distinguish two worlds, even though each individual agent in the group is unable to distinguish them. For that purpose, we generalize beyond simplicial complexes and consider instead simplicial sets. By doing so, we define a new semantics for epistemic logic with distributed knowledge. As it turns out, these models are the geometric counterpart of a generalization of Kripke models, called "pseudo-models". We identify various interesting sub-classes of these models, encompassing all previously studied variants of simplicial models; and give a sound and complete axiomatization for each of them.