PSPACE-completeness of bimodal transitive weak-density logic

Abstract Windows have been introduced in [1] as a tool for designing polynomial algorithms to check satisfiability of a bimodal logic of weak-density. In this paper, after revisiting the “folklore” case of bimodal ${\textbf{K}}4$ already treated in [8], but which is worth a fresh review, we show that windows allow to polynomially solve the satisfiability problem when adding transitivity to weak-density, by mixing algorithms for bimodal ${\textbf{K}}4$ together with windows-approach. The conclusion is that both satisfiability and validity are $\textbf{PSPACE}$-complete for these logics.