Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....function (see Fig. 1(iii) 22 [ 21 ; 11 ) 12 ; where [ denotes union of graph relations, and ( 11 ) denotes n 0 ( 11 ) The above formula for computing the application is essentially the Execution Formula from Girard s Geometry of Interaction ([Gir89]) 5 Q Q Q (ii) 11 12 22 21 (i) t; t (n) iii) Fig. 1. Geometrical description of linear application. The de nition of the operation on our applicative structure is quite simple. The operation is intended to produce, from a single copy of , ....

J.Y.Girard. Towards a Geometry of Interaction, Contemporary Mathematics 92, 1989, 69-108.

....also be of interest as a worked example of an application of s substructural type system. Indeed, the results of this paper provided the initial spur for the calculus and the logic BI of the author and David Pym [17] This spur came about originally from attempts [8, 9] to apply linear logic [5, 6], which had brought to the fore the signi cance of restricting Contraction; the speci c requirements of the application, however, required a di erent approach to controlling the structural rules. The central elements of the resulting approach, as represented in or BI, can and should be ....

J.-Y. Girard. Towards a geometry of interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 69-108. American Mathematical Society, 1989.

....fashion. ffl Game Semantics mediates between traditional operational and denotational semantics, combining the good structural properties of one with the ability to model computational fine structure of the other. This is similar to the motivation for the Geometry of Interaction programme [Gir89b, Gir89a, AJ92a]; indeed, we shall exhibit strong connections between our semantics and the Geometry of Interaction. 1.1 Overview of Results Blass has recently described a Game semantics for Linear Logic [Bla92b] This has good claims to be the most intuitively appealing semantics for Linear Logic presented so ....

....play. Thus such a strategy is induced by a partial function on the set of moves in the game. The interpretation of proofs in MLL MIX by strategies, when analysed in terms of these underlying functions on moves, turns out to be very closely related to the Geometry of Interaction interpretation [Gir89b, Gir89a, Gir88]. The contents of the reminder of this paper are as follows. Section 2 reviews MLL MIX. Section 3 describes our game semantics for MLL MIX. Section 4 is devoted to the proof of the Full Completeness Theorem. Section 5 outlines how our semantics can be extended to full Classical Linear Logic. ....

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J.-Y. Girard. Towards a geometry of interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.

....a coin in my pocket, and if you give me another, then I will have enough to buy a chocolate. However, coin 6 coin choc because a single coin is not enough to buy a chocolate. It is here that the reader will detect similarity with Girard s Marlboro s and Camels reading of linear logic [21]. However, the divergences are both more interesting than the similarities and illustrate how great is the difference between BI and linear logic. First, and foremost, Girard s reading is about proofs asactions where, for example, choc: the (type of the) act of buying a chocolate. In ....

J.-Y. Girard. Towards a geometry of interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 69--108. American Mathematical Society, 1989.

....every heap. To see why Weakening fails, consider (x 2, 3)# (y 4, 5) For this to be true the current heap must have size two, and 2, 3) cannot then hold because it requires the current heap to have size one. The importance of restricting Contraction was brought to the fore by linear logic [12, 13]. But it is important to realize that BI takes a very di#erent approach to the surrounding additive connectives. To see this, consider that P #Q Q always holds in linear logic, using the decomposition Q = P #Q and the rule of Dereliction for . But here, 1, 2) # false false ....

....be interesting to attempt to apply these ideas in related situations where aliasing is prevalent, such as # calculus or object calculi. In the linear logic literature there have been numerous hints, suggesting that substructural logic can be used to specify and reason about actions locally (e.g. [13, 21]) While this proposal was tantalyzing, it has not subsequently been developed very far, certainly not as far as a program logic for pointers. Encodings of the semantics of imperative languages, e.g. 9] are important and useful, but fall well short of program logic. The results of this paper ....

Girard, J.-Y. Towards a geometry of interaction. In Categories in Computer Science and Logic (1989), American Mathematical Society, pp. 69--108. Contemporary Mathematics Volume 92.

....a coin in my pocket, and if you give me another, then I will have enough to buy a chocolate. However, coin 6 coin choc because a single coin is not enough to buy a chocolate. It is here that the reader will detect similarity with Girard s Marlboro s and Camels reading of linear logic [20]. However, the divergences are both more interesting than the similarities and illustrate how great is the difference between BI and linear logic. First, and foremost, Girard s reading is about proofs asactions where, for example, choc: the (type of the) act of buying a chocolate. 26 In ....

J.-Y. Girard. Towards a geometry of interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 69--108. American Mathematical Society, 1989.

....analogies between game semantics and concurrency semantics, and [Abr94] for other aspects. We now describe composition in terms of the functions inducing strategies. Say we have f : A B; g : B C. We want to nd h such that f ; g = h . We shall compute h by the execution formula [Gir89b, Gir89a, Gir88]. Before giving the formal de nition, let us explain the idea, which is rather simple. We want to hook the strategies up so that Player s moves in B under get turned into Opponent s moves in B for , and vice versa. Consider the following picture: 11 ....

Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69-108. American Mathematical Society, 1989.

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Jean-Yves Girard. Towards a geometry of interaction. In Categories in computer science and logic (Boulder, CO, 1987), pages 69-108. Amer. Math. Soc., Providence, RI, 1989.

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Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.

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J.-Y. Girard. Towards a geometry of interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.

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Jean-Yves Girard. Towards a geometry of interaction. Contemporary Mathematics, 92:69--108.

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Jean-Yves Girard. Towards a geometry of interaction. Contemporary Mathematics, 92:69--108.

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Girard, J.-Y. Towards a geometry of interaction. In Categories in Computer Science and Logic (1989), American Mathematical Society, pp. 69-108. Contemporary Mathematics Volume 92.

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J.-Y. Girard. Towards a geometry of interaction. Contemp. Math., 92:69-108, 1989.

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J.-Y. Girard. Towards a geometry of interaction. In: Contemporary Math. 92, Amer. Math. Soc., 1989. 69-108.

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J.-Y. Girard. Towards a geometry of interaction. In: Contemporary Math. 92, Amer. Math. Soc., 1989. 69-108.

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J.-Y. Girard. Towards a geometry of interaction. In: Contemporary Math. 92, Amer. Math. Soc., 1989. 69-108.

No context found.

Jean-Yves Girard, Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108, American Mathematical Society, 1989.

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Jean-Yves Girard. Towards a geometry of interaction. Contemporary Mathematics, 92:69--108.

No context found.

Jean-Yves Girard. Towards a geometry of interaction. Contemporary Mathematics, 92:69--108.

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Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69 - 108. American Mathematical Society, 1989.

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J.-Y. Girard, Towards a geometry of interaction, in: John W. Gray, Andre Scedrov (eds.), Categories in computer science and logic, Proceedings of the AMS-IMS-SIAM joint summer research conference held June 14-20, 1987 (University of Colorado, Boulder) with support from the National Science Foundation, Providence 1989 [Contemporary Mathematics 92], p. 69--108

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J.-Y. Girard. Towards a geometry of interaction. In John W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69-108, Providence, Rhode Island, 1989. American Mathematical Society.

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Girard, J.-Y. Towards a geometry of interaction. In Categories in Computer Science and Logic (1989), American Mathematical Society, pp. 69-108. Contemporary Mathematics Volume 92.

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J.-Y. Girard. Towards a geometry of interaction. In Categories in Computer Science and Logic, Contemporary Mathematics 92, pages 69--108. AMS Publications, 1989.

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