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Pierre Clairambault

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8 papers

2 author rows

CSL Conference 2026 Conference Paper

Towards A Rosetta Stone of Interactive and Quantitative Semantics (Invited Talk)

Pierre Clairambault

Quantitative semantics are those denotational semantics that inherit from linear logic [Jean-Yves Girard, 1987] a sensitivity to the multiplicity of resources involved in computation. Those include the relational model [Jean-Yves Girard, 1987] and its numerous variations (such as finiteness spaces [Thomas Ehrhard, 2005], weighted relational models [Jim Laird et al. , 2013] and their extensions [Thomas Ehrhard et al. , 2011; Thomas Ehrhard, 2002], generalized species of structure [Fiore et al. , 2008], span models [Paul-André Melliès, 2019; Pierre Clairambault and Simon Forest, 2023], etc), as well as related syntactic methods such as non-idempotent intersection types [Daniel de Carvalho, 2018] and Taylor expansion of lambda-terms [Thomas Ehrhard and Laurent Regnier, 2003]. Interactive semantics are usually also quantitative, but in addition they present the interactive behaviour of proofs and programs, generally organized chronologically - those include the many variants of game semantics (starting with [J. M. E. Hyland and C. -H. Luke Ong, 2000; Samson Abramsky et al. , 2000]), and other frameworks such as Geometry of Interaction [Girard, 1989] or ludics [Jean-Yves Girard, 2001]. Both families are cornerstones of modern denotational semantics, and both have associated Alonzo Church awards: game semantics in 2017, and quantitative semantics (in particular, differential linear logic and the differential λ-calculus) in 2024. It has more or less always been clear to the experts that the two, sharing an origin in linear logic, are conceptually related. Yet there are differences, which seem fundamental: in particular, while quantitative models compose relationally, the composition of strategies follows an intricate "parallel interaction plus hiding" process inspired from concurrency theory [Abramsky, 1997]. The two families of models have also historically targeted different kinds of languages: whereas quantitative semantics focused on theoretical calculi (and the λ-calculus in particular), game semantics is known for fully abstract models for languages with elaborate combinations of effects including local state [Samson Abramsky and Guy McCusker, 1996], control operators [James Laird, 1997], and concurrent primitives [Dan R. Ghica and Andrzej S. Murawski, 2008]. Early on, researchers have explored the relationship between the two [Thomas Ehrhard, 1996; Patrick Baillot et al. , 1997], and investigations on this question have spanned decades [Pierre Boudes, 2009; Ana C. Calderon and Guy McCusker, 2010; Takeshi Tsukada and C. -H. Luke Ong, 2016; C. -H. Luke Ong, 2017]. In particular, Melliès' work on asynchronous games [Paul-André Melliès, 2006; Paul-André Melliès, 2005] made significant conceptual contributions, showing that the issue was enlightened by adopting a positional formulation of game semantics, where points in the relational model simply arise as certain positions. This talk surveys recent developments in this line of work, shedding light on the connection between those two families. Our work is set in so-called "thin concurrent games" [Simon Castellan et al. , 2019; Pierre Clairambault, 2024], an extension with symmetry of Rideau and Winskel’s concurrent games on event structures [Silvain Rideau and Glynn Winskel, 2011]. Event structures being one of the main "truly concurrent" models of concurrency [Glynn Winskel, 1986], it is perhaps expected that thin concurrent games can model concurrent languages: they provide a truly concurrent refinement of Ghica and Murawski’s fully abstract model of Idealized Concurrent Algol [Simon Castellan and Pierre Clairambault, 2024; Pierre Clairambault, 2024]. But beyond the semantics of concurrency, thin concurrent games are also a deep reworking on game semantics built from causal principles, inheriting from asynchronous games a positional flavour. In thin concurrent games, strategies have a dual nature: an event-based nature where they appear as certain event structures composed via parallel interaction plus hiding; or a positional nature where they appear as certain spans of groupoids, composed by pullback (modulo a technical condition on strategies called visibility) - they can be regarded both as a games and a relational model! Leveraging this dual nature, in a sequence of papers with Castellan, de Visme, Olimpieri and Paquet, we have been able to link the single framework of thin concurrent games with numerous other models. This includes various traditional alternating or non-alternating games models [Simon Castellan and Pierre Clairambault, 2024; Pierre Clairambault, 2024], the weighted relational model [Pierre Clairambault and Hugo Paquet, 2021], the quantum relational model [Pierre Clairambault and Marc de Visme, 2020], generalized species of structure [Pierre Clairambault et al. , 2023], and - going beyond quantitative semantics - the linear Scott model [Clairambault, 2025], a linear decomposition of standard Scott domain semantics [Thomas Ehrhard, 2012]. All these distinct models are obtained by projecting away certain aspects of thin concurrent games, giving some support to the claim that thin concurrent games are a Rosetta stone for interactive and quantitative semantics.

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FSCD Conference 2023 Conference Paper

Strategies as Resource Terms, and Their Categorical Semantics

Lison Blondeau-Patissier
Pierre Clairambault
Lionel Vaux Auclair

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.

