A new hope

Knowledge has long been identified as an inherent component of agents' decision-making in distributed systems. However, for agents in fault-tolerant distributed systems with fully byzantine agents, achieving knowledge is, in most cases, unrealistic. If agents can both lie and themselves be mistaken, then a message received is generally not sufficient to create knowledge. This problem is adequately addressed by an epistemic modality named hope, which has already been axiomatized. In this paper, we propose an alternative complete axiomatization for the hope modality by removing the reliance on designated atoms denoting correctness of individual agents and show that hope can be described as a KB4_n system. This additionally bringsa more streamlined presentation of the common hope modality (the hope analog of common knowledge). We also combine KB4_n hope modalities with S5_n knowledge modalities traditionally used in the epistemic analysis of fault-free distributed systems and present a logic enriched with both common knowledge and common hope. In these logics we formalize as frame-characterizable axioms some of the main properties of fully byzantine distributed systems: bounds on the number of faulty agents and the epistemic limitations due to agents' inability to rule out brain-in-a-vat scenarios.