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Implicit Automata in λ-calculi III: Affine Planar String-to-string Functions

Cécilia Pradic Orcid Logo, Ian Price

Electronic Notes in Theoretical Informatics and Computer Science, Volume: Volume 4 - Proceedings of MFPS XL

Swansea University Authors: Cécilia Pradic Orcid Logo, Ian Price

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DOI (Published version): 10.46298/entics.14804

Abstract

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...

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Published in: Electronic Notes in Theoretical Informatics and Computer Science
ISSN: 2969-2431
Published: Oxford Centre pour la Communication Scientifique Directe (CCSD) 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa68303
Abstract: 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.
Keywords: non-commutative linear logic, transducers, λ-calculus, automata theory, Church encodings
College: Faculty of Science and Engineering

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