Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloquium 88, volume 127 of Studies in Logic and the Foundations of Mathematics, pages 221-260. North Holland Publishing Company, Amsterdam, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of an interthread communication model, and allow to incorporate speci c strategies for dividing the computation of the execution path into smaller tasks. 1 Introduction This paper proposes novel parallel implementation techniques for the calculus based on the geometry of interaction (GoI) [6, 5, 7]. GoI based implementation is quite di erent from other techniques: it uses a graph representation of each term, from which its value is derived by performing path computations, which can be done locally and asynchronously. This encompasses both reduction and the variable substitution ....

....and in Sect. 8 we conclude with some comments about implementing these ideas. The long version of this paper contains proofs and examples of execution. 2 Background Our treatment of the theory here is necessarily super cial; for a more thorough introduction to GoI (including VR and DVR) see [6, 5, 2, 3]. The Language. We will use a typed calculus with a single base type. The syntax of our terms (ranged over by t; u; v) will be (with x; y; z variables, n an integer constant and S the successor function) t : n j S j x j uv j x:u The typing rules are the standard ones: if with x : we ....

[Article contains additional citation context not shown here]

Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloquium 88, volume 127 of Studies in Logic and the Foundations of Mathematics, pages 221-260. North Holland Publishing Company, Amsterdam, 1989.

....of an inter thread communication model, and allow to incorporate speci c strategies for dividing the computation of the execution path into smaller tasks. 1 Introduction This paper proposes novel parallel implementation techniques for the calculus based on the geometry of interaction (GoI) [6, 5, 7]. GoI based implementation is quite di erent from other techniques: it is based on a graph representation of each term, from which its value is derived by performing path computations, which can be done locally and asynchronously. This encompasses both reduction and the variable substitution ....

Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloquium 88, volume 127 of Studies in Logic and the Foundations of Mathematics, pages 221-260. North Holland Publishing Company, Amsterdam, 1989.

....the additives and the interaction between additives and exponentials by means of weights [7] We describe the interpretation by a token machine which allows us to recover the usual MELL case by forgetting all the additive information. The Geometry of Interaction (GoI) introduced by Girard [5], is an interpretation of proofs (programs) by bideterministic automatons, turning the global cut elimination steps ( reduction) into local transitions [4] One of the main results of the MELL GoI is that it gives an algebraic characterization of the persistent paths of a proof, that is the ....

....soundness result for LL proofs of 1 1. Proofs of 1 1 give an encoding of booleans since there are exactly two normal proofs of this sequent in LL. The restriction to these boolean results is very drastic but sucient, from a computational point of view, to distinguish di erent results (see [5] for a longer discussion) Without clearly decomposing LL cut elimination into these two steps ( reduction and extraction) the study of the modi cations of the interpretation during LL reduction would be very complicated and the results very dicult to set out and to prove. Moreover this precise ....

Jean-Yves Girard. Geometry of interaction I: an interpretation of system F . In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88. North-Holland, 1988.

....coming from categorical models of the calculus; and SKI combinators coming from combinatory logic. In [9] the second author gave a different kind of compilation of a simple calculus based functional programming language coming directly from Girard s Geometry of Interaction semantics [7, 6]; a semantics of computation capturing the actual reduction process. One way of understanding the Geometry of Interaction is to think of a program represented as a graph. The meaning of a program is then given by the set of paths in the graph. Of course not all paths, but some particular ones ....

Jean-Yves Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloquium 88, Amsterdam, 1989. North Holland.

.... ij 2 A, such that a coefficient e ij ( S) S 0 ) iff there is a successful run starting in the i th conclusion with the initial pair ( S) and ending in the j th conclusion with the final pair ( S 0 ) This is Girard s execution formula rephrased as appropriate in our framework; see [8, 9, 10] for the original presentation. This may seem a formidable thing to compute, but remember that all actions are finitely representable, and then it is easy to come up with a finitary formulation of this ex(R) Note also that by bi determinicity ex(R) is a self dual action matrix. Now, take note, ....

Jean-Yves Girard. Geometry of interaction I: an interpretation of system F . In Ferro & al., editor, Proceedings of A.S.L. Meetings. North-Holland, 1988.

....Luminy Marseille October 12, 1996 Abstract This note defines a new graphical local calculus, directed virtual reductions. It is designed to compute Girard s execution formula EX, an invariant of closed functional evaluation obtained from the geometry of interaction interpretation of calculus [4]. The calculus is obtained by synchronizing another graphical local calculus presented in local and asynchronous beta reduction : virtual reductions [3] This synchronization makes it easier to mechanize than general virtual reductions. In undirected virtual reductions the consistency of the ....

