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, Reg92]). It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the -term/net. Remarkably, the proof puts to use for the first time the interpretation of -terms/nets as operators on the Hilbert space. 1 Presentation -Calculus is simple but not completely convincing as a real machine-language. Real machine instructions have a fixed run-time; a fi-reduction step does not. Some implementations do map fi-reductions into sequences of real elementary steps (as in environment machines for example) but they use a global time t...

. We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme. 1 Introduction A fundament...

We study a general algebraic framework which underlies a wide range of computational formalisms...

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1

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.

There are two quite different possibilities for implementing linear head reduction in -calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former. Keywords: -calculus, abstract machines, geometry of interaction, reversible computations. 1 Introduction Notation. We denote the application of U to V by (U)V , e.g., the Church integer ¯ 2 will be fx (f)(f)x. Linear head reduction. But what is exactly linear head reduction, to begin with. It is a variant of head reduction, where one substitutes at each step the leftmost occurrence of c fl1996 Elsevier Science B. V. Danos & Regnier variable whenever it is engaged into a redex, as in: (f (f )(f)x)y y ! (f(y y)(f)x)y y ! (f(y (f )x)(f)x)y y ! (f(y (y y)...

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of di#erent and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

. We develop a theory of structured objects and their relational manipulation as a basis of theories of processes in which connection and interaction among computing agents at multiple interface points are the fundamental elements. Starting from a simple algebraic scheme for "processes with (port) names" as found in usual process calculi, an analysis of an internal structure of name-based processes leads to a leaner, but equivalent, presentation of processes, which is more geometricand which dispenses with names. It is shown that a coherent set-like theory of structured objects and their manipulation, which does underlie the preceding name-free formalisms of interaction such as proof nets, can be developed based on the namefree presentation of processes, where the notion of symmetries plays a fundamental role. The equivalence between the name-based universe and the name-free universe is established in a general context, allowing us to movebetween two universes without losing essential ...

There are two quite different possibilities for implementing linear head reduction in -calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former. Keywords: -calculus, abstract machines, geometry of interaction, reversible computations. 1 Introduction Notation. We denote the application of U to V by (U )V , e.g., the Church integer 2 will be fx (f)(f)x. Linear head reduction. But what is exactly linear head reduction, to begin with. It is a variant of head reduction, where one substitutes at each step the leftmost occurrence of variable whenever it is engaged into a redex, as in: (f (f )(f)x)y y ! (f(y y)(f)x)y y ! (f(y (f)x)(f)x)y y ! (f(y (y y)x)(f)x)y y ! (f(y (y x)x)(f)x)y y where the succ...