# 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])....

[...]

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....

[...]

...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....

[...]

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....

[...]

37 citations

##### References

^{1}

275 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]....

[...]

257 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]....

[...]

151 citations

142 citations

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

...But λσ does not preserve β-strong normalization as shown by Mellies, who exhibited a well-typed term which, due to the substitution composition rules in λσ, is not λσ-strongly normalizing [18]....

[...]

129 citations

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

...Meanwhile, in the linear logic community, many studies focused of the connection between λ-calculus (without explicit substitutions) and proof nets, trying to find the proper variant or extension of proof nets that could be used to cleanly simulate β-reduction, like in [7]....

[...]