Asynchronous Template Games and the Gray Tensor Product of 2-Categories

Melliès, Paul-André

doi:10.1109/lics52264.2021.9470758

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…The work is specific to concurrent games, but the algebraic perspective could facilitate the transfer of ideas to other kinds of games, e.g. [12].…”

Section: Discussionmentioning

confidence: 99%

Bi-invariance for Uniform Strategies on Event Structures

Paquet¹

2023

Electronic Notes in Theoretical Informatics and Computer Science

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A recurring problem in game semantics is to enforce uniformity in strategies. Informally, a strategy is uniform when the Player's behaviour does not depend on the particular indexing of moves chosen by the Opponent. In game semantics, uniformity is used to define a resource modality !, that can be exploited for the semantics of programming languages. In this paper we give a new account of uniformity for strategies on event structures. This work is inspired by an older idea due to Melli\`es, that uniformity should be expressed as "bi-invariance" with respect to two interacting group actions. We explore the algebraic foundations of bi-invariance, adapt this idea to the language of event structures and define a general notion of uniform strategy in this context. Finally we revisit an existing approach to uniformity, and show how this arises as a special case of our constructions.

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“…The work is specific to concurrent games, but the algebraic perspective could facilitate the transfer of ideas to other kinds of games, e.g. [12].…”

Section: Discussionmentioning

confidence: 99%

Bi-invariance for Uniform Strategies on Event Structures

Paquet¹

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…This paper develops abstract categorical notions, but these are intended to be immediately practical. Specifically, the literature contains no satisfying account of call-by-value languages in bicategories of games ( [31,3]), spans [6], or profunctors [7], and this work offers a technical basis to fill that gap. Our next steps will be in this direction.…”

Section: Discussionmentioning

confidence: 99%

“…Bicategories have recently found prominence as models of computational processes: see e.g. [31,6,12,1]. We illustrate this with two simple examples: spans of sets, and graded monads.…”

Section: Bicategorical Modelsmentioning

confidence: 94%

“…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%

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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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“…A detailed comparison of this class of asynchronous automaton with the more liberal notion of automaton with concurrency relations [DS02, GM10] appears in [Mor05]. The ID property implies that every NFDA over F B Σ /D satisfies the cube property (CP) discussed in [HM00] and thus defines what is called an asynchronous graph in [MM07,Mel21]. We leave for future work the connection with Zielonka's theorem which is based on a more concrete and process-based notion of concurrent automaton [Zie87, BM06].…”

Section: Example 211 (Product Automata) the Product Of Two Finitary U...mentioning

confidence: 99%

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

Melliès¹,

Zeilberger²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We begin by explaining how any context-free grammar encodes a functor of operads from a freely generated operad into a certain "operad of spliced words". This motivates a more general notion of CFG over any category $C$, defined as a finite species $S$ equipped with a color denoting the start symbol and a functor of operads $p : Free[S] \to W[C]$ into the operad of spliced arrows in $C$. We show that many standard properties of CFGs can be formulated within this framework, and that usual closure properties of CF languages generalize to CF languages of arrows. We also discuss a dual fibrational perspective on the functor $p$ via the notion of "displayed" operad, corresponding to a lax functor of operads $W[C] \to Span(Set)$. We then turn to the Chomsky-Sch\"utzenberger Representation Theorem. We describe how a non-deterministic finite state automaton can be seen as a category $Q$ equipped with a pair of objects denoting initial and accepting states and a functor of categories $Q \to C$ satisfying the unique lifting of factorizations property and the finite fiber property. Then, we explain how to extend this notion of automaton to functors of operads, which generalize tree automata, allowing us to lift an automaton over a category to an automaton over its operad of spliced arrows. We show that every CFG over a category can be pulled back along a ND finite state automaton over the same category, and hence that CF languages are closed under intersection with regular languages. The last important ingredient is the identification of a left adjoint $C[-] : Operad \to Cat$ to the operad of spliced arrows functor, building the "contour category" of an operad. Using this, we generalize the C-S representation theorem, proving that any context-free language of arrows over a category $C$ is the functorial image of the intersection of a $C$-chromatic tree contour language and a regular language.

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Asynchronous Template Games and the Gray Tensor Product of 2-Categories

Cited by 4 publications

References 21 publications

Bi-invariance for Uniform Strategies on Event Structures

Bi-invariance for Uniform Strategies on Event Structures

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

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

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