Abstract

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

中文翻译：

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非正态模态和道义逻辑的超继计算：反模型和最佳复杂性

摘要

我们为经典立方的所有系统及其公理为$ {T} $，$ {P} $和$ {D} $的扩展给出一些超序计算，对于每个$ n \ geq 1 $，规则$ {RD} _n ^ + $。结石是内部的，因为它们仅使用逻辑语言以及其他结构性连接词。我们通过切消消除的句法证据表明，结石相对于相应的公理化是完整的。然后，我们在超序结石中定义了一种终止证明搜索策略，并表明它对于coNP完全逻辑是最佳的。而且，我们表明，从公式或每个公式的每个失败的证明中，有可能直接在多项式大小的双邻域语义中为coNP逻辑提取正反模型，在规则逻辑中也可以从关系语义中提取其反模型。规则立方体的标记系统中的规则应用和派生。