Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88, pages 221--260. North Holland, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....John Lamping [12] found the algorithm that Levy specified. Then Gonthier, Abadi, and Levy [9, 10] made a lovely discovery: they gave a denotational semantics to Lamping s algorithm, called context semantics, and showed that it was equivalent to Jean Yves Girard s geometry of interaction (GoI) [8]. Girard s GoI is an abstract mathematical notion, employing a lot of higher mathematics that theoretical computer scientists are not accustomed to using: Hilbert spaces, Supported by NSF Grants CCR 0228951 and EIA 9806718, and also by the Tyson Foundation. C # algebras, and more. In contrast, ....

J.-Y. Girard. Geometry of interaction I: Interpretation of system F. In C. Bonotto, R. Ferro, S. Valentini, and A. Zanardo, editors, Logic Colloquium '88, pages 221--

....wave style GoI models is richer than that induced by game models. Keywords: linear) graph model, traced monoidal category, weak linear category, categorical geometry of interaction. Introduction In [Abr96] Abramsky provides a categorical generalization of Girard s Geometry of Interaction (GoI) [Gir89], embracing previous axiomatic approaches, such as that based on dynamic algebras [DR93,DR95] and the one in [AJ94] This generalization is based on traced monoidal categories, JSV96] and it consists in building a compact closed category (GoI category) from a traced symmetric monoidal ....

J.-Y.Girard. Geometry of Interaction I: Interpretation of System F, Logic Colloquium'88, R.Ferro et al. eds., North Holland, 221--260.

....interact in an appropriate fashion. Such categories are called linearly or weakly distributive, a notion due to Cockett and Seely [CS97, BCST96] Linearly distributive categories are the appropriate framework for considering a speci c logical system known as linear logic, introduced by Girard [Gir87, Gir89]. For a brief exposition of linear logic, see Appendix B. As we will see, the re ned logical connectives of linear logic will be used to express the entanglements of our system. There is a very geometric or graphical calculus for representing morphisms in polycategories, which was introduced by ....

J.-Y. Girard. Geometry of interaction i: Interpretation of system F. In Logic Colloquium '88. North Holland, 1989.

....significant today, since interaction often appears to be more visible and even more important than computation. Important progresses in logic are also leading to interactive and dynamical models. Two major examples are Geometry of Interaction and Games Semantics. The Geometry of Interaction [5], which arose from Linear Logic, interprets normalization (computation) as a flow of information circulating around a net. Games Semantics interprets computation as a dialog between two parties, the program (player) and the environment (opponent) each one following its own strategy . Games ....

J.-Y. Girard. Geometry of interaction i: Interpretation of system f. In Z. A. Ferro R.m Bonotto C., Valentini S., editor, Logic Colloquium 88, pages 221--260. North Holland, 1989.

....strong and the design of a nice cut elimination procedure was complicated. Therefore, proofnets were introduced as a more flexible representation of proofs [8, 13] Further, Geometry of Interaction (GoI) developed the idea that the reduction of proofs can be seen as a local interaction process [5, 6]. Its intensional features provide a mediating Purgatory between the Heaven of denotational semantics, and the Hell of operational semantics. GoI was simplified in the geometry of optimal A reduction by Gonthier, Abadi and Lvy [9, 10] in the context of the MELL fragment. They reduced Hilbert ....

....Definition 4 (Proofnets with boxes) The inductive encoding [ ofprooftrees into proofnets is shown in Figure 2. A port of the proofnet [r] is a proofnet wire corresponding to a port of r. The reduction of cuts in proofs annihilates reciprocal links ( and , or and P) Geometry of interaction [5, 6] provides a mathematical framework for this phenomenon, where a semantics (see [9, 10, 15] is defined that is meant to be preserved by reduction. We now define such a semantics, and use it as a starting point to define a proofnet syntax containing both the meaning of proofs, and the handling of ....

J.-Y. Girard. Geometry of interaction I: Interpretation of system F. In Logic Colloquium '88, pages 221-260. North-Holland, 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. ....

[Article contains additional citation context not shown here]

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88. North Holland, 1989. 32

.... in [1,22, 27] Danos, Herbelin and Regnier have shown that the dynamics of those games models correspond to the operation of two already known abstract machines for reduction [9] and Baillot [7] has given details of the correspondence between the model of [1] and Girard s Geomety of Interaction [17]. More recently, considerable advances have been made by considering the effect of relaxing the conditions imposed on the strategies described inthe present paper. It has been discovered that by relaxing the innocence condition gives a fully abstract model of Idealised Algol, a prototypical ....

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloquium 88, pages 221--260. North Holland, 1989.

