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.
翻译:我们引入了在时态逻辑背景下显示微积分证据和标记微积分证据之间的翻译。 首先, 我们显示显示显示微积分中以普通路径轴轴延伸的最小时态逻辑 Kt 的每一种衍生物都可以有效地转换成相应的标记微积分的衍生物。 关于反向翻译, 我们显示对于 Kt 来说, 以路径轴轴延伸, 相应的标记微积分中的每一种衍生物都可以被置于一种特殊的形式中, 可以转换到相关的显示微积分中的衍生物。 这个反向翻译中的关键洞察力是将显示序列的直观表示为标签多条树。 标签多条树是显示序列的等同类, 代表显示序列的摩杜卢显示后缀的等值, 也为相关时态逻辑的对应结果提供了光亮。