engleski

n-bisimulations for generalised Veltman semantics

As incompleteness results emerged in modal logic, logicians started to investigate what modal languages actually are, or said differently, what is their position in the logical universe. Van Benthem’s characterisation theorem shows that modal languages correspond to the bisimulation invariant fragment of first-order languages. One can prove that result with use of classical methods of first-order model theory. However, many problems arised when one tries to use such methods to prove a characterisation theorem over the provability logic GL. Because of that, A. Dawar and M. Otto develop a models-for-games method, which provides conditions from which a characterisation theorem over particular class of models immediately follows. Using that, not only that characterisation theorem for provability logic GL was proved, but also M. Vuković and T. Perkov proved that this result can be extended to Veltman models for the interpretability logic IL. To prove that, they used bisimulation games on Veltman models for interpretability logic. Since Veltman semantics is not fine-grained enough for certain application, the notion of generalised Veltman semantics emerged to obtain certain nonderivability results. It has turned out that this semantics has various good properties. One question that arises is can models-for-games method be used to prove a characterisation theorem with respect to generalised Veltman semantics. In order to do that, one needs to define n-bisimulation and n-bisimulation games for generalised Veltman semantics. We carry that out in this work. Also, it is easy to show that bisimilar worlds are modally equivalent. But what about other direction, that is, are modally equivalent worlds bisimilar? We negatively answer that question by using counterexamples for Veltman semantics and turning them into counterexamples for generalised Veltman semantics. Finally, we define n-bisimulation games for generalised Veltman semantics and prove the equivalence between the existence of a winning strategy in the n-bisimulation game and the existence of an n-bisimulation.

Generalised Veltman semantics ; bisimulation ; bisimulation games ; modal equivalence

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