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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 ea ..."
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Cited by 61 (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...
Interaction Combinators
 Information and Computation
, 1995
"... This paper is the continuation of the author 's work on interaction nets, inspired by Girard's proof nets for linear logic, but no preliminary knowledge of these topics is required for its reading. Introduction ..."
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Cited by 45 (3 self)
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This paper is the continuation of the author 's work on interaction nets, inspired by Girard's proof nets for linear logic, but no preliminary knowledge of these topics is required for its reading. Introduction
Elementary Complexity and Geometry of Interaction
, 2000
"... We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of prog ..."
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Cited by 28 (6 self)
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We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs. Keywords: Elementary Linear Logic, Geometry of interaction, Complexity, Semantics.
Context semantics, linear logic and computational complexity
 In Proc. 21th IEEE Syposium on Logic in Computer Science
, 2006
"... We show that context semantics can be fruitfully used to estimate the computational cost of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proofnet in such a way that the time needed to normalize a given proof is deeply related to its weight: th ..."
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Cited by 19 (7 self)
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We show that context semantics can be fruitfully used to estimate the computational cost of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proofnet in such a way that the time needed to normalize a given proof is deeply related to its weight: the time needed to normalize a proofnet is bounded by a polynomial on its weight, while there are strategies such that the weight is a lower bound to normalization time. 1
Category theory for linear logicians
 Linear Logic in Computer Science
, 2004
"... This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categori ..."
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Cited by 13 (2 self)
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This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0
A Token Machine for Full Geometry of Interaction
 In TLCA ’01, SLNCS 2044
, 2001
"... We present an extension of the Interaction Abstract Machine (IAM) [4] to full Linear Logic with Girard's Geometry of Interaction (GoI) [6]. We propose a simplied way to interpret the additives and the interaction between additives and exponentials by means of weights [7]. We describe the interp ..."
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We present an extension of the Interaction Abstract Machine (IAM) [4] to full Linear Logic with Girard's Geometry of Interaction (GoI) [6]. We propose a simplied way to interpret 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 MELLGoI is that it gives an algebraic characterization of the persistent paths of a proof, that is the paths in the graph structure of the proof that are not broken by cut elimination (see [3]). More abstractly, the presentation of the GoI may be generalized in the framework of traced monoidal categories [10]. Because of its local f...
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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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
Proof Nets and Explicit Substitutions
 Mathematical Structures in Computer Science
, 2000
"... We refine the simulation technique introduced in [10] to show strong normalization of calculi with explicit substitutions via termination of cut elimination in proof nets [12]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimina ..."
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We refine the simulation technique introduced in [10] to show strong normalization of calculi with explicit substitutions via termination of cut elimination in proof nets [12]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the l  calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed l to proof nets. Finally, we propose a version of typed l with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the l calculus with de Bruijn indices [8]. 1
Linear Logic by Levels and Bounded Time Complexity
, 2009
"... This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity b ..."
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Cited by 9 (2 self)
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This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity bound on the cutelimination procedure. Here, we propose a more flexible control principle, that of indexing, which allows us to extend Girard’s systems while keeping the same complexity properties. A consequence of the higher flexibility of indexing with respect to stratification is the absence of boxes for handling the § modality. We finally propose a variant of our polytime system in which the § modality is only allowed on atoms, and which may thus serve as a basis for developing λcalculus type assignment systems with more efficient typing algorithms than existing ones.
A Categorical Model for the Geometry of Interaction
, 2004
"... We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard's original approach to GoI ..."
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Cited by 8 (1 self)
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We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.