Characterization of First-Order String Transductions via Planar Affine λ-Calculus

Core Concepts

Affine string-to-string functions definable in the planar affine λ-calculus λ℘ and first-order string transductions coincide.

Abstract

The paper investigates the expressiveness of the planar affine λ-calculus λ℘ in computing string-to-string functions. It proves that the class of affine λ℘-definable string-to-string functions is exactly the class of first-order string transductions.
The key insights are:

λ℘-transducers, which are λ℘-terms of a specific form, can be interpreted as two-way planar reversible finite transducers (2PRFTs), a variant of two-way transducers that capture first-order transductions.
This interpretation is done by defining an interpretation of purely affine λ℘-terms in the category of planar diagrams TransDiagΓ, which is shown to be a strict monoidal-closed poset-enriched category.
The interpretation maps β-reductions to inequalities in TransDiagΓ, allowing to show that every λ℘-transducer can be compiled into an equivalent 2PRFT.
The converse direction, that every first-order transduction is λ℘-definable, follows from a Krohn-Rhodes-style decomposition lemma and the closure properties of affine λ℘-definable functions.

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Key Insights Distilled From

Implicit automata in λ-calculi III

by Céci... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03985.pdf

Deeper Inquiries

What other classes of string-to-string functions can be characterized via variants of the affine λ-calculus

In addition to affine λ-calculus, other classes of string-to-string functions can be characterized using variants such as linear logic and non-commutative linear logic. Linear logic, which restricts the use of resources, has been used to capture complexity classes of functions. Non-commutative linear logic, on the other hand, enforces a specific order of operations, which can be useful in characterizing certain types of string-to-string functions. By leveraging these different variants of λ-calculus, researchers can explore and define various classes of string-to-string functions with different computational properties.

Can the techniques used here be extended to capture more expressive classes of tree transductions

The techniques used in this work can potentially be extended to capture more expressive classes of tree transductions. By adapting the concepts and methodologies from affine λ-calculus and planar diagrams to the domain of tree transductions, researchers can explore the computational complexity and expressiveness of functions that operate on tree structures. This extension may involve developing new interpretations, transformations, and formalisms that are tailored to the characteristics and challenges of tree transductions.

How do the insights from this work relate to the geometry of interaction interpretation of the λ-calculus

The insights from this work can be related to the geometry of interaction interpretation of the λ-calculus. The geometry of interaction is a framework that studies the interaction between proofs and programs in a computational setting. By exploring the connections between affine λ-calculus, planar diagrams, and the geometry of interaction, researchers can gain a deeper understanding of the computational processes underlying these formal systems. This interdisciplinary approach can lead to new insights into the relationship between logic, computation, and geometry in the context of functional programming and automata theory.