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 Girard-style and Abramsky-Jagadeesan-style versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girard-style GoI was dubbed "particle-style", since it concerns information particles or tokens flowing around a network, while the Abramsky-Jagadeesan style GoI was dubbed "wave-style", 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 coproduct-based (i.e. our "particle-style") and "multiplicative" for product-based (i.e. our "wave-style"); this is not suitable for our purposes, because of the clash with Linear Logic term...

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

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.

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

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 MELL-GoI 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...

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

An examination of Girard's execution formula suggests implementations of the Geometry of Interaction at the syntactic level. In this paper we limit our scope to groundtype terms and study the parallel aspects of such implementations, by introducing a family of abstract machines which can be directly implemented. These machines address all the important implementation issues such as the choice of an inter-thread communication model, and allow to incorporate specic strategies for dividing the computation of the execution path into smaller tasks. 1

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 cut-elimination, 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 order-continuity and associativity. From this we expect a definite break with essentialism à la Tarski: an interpretation of logic which does not presuppose logic! 1

We define an observational equivalence for Lafont’s interaction combinators, which we prove to be the least discriminating non-trivial congruence on total nets (nets admitting a deadlock-free normal form) respecting reduction. More interestingly, this equivalence enjoys an internal separation property similar to that of Böhm’s Theorem for the λ-calculus.

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the reduction strategy) and a lower bound to the number of steps to normal form (for certain reduction strategies). Weights are then exploited in proving strong soundness theorems for various subsystems of linear logic, namely elementary linear logic, soft linear logic and light linear logic. 1