This short note establishes the equivalence of the indexed semantics by Wang (2006) and the epistemic world semantics of Kaneko and Suzuki (2003) for epistemic logic of bounded depths.
In computer science, various logical languages are defined to analyze the properties of systems. One way to pinpoint the essential differences between those logics is to compare their expressivity in terms of distinguishing power and expressive power. In this paper, we study those two concepts by regarding the latter notion as the former lifted to classes of models. We show some general results on lifting an invariance relation on models to one on classes of models, such that when the former corresponds to the distinguishing power of a logic, the latter corresponds to its expressive power, given certain compactness requirements. In particular, we introduce the notion of class bisimulation to capture the expressive power of modal logics. We demonstrate the application of our results by revisiting modal definability with our new insights.