We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics.

中文翻译：

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显示到标记的证明并再次返回以进行紧张的逻辑

我们在时态逻辑的上下文中介绍了显示微积分证明和标记微积分证明之间的翻译。首先，我们表明，用一般路径公理扩展的最小时态逻辑 Kt 的显示演算中的每个推导都可以有效地转换为相应标记演算中的推导。关于逆转换，我们表明对于用路径公理扩展的 Kt，相应标记演算中的每个推导都可以放入一种特殊形式，该形式可转换为相关显示演算中的推导。这种逆向翻译的一个关键见解是将显示序列的规范表示为标记的多树。表示显示序列模显示假设的等价类的标记多树也阐明了时态逻辑的相关对应结果。