We characterize regular string transductions as programs in a linear $\lambda$-calculus with additives. One direction of this equivalence is proved by encoding copyless streaming string transducers (SSTs), which compute regular functions, into our $\lambda$-calculus. For the converse, we consider a categorical framework for defining automata and transducers over words, which allows us to relate register updates in SSTs to the semantics of the linear $\lambda$-calculus in a suitable monoidal closed category. To illustrate the relevance of monoidal closure to automata theory, we also leverage this notion to give abstract generalizations of the arguments showing that copyless SSTs may be determinized and that the composition of two regular functions may be implemented by a copyless SST. Our main result is then generalized from strings to trees using a similar approach. In doing so, we exhibit a connection between a feature of streaming tree transducers and the multiplicative/additive distinction of linear logic. Keywords: MSO transductions, implicit complexity, Dialectica categories, Church encodings

中文翻译：

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类型化 $\lambda$-calculi II 中的隐式自动机：流式传感器与分类语义

我们将常规字符串转导描述为带有添加剂的线性 $\lambda$ 演算中的程序。这种等价的一个方向是通过将计算正则函数的无副本流字符串转换器 (SST) 编码到我们的 $\lambda$ 演算中来证明的。相反，我们考虑了一个用于定义单词上的自动机和转换器的分类框架，它允许我们将 SST 中的寄存器更新与合适的幺半群闭类别中的线性 $\lambda$-演算的语义相关联。为了说明幺半群闭包与自动机理论的相关性，我们还利用这个概念给出了论证的抽象概括，表明无副本 SST 可以被确定，并且两个正则函数的组合可以由无副本 SST 实现。然后使用类似的方法将我们的主要结果从字符串推广到树。在这样做时，我们展示了流树传感器的特征与线性逻辑的乘法/加法区别之间的联系。关键词：MSO 转导，隐含复杂性，辩证法范畴，Church 编码