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How to Play the Accordion: Uniformity and the (Non-)Conservativity of the Linear Approximation of the λ-Calculus

Authors Rémy Cerda , Lionel Vaux Auclair

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  • A short abstract of a preliminary version of this work has been presented at the 7th International Workshop on Trends in Linear Logic and its Applications (TLLA'23). Most of this work has also appeared in the first author’s PhD thesis, with the noticeable difference that Theorem 27 was only proved in a qualitative setting.
  • Previous Version (Short abstract presented at TLLA'23) https://lipn.univ-paris13.fr/TLLA/2023/abstracts/05-Final.pdf
  • Previous Version (PhD thesis version, see Chapter 5) https://hal.science/tel-04664728

Subject Classification

ACM Subject Classification
  • Theory of computation → Rewrite systems
  • Theory of computation → Lambda calculus
  • Theory of computation → Proof theory
  • Theory of computation → Linear logic
Keywords
  • program approximation
  • quantitative semantics
  • lambda-calculus
  • linear approximation
  • Taylor expansion
  • conservativity

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Abstract

Twenty years after its introduction by Ehrhard and Regnier, differentiation in λ-calculus and in linear logic is now a celebrated tool. In particular, it allows to write the Taylor formula in various λ-calculi, hence providing a theory of linear approximations for these calculi. In the standard λ-calculus, this linear approximation is expressed by results stating that the (possibly) infinitary β-reduction of λ-terms is simulated by the reduction of their Taylor expansion: in terms of rewriting systems, the resource reduction (operating on Taylor approximants) is an extension of the β-reduction.
In this paper, we address the converse property, conservativity: are there reductions of the Taylor approximants that do not arise from an actual β-reduction of the approximated term? We show that if we restrict the setting to finite terms and β-reduction sequences, then the linear approximation is conservative. However, as soon as one allows infinitary reduction sequences this property is broken. We design a counter-example, the Accordion. Then we show how restricting the reduction of the Taylor approximants allows to build a conservative extension of the β-reduction preserving good simulation properties. This restriction relies on uniformity, a property that was already at the core of Ehrhard and Regnier’s pioneering work.

Cite As Get BibTex

Rémy Cerda and Lionel Vaux Auclair. How to Play the Accordion: Uniformity and the (Non-)Conservativity of the Linear Approximation of the λ-Calculus. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.23

Author Details

Rémy Cerda
  • Aix-Marseille Université, CNRS, I2M, France
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
Lionel Vaux Auclair
  • Aix-Marseille Université, CNRS, I2M, France

Funding

  • Cerda, Rémy: Partially supported by the French ANR project RECIPROG (ANR-21-CE48-019).
  • Vaux Auclair, Lionel: Partially supported by the French ANR projects LambdaComb (ANR-21-CE48-0017), RECIPROG (ANR-21-CE48-019), and PPS (ANR-19-CE48-0014).

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