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22
The Geometry of Optimal Lambda Reduction
, 1992
"... Lamping discovered an optimal graphreduction implementation of the calculus. Simultaneously, Girard invented the geometry of interaction, a mathematical foundation for operational semantics. In this paper, we connect and explain the geometry of interaction and Lamping's graphs. The geometry of int ..."
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Cited by 97 (2 self)
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Lamping discovered an optimal graphreduction implementation of the calculus. Simultaneously, Girard invented the geometry of interaction, a mathematical foundation for operational semantics. In this paper, we connect and explain the geometry of interaction and Lamping's graphs. The geometry of interaction provides a suitable semantic basis for explaining and improving Lamping's system. On the other hand, graphs similar to Lamping's provide a concrete representation of the geometry of interaction. Together, they offer a new understanding of computation, as well as ideas for efficient and correct implementations.
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Linear Logic Without Boxes
, 1992
"... Girard's original definition of proof nets for linear logic involves boxes. The box is the unit for erasing and duplicating fragments of proof nets. It imposes synchronization, limits sharing, and impedes a completely local view of computation. Here we describe an implementation of proof nets withou ..."
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Cited by 53 (0 self)
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Girard's original definition of proof nets for linear logic involves boxes. The box is the unit for erasing and duplicating fragments of proof nets. It imposes synchronization, limits sharing, and impedes a completely local view of computation. Here we describe an implementation of proof nets without boxes. Proof nets are translated into graphs of the sort used in optimal calculus implementations; computation is performed by simple graph rewriting. This graph implementation helps in understanding optimal reductions in the calculus and in the various programming languages inspired by linear logic. 1 Beyond the calculus The calculus is not entirely explicit about the operations of erasing and duplicating arguments. These operations are important both in the theory of the  calculus and in its implementations, yet they are typically treated somewhat informally, implicitly. The proof nets of linear logic [1] provide a refinement of the calculus where these operations become explici...
Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by S ..."
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Cited by 44 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 41 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
Geometry of Interaction III: Accommodating the Additives
 In: Advances in Linear Logic, LNS 222,CUP, 329–389
, 1995
"... The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution f ..."
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Cited by 29 (5 self)
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The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponentialfree conclusions. 1
First Order Linear Logic without Modalities Is NEXPTIMEHard
 Theoretical Computer Science
, 1994
"... The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace ..."
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Cited by 15 (11 self)
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The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace hardness of quantified boolean formula validity, utilizing some of the surprisingly powerful and expressive machinery of linear logic. 1 Introduction Linear logic, introduced by Girard, is a resourcesensitive refinement of classical logic [10, 29]. Linear logic gains its expressive power by restricting the "structural" proof rules of contraction (copying) and weakening (erasing). The contraction rule makes it possible to reuse any stated assumption as often as desired. The weakening rule makes it possible to use dummy assumptions, i.e., it allows a deduction to be carried out without using all of the hypotheses. Because contraction and weakening together make it possible to use an assu...
Linear Logic, Comonads and Optimal Reductions
 Fundamentae Informaticae
, 1993
"... The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a ..."
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Cited by 7 (3 self)
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The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a "local implementation" of naturality laws, that is as the broadcasting of some information from the output to the inputs of a term, following its connected structure. 1 Introduction More than fifteen years ago, L'evy [Le78] proposed a theoretical notion of optimality for calculus normalization. Roughly speaking, a reduction technique is optimal if it is able to profit of all the sharing expressed in initial term, avoiding useless duplications. For a long time, no implementation was able to achieve L'evy's performance (see [Fie90] for a quick survey). People started already to doubt of the existence of optimal evaluators, when Lamping and Kathail independently found a solution [Lam90,Ka90]...
On categorical models of classical logic and the geometry of interaction
, 2005
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this co ..."
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Cited by 5 (0 self)
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this collapse by interpreting cutreduction by a posetenrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the settheoretic product. In this article, which is selfcontained, we present an improved axiomatization of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negationfree models called Dummett categories. Examples include, besides the classical categories above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from homsemilattices which have a straightforward prooftheoretic definition. Moreover, we show that the GeometryofInteraction construction can be extended from multiplicative linear logic to classical logic, by applying it to obtain a classical
Geometry of Interaction IV: the Feedback Equation
, 2005
"... The first three papers on Geometry of Interaction [9, 10, 11] did establish the universality of the feedback equation as an explanation of logic; this equation corresponds to the fundamental operation of logic, namely cutelimination, i.e., logical consequence; this is also the oldest approach to lo ..."
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Cited by 5 (1 self)
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The first three papers on Geometry of Interaction [9, 10, 11] did establish the universality of the feedback equation as an explanation of logic; this equation corresponds to the fundamental operation of logic, namely cutelimination, i.e., logical consequence; this is also the oldest approach to logic, syllogistics! But the equation was essentially studied for those Hilbert space operators coming from actual logical proofs. In this paper, we take the opposite viewpoint, on the arguable basis that operator algebra is more primitive than logic: we study the general feedback equation of Geometry of Interaction, h(x⊕y) = x ′ ⊕σ(y), where h,σ are hermitian, �h � ≤ 1, and σ is a partial symmetry, σ 3 = σ. We show that the normal form which yields the solution σ�h�(x) = x ′ in the invertible case can be extended in a unique way to the general case, by various techniques, basically ordercontinuity and associativity. From this we expect a definite break with essentialism à la Tarski: an interpretation of logic which does not presuppose logic! 1