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The Lambda Calculus Is Quantifiable

Authors Valentin Maestracci , Paolo Pistone

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ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Program semantics
Keywords
  • Lambda-calculus
  • Scott semantics
  • Partial metric spaces
  • Böhm trees
  • Taylor expansion

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Abstract

In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from Scott domains. First, we study quantitative variants, based on program distances, of sensible equational theories for the λ-calculus, like those arising from Böhm trees and from the contextual preorder. Then, we introduce applicative distances capturing higher-order Scott topologies, including reflexive objects like the D_∞ model. Finally, we provide a quantitative insight on the well-known connection between the Böhm tree of a λ-term and its Taylor expansion, by showing that the latter can be presented as an isometric transformation.

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Valentin Maestracci and Paolo Pistone. The Lambda Calculus Is Quantifiable. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 34:1-34:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.34

Author Details

Valentin Maestracci
  • I2M, Université d'Aix-Marseille, France
Paolo Pistone
  • LIP, Université Claude Bernard Lyon 1, France

Funding

  • Pistone, Paolo: Research has been funded by the French project ANR-23-CPJ1-0054-01.

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