Game Semantics for Modal Logic with Counting

Game semantics provides an alternative perspective in the understanding of standard logical concepts such as truth, model similarity and consistency. In the game, one player claims that certain logical property holds (e.g. a formula is true in a model, or two models are similar), and the other player claims the opposite. The winning strategy of the first player is used to characterize the truth of a formula/similarity between two models. Modal logic of counting is the extension of modal logic by adding formulas of the form #phi greater than or similar to #psi stating that the number of phi-successors is greater than or equal to the number of psi-successors. In this paper, we define the evaluation game and the model comparison game for modal logic with counting. In the evaluation game, we make use of the fact that if the number kappa(1) of phi-successors is greater than or equal to the number kappa(2) of psi-successors, then there is a cardinal number kappa(3) such that kappa(1) >= kappa(3) >= kappa(2). In the model comparison game, we define #-bisimulation and n-#-bisimulation, which are the basic notions in the definition of the model comparison game for modal logic with counting.