We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an $\omega $ -rule, and prove that the derivability problem in this calculus is $\Pi _1^0$ -hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).

中文翻译：

---

具有 KLEENE STAR 的无限 LAMBEK 微积分的复杂性

我们考虑 Lambek 演算，或非交换乘法直觉线性逻辑，用迭代扩展，或 Kleene 星，通过一个公理化$\欧米茄$-规则，并证明这个微积分中的可推导问题是$\Pi _1^0$-难的。这解决了 Buszkowski (2007) 留下的一个问题，他为无限动作逻辑获得了相同的复杂性界限，其中还包括加法合取和析取。作为副产品，我们证明任何没有空词的上下文无关语言都可以通过具有唯一类型分配的 Lambek 语法生成，而无需施加 Lambek 的非空性限制（参见 Safiullin，2007）。