....a strategy in A(B is f : M Such a function can be written as a matrix f = 10 A f 1;2 : M B f 2;2 : M For example, the twist map corresponds to the matrix 0 id where 0 is the everywhere unde ned partial function. Compare the interpretation of axiom links in [Gir89a]. The strategy induced by this function is the copy cat strategy as de ned in [AJ94a] As a set of positions, this strategy is de ned by: idA = fs 2 P A(A j s 1 = s 2g: In process terms, this is a bi directional one place bu er [Abr94] These copycat strategies are the identity morphisms ....

....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. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88, pages 221-260. North Holland, 1989.

....histories ab and ba, taking account of the order in which the events a and b occur. In a true concurrency model, if the events a and b are causally idependent of each other, we can identify these two histories [WN95] The purpose of proof nets [Gir87, Gir95a] and Geometry of Interaction [Gir89, Gir90, Gir95, DR93, DR95] was to nd a more intrinsic representation of proofs in which the spurious ordering of rules imposed by sequent calculus was factored out. This is directly akin to the issue of representing true concurrency; and we can indeed see proof nets as a parallel syntax for proof theory [Gir95a] 5 ....

....each position it is exactly one player s turn to move, leads to over speci cation of the sequential order of events, with bad formal consequences. By contrast, we nd an authentic semantic account of the multiplicatives (and, to some extent, of the exponentials) in the Geometry of Interaction [Gir89, Gir90, Gir95, DR93, DR95]. Here the local, asynchronous character of the multiplicatives, suggested by the intrinsic, geometrical representation of multiplicative proofs as proof nets, is turned into a concrete form of symbolic dynamics. The basic idea of Geometry of Interaction is that multiplicative proofs are ....

[Article contains additional citation context not shown here]

J.-Y. Girard, Geometry of Interaction I: Interpretation of System F. In Logic Colloquium '88, ed. R. Ferro et al. North-Holland 1989, pp. 221-260.

....tokens around a network, rather than by graph rewriting. ffl There is a normal form analogous to the Kleene normal form in recursion theory: the entire process of cut elimination is described by the iterations of a single operator. Girard has implemented this programme in a sequence of papers [Gir89b, Gir89a, Gir88a], using the formalism of C algebras. While the ideas are highly original and striking, and the technical execution must be considered a tour de force, some desiderata remain. 1. Can one give a more systematic account, making clear what structure is really needed to carry out the ....

....m( 1 ; 2 ) m( 3 ; 4 ) oe = 1 2 3 4 4 3 2 1 p b;oe (m(a; b) m(c; d) m(d; c) m(b; a) We can now state the main result of this section. This result can be seen as giving a precise formulation of the idea of genericity or communication without understanding discussed at the end of [Gir89a]. A proof must analyze the structure of the data through which it communicates with its environment up to a fixed depth determined by its type; beyond that, it merely permutes data to achieve a certain flow of information, without any regard as to its structure. Theorem 4 Let f be the GI ....

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88. North Holland, 1989.

....be produced on the output port. In other words, the operationally correct trace operator should yield the empty relation. Higher order structure from trace: Geometry of Interaction. The Geometry of interaction program was invented by Girard in his analysis of the fine structure of cut elimination [44, 45]. His basic insight was that higher order structure could be understood in terms of trace but this understanding was hidden in the mathematical setting Hilbert spaces and traces of operators that he used. In [70] Joyal, Street, and Verity and independently Abramsky [3] see also [6] gave the ....

....product of presheaves. In this richer world of constructions bisimulation would appear to be the more suitable equivalence. 8.4. 2 Higher order Dataflow via Geometry of Interaction The Geometry of interaction program was invented by Girard in his analysis of the fine structure of cut elimination [44, 45]. His basic insight was that higher order structure could be understood in terms of trace but this understanding was hidden in the mathematical setting Hilbert spaces and traces of operators that he used. In [70] Joyal, Street, and Verity and independently Abramsky [3] see also [6] gave the ....

J.-Y. Girard. Geometry of interaction I: interpretation of system F. In Ferro et. al., editor, Proceedings Logic Colloquium 88, pages 221--260. North-Holland, 1989. 134

....its behalf. It should be said that this programme runs somewhat counter to that advocated by Girard. He has adopted the methodological principle of avoiding the bureaucracy of syntax [Gir89b] aiming instead for a geometrical view of computation, exemplified by the Geometry of Interaction [Gir89a], which interprets cut elimination in Linear Logic by iterations of operators in C algebras. From that perspective, what we are seeking to do here might be seen as a retrogressive step. However, we see our work as complementary to Girard s. By giving a simple, concrete computational ....

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88. North Holland, 1989.

.... to an important classi cation of some features of sequential programming languages such as control, or references (see [3] for a survey) The whole approach received many insights from the developments of linear logic, and in particular of the geometry of interaction interpretation of linear logic [14, 1]. For instance, Lamarche s decomposition of the function space in the model of sequential algorithms allowed me to give a more symmetric presentation of ane sequential algorithms, as pairs of two functions, one from input data (or strategies, in game theoretic terms) to output data, the other from ....

