[Paper Review] On expressive power and class invariance

Yanjing Wang, Francien Dechesne|ArXiv.org|May 27, 2009

Formal Methods in Verification9 references5 citations

TL;DR

This paper redefines expressive power in modal logics as distinguishing power generalized to classes of models, introducing class bisimulation as a precise characterization of expressive power under compactness. It establishes that for compact classes, class invariance via bisimulation exactly captures definability, offering a new framework for Lindström-type characterizations of logics like ML and ML∞.

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.

Motivation & Objective

• To clarify the distinction between distinguishing power and expressive power in logical languages.
• To generalize model-level invariance relations (e.g., bisimulation) to classes of models to capture expressive power.
• To establish conditions under which lifted invariance relations precisely characterize expressive power.
• To apply the framework to modal logics, particularly via class bisimulation, to re-derive results on modal definability.

Proposed method

• Lifts model-level structural equivalences (e.g., bisimulation) to relations on classes of models, defining class invariance.
• Introduces class bisimulation as a key tool to characterize expressive power of modal logics.
• Applies compactness conditions to ensure that class invariance corresponds precisely to expressive power.
• Uses m-saturated classes as a core structure to identify maximal collections where class bisimulation and modal equivalence coincide.
• Employs Lindström-type theorems as a framework to analyze expressive power in compact classes.
• Demonstrates that for compact classes, expressive power corresponds exactly to class-indistinguishability under class bisimulation.

Experimental results

Research questions

• RQ1 How can expressive power be formally understood as distinguishing power generalized to classes of models?
• RQ2 What is the appropriate notion of invariance on classes of models that captures expressive power?
• RQ3 Under what conditions does class bisimulation precisely characterize the expressive power of a logic?
• RQ4 Can the framework be used to re-derive or simplify results on modal definability?
• RQ5 What role does compactness play in ensuring that class invariance relations correspond to expressive power?

Key findings

• For compact classes of models, class bisimulation corresponds exactly to expressive power, meaning two classes are indistinguishable under a logic iff they are bisimilar as classes.
• The paper establishes that class invariance relations lifted from model-level equivalences (e.g., bisimulation) precisely capture expressive power under compactness.
• It is shown that class bisimulation captures modal definability: a class of models is definable in ML iff it is closed under class bisimulation.
• The failure of the equivalence $\mathcal{C}_1 \asymp_{\text{ML}_\infty} \mathcal{C}_2 \Leftrightarrow \mathcal{C}_1 \mathrel{\underline{\leftrightarrow}}^E_{\mathcal{C}} \mathcal{C}_2$ is demonstrated via a counterexample involving reversed ordinals and their modifications.
• The class of m-saturated model classes forms the maximal collection for which class bisimulation and modal equivalence coincide, as formalized in Theorem 36.
• Compactness is identified as a critical condition: without it, distinguishing power and expressive power may diverge, limiting the applicability of Lindström-type characterizations.