# Proof Nets and Explicit Substitutions

##### Citations

9 citations

### Cites background or methods or result from "Proof Nets and Explicit Substitutio..."

...Furtherly, as can be deducted from [9], this can be the ground for new work on calculi with explicit substitutions: whether by extending some results to untyped calculi; or by considering explicit substitutions for nondeterministic calculi akin to Boudol’s λ-calculus with resources (see [6])....

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...Our system of reductions and equivalences bears close resemblance to the one developed for MELL in [9]....

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...We thus generalize the equivalences and reductions of [9], providing as a byproduct the first proof of confluence1 for such LL proof nets with equivalences in the completely pure case, as previous works concentrated on normalization in the typed one....

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...The push equivalence9 has already been studied in the literature on proof nets and explicit substitutions [8,9]....

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...In fact, stripping DiLL of all its differential features, the only difference is the absence in [9] of anything related to the p-reduction....

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7 citations

6 citations

### Cites background or result from "Proof Nets and Explicit Substitutio..."

...After the first results stated in Ehrhard and Regnier (2006) but restricted to the promotion-free fragment, other works have strengthened this claim: as already mentioned, the first author showed WN of simply typed DiLL in Pagani (2009) (see also Gimenez (2011) for a different proof, based on reducibility candidates), the second showed confluence (or rather Church–Rosser modulo, as we will have occasion to explain) of the untyped case in Tranquilli (2009a), and we here add yet another brick to the solidity of the system....

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...After the first results stated in Ehrhard and Regnier (2006) but restricted to the promotion-free fragment, other works have strengthened this claim: as already mentioned, the first author showed WN of simply typed DiLL in Pagani (2009) (see also Gimenez (2011) for a different proof, based on reducibility candidates), the second showed confluence (or rather Church–Rosser modulo, as we will have occasion to explain) of the untyped case in Tranquilli (2009a), and we here add yet another brick to the solidity of the system. Confluence in presence of a formal sum modelling nondeterminism means that such a nondeterministic choice is completely internal and does not depend on the reduction strategy. Finally, an enticing point of interest is the link with concurrency. Differentiation, as modelled by the codereliction rule, can be in fact seen as the introduction of a one-use entity that goes to only one among the possibly many who are asking for it. This has been employed by Ehrhard and Laurent (2007) to model concurrent communication, where only one among many may answer to a given signal....

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...The push equivalence7 has already been studied in the literature on proof nets and explicit substitutions (Di Cosmo and Guerrini 1999; Di Cosmo et al. 2003)....

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...Unlike LL, where such semantic equivalences can be added and studied a posteriori (as, for example, in Di Cosmo et al. (2003)), in DiLL they must be included in order to assure confluence (Tranquilli 2009a)....

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6 citations

5 citations

### Cites background from "Proof Nets and Explicit Substitutio..."

...Such equivalences where already studied in literature about explicit substitutions [13], or for differential nets [14](3)....

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##### References

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2,304 citations

### "Proof Nets and Explicit Substitutio..." refers background or methods in this paper

...In this paper we refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]....

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...While we refer the interested reader to [13] for more details on linear logic in general, we give here a one-sided presentation of the sequent calculus for MELL:...

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1,480 citations

577 citations

### "Proof Nets and Explicit Substitutio..." refers background in this paper

...The pioneer calculus with explicit substitutions, λσ, was introduced in [1] as a bridge between the classical λ-calculus and concrete implementations of functional programming languages....

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321 citations

### "Proof Nets and Explicit Substitutio..." refers methods in this paper

...1 Using various translations of the λ-calculus into proof nets, new abstract machines have been proposed, exploiting the Geometry of Interaction and the Dynamic Algebras [14, 2, 5], leading to the works on optimal reduction [15, 17]....

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