Implicit automata in \lambda-calculi III: affine planar string-to-string functions
<|reference_start|>Implicit automata in \lambda-calculi III: affine planar string-to-string functions: 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.<|reference_end|>
@article{pradic2024implicit,
    title={Implicit automata in {\lambda}-calculi III: affine planar
  string-to-string functions},
    author={C'ecilia Pradic and Ian Price},
    journal={arXiv preprint arXiv:2404.03985},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.03985},
    primaryClass={cs.LO cs.FL}
}