In this paper, we consider the logic ${\textsf{ITL}}^{e}$, a variant of intuitionistic linear temporal logic that is interpreted over the class of dynamic Kripke frames. These are bi-relational structures of the form $ \langle{W, \preccurlyeq , f}\rangle $ where $\preccurlyeq $ is a partial order on $W$ and $f: W \to W$ is a $\preccurlyeq $-monotone function. Our main result answers a question recently raised by Boudou et al. (2017, A decidable intuitionistic temporal logic. In Computer Science Logic 2017, pp. 14:1–14:17. Vol. 82 of LIPIcs) about axiomatizing this logic. We provide an axiomatization of ${\textsf{ITL}}^{e}$ and prove its strong completeness with respect to the class of all dynamic Kripke frames. The proposed axiomatization is infinitary; it has two derivation rules with countably many premises and one conclusion. It should be mentioned that ${\textsf{ITL}}^{e}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete.

中文翻译：

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直觉主义时序逻辑的强完整公理化

在本文中，我们考虑逻辑 ${\textsf{ITL}}^{e}$，它是在动态 Kripke 框架类上解释的直觉线性时间逻辑的变体。这些是形式为 $ \langle{W, \preccurlyeq , f}\rangle $ 的双关系结构，其中 $\preccurlyeq $ 是 $W$ 和 $f 的偏序：W \to W$ 是 $\preccurlyeq $-单调函数。我们的主要结果回答了 Boudou 等人最近提出的一个问题。（2017，A 可判定的直觉时间逻辑。在 Computer Science Logic 2017，第 14:1-14:17。第 82 卷 LIPIcs）关于公理化这种逻辑。我们提供了 ${\textsf{ITL}}^{e}$ 的公理化，并证明了它对所有动态 Kripke 框架类的强完备性。建议的公理化是无限的；它有两条推导规则，有可数个前提和一个结论。