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Comparing Hierarchies of Types in Models of Linear Logic
, 2003
"... We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distri ..."
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Cited by 6 (3 self)
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We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of backandforth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: errorfree vs. erroraware, alternated vs. nonalternated, backtracking vs. repetitive, uniform vs. nonuniform.
From Asynchronous Games to Concurrent Games
, 2008
"... Game semantics was introduced in order to capture the dynamic behaviour of proofs and programs. In these semantics, the interaction between a program and its environment is modeled by a series of moves exchanged between two players in a game. Every program thus induces a strategy describing how it r ..."
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Cited by 1 (0 self)
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Game semantics was introduced in order to capture the dynamic behaviour of proofs and programs. In these semantics, the interaction between a program and its environment is modeled by a series of moves exchanged between two players in a game. Every program thus induces a strategy describing how it reacts when it is provided information by its environment. Traditionally, strategies considered in game semantics are alternating: the two protagonists play a move one after the other. This property is very natural when modeling sequential programming languages, but is not desirable for programs with concurrent features, since interactions cannot be synchronized globally anymore. Extending fundamental notions of game semantics to a nonalternating setting is far from being straightforward and requires to deeply rethink the definition of strategies. Recently, a series of interactive models, such as concurrent games where strategies are closure operators, were introduced in order to give denotational semantics of programming languages or logics with concurrent features. However, these models were poorly connected with traditional game semantics. We show here that asynchronous games, which combine true concurrency and game semantics, can be used to provide a precise link between these two kind of interactive semantics, thus laying foundations for game semantics of concurrent systems. 1
On dialogue games and coherent strategies ˚
"... We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended w ..."
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We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature.
Realizability for Peano Arithmetic with Winning Conditions in HON Games
, 2013
"... We build a realizability model for Peano arithmetic based on winning conditions for HON games. First we define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winni ..."
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We build a realizability model for Peano arithmetic based on winning conditions for HON games. First we define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winning condition corresponding to the formula. Finally we apply this to Peano arithmetic with relativized quantifications and give the example of witness extraction for Π 0 2formulas.