Games and Strategies as Event Structures

Castellan, Simon; Clairambault, Pierre; Rideau, Silvain; Winskel, Glynn

doi:10.23638/lmcs-13(3:35)2017

Cited by 14 publications

(6 citation statements)

References 28 publications

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“…In many situations-in particular where composition is defined by a universal property-these laws only hold up to coherent isomorphism: the resulting structure is that of a bicategory. Bicategories are rife in mathematics and theoretical computer science, arising for instance in algebra [1], [2], semantics of computation [3], [4], datatype models [5], [6], categorical logic [7], [8], and categorical algebra [9], [10], [11].…”

Section: Introductionmentioning

confidence: 99%

A type theory for cartesian closed bicategories (Extended Abstract)

Fiore

Saville

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Index Terms-typed lambda calculus, higher category theory, Curry-Howard-Lambek correspondence, cartesian closed bicategoriesAbstract-We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal property, thereby lifting the Curry-Howard-Lambek correspondence to the bicategorical setting. Our approach is principled and practical. Weak substitution structure is constructed using a bicategorification of the notion of abstract clone from universal algebra, and the rules for products and exponentials are synthesised from semantic considerations. The result is a type theory that employs a novel combination of 2-dimensional type theory and explicit substitution, and directly generalises the Simply-Typed Lambda Calculus. This work is the first step in a programme aimed at proving coherence for cartesian closed bicategories.

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Section: Introductionmentioning

confidence: 99%

A type theory for cartesian closed bicategories (Extended Abstract)

Fiore

Saville

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…As a matter of fact, there is. In [3], Blondeau-Patissier and Clairambault introduced a causal representation of innocent strategies (inspired from concurrent games [10,11] -see also [20]) as a technical tool to prove a positional injectivity theorem for innocent strategies. There a strategy is not a set of plays, but instead gathers diagrams as in Figure 6 in which the trained eye can read exactly the same data as in the resource term s : the model replaces the chronological plays of game semantics with causal structures called augmentations, of which the plays are just particular linearizations.…”

Section: Resource Terms As Augmentationsmentioning

confidence: 99%

“…Intuitively, a state is a sub-tree of A but where each branch may be explored multiple times -such structures were first introduced by Boudes [6] under the name thick subtrees. Here, by analogy with concurrent games [10], we call them configurations: and the display map ∂…”

Section: Constructions We Writementioning

confidence: 99%

“…We rely on a representation of plays quotienting out Opponent's scheduling, recently introduced by the first two authors [3]. This was inspired from concurrent games [10] -similar causal structures existed before, first suggested in [16], and fleshed out more in [20]. In [3], plays are replaced by so-called augmentations, which augment valid states of the game with causal constraints imposed by the program.…”

Section: Introductionmentioning

confidence: 99%

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Strategies as Resource Terms, and their Categorical Semantics

Blondeau-Patissier¹,

Clairambault²,

Auclair³

2023

Preprint

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As shown by Tsukada and Ong, normal (extensional) simply-typed 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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“…The broader context for this work is the recent occurrence of bicategories in the semantics of programming languages. Bicategories of profunctors are now prominent in the analysis of linear logic and the λ -calculus ( [7,11,22]), and game semantics employs a variety of span-like constructions that compose weakly ( [31,3]). These models have also influenced the development of 2-dimensional type theories ( [8,34]).…”

Section: Bicategorical Modelsmentioning

confidence: 99%

Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

Paquet,

Saville

2023

Electron. Proc. Theor. Comput. Sci.

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Premonoidal categories and Freyd categories provide an encompassing framework for the semantics of call-by-value programming languages. Premonoidal categories are a weakening of monoidal categories in which the interchange law for the tensor product may not hold, modelling the fact that effectful programs cannot generally be re-ordered. A Freyd category is a pair of categories with the same objects: a premonoidal category of general programs, and a monoidal category of 'effect-free' programs which do admit re-ordering.Certain recent innovations in semantics, however, have produced models which are not categories but bicategories. Here we develop the theory to capture such examples by introducing premonoidal and Freyd structure in a bicategorical setting. The second dimension introduces new subtleties, so we verify our definitions with several examples and a correspondence theorem-between Freyd bicategories and certain actions of monoidal bicategories-which parallels the categorical framework.

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Games and Strategies as Event Structures

Cited by 14 publications

References 28 publications

A type theory for cartesian closed bicategories (Extended Abstract)

A type theory for cartesian closed bicategories (Extended Abstract)

Strategies as Resource Terms, and their Categorical Semantics

Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

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