Involutive Nonassociative Lambek Calculus (InNL) is a nonassociative version of Noncommutative Multiplicative Linear Logic (MLL) (Abrusci in J Symb Log 56:1403–1451, 1991), but the multiplicative constants are not admitted. InNL adds two linear negations to Nonassociative Lambek Calculus (NL); it is a strongly conservative extension of NL (Buszkowski in Amblard, de Groote, Pogodalla, Retoré (eds) Logical aspects of computational linguistics. LNCS, vol 10054. Springer, Berlin, pp 68–84, 2016). Here we also add unary modalities satisfying the residuation law and De Morgan laws. For the resulting logic InNLm, we define and study phase spaces (some frame models, typical for linear logics). We use them to prove the cut elimination theorem for a one-sided sequent system for InNLm, introduced here. Phase spaces are also employed in studying auxiliary systems InNLm(k), assuming the k-cyclic law for negation. The latter behave similarly as Classical Nonassociative Lambek Calculus, studied in de Groote and Lamarche (Stud Log 71(3):355–388, 2002) and Buszkowski (2016). We reduce the provability in InNLm to that in InNLm(k). This yields the equivalence of type grammars based on InNLm with ($$\epsilon $$ϵ-free) context-free grammars and the PTIME complexity of InNLm.

中文翻译：

---

关于对合非关联兰贝克微积分

对合非关联兰贝克微积分 (InNL) 是非交换乘法线性逻辑 (MLL) 的非关联版本（J Symb Log 56:1403–1451, 1991 中的 Abrusci），但不允许乘法常数。InNL 在非关联兰贝克微积分 (NL) 中添加了两个线性否定；它是 NL 的一个非常保守的扩展（Buszkowski in Amblard, de Groote, Pogodalla, Retoré (eds) 计算语言学的逻辑方面。LNCS，第 10054 卷。Springer，柏林，第 68-84 页，2016 年）。在这里，我们还添加了满足残差定律和德摩根定律的一元模态。对于生成的逻辑 InNLm，我们定义并研究了相空间（一些框架模型，典型的线性逻辑）。我们使用它们来证明此处介绍的 InNLm 单边序列系统的消减定理。相空间也用于研究辅助系统 InNLm(k)，假设 k 循环定律为否定。后者的行为与 de Groote 和 Lamarche (Stud Log 71(3):355–388, 2002) 和 Buszkowski (2016) 中研究的经典非关联兰贝克微积分相似。我们将 InNLm 中的可证明性降低到 InNLm(k) 中的可证明性。这产生了基于 InNLm 的类型文法与 ($$\epsilon $$ϵ-free) 上下文无关文法和 InNLm 的 PTIME 复杂性的等价性。