Parameterized complexity of n-dense modal logics

2026-04-13

AI Summary

This study addresses the exact computational complexity of the satisfiability problem for n-dense modal logics, which has been known to lie between PSPACE and EXPSPACE. By taking modal depth as a parameter, the work introduces a recursive window method—an extension of the classical window analysis technique—and combines it with tools from parameterized complexity theory and modal semantics to devise a polynomial-space algorithm. This result refines the previously established NEXPTIME upper bound to para-PSPACE, thereby establishing a tight bound for the problem within the framework of parameterized complexity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

Exact tight bounds of the complexity of the satisfiability problem for dense modal logics is a difficult question, likely somewhere between $\PSPACE$ and $\EXPSPACE$ depending of the logic under question. For a class of them, called here $n$-dense logics (characterized by axioms $\Box^n p\rightarrow \Box p$), we refine the known results -- membership in $\NEXPTIME$ -- in the light of parameterized complexity, as introduced in \cite{Downey}, and prove that they belong to the parameterized class para-$\PSPACE$: there exists a poly-space algorithm once the modal depth of the input is considered as a parameter. This is done by generalizing the novel analysis tool introduced in \cite{BalGasq25}, and therein called windows, to \emph{recursive windows}.

Problem

• parameterized complexity
• modal logic
• satisfiability problem
• n-dense logics
• modal depth

Innovation

• parameterized complexity
• n-dense modal logics
• para-PSPACE
• recursive windows
• satisfiability problem