A new notion of bisimulations of Verbrugge semantics

The basic semantics for the interpretability logics are Veltman models defined by D. de Jongh and F. Veltman in 1988. They were used by V. Švejdar in 1991. to prove some independence results. De Jongh tried to generalize Švejdar's arguments and came up with the notion of generalized Veltman semantics. Since R. Verbrugge worked this out in an unpublished note, this semantics has recently been named after her. The basic equivalence between Veltman models are bisimulations. M. Vuković defined bisimulations (and their finite approximations called n- bisimulations) for Verbrugge semantics. It was proved by M. Vuković and D. Vrgoč that n-bisimilar worlds are modally n-equivalent, i.e. they satisfy the same IL- formulas of modal depth up to n. We have shown that the converse is generally not true. So we have defined a new notion of bisimulations for Verbrugge semantics called w- bisimulations. In this talk we will present them and show the desired converse: n-equivalent worlds are n-w-bisimilar. In order to do that we define Verbrugge model comparison games called w-games and show that w-bisimulation relations may be understood as descriptions of winning strategies for one player in a w-game.