It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type calculus (ETTC). ETTC is a calculus of specific typed terms, which represent tuples of strings, more precisely bipartite graphs decorated with strings. Types are derived from linear logic formulas, and rules correspond to concrete operations on these string-labeled graphs, so that they can be conveniently visualized. This provides the above mentioned fragment of MLL1 that is relevant for language modeling not only with some alternative syntax and intuitive geometric representation, but also with an intrinsic deductive system, which has been absent. In this work we consider a non-trivial notationally enriched variation of the previously introduced ETTC, which allows more concise and transparent computations. We present both a cut-free sequent calculus and a natural deduction formalism.

翻译：已知不同范畴语法在一阶乘法线性逻辑（MLL1）的一个片段中具有表层表示。我们证明该相关片段等价于近期提出的扩展张量类型演算（ETTC）。ETTC是一种特定类型项（表示字符串元组，更精确地说是由字符串装饰的二分图）的演算。其类型源自线性逻辑公式，规则对应这些带标签字符串图上的具体操作，使得这些操作可被直观可视化。这为语言建模提供了上述MLL1片段——不仅包含替代语法和直观几何表示，更具备此前缺失的内在演绎系统。本研究对先前提出的ETTC进行了非平凡符号化扩展，实现了更简洁透明的计算。我们同时给出了无切割序贯演算和自然演绎形式化体系。