The basic semantics for interpretability logics is Veltman semantics. R. Verbrugge defined a new relational semantics for interpretability logics, which today is called Verbrugge semantics in her honor. It has turned out that this semantics has various good properties (see e.g. [2]). Bisimulations are the basic equivalence relations between Veltman models. M. Vuković in [4] defined bisimulations and their finite approximations called \(n\)-bisimulations for Verbrugge semantics. M. Vuković and T. Perkov in [3] used bisimulations and bisimulation games to prove the van Benthem’s characterization theorem with respect to Veltman semantics. However, we have proved in [1] that \(n\)-modally worlds do not necessarily have to be \(n\)-bisimilar. Therefore, in [1] we gave a new version of bisimulations, which we called weak bisimulations.

In this talk, we will make an overview of the results that can be obtained using weak bisimulations and the corresponding weak bisimulation games. We will show that \(n\)-modally equivalent worlds are \(n\)-weak bisimilar. Also, we will emphasize results about the finite model property and the van Benthem’s characterization theorem with respect to Verbrugge semantics.

Bibliography

1. S. Horvat, T. Perkov, M. Vukovi’c, Bisimulations and bisimulations games for Vebrugge semantics, Mathematical Logic Quarterly, vol. 69 (2023), no. 2, pp. 231–243.
2. L. Mikec, M. Vuković,Interpretability logics and generalized Veltman semantics, The Journal of Symbolic Logic, vol. 85 (2020), no. 2, pp. 749–772.
3. T. Perkov, M. Vuković,A bisimulation characterization for interpretability logic,Logic Journal of the IGPL,vol. 22 (2014), no. 6, pp. 872–879.
4. M. Vuković,Bisimulations between generalized Veltman models and Veltman models,Mathematical Logic Quarterly,vol. 54 (2008), no. 4, pp. 368–373.