Implicit automata in λ-calculi III: affine planar string-to-string functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917870574764032 |
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| author | Pradic, Cécilia Price, Ian |
| author_facet | Pradic, Cécilia Price, Ian |
| contents | We prove a characterization of first-order string-to-string transduction via $λ$-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 $λ$-term using a known Krohn-Rhodes-style decomposition lemma. The converse direction is given by compiling $λ$-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 $λ$-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 $β$-equivalent terms, but it does turn $β$-reductions into inequalities in a poset-enrichment of the category of diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Implicit automata in λ-calculi III: affine planar string-to-string functions Pradic, Cécilia Price, Ian Logic in Computer Science Formal Languages and Automata Theory 68N18, 03B40, 68Q55, 68Q45, 03B47 We prove a characterization of first-order string-to-string transduction via $λ$-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 $λ$-term using a known Krohn-Rhodes-style decomposition lemma. The converse direction is given by compiling $λ$-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 $λ$-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 $β$-equivalent terms, but it does turn $β$-reductions into inequalities in a poset-enrichment of the category of diagrams. |
| title | Implicit automata in λ-calculi III: affine planar string-to-string functions |
| topic | Logic in Computer Science Formal Languages and Automata Theory 68N18, 03B40, 68Q55, 68Q45, 03B47 |
| url | https://arxiv.org/abs/2404.03985 |