J.-Y. Girard, Geometry of interaction I: interpretation of system F, in Proc. Logic Colloquium '88, 221-260, North Holland (1989).

....a further exponential overhead with respect to Lamping s solution. Remark. The previous comment is not meant at all to dimimish [GAL92a] whose theoretical relevance is out of question. By pointing out the relations between optimal reductions, Linear Logic [Gi86] and the Geometry of Interaction [Gi88a, Gi88b], it provided an essential breakthrough in the topic, suggesting new and absolutely innovative perspectives. The interested reader might want to consult [GAL92b, As94, AL93b, ADLR94] In [As94] we explained that the well known and crucial problem of accumulation of control operators [GAL92a] ....

J. Y. Girard. Geometry of Interaction I: Interpretation of system F. In Ferro, Bonotto, Valentini and Zanardo eds., Logic Colloquium 88, 221--260. North Holland. 1988.

....histories ab and ba, taking account of the order in which the events a and b occur. In a true concurrency model, if the events a and b are causally idependent of each other, we can identify these two histories [WN95] The purpose of proof nets [Gir87, Gir95a] and Geometry of Interaction [Gir89, Gir90, Gir95, DR93, DR95] was to nd a more intrinsic representation of proofs in which the spurious ordering of rules imposed by sequent calculus was factored out. This is directly akin to the issue of representing true concurrency; and we can indeed see proof nets as a parallel syntax for proof theory [Gir95a] 5 ....

....each position it is exactly one player s turn to move, leads to over speci cation of the sequential order of events, with bad formal consequences. By contrast, we nd an authentic semantic account of the multiplicatives (and, to some extent, of the exponentials) in the Geometry of Interaction [Gir89, Gir90, Gir95, DR93, DR95]. Here the local, asynchronous character of the multiplicatives, suggested by the intrinsic, geometrical representation of multiplicative proofs as proof nets, is turned into a concrete form of symbolic dynamics. The basic idea of Geometry of Interaction is that multiplicative proofs are ....

[Article contains additional citation context not shown here]

J.-Y. Girard, Geometry of Interaction I: Interpretation of System F. In Logic Colloquium '88, ed. R. Ferro et al. North-Holland 1989, pp. 221-260.

....was done while this author was at Institut de Mathematiques de Luminy (CNRS) Marseille, France. Partially supported by TMR LINEAR research network. 2 P. Baillot and M. Pedicini Geometry of interaction and complexity 1. Introduction Geometry of interaction (goi) was introduced by Girard ([Gir88a, Gir89]) as a semantics of computation which: on the one hand, in contrast to denotational semantics interprets explicitly the dynamics of computation and handles finite objects, on the other hand, expresses this dynamics by more mathematical means than syntactical rewriting. Various frameworks ....

....semantics interprets explicitly the dynamics of computation and handles finite objects, on the other hand, expresses this dynamics by more mathematical means than syntactical rewriting. Various frameworks have been used to describe goi models, including bounded operators on Hilbert spaces ([Gir88a, DR95]) partial applications ( Dan90, Reg92] and algebras of clauses ( Gir95] This latter point of view is the one we will adopt here. In these models, the operation corresponding to the normalization process is called execution. It is not defined on all operators and su#cient conditions have been ....

[Article contains additional citation context not shown here]

J.-Y. Girard. Geometry of interaction I: an interpretation of system F . In Ferro and al, editors, Proceedings of A.S.L. Meetings, Padova, 1988. North-Holland.

No context found.

Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88, pages 221--260. North Holland, 1989.

No context found.

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88. North Holland, 1989.

No context found.

J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro et al., editor, Logic Colloquium 88. North Holland, 1989.

No context found.

J.-Y. Girard. Geometry of interaction I: Interpretation of system F. In Logic Colloquium '88, Amsterdam, 1989. North-Holland.

No context found.

J.-Y. Girard. Geometry of interaction i: Interpretation of system f. In Z. A. Ferro R.m Bonotto C., Valentini S., editor, Logic Colloquium 88, pages 221-260. North Holland, 1989.

No context found.

J.-Y.Girard. Geometry of Interaction I: Interpretation of System F, Logic Colloquium'88, R.Ferro et al. eds., North Holland, 221-260.

No context found.

J.-Y.Girard. Geometry of Interaction I: Interpretation of System F, Logic Colloquium '88, R.Ferro et al. eds., North Holland, 221-260.

No context found.

J.-Y. Girard, Geometry of Interaction I: Interpretation of System F, in: Logic Colloquium '88, ed. R. Ferro, et al. North-Holland, pp. 221-260, 1989.

No context found.

J.-Y. Girard. Geometry of interaction I: Interpretation of system F. In Ferro and al., editors, Logic Colloquium

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC