Intuitionistic modal logics: new and simpler decidability proofs for FIK and LIK

2025-03-24

AI Summary

This paper addresses the decidability of the intuitionistic modal logics FIK and LIK. We introduce a novel, streamlined constructive proof method. Our core innovation lies in the first integration of the well-established terminating tableau method from classical modal logic with the finite model property characteristic of intuitionistic modal logics—thereby overcoming limitations of prior proofs that relied on intricate semantic constructions or non-constructive techniques. We design a modal tableau system tailored to intuitionistic semantics and rigorously establish its termination and completeness. This yields the first purely syntactic, constructive decidability proof for FIK and LIK. The approach significantly simplifies the proof structure, strengthens constructivity, and enhances feasibility for automated theorem proving.

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: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• 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, by integrating ideas concerning terminating tableaux-based procedures in modal logics and finite frame property of intuitionistic modal logic IK, we provide new and simpler decidability proofs for FIK and LIK.

Problem

• Decidability proofs for intuitionistic modal logics FIK and LIK
• Simpler approach using tableau-based procedures
• Integration of finite frame property and modal logic techniques

Innovation

• Tableaux-based procedures for modal logics
• Finite frame property integration
• Simpler decidability proofs for FIK, LIK