# Intuitionistic differential nets and lambda-calculus

##### Citations

49 citations

43 citations

### Cites background from "Intuitionistic differential nets an..."

...The outer -reduction o −→ is the closure to linear contexts of the steps given in Definition 3....

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...Definition 5....

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...Partially founded by the Italian MIUR project CONCERTO, and the French ANR projet blanc CHOCO, ANR-07-BLAN-0324. showed a Curry-Howard correspondence between this calculus and Ehrhard and Regnier’s differential nets [3]....

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...This paper deals extensively with the weakest notion of solvability, which asks that a term is solvable whenever a suitable context filled with it reduces to a sum, where at least one addend is the identity....

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...head-normal form....

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34 citations

34 citations

### Cites background from "Intuitionistic differential nets an..."

...[38] Paolo Tranquilli (2009): Nets Between Determinism and Nondeterminism....

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...Lionel Vaux [40] and Paolo Tranquilli [38, 39] study the relationship between the differential λ-calculus and differential proof nets....

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...[39] Paolo Tranquilli (2011): Intuitionistic differential nets and lambda-calculus....

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33 citations

### Cites background or methods from "Intuitionistic differential nets an..."

...In [4,10] this reduction is called the giant-step one (hence the name of the rule) to distinguish it from the baby-step one we will discuss in Section 5....

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...Indeed, the second author shows in [4] that resource calculus corresponds to the intuitionistic minimal fragment of differential nets with promotion [5], exactly as λ-calculus corresponds to the intuitionistic minimal fragment of linear logic proof-nets [6]....

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...We conclude the paper by discussing another, more atomic reduction of resource terms, called baby-step reduction in [4] (here Definition 10)....

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...This way of writing the application comes from Girard’s linear logic [6]: indeed !-marked arguments (called perpetual) correspond exactly to exponential boxes (see [4]), the synchronized areas of proofs viable for non-linear operations (duplication and erasing)....

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##### References

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2,304 citations

### "Intuitionistic differential nets an..." refers background in this paper

...Introduction Twenty years ago Jean-Yves Girard introduced Linear Logic (LL, [13]) starting from a fine analysis of the coherent semantics he had introduced for system F....

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1,480 citations

402 citations

### "Intuitionistic differential nets an..." refers background in this paper

...Though structures already have computational meaning, we define the correctness criterion following the Danos-Regnier one for LL proof nets [6]....

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307 citations

### "Intuitionistic differential nets an..." refers background in this paper

...Non linear and linear substitutions enjoy the same properties found in [11], though due to the simpler syntax proofs are somewhat easier....

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...Differential λ-calculus and differential nets Anatural direction of investigation arising from [12] and [11] is the questionwhether differential λ-calculus can be translated into differential nets....

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...The second problem arises from the particular syntax presented in [11]....

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...The substitutions employed are those found in differential λ-calculus [11], most notably the linear one....

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...Recently Ehrhard has refined the coherent semantics by means of topological vector spaces and continuous linear maps [9,10], and again from such semantical refinement the same author and Regnier presented extensions with syntactic differential operators for both Linear Logic [12] and λ-calculus [11]....

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