Beyond Regularity for Presburger Modal Logics

Satisfiability problem for modal logic K with quantifier-free Presburger and regularity constraints (EML) is known to be pspace-complete. In this paper, we consider its extension with nonregular constraints, and more specifically those expressed by visibly pushdown languages (VPL). This class of languages behaves nicely, in particular when combined with Propositional Dynamic Logic (PDL). By extending EML, we show that decidability is preserved if we allow at most one positive VPL-constraint at each modal depth. However, the presence of two VPL-contraints or the presence of a negative occurrence of a single VPL-constraint leads to undecidability. These results contrast with the decidability of PDL augmented with VPL-constraints.