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15
Proofnets and the Hilbert space
 Advances in Linear Logic
, 1995
"... Girard's execution formula (given in [Gir88a]) is a decomposition of usual fireduction (or cutelimination) 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 ..."
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Cited by 52 (3 self)
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Girard's execution formula (given in [Gir88a]) is a decomposition of usual fireduction (or cutelimination) 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 machinelanguage. Real machine instructions have a fixed runtime; a fireduction step does not. Some implementations do map fireductions into sequences of real elementary steps (as in environment machines for example) but they use a global time t...
A Relational Model of NonDeterministic Dataflow
 In CONCUR'98, volume 1466 of LNCS
, 1998
"... . 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 ..."
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Cited by 27 (13 self)
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. 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...
Elementary Structures in Process Theory (1) Sets with Renaming
, 1997
"... We study a general algebraic framework which underlies a wide range of computational formalisms... ..."
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Cited by 19 (6 self)
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We study a general algebraic framework which underlies a wide range of computational formalisms...
A Structural Approach to Reversible Computation
 Theoretical Computer Science
, 2001
"... 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 lowlevel machine models. By contrast, we develop ..."
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Cited by 18 (3 self)
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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 lowlevel machine models. By contrast, we develop a more structural approach. We show how highlevel functional programs can be mapped compositionally (i.e. in a syntaxdirected 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
Reversible, Irreversible and Optimal lambdamachines (Extended Abstract)
, 1996
"... 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 ..."
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Cited by 15 (1 self)
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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)...
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 15 (4 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.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 13 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
Categorical Models for Concurrency: Independence, Fairness and Dataflow
 BRICS DISSERTATION SERIES DS001
, 2000
"... 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 t ..."
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Cited by 6 (4 self)
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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 different 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. The work
Formalising Business Process Execution with Bigraphs and Reactive XML
, 2006
"... Bigraphical Reactive Systems have been proposed as a meta model for global ubiquitous computing generalising process calculi for mobility such as the picalculus and the Mobile Ambients calculus as well as graphical models for concurrency such as Petri Nets. We investigate in this paper how Bigrap ..."
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Cited by 4 (3 self)
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Bigraphical Reactive Systems have been proposed as a meta model for global ubiquitous computing generalising process calculi for mobility such as the picalculus and the Mobile Ambients calculus as well as graphical models for concurrency such as Petri Nets. We investigate in this paper how Bigraphical Reactive Systems represented as Reactive XML can be used to provide a formal semantics as well as an extensible and mobile platform independent execution format for XML based business process and workflow description languages such as WSBPEL and XPDL. We propose to extend the formalism with primitives for XPath evaluation and higherorder reaction rules to allow for a very direct and succinct semantics.
Process Structures
, 1997
"... . 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 (po ..."
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Cited by 1 (0 self)
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. 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 namebased processes leads to a leaner, but equivalent, presentation of processes, which is more geometricand which dispenses with names. It is shown that a coherent setlike theory of structured objects and their manipulation, which does underlie the preceding namefree 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 namebased universe and the namefree universe is established in a general context, allowing us to movebetween two universes without losing essential ...