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FSCD Conference 2021 Conference Paper

Positional Injectivity for Innocent Strategies

Lison Blondeau-Patissier
Pierre Clairambault

In asynchronous games, Melliès proved that innocent strategies are positional: their behaviour only depends on the position, not the temporal order used to reach it. This insightful result shaped our understanding of the link between dynamic (i. e. game) and static (i. e. relational) semantics. In this paper, we investigate the positionality of innocent strategies in the traditional setting of Hyland-Ong-Nickau-Coquand pointer games. We show that though innocent strategies are not positional, total finite innocent strategies still enjoy a key consequence of positionality, namely positional injectivity: they are entirely determined by their positions. Unfortunately, this does not hold in general: we show a counter-example if finiteness and totality are lifted. For finite partial strategies we leave the problem open; we show however the partial result that two strategies with the same positions must have the same P-views of maximal length.

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Highlights Conference 2019 Conference Abstract

On the expressivity of linear recursion schemes

Pierre Clairambault
Andrzej Murawski.

We investigate the expressive power of higher-order recursion schemes (HORS) restricted to linear types. Two formalisms are considered: multiplicative additive HORS (MAHORS), which feature both linear function types and products, and multiplicative HORS (MHORS), based on linear function types only. For MAHORS, we establish an equi-expressivity result with a variant of tree-stack automata. Consequently, we can show that MAHORS are strictly more expressive than first-order HORS, that they are incomparable with second-order HORS, and that the associated branch languages lie at the third order of the collapsible pushdown hierarchy. In the multiplicative case, we show that MHORS are equivalent to a special kind of pushdown automata. It follows that any MHORS can be translated to an equivalent first-order MHORS in polynomial time. Further, we show that MHORS generate regular trees and can be translated to equivalent order-0 HORS in exponential time. Consequently, MHORS turn out to have the same expressive power as 0-HORS but they can be exponentially more concise. Our results are obtained through a combination of techniques from game semantics, the geometry of interaction and automata theory.

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MFCS Conference 2019 Conference Paper

On the Expressivity of Linear Recursion Schemes

Pierre Clairambault
Andrzej S. Murawski

We investigate the expressive power of higher-order recursion schemes (HORS) restricted to linear types. Two formalisms are considered: multiplicative additive HORS (MAHORS), which feature both linear function types and products, and multiplicative HORS (MHORS), based on linear function types only. For MAHORS, we establish an equi-expressivity result with a variant of tree-stack automata. Consequently, we can show that MAHORS are strictly more expressive than first-order HORS, that they are incomparable with second-order HORS, and that the associated branch languages lie at the third level of the collapsible pushdown hierarchy. In the multiplicative case, we show that MHORS are equivalent to a special kind of pushdown automata. It follows that any MHORS can be translated to an equivalent first-order MHORS in polynomial time. Further, we show that MHORS generate regular trees and can be translated to equivalent order-0 HORS in exponential time. Consequently, MHORS turn out to have the same expressive power as 0-HORS but they can be exponentially more concise. Our results are obtained through a combination of techniques from game semantics, the geometry of interaction and automata theory.

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CSL Conference 2018 Conference Paper

Fully Abstract Models of the Probabilistic lambda-calculus

Pierre Clairambault
Hugo Paquet

We compare three models of the probabilistic lambda-calculus: the probabilistic Böhm trees of Leventis, the probabilistic concurrent games of Winskel et al. , and the weighted relational model of Ehrhard et al. Probabilistic Böhm trees and probabilistic strategies are shown to be related by a precise correspondence theorem, in the spirit of existing work for the pure lambda-calculus. Using Leventis' theorem (probabilistic Böhm trees characterise observational equivalence), we derive a full abstraction result for the games model. Then, we relate probabilistic strategies to the weighted relational model, using an interpretation-preserving functor from the former to the latter. We obtain that the relational model is also fully abstract.

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CSL Conference 2018 Conference Paper

The True Concurrency of Herbrand's Theorem

Aurore Alcolei
Pierre Clairambault
Martin Hyland
Glynn Winskel

Herbrand's theorem, widely regarded as a cornerstone of proof theory, exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a first-order purely existential formula to that of a finite disjunction. In the general case, it reduces first-order validity to propositional validity, by understanding the structure of the assignment of first-order terms to existential quantifiers, and the causal dependency between quantifiers. In this paper, we show that Herbrand's theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems, thereby exposing the concurrency underlying the computational content of classical proofs. The causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers implicit in proofs. Furthermore concurrent strategies can be composed, yielding a compositional proof of Herbrand's theorem, simply by interpreting classical sequent proofs in a well-chosen denotational model.

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MFCS Conference 2017 Conference Paper

Distributed Strategies Made Easy

Simon Castellan
Pierre Clairambault
Glynn Winskel

Distributed/concurrent strategies have been introduced as special maps of event structures. As such they factor through their "rigid images, " themselves strategies. By concentrating on such "rigid image" strategies we are able to give an elementary account of distributed strategies and their composition, resulting in a category of games and strategies. This is in contrast to the usual development where composition involves the pullback of event structures explicitly and results in a bicategory. It is shown how, in this simpler setting, to extend strategies to probabilistic strategies; and indicated how through probability we can track nondeterministic branching behaviour, that one might otherwise think lost irrevocably in restricting attention to "rigid image" strategies.

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