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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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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
Proof Nets and the λcalculus
 Linear Logic in Computer Science, 65–118
, 2004
"... In this survey we shall present the main results on proof nets for the Multiplicative and Exponential fragment of Linear Logic (MELL) and discuss their connections with λcalculus. The survey ends with a short introduction to sharing reduction. The part on proof nets and on the encoding of λterms ..."
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In this survey we shall present the main results on proof nets for the Multiplicative and Exponential fragment of Linear Logic (MELL) and discuss their connections with λcalculus. The survey ends with a short introduction to sharing reduction. The part on proof nets and on the encoding of λterms is selfcontained and the proofs of the main theorems are given in full details. Therefore, the survey can be also used as a tutorial on that topics. 1
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