Hypergraph Semantics for Doxastic Logics

Simplicial models have become a crucial tool for studying distributed computing. These models, however, are only able to account for the knowledge, but not for the beliefs of agents. We present a new semantics for logics of belief. Our semantics is based on directed hypergraphs, a generalization of ordinary directed graphs in which edges are able to connect more than two vertices. Directed hypergraph models preserve the characteristic features of simplicial models for epistemic logic, while also being able to account for the beliefs of agents. We provide systems of both consistent belief and merely introspective belief. The completeness of our axiomatizations is established by the construction of canonical hypergraph models. We also present direct conversions between doxastic Kripke models and directed hypergraph models.