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 \(\#\varphi \succsim \#\psi \) stating that the number of \(\varphi \) -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 \(\varphi \) -successors is greater than or equal to the number \(\kappa _2\) of \(\psi \) -successors, then there is a cardinal number \(\kappa _3\) such that \(\kappa _1\ge \kappa _3\ge \kappa _2\) . 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.

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Game Semantics for Modal Logic with Counting

  • Xiaoxuan Fu,
  • Zhiguang Zhao

摘要

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 \(\#\varphi \succsim \#\psi \) stating that the number of \(\varphi \) -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 \(\varphi \) -successors is greater than or equal to the number \(\kappa _2\) of \(\psi \) -successors, then there is a cardinal number \(\kappa _3\) such that \(\kappa _1\ge \kappa _3\ge \kappa _2\) . 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.