Bisimulations and bisimulation games between Verbrugge models

Interpretability logic is a modal formalization of relative interpretability between first-order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w-bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.