Nondeterminism in Game Semantics via Sheaves

Tsukada, Takeshi; Ong, C.-H. Luke

doi:10.1109/lics.2015.30

Cited by 22 publications

(33 citation statements)

References 22 publications

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“…7.17. On the other hand, the notion of positionality used in §7.2.3 may be preserved by composition for non-deterministic innocent strategies, in the vein of [HP12,TO15]. We do not know yet how such notions of non-deterministic strategies behave w.r.t.…”

Section: Resultsmentioning

confidence: 99%

Monoidal-closed categories of tree automata

Riba

2020

Math. Struct. Comp. Sci.

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This paper surveys a new perspective on tree automata and Monadic second-order logic (MSO) on infinite trees. We show that the operations on tree automata used in the translations of MSO-formulae to automata underlying Rabin’s Tree Theorem (the decidability of MSO) correspond to the connectives of Intuitionistic Multiplicative Exponential Linear Logic (IMELL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fibered monoidal-closed structure which in particular handles a linear complementation of alternating automata. Moreover, this monoidal structure is actually Cartesian on non-deterministic automata, and an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(–) exponential modality of IMELL. (But this operation is unfortunately not a functor because it does not preserve composition.) Our model of IMLL consists in categories of games which are based on usual categories of two-player linear sequential games called simple games, and which generalize usual acceptance games of tree automata. This model provides a realizability semantics, along the lines of Curry–Howard proofs-as-programs correspondence, of a linear constructive deduction system for tree automata. This realizability semantics, which can be summarized with the slogan “automata as objects, strategies as morphisms,” satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it makes it possible to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata.

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Section: Resultsmentioning

confidence: 99%

Monoidal-closed categories of tree automata

Riba

2020

Math. Struct. Comp. Sci.

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“…It is only recently that some new progress has been made with the discovery that some of these effects could be reconciled in a more refined, more intensional games framework. For instance, in [6,16] innocence is reconciled with non-determinism, and in [15] with probabilities. In [7], innocence is reconciled with concurrency.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, there has been a lot of activity in the past five years or so around games model that do observe the branching information. It is a feature of Hirschowitz's work presenting strategies as presheaves or sheaves on certain categories of cospans [12]; of Tsukada and Ong's work on nondeterministic innocence via sheaves [16]; and of our own line of work presenting strategies as certain event structures [5,7,14].…”

Section: Introductionmentioning

confidence: 99%

Non-angelic Concurrent Game Semantics

Castellan

Clairambault

Hayman

et al. 2018

Lecture Notes in Computer Science

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The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten. We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation. Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism, and still get a category-this partial hiding remains sound up to weak bisimulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences.

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“…We will here mainly be concerned with the numerous variations on arena games. This comprises, e.g., the original dialogue game model of PCF [20,33], Abramsky et al 's model for general references [5], Harmer and McCusker's model for nite nondeterminism [15,17], Laird's model for control operators [24], and the recent model by Tsukada and Ong [37]. We will also brie y consider other models of PCF [6] and of linear logic [10].…”

Section: Introductionmentioning

confidence: 99%

“…E.g., depending on the considered language, various constraints are imposed upon strategies, like innocence or well-bracketing. Further sources of diversity have appeared with recent extensions, e.g., sheaf-based innocence [37], nominal models [32], tensorial logic [30], or concurrent extensions [18,34].…”

Section: Introductionmentioning

confidence: 99%

What's in a game?

Eberhart

Hirschowitz

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly di cult proofs of such basic results as associativity of composition of strategies. We set out to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We nally show that our constructions cover many concrete cases, mainly among the early models [6, 20] and the very recent sheaf-based ones [37].

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Nondeterminism in Game Semantics via Sheaves

Cited by 22 publications

References 22 publications

Monoidal-closed categories of tree automata

Monoidal-closed categories of tree automata

Non-angelic Concurrent Game Semantics

What's in a game?

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