# Proof Nets and Explicit Substitutions

##### Citations

77 citations

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

...Explicit substitutions (ES) have been connected to linear logic by Kesner and co-authors in a sequence of works [26, 32, 33], culminating in the linear substitution calculus (LSC), a new formalism with ES behaviorally isomorphic to proof nets (introduced in [6], developed in [1, 3, 4, 7, 10], and bearing similarities with calculi by De Bruijn [25], Nederpelt [42], and Milner [41])....

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53 citations

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

...But the λws-calculus has a complicated syntax and its named version [13] is even less intelligible....

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...However, the strong normalisation proof for λws given in [13] reveals a natural semantics for composition of ES via Linear Logic’s proof-nets [19], suggesting that weakening (explicit erasure) and contraction (explicit duplication) can be added to the calculus without losing strong normalisation....

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49 citations

44 citations

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

...Solutions proposed in LL include – adopting a syntax which identifies contractions made at several exponential depths, as in [23] – for now it seems hard to apply it in differential nets with boxes, we will see how the rule of codereliction against box introduces many difficulties; – using such an identification as an equivalence relation, as hinted in [12] for DINs and investigated in [3,8] for LL proof nets – an elegant solution, though it is less so with respect to freely moving around weakenings, as it may generate infinite trees with weakened leaves; – using it as a set of reductions, as in [4] – which is is the way we are adopting here....

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37 citations

##### References

129 citations

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

...As customary in explicit substitutions calculi with names [3], we work modulo α-conversion, so that we can suppose that in the rule Weak the set ∆ does not contain variables that are bound in M ....

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111 citations

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

...Finally, in [10], the first two authors of this work showed for the first time that explicit substitutions could be tightly related to linear logic’s proof nets, by providing a translation into a variant of proof nets from λx [19, 4], a simple calculus with explicit substitutions and named variables, but no composition....

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98 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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49 citations

47 citations