On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

Olimpieri, Federico; Auclair, Lionel Vaux

doi:10.46298/lmcs-18(1:1)2022

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Of course, much remains to be done: notably, we would like to understand better the links between thin concurrent games and generalized species of structure [FGHW08]. Much of the present development is also reminiscent of issues related to rigid resource terms and the Taylor development of λ-terms [OA20].…”

Section: Discussionmentioning

confidence: 95%

The Quantitative Collapse of Concurrent Games with Symmetry

Clairambault,

Paquet

2021

Preprint

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We explore links between the thin concurrent games of Castellan, Clairambault and Winskel, and the weighted relational models of linear logic studied by Laird, Manzonetto, McCusker and Pagani. More precisely, we show that there is an interpretationpreserving "collapse" functor from the former to the latter. On objects, the functor defines for each game a set of possible execution states. Defining the action on morphisms is more subtle, and this is the main contribution of the paper.Given a strategy and an execution state, our functor needs to count the witnesses for this state within the strategy. Strategies in thin concurrent games describe non-linear behaviour explicitly, so in general each witness exists in countably many symmetric copies. The challenge is to define the right notion of witnesses, factoring out this infinity while matching the weighted relational model. Understanding how witnesses compose is particularly subtle and requires a delve into the combinatorics of witnesses and their symmetries.In its basic form, this functor connects thin concurrent games and a relational model weighted by N ∪ {+∞}. We will additionally consider a generalised setting where both models are weighted by elements of an arbitrary continuous semiring; this covers the probabilistic case, among others. Witnesses now additionally carry a value from the semiring, and our interpretation-preserving collapse functor extends to this setting.

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Section: Discussionmentioning

confidence: 95%

The Quantitative Collapse of Concurrent Games with Symmetry

Clairambault,

Paquet

2021

Preprint

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“…Indeed, in their seminal results [ER06; ER08], Ehrhard and Regnier exploited a uniformity property to show that the normalization (rather than an arbitrary reduction) of a sum of resource terms obtained by Taylor expansion does not generate sums of coefficients: each term occurring in the normal form is generated by a single term of the original sum. It is then possible to deduce the quantitative Commutation theorem from the qualitative one: this was essentially the path followed by Ehrhard and Regnier, and revisited by Olimpieri and the second author [OV22]. In the latter work, the qualitative Commutation theorem was established quite straightforwardly, by proving that Taylor expansion commutes with a variant of hereditary head reduction (the reduction strategy underlying the definition of Böhm trees).…”

Section: Discussionmentioning

confidence: 96%

Finitary Simulation of Infinitary $\beta$-Reduction via Taylor Expansion, and Applications

Cerda,

Auclair

2023

Logical Methods in Computer Science

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Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus.

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On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

Cited by 2 publications

References 19 publications

The Quantitative Collapse of Concurrent Games with Symmetry

The Quantitative Collapse of Concurrent Games with Symmetry

Finitary Simulation of Infinitary $\beta$-Reduction via Taylor Expansion, and Applications

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