Epistemic ensembles in semantic, symbolic, and distributed environments

Abstract Epistemic ensembles are systems of knowledge-based agents capable of accessing, sharing, and updating information about themselves and their peers. These agents can operate on shared or local epistemic states through actions that dynamically alter the knowledge of some or all members of the ensemble. To support abstract reasoning over such systems, we introduce the notion of focus set — a selected set of logical formulæ that provide an abstraction from the underlying system state. Based on this abstraction, we define global and distributed symbolic representations of epistemic states, along with representable epistemic actions that enable efficient symbolic updates. For formal analysis, we define a generic operational semantics and develop and relate three complementary semantic frameworks: (1) a semantic environment, where system states are modelled as epistemic states, (2) a symbolic environment, where knowledge is represented as sets of logical formulæ, and (3) a distributed environment, represented by a family of local knowledge bases where each agent has its own local symbolic state. We establish a correspondence between these environments via a notion of relative elementary equivalence. Our main result demonstrates that equivalent configurations simulate each other’s behaviour and satisfy the same dynamic epistemic formulæ, ensuring representational consistency across all three perspectives. This provides a robust foundation for reasoning about distributed knowledge and belief dynamics in cooperative multi-agent systems.