Jean-Yves Girard. Geometry of interaction I: an interpretation of system F . In Ferro & al., editor, Proceedings of A.S.L. Meetings. North-Holland, 1988. 12

....principle , is fundamental and very fruitful. The Curry Howard isomorphism establishes a deep correspondence between the notion of proof and the notion of computation. It is this correspondence that leads to various semantics of proofs , the most recent one being Girard s geometry of interaction [10]. However, a discussion of this subject would take us beyond the scope of this paper, and we will restrict ourselves to a (thorough) discussion of the notion of proof normalization. The idea of proof normalization goes back to Gentzen ( 6] 1935) Gentzen noted that (formal) proofs can contain ....

Jean-Yves Girard. Geometry of interaction I: Interpretation of system F. In Ferro Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221--260. North-Holland, Elsevier, 1989.

....for neural networks as an inverse problem which may be solved (or optimised) in an iterative way to proof reduction mechanisms in logic. In particular, Girard s approach to proof normalisation in linear logic is also based on the Neumann series to calculate the inverse of a proof operator [Girard, 1989, Girard, 1988, Girard, 1995] 5.3. Mathematical Analysis. Providing a useful arena for mathematical analysis is another relevant feature of the above framework. C algebras not just give us a well defined semantic for neural networks which associates to a neural network (specification) an ....

Jean-Yves Girard. Geometry of Interaction 1: Interpretation of System F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221--260, North Holland, 1989.

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

....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 oe f : A B; oe g : B C. We want to find h such that oe f ; oe g = oe h . We shall compute h by the execution formula [Gir89b, Gir89a, Gir88]. Before giving the formal definition, let us explain the idea, which is rather simple. We want to hook the strategies up so that Player s moves in B under oe get turned into Opponent s moves in B for , and vice versa. Consider the following picture: oe Delta Delta Delta Delta Delta ....

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.

.... C algebras: Mulvey s Quantales [Mulvey, 1986, Borceux et al. 1989, Rosenthal, 1990] are an abstraction of the lattice of closed right ideals of a C algebra; in Girard s Geometry of Interaction (linear) proofs are represented and characterised within a C algebra setting [Girard, 1989b, Girard, 1989a, Girard, 1988, Girard, 1994, Girard et al. 1995] and finally there exists an interesting characterisation of (the Lindenbaum algebras, i.e. MV algebras, of) multi valued logics by Mundici based on the K Theory of C algebras [Mundici, 1986, Mundici, 1989] C algebras are an abstraction ....

Jean-Yves Girard. Geometry of Interaction 1: Interpretation of System F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221--260, North Holland, 1989.

.... [L ev80] Several refinements of sharing graphs have been successively proposed by Gonthier et al. GAL92a,GAL92b] and by Asperti and Laneve [AL94,Asp95] The work of Gonthier et al. addressed how Lamping s formalism can be interpreted inside the so called Geometry of Interaction (GOI) of Girard [Gir89]; Asperti presented a more categorical justification of Gonthier s technique; Asperti and Laneve gave a generalization of the methodology to the so called Interaction Systems, the subclass of the Combinatory Reduction Systems [Klo80] for which it is possible to find a Curry Howard analogy with a ....

Jean-Yves Girard. Geometry of interaction 1: Interpretation of system F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloqium `88, pages 221--260, 1989. Elsevier (North-Holland).

....model of quantum computation. 4.2 Linear Logic Another interesting example of an abstract Hilbert machine can be found in Girard s Geometry of Interaction . In a series of articles J. Y. Girard gives a semantics of proofs in linear logic which is based on a Hilbert space construction [16, 15, 14, 17]. Its quite simple to embed the common logical descriptions of facts or states into a Hilbert space structure. Let us fix a certain (countable) set of qualities or predicates p i . A predicate formula F e.g. in disjunctive normal form, i.e. F = f 1 f 2 : f k with f i = p i 1 p i 2 : ....

....approximate it as close as desired (even if this might not be a numerically favourable or stable way to do it) 5. 2 Proof Normalisation Let us look more concretely at Girard s Geometry of Interaction we introduced above, where every proof (procedure) is encoded as an abstract C operator [16, 15, 14, 17]. Proof normalisation is as said implemented by the so called execution formula EX(U; oe) U P (oeU) n . In the light of what we said about the iterative (approximate) solution to linear equations, this formula can be seen as constructing the solution of linear equation, namely: ....

