Decidability of intuitionistic modal logics with transitive modal relations (IK4 and IS4)

Establish the decidability of intuitionistic modal logics whose modal accessibility relation is transitive, including the systems IK4 and IS4.

Background

The paper develops a terminating proof-search methodology for intuitionistic tense logics using nested sequents, including loop-checking and finite counter-model extraction, thereby proving the finite model property and decidability for logics IK_t with combinations of T, B, and (d).

While these results cover a broad class of intuitionistic tense logics, the decidability of intuitionistic modal logics featuring transitive modal relations—most notably IK4 and IS4—has not been settled. The authors note that their techniques address core obstacles for such logics (loop-checking and non-invertibility) and that their systems can be modularly adapted to settings with transitivity, suggesting a potential path toward resolving the open problem.