We prove a characterization of first-order string-to-string transduction via $\lambda$-terms typed in non-commutative affine logic that compute with Church encoding, extending the analogous known characterization of star-free languages. We show that every first-order transduction can be computed by a $\lambda$-term using a known Krohn-Rhodes-style decomposition lemma. The converse direction is given by compiling $\lambda$-terms into two-way reversible planar transducers. The soundness of this translation involves showing that the transition functions of those transducers live in a monoidal closed category of diagrams in which we can interpret purely affine $\lambda$-terms. One challenge is that the unit of the tensor of the category in question is not a terminal object. As a result, our interpretation does not identify $\beta$-equivalent terms, but it does turn $\beta$-reductions into inequalities in a poset-enrichment of the category of diagrams.

翻译：我们证明了通过非交换仿射逻辑中类型化的λ项（采用Church编码计算）实现的一阶字符串到字符串转换的刻画，扩展了已知的无星号语言的类似刻画。我们证明每个一阶转换均可通过λ项计算，其中使用了已知的Krohn-Rhodes式分解引理。反方向通过将λ项编译为双向可逆平面转换器实现。该翻译的正确性需要证明这些转换器的转移函数存在于一个幺半群闭图表范畴中，该范畴可解释纯仿射λ项。一个挑战在于该范畴张量积的单位元不是终对象。因此，我们的解释虽不识别β等价项，但能将β归约转化为图表范畴偏序富化结构中的不等式关系。