Parameterized complexity of n-dense modal logics

TL;DR

This paper investigates the parameterized complexity of n-dense modal logics, showing they belong to para-PSPACE with a poly-space algorithm when considering modal depth as a parameter.

Contribution

It refines existing complexity bounds for n-dense modal logics by introducing recursive windows, linking their complexity to para-PSPACE.

Findings

n-dense modal logics are in para-PSPACE when parameterized by modal depth.

Introduces recursive windows as a new analysis tool.

Provides refined complexity bounds between PSPACE and EXPSPACE.

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}.