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Soft Linear Logic and Polynomial Time
 THEORETICAL COMPUTER SCIENCE
, 2002
"... We present a subsystem of second order Linear Logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and viceversa. ..."
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We present a subsystem of second order Linear Logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and viceversa.
Encoding Linear Logic with Interaction Combinators
 Information and Computation
, 2002
"... The purpose of this paper is to demonstrate how Lafont’s interaction combinators, a system of three symbols and six interaction rules, can be used to encode linear logic. Specifically, we give a translation of the multiplicative, exponential and additive fragments of linear logic together with a str ..."
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The purpose of this paper is to demonstrate how Lafont’s interaction combinators, a system of three symbols and six interaction rules, can be used to encode linear logic. Specifically, we give a translation of the multiplicative, exponential and additive fragments of linear logic together with a strategy for cutelimination which can be faithfully simulated. Finally, we show briefly how this encoding can be used for evaluating �terms. In addition to offering a very simple, perhaps the simplest, system of rewriting for linear logic and the �calculus, the interaction net implementation that we present has been shown by experimental testing to offer a good level of sharing, in terms of the number of cutelimination steps (resp. ¬reduction steps). In particular it performs better than all extant finite systems of interaction nets.
Coinductive Techniques for Operational Equivalence of Interaction Nets
, 1998
"... In this paper we study a notion of operational equivalence for interaction nets, following the recent success of applying methods based on bisimulation to functional and object oriented programming languages. We set up notions of contextual equivalence and bisimilarity and show that they coincide. A ..."
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In this paper we study a notion of operational equivalence for interaction nets, following the recent success of applying methods based on bisimulation to functional and object oriented programming languages. We set up notions of contextual equivalence and bisimilarity and show that they coincide. A coinduction principle then gives a simple and robust way of showing when two interaction nets are contextually equivalent. We include several examples to demonstrate the usefulness of the approach, in particular for optimizing interaction nets. 1 Introduction One of the most fundamental notions in programming languages is that of program equivalence: when can one program fragment be replaced by another. A notion of equivalence should be substitutive so that programs remain equivalent in all contexts. Amongst other applications, this facilitates proving properties of programs, and gives a sound basis for program optimization. Interaction nets provide an interesting new perspective as both a...
Multiport interaction nets and concurrency
 In Proceedings of CONCUR 2005, number 3653 in Lecture Notes in Computer Science
, 2005
"... Interaction Nets (IN) are a model of distributed computation introduced by Lafont [Laf90], which can be seen as a generalization of Girard's multiplicative proofnets [Gir87]. They admit an extremely simple system of universal combinators [Laf97], which has a very natural algebraic semantics ..."
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Interaction Nets (IN) are a model of distributed computation introduced by Lafont [Laf90], which can be seen as a generalization of Girard's multiplicative proofnets [Gir87]. They admit an extremely simple system of universal combinators [Laf97], which has a very natural algebraic semantics in the style of the socalled Geometry of Interaction (GoI, [Gir88]). Even though IN are Turingcomplete, their strong determinism prevents them from expressing concurrent behavior. In his Ph.D. thesis [Ale99], Vladimir Alexandriev has dened several nondeterministic extensions of IN. We consider here what he called Interaction Nets with Multiple Principal Ports (INMPP), and we show that they are a very expressive model of concurrent computation by encoding within them the calculus (without sums or match). We also show that INMPP too admit a surprisingly simple system of universal combinators, which is an extension of Lafont's system. These combinators may be the key to the
Type Assignment and Termination of Interaction Nets
"... Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisi ..."
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Cited by 7 (4 self)
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Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisimple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...
A Graph Rewriting Semantics for the Polyadic πCalculus (Extended Version)
, 2000
"... We give a hypergraph rewriting semantics for the polyadic πcalculus, based on rewriting rules equivalent to those in the doublepushout approach. The structural congruence of the πcalculus is replaced by hypergraph isomorphism. The correctness of the encoding from the graphbased notation into πc ..."
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Cited by 5 (2 self)
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We give a hypergraph rewriting semantics for the polyadic πcalculus, based on rewriting rules equivalent to those in the doublepushout approach. The structural congruence of the πcalculus is replaced by hypergraph isomorphism. The correctness of the encoding from the graphbased notation into πcalculus can be shown by using an intermediate notation, socalled namebased graph terms, which form a bridge from graphs with explicit connections (by fusing nodes) to process calculi with implicit connections (by common channel names).
Symmetric Action Calculi
 Theoretical Computer Science
, 1999
"... Many calculi for describing interactive behaviour involve names, nameabstraction and namerestriction. Milner's reflexive action calculi provide a framework for exploring such calculi. It is based on names and nameabstraction. We introduce an alternative framework, the symmetric action calcul ..."
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Many calculi for describing interactive behaviour involve names, nameabstraction and namerestriction. Milner's reflexive action calculi provide a framework for exploring such calculi. It is based on names and nameabstraction. We introduce an alternative framework, the symmetric action calculi, based on names, conames and namerestriction (or hiding). Nameabstraction is intepreted as a derived operator. The symmetric framework conservatively extends the reflexive framework. It allows for a natural intepretation of a variety of calculi: we give interpretations for the calculus, the I calculus and a variant of the fusion calculus. We then give a combinatory version of the symmetric framework, in which namerestriction also is expressed as a derived operator. This combinatory account provides an intermediate step between our nonstandard use of names in graphs, and the more standard graphical structure arising from category theory. To conclude, we briey illustrate the connection ...
Light logics and optimal reduction: Completeness and complexity. Extended Version. Available at http://www.arxiv.org/ abs/0704.2448
, 2007
"... Typing of lambdaterms in Elementary and Light Affine Logic (EAL, LAL resp.) has been studied for two different reasons: on the one hand the evaluation of typed terms using LAL (EAL resp.) proofnets admits a guaranteed polynomial (elementary, resp.) bound; on the other hand these terms can also b ..."
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Typing of lambdaterms in Elementary and Light Affine Logic (EAL, LAL resp.) has been studied for two different reasons: on the one hand the evaluation of typed terms using LAL (EAL resp.) proofnets admits a guaranteed polynomial (elementary, resp.) bound; on the other hand these terms can also be evaluated by optimal reduction using the abstract version of Lamping’s algorithm. The first reduction is global while the second one is local and asynchronous. We prove that for LAL (EAL resp.) typed terms, Lamping’s abstract algorithm also admits a polynomial (elementary, resp.) bound. We also show its soundness and completeness (for EAL and LAL with type fixpoints), by using a simple geometry of interaction model (context semantics).
Memoryful Geometry of Interaction From Coalgebraic Components to Algebraic Effects
"... Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to s ..."
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Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to state machines—“GoI implementation, ” so to speak—has been worked out by Mackie, Ghica and others. In this paper, we develop Abramsky’s idea of resumption based GoI systematically into a generic framework that accounts for computational effects (nondeterminism, probability, exception, global states, interactive I/O, etc.). The framework is categorical: Plotkin & Power’s algebraic operations provide an interface to computational effects; the framework is built on the categorical axiomatization of GoI by Abramsky, Haghverdi and Scott; and, by use of the coalgebraic formalization of component calculus, we describe explicit construction of state machines as interpretations of functional programs. The resulting interpretation is shown to be sound with respect to equations between algebraic operations, as well as to Moggi’s equations for the computational lambda calculus. We illustrate the construction by concrete examples.