On expressive power and class invariance

TL;DR

This paper explores the relationship between logical expressivity and invariance by extending model-based invariance concepts to classes of models, introducing class bisimulation to analyze modal logic expressiveness.

Contribution

It introduces a framework for lifting invariance relations to classes of models and applies this to modal logic, providing new insights into expressiveness and definability.

Findings

Established general results on lifting invariance to classes of models

Introduced the notion of class bisimulation for modal logics

Revisited modal definability using the new framework

Abstract

In computer science, various logical languages are defined to analyze 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.