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Least and Greatest Fixpoints in Game Semantics
, 2009
"... We show how solutions to many recursive arena equations can be computed in a natural way by allowing loops in arenas. We then equip arenas with winning functions and total winning strategies. We present two natural winning conditions compatible with the loop construction which respectively provide i ..."
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We show how solutions to many recursive arena equations can be computed in a natural way by allowing loops in arenas. We then equip arenas with winning functions and total winning strategies. We present two natural winning conditions compatible with the loop construction which respectively provide initial algebras and terminal coalgebras for a large class of continuous functors. Finally, we introduce an intuitionistic sequent calculus, extended with syntactic constructions for least and greatest fixed points, and prove it has a sound and (in a certain weak sense) complete interpretation in our game model.
Game semantics for firstorder logic
, 2010
"... We refine HO/N game semantics with an additional notion of pointer (mupointers) and extend it to firstorder classical logic with completeness results. We use a Church style extension of Parigot’s lambdamucalculus to represent proofs of firstorder classical logic. We present some relations with ..."
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Cited by 4 (0 self)
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We refine HO/N game semantics with an additional notion of pointer (mupointers) and extend it to firstorder classical logic with completeness results. We use a Church style extension of Parigot’s lambdamucalculus to represent proofs of firstorder classical logic. We present some relations with Krivine’s classical realizability and applications to type isomorphisms.
Totality in arena games
, 2009
"... We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We ..."
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Cited by 2 (2 self)
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We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We discuss how this theorem relates to normalization of linear head reduction in simplytyped λcalculus, leading us to a semantic realizability proof à la Kleene of our theorem. We then present another proof of a more combinatorial nature. Finally, we discuss the exact class of strategies to which our theorems apply.
Under consideration for publication in Math. Struct. in Comp. Science Currystyle Type Isomorphisms and Game Semantics
, 2007
"... Currystyle system F, i.e. system F with no explicit types in terms, can be seen as a core presentation of polymorphism from the point of view of programming languages. This paper gives a characterisation of type isomorphisms for this language, by using a game model whose intuitions comes both from ..."
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Currystyle system F, i.e. system F with no explicit types in terms, can be seen as a core presentation of polymorphism from the point of view of programming languages. This paper gives a characterisation of type isomorphisms for this language, by using a game model whose intuitions comes both from the syntax and from the game semantics universe. The model is composed of: an untyped part to interpret terms, a notion of arena to interpret types, and a typed part to express the fact that an untyped strategyσplays on an arena A. By analysing isomorphisms in the model, we prove that the equational system corresponding to type isomorphisms for Currystyle system F is the extension of the equational system for Churchstyle isomorphisms with a new, nontrivial equation: ∀X.A≃ε A[∀Y.Y/X] if X appears only positively in A. 1.