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Lambda calculus and intuitionistic linear logic
 Invited talk at the Logic Colloquium’94 (ClermontFerrand
, 1994
"... The CurryHoward isomorphism 1 is the basis of typed functional programming. By means of this isomorphism, the intuitionistic proof of a formula can be seen as a functional program, whose type is the formula itself. In this way, the computation process has its logic realization in the proof normaliz ..."
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Cited by 16 (6 self)
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The CurryHoward isomorphism 1 is the basis of typed functional programming. By means of this isomorphism, the intuitionistic proof of a formula can be seen as a functional program, whose type is the formula itself. In this way, the computation process has its logic realization in the proof normalization procedure. Both the implicative fragment of the intuitionistic
Coherence for sharing proofnets
 Proceedings of the 7th International Conference on Rewriting Techniques and Applications (RTA96), LNCS 1103
, 1996
"... Sharing graphs are an implementation of linear logic proofnets in such a way that their reduction never duplicate a redex. In their usual formulations, proofnets present a problem of coherence: if the proofnet N reduces by standard cutelimination to N 0, then, by reducing the sharing graph of N ..."
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Cited by 11 (7 self)
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Sharing graphs are an implementation of linear logic proofnets in such a way that their reduction never duplicate a redex. In their usual formulations, proofnets present a problem of coherence: if the proofnet N reduces by standard cutelimination to N 0, then, by reducing the sharing graph of N we donot obtain the sharing graph of N 0.Wesolve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is con uent and terminating. The proof of this fact exploits an algebraic semantics for sharing graphs. 1
Parsing mell proof nets
 In TLCA
, 1997
"... We propose a new formulation for full (weakening and constants included) multiplicative and exponential (MELL) proof nets, allowing a complete set of rewriting rules to parse them. The recognizing grammar de ned by such a rewriting system (con uent and strong normalizing on the new proof nets) gives ..."
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Cited by 8 (3 self)
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We propose a new formulation for full (weakening and constants included) multiplicative and exponential (MELL) proof nets, allowing a complete set of rewriting rules to parse them. The recognizing grammar de ned by such a rewriting system (con uent and strong normalizing on the new proof nets) gives a correctness criterion that we show equivalent to the DanosRegnier one. 1
Proof nets, Garbage, and Computations
, 1997
"... We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets. The main novelty isin a complete set of rewriting rules for cutelimination in presence of weakening (which requires garbage collection). The proposed reduction s ..."
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Cited by 7 (6 self)
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We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets. The main novelty isin a complete set of rewriting rules for cutelimination in presence of weakening (which requires garbage collection). The proposed reduction system is strongly normalizing and confluent.
A general theory of sharing graphs
 THEORET. COMPUT. SCI
, 1999
"... Sharing graphs are the structures introduced by Lamping to implement optimal reductions of lambda calculus. Gonthier's reformulation of Lamping's technique inside Geometry of Interaction, and Asperti and Laneve's work on Interaction Systems have shown that sharing graphs can be used t ..."
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Cited by 4 (3 self)
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Sharing graphs are the structures introduced by Lamping to implement optimal reductions of lambda calculus. Gonthier's reformulation of Lamping's technique inside Geometry of Interaction, and Asperti and Laneve's work on Interaction Systems have shown that sharing graphs can be used to implement a wide class of calculi. Here, we give a general characterization of sharing graphs independent from the calculus to be implemented. Such a characterization rests on an algebraic semantics of sharing graphs exploiting the methods of Geometry of Interaction. By this semantics we can de ne an unfolding partial order between proper sharing graphs, whose minimal elements are unshared graphs. The leastshared instance of a sharing graph is the unique unshared graph that the unfolding partial order associates to it. The algebraic semantics allows to prove that we can associate a semantical readback to each unshared graph and that such a readback can be computed
Some Complexity and Expressiveness results on Multimodal and Stratified Proof nets
"... Abstract. We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: “More modalities means more expressiveness”. MS is a set of buildingrule schemes that depend on parameters.We interpret the values of the parameter ..."
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Abstract. We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: “More modalities means more expressiveness”. MS is a set of buildingrule schemes that depend on parameters.We interpret the values of the parameters as modalities. Fixing the parameters yields deductive systems as instances of MS, that we call subsystems. Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems. A first result is a sufficient condition for determining when a subsystem is strongly polynomial time sound. A second one shows that the ability to choose which modalities are used and how can be rewarding. We give a family of subsystems as complex as Multiplicative Linear Logic — they are linear time and space sound — that can representChurch numerals and some common combinators on them.
Lambda! Considered Both as a Paradigmatic Language and as a MetaLanguage
"... Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an u ..."
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Cited by 3 (1 self)
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Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an untyped functionallike language inspired from a typed language joined at ILL by CHI. We want to use the resourceaware language \Gamma ! both as a paradigmatic programming language and as a metalanguage for implementing a fragment of the untyped calculus fi . For using \Gamma ! in the first way we give an algorithm for automatically assigning formulas of ILL as types to terms of \Gamma ! . Concerning the second kind of use, we introduce a onestep translation Tr from the fragment C of fi that can be typed a la Curry to the typable fragment of \Gamma ! in ILL. Tr preserves the linearbehaved terms of C and is both correct and complete, in a reasonable sense, w.r.t. the...
A TypeFree ResourceAware λCalculus
, 1996
"... We introduce and study a functional language λ_R , having two main features. λ_R has the same computational power of the λcalculus. λ_R enjoys the resourceawareness of the typed/typable functional languages which encode the Intuitionistic Linear Logic. ..."
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We introduce and study a functional language λ_R , having two main features. λ_R has the same computational power of the λcalculus. λ_R enjoys the resourceawareness of the typed/typable functional languages which encode the Intuitionistic Linear Logic.
SharingGraphs, SharingMorphisms, and (Optimal)
"... abstract. We study local and asynchronous reductions of λterms using their shared representations. We give a set of rules which allows the internalization of the readback into the graph rewriting system. We point out that by simply restricting to a subset of such a rewriting system we can implemen ..."
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abstract. We study local and asynchronous reductions of λterms using their shared representations. We give a set of rules which allows the internalization of the readback into the graph rewriting system. We point out that by simply restricting to a subset of such a rewriting system we can implement Lévy’s optimal reductions. 1