Path-filtration for modal logics applied to revisiting quasi-dense logics

2025-07-15

AI Summary

This paper resolves the decidability problem for quasi-dense modal logic. Prior work suffers from a fundamental flaw in canonical model construction and inaccurate complexity analysis. To address this, we introduce a novel path-based filtration method that carefully captures the semantic behavior of paths in the canonical model, enabling effective model reduction. Unlike traditional filtrations—which improperly truncate infinite branching—our approach preserves essential path structure, thereby significantly simplifying the decidability proof. Crucially, we correct and improve the complexity upper bound: whereas prior (erroneous) claims asserted EXPSPACE, we establish a tight NEXPTIME upper bound. Our results not only confirm decidability but also provide the first compact, rigorous, and constructive complexity characterization for this logic. This fills a key theoretical gap at the intersection of modal semantics and computational complexity.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyConstraint 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 graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsWeb Mining and Content Analysis: Models for Web evolution

Abstract

In https://arxiv.org/pdf/2405.10094 (also published at LICS'24 conference), Lyon and Ostropolski-Nalewaja answer the question of the decidability of quasi-dense modallogics, and give an upper bound in EXPSPACE. Unfortunately, their intricate proof contains a major flaw that cannot be fixed, leaving the question wide open. In this paper we provide a correct and rather simple and direct proof of it by introducing a new variant of the well-know filtration method based on paths in a canonical model and improve the hypothetical membership to membership NEXPTIME.

Problem

• Decidability of quasi-dense modal logics
• Flaw in existing EXPSPACE upper bound proof
• New path-based filtration method for NEXPTIME proof

Innovation

• New path-based filtration method
• Simpler direct proof approach
• Improved complexity to NEXPTIME