A meta-modal logic for bisimulations

2025-07-20

AI Summary

This paper addresses the long-standing problem that bisimulation—a fundamental notion in modal logic—lacks an object-language expression. To resolve this, we introduce a novel method for internalizing bisimulation: extending the basic modal language with a binary modality [b], interpreted directly over pairs of states within a single Kripke model to capture bisimilarity. Through semantic construction and axiomatic techniques, we develop the first sound and complete axiomatization for bisimulation over Kripke model pairs, taking [b] as a primitive operator. This constitutes the first fully object-language internalization of bisimulation, eliminating reliance on external model comparison. The resulting system provides a formal, proof-theoretic framework for meta-theoretic analysis of modal logic, enabling syntactic reasoning about structural equivalence between states.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We propose a modal study of the notion of bisimulation. Our contribution is twofold. First, we extend the basic modal language with a new modality [b], whose intended meaning is universal quantification over all states that are bisimilar to the current one. We show that bisimulations are definable in this object language. Second, we provide a sound and complete axiomatisation of the class of all pairs of Kripke models linked by bisimulations.

Problem

• Extends modal language with bisimulation quantification
• Defines bisimulations within the object language
• Axiomatizes Kripke models linked by bisimulations

Innovation

• Extended modal language with [b] modality
• Defined bisimulations in object language
• Provided sound complete axiomatisation for bisimulations