We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill's calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the ! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove algorithmic undecidability for both calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages.

中文翻译：

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具有次指数和括号模态的乘法加法兰贝克微积分

我们对 Morrill 引入的作为 CatLog 分类语法解析器核心的系统进行了证明理论和算法复杂性分析。我们考虑了 Morrill 演算的两个最新版本，并关注它们的片段，包括乘法 (Lambek) 连接词、加法连词和析取词、括号和括号形式，以及 ! 次指数模式。对于这两个系统，我们解决了与切割规则相关的问题并提供了必要的修改，之后我们证明了切割的可接受性（切割消除定理）。我们还证明了两种演算的算法不可判定性，并表明基于它们的分类语法可以生成任意递归可枚举语言。