Modal definability in Euclidean modal logics

2025-08-14

AI Summary

This paper investigates the computability boundary of frame definability in Euclidean modal logics. While frame definability is known to be undecidable in general modal logic, we establish the first systematic characterization of undecidability within the Euclidean case: the frame definability problem is recursively unsolvable for any Euclidean modal logic whose class of frames contains an infinitely branching Euclidean tree—i.e., a converse tree satisfying both transitivity and symmetry. Methodologically, we combine modal semantic analysis, frame definability theory, and recursion-theoretic reduction techniques to construct an effective many-one reduction from the Post Correspondence Problem to frame definability. Our results yield a precise decidability/undecidability dichotomy criterion for Euclidean modal logics and reveal a fundamental connection between logical structure—specifically, branching complexity—and computational complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

This paper is about the computability of the modal definability problem in classes of frames determined by Euclidean modal logics. We characterize those Euclidean modal logics such that the classes of frames they determine give rise to an undecidable modal definability problem.

Problem

• Computing modal definability in Euclidean logics
• Characterizing Euclidean logics with undecidable definability
• Determining frame classes causing undecidability problems

Innovation

• Characterizing Euclidean modal logics
• Determining frame classes properties
• Identifying undecidable definability problems