We introduce infinitary action logic with exponentiation---that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is $\Pi_1^1$-complete. Second, we show that the closure ordinal of its derivability operator is $\omega_1^{\mathrm{CK}}$.

中文翻译：

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带幂的无穷动作逻辑

我们引入了带幂的无限动作逻辑——即，乘法加法兰贝克演算扩展了 Kleene 星和一系列次指数模态，它允许一些结构规则（收缩、弱化、置换）。该逻辑以无穷连续演算的形式呈现。我们证明了切割消除，并且在至少一个次指数允许非局部收缩的情况下，在两种意义上建立精确的复杂性边界。首先，我们证明这个逻辑的可推导性问题是 $\Pi_1^1$-complete。其次，我们证明其可导性算子的闭包序数是 $\omega_1^{\mathrm{CK}}$。