When do modal definability and preservation theorems transfer to the finite?

2026-03-12

AI Summary

This study addresses the long-standing open question of whether classical modal definability and preservation theorems hold in the finite, a setting where many first-order transfer results fail. By systematically investigating the validity of key semantic characterizations in modal logic—such as bisimulation safety and the Goldblatt–Thomason theorem—within finite structures, the paper combines model-theoretic techniques, bisimulation methods, and frame-algebraic analysis to establish, for the first time, that the bisimulation safety theorem remains valid in the finite. It further demonstrates that certain first-order preservation theorems do not carry over to this restricted domain, thereby advancing the theory of modal correspondence in finite models.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study which classic modal definability and preservation results survive when attention is restricted to finite structures, where many first-order transfer theorems are known to break down. Several semantic characterizations for modal formula classes survive the passage to the finite, while a number of first-order preservation theorems for basic frame operations fail. Our main positive result is that the Bisimulation Safety Theorem does transfer to finite structures. We also discuss computability aspects, and analogues in the finite for the Goldblatt-Thomason theorem and for modal correspondence theory.

Problem

• modal definability
• preservation theorems
• finite structures
• Bisimulation Safety Theorem
• Goldblatt-Thomason theorem

Innovation

• modal definability
• preservation theorems
• finite structures
• bisimulation safety
• Goldblatt-Thomason theorem