补全图景：带逆模态的分级模态逻辑的复杂性
Completing the Picture: Complexity of Graded Modal Logics with Converse
AI总结：
本文填补了带分级逆模态的分级模态逻辑在传统框架类上的复杂性分类空白，证明了其在欧几里得框架上的NExpTime完全性，并证实了其变体在传递框架上的可判定性。
摘要：
对于分级模态语言在传统框架类上的局部和全局可满足性问题的复杂性，已经建立了完整的分类。所谓“传统”框架类，是指由自反性、连续性、对称性、传递性和欧几里得性质的任意正组合所刻画的那些类。在本文中，我们填补了带分级逆模态的分级模态语言在类似分类中剩余的空白。特别地，我们证明了该语言在欧几里得框架类上是NExpTime完全的，从而表明在该框架类上，所考虑的语言比不带分级模态或不带逆模态的语言更难。我们还考虑了其变体，该变体不允许分级逆模态，但仍允许基本的逆模态。对于该变体，我们最重要的结果是证实了一个早期的猜想，即它在传递框架上可判定。这与带分级逆模态的语言的不可判定性形成对比。
英文摘要：
A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames have 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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