Recently, much work has been carried out to study simplicial interpretations of modal logic. While notions of (distributed) knowledge have been well investigated in this context, it has been open how to model belief in simplicial models. We introduce polychromatic simplicial complexes, which naturally impose a plausibility relation on states. From this, we can define various notions of belief. We further explore how these complexes support a somebody-knows modality.

Simplicial complexes are a successful model for distributed computing. They have recently been observed to provide an interesting model for epistemic multi-agent logic where the agents' local states are the main building blocks (instead of the global states). A natural generalization is to study epistemic logic on semi-simplicial sets. However, finding the appropriate modal logic for semi-simplicial models has been an open question. We answer this by introducing the logic of synergistic knowledge and establishing its soundness and completeness.

Abstract We introduce the probabilistic two-agent justification logic $\textsf {IPJ}$, a logic in which we can reason about agents that perform interactive proofs. In order to study the growth rate of the probabilities in $\textsf {IPJ}$, we present a new method of parametrizing $\textsf {IPJ}$ over certain negligible functions. Further, our approach leads to a new notion of zero-knowledge proofs.