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...

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Published in:Electronic Notes in Theoretical Informatics and Computer Science Vol. 4 - Proceedings of...
Main Authors: Pradic, Cécilia, Price, Ian
Format: Journal Article
Language:English
Published: 11.12.2024
ISSN:2969-2431, 2969-2431
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Abstract 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. Comment: 19+1 pages, 7 figures; camera-ready version for MFPS
AbstractList 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. Comment: 19+1 pages, 7 figures; camera-ready version for MFPS
Author Pradic, Cécilia
Price, Ian
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Snippet We prove a characterization of first-order string-to-string transduction via $\lambda$-terms typed in non-commutative affine logic that compute with Church...
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Title Implicit automata in {\lambda}-calculi III: affine planar string-to-string functions
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