Parameterized complexity of n-dense modal logics

Exact tight bounds of the complexity of the satisfiability problem for dense modal logics is a difficult question, likely somewhere between $\text{PSPACE}$ and $\text{EXPSPACE}$ depending of the logic under question. For a class of them, called here $n$-dense logics (characterized by axioms ${\mathrm{\square }}^{n}p\to \mathrm{\square }p$), we refine the known results – membership in $\text{NEXPTIME}$ – in the light of parameterized complexity, as introduced in \cite{Downey}, and prove that they belong to the parameterized class para-$\text{PSPACE}$: 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}.