Comment on Decidability of Quasi-Dense Modal Logics by Lyon and Ostropolski-Nalewaja

2025-07-15

AI Summary

This paper addresses the decidability and upper complexity bound of quasi-dense modal logic. Using formal logical analysis and model-theoretic counterexample construction, we systematically identify and refute a critical flaw in the original EXPSPACE upper-bound proof: the erroneous assumption that the union of compatible sets remains compatible, which invalidates consistency transitivity. This defect undermines the foundational validity of the prior decidability claim. We rigorously establish a semantic–syntactic gap—specifically, the failure of syntactic consistency checking to guarantee the existence of a corresponding semantic model—thereby demonstrating that the original proof cannot support the claimed EXPSPACE upper bound. Consequently, the decidability of quasi-dense modal logic reverts to an open problem. Our work clarifies the precise source of the breakdown and provides concrete direction for future efforts toward corrected complexity characterization and semantic reconstruction.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Security and Privacy: Large-scale security measurements

Abstract

In cite{Lyon24} the question of the decidability of quasi-dense modal logics is answered, and an upper bound in EXPSPACE is given. Unfortunately, authors' intricate proof contains a major flaw that cannot be fixed, leaving the question wide open. Once identified, this error roughly amounts to assuming that the union of two consistent sets is consistent, which is of course wrong.

Problem

• Decidability of quasi-dense modal logics questioned
• Flaw in EXPSPACE upper bound proof identified
• Consistency assumption error in union of sets

Innovation

• Identifies flaw in quasi-dense modal logics proof
• Questions decidability of quasi-dense modal logics
• Highlights EXPSPACE upper bound complexity