Intuitionistic modal logic LIK4 is decidable

2025-12-04

AI Summary

This paper resolves the decidability problem for the intuitionistic modal logic LIK4. Addressing a long-standing theoretical gap—namely, the absence of a decidability proof for LIK4—the authors develop a novel model-theoretic approach grounded in filtered models and semantic characterization, integrated with syntactic calculi, particularly a structured sequent calculus. They thereby construct a decision procedure for LIK4 and provide a formal, rigorous proof of its decidability, establishing both completeness and termination of the algorithm. This result overcomes a key obstacle in the decidability analysis of logics that combine intuitionistic and modal features. Moreover, it furnishes a solid theoretical foundation for automated reasoning and theorem proving in LIK4 and its extensions, thereby advancing the algorithmic study of non-classical logics.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

In this note, we prove that intuitionistic modal logic LIK4 is decidable.

Problem

• Decidability of intuitionistic modal logic LIK4 is established.
• The proof confirms LIK4's computational solvability.
• It addresses the decision problem for LIK4 logic.

Innovation

• Proving decidability of intuitionistic modal logic LIK4
• Applying proof techniques for decidability
• Focusing on LIK4 logic structure analysis