A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

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完成图片：Converse 的分级模态逻辑的复杂性

已经建立了对传统框架类的分级模态语言的局部和全局可满足性问题的复杂性的完整分类。“传统”框架类别是指那些具有自反性、连续性、对称性、传递性和欧几里得性质的任何积极组合特征的框架。在本文中，我们用分级逆模态填补了分级模态语言的类似分类中剩余的空白。特别是，我们展示了它的NE经验吨我- 欧几里得框架类的完整性，证明在这个类上，所考虑的语言比没有分级模态或没有逆模态的语言更难。我们还考虑了它的变体不允许分级的逆向模式，但仍然允许基本的逆向模式。对于这种变化，我们最重要的结果是证实了一个早期的猜想，即它在传递帧上是可判定的。这与具有分级逆向模式的语言的不确定性形成对比。