[Article contains additional citation context not shown here]

Jean-Yves Girard. Geometry of Interaction 1: Interpretation of System F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221--

.... This paper develops for the third time a semantics of computation free from the twin drawbacks of reductionism (which leads to static modelisation) and subjectivism (which leads to syntactical abuses, in other terms bureaucracy) Such a semantics was developed previously by Jean Yves Girard [Gir89, Gira] and by John Lamping [Lam90] Girard is a logician INRIA Rocquencourt. y Digital Equipment Corporation, Systems Research Center. 0 and Lamping is an autodidactic engineer. It is no surprise that they never read one another although they were working on the same problem from different ....

Jean-Yves Girard. Geometry of interaction I: Interpretation of system F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221--260. Elsevier Science Publishers B.V. (North Holland) , 1989.

....and the Hilbert space V. Danos L. Regnier Abstract Girard s execution formula (given in [Gir88a]) is a decomposition of usual fi reduction (or cut elimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a term or a net, as the sum of maximal paths on the term net that are not cancelled by the algebra L (as was done in [Dan90, ....

....ffl some mathematics dealing with syntax is uncovered. Goal and organization of this paper. A persistent path is a path on a term which survives the action of any reduction (defined in [Reg92] A regular path is a path which is not cancelled by Girard s algebraic device L (defined in [Gir88a]) The paper mainly aims at proving the equivalence of these two definitions and proceeds in the following order: first the geometric viewpoint on reduction is given, yielding a V. Danos L. Regnier definition of persistence; second the algebraic viewpoint, yielding a definition of regularity; ....

[Article contains additional citation context not shown here]

Jean-Yves Girard. Geometry of interaction I: interpretation of system F . In Ferro, Bonotto, Valantini, and Zanardo, editors, Proceedings of the Logic Colloquium 88, pages 221--260, Padova, Italy, 1988. North Holland.

....de Luminy Marseille Abstract. This note defines a new graphical local calculus, directed virtual reductions. It is designed to compute Girard s execution formula EX, an invariant of closed functional evaluation obtained from the geometry of interaction interpretation of calculus [5]. The calculus is obtained by synchronizing another graphical local calculus presented in local and asynchronous beta reduction : virtual reductions [4] This synchronization makes it easier to mechanize than general virtual reductions. In undirected virtual reductions the consistency of the ....

Jean-Yves Girard. Geometry of interaction I: an interpretation of system F . In Ferro & al., editor, Proceedings of A.S.L. Meetings. North-Holland, 1988.

.... of terms [L ev80] Several refinements of them where successively proposed by Gonthier et al. GAL92a,GAL92b] and by Asperti and Laneve [AL94,Asp95] The work of Gonthier et al. addressed how Lamping s formalism could be interpreted inside the so called Geometry of Interaction (GOI) of Girard [Gir89]; Asperti presented a more categorical justification of the technique; Asperti and Laneve gave a generalization of the methodology to the so called Interaction Systems, the subclass of the Combinatory Reduction Systems for which it is possible to find a Curry Howard analogy with a suitable ....

Jean-Yves Girard. Geometry of interaction 1: Interpretation of system F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, editors, Logic Colloqium `88, pages 221--260, Amsterdam, The Netherlands, 1989. Elsevier (North-Holland).

....studies ( 3] 8] Girard s seminal execution formula EX, as the computation of regular paths in the graph representation (inherited from proof nets of Linear Logic) of terms. EX is a scalar , attached to any such graph, and belonging to the dynamic algebra L (EX and L both are defined in [5]) on which one can read back the result of the ordinary fi computation (whenever there is one) in case of data types (typically Church numerals) The mere fact that objects with such a geometrical touch (I mean those regular paths) lurk somewhere under the skirts of the old lady (fi reduction) ....

Jean-Yves Girard. Geometry of interaction 1: an interpretation of system F . In Proceedings of A.S.L. Meetings, 1988.

No context found.

Jean-Yves Girard. 1989. Geometry of interaction I: Interpretation of system F. In C. Bonotto, R. Ferro, S. Valentini, and A. Zanardo, editors, Logic Colloquium '88, pages 221--260. NorthHolland.

No context found.

Jean-Yves Girard. 1989. Geometry of interaction I: Interpretation of system F. In C. Bonotto, R. Ferro, S. Valentini, and A. Zanardo, editors, Logic Colloquium '88, pages 221--260. North-Holland.

No context found.

Jean-Yves Girard. 1989. Geometry of interaction I: Interpretation of system F. In C. Bonotto, R. Ferro, S. Valentini, and A. Zanardo, editors, Logic Colloquium '88, pages 221--260. NorthHolland.

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