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Strategies as Resource Terms, and Their Categorical Semantics

Authors Lison Blondeau-Patissier, Pierre Clairambault , Lionel Vaux Auclair

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Author Details

Lison Blondeau-Patissier
  • Aix Marseille Univ, CNRS, I2M and LIS, Marseille, France
Pierre Clairambault
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Lionel Vaux Auclair
  • Aix Marseille Univ, CNRS, I2M, Marseille, France

Funding

  • Clairambault, Pierre: Supported by the ANR projects DyVerSe (ANR-19-CE48-0010-01) and by the Labex MiLyon (ANR-10-LABX-0070) of Université de Lyon.
  • Vaux Auclair, Lionel: Supported by the ANR projects PPS (ANR-19-CE48-0014), LambdaComb (ANR-21-CE48-0017) and Reciprog (ANR-21-CE48-0019).

Cite As Get BibTex

Lison Blondeau-Patissier, Pierre Clairambault, and Lionel Vaux Auclair. Strategies as Resource Terms, and Their Categorical Semantics. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSCD.2023.13

Abstract

As shown by Tsukada and Ong, simply-typed, normal and η-long resource terms correspond to plays in Hyland-Ong games, quotiented by Melliès' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model {w.r.t.} both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization.
In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential λ-calculus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Denotational semantics
  • Theory of computation → Categorical semantics
Keywords
  • Resource calculus
  • Game semantics
  • Categorical semantics

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