The language X : circuits, computations and classical logic

The language X : circuits, computations and classical logic (2005)

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by Steffen Van Bakel , Stéphane Lengr , Pierre Lescanne , Fife Ky Ss , École Normale Supérieure De Lyon

Venue:	Proceedings of Ninth Italian Conference on Theoretical Computer Science (ICTCS’05
Citations:	22 - 17 self

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@INPROCEEDINGS{Bakel05thelanguage,
    author = {Steffen Van Bakel and Stéphane Lengr and Pierre Lescanne and Fife Ky Ss and École Normale Supérieure De Lyon},
    title = {The language X : circuits, computations and classical logic},
    booktitle = {Proceedings of Ninth Italian Conference on Theoretical Computer Science (ICTCS’05},
    year = {2005},
    pages = {81--96},
    publisher = {Springer-Verlag}
}

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Abstract

Abstract. We present the syntax and reduction rules for X, an untyped language that is well suited to describe structures which we call “circuits ” and which are made of parts that are connected by wires. To demonstrate that X gives an expressive platform, we will show how, even in an untyped setting, that we can faithfully embed algebraic objects and elaborate calculi, like the naturals, the λcalculus, Bloe and Rose’s calculus of explicit substitutions λx, and Parigot’s λµ. 1

Citations

359	Explicit Substitutions - Abadi, Cardelli, et al. - 1992
1	P �α † �xQ → A (act-L) P �α † �xQ, and the result follows from the above. 2. Similar. Lemma 9. 1. P � δ † �z〈z.α〉 → P[α/δ], if P is pure. 2. 〈z.α〉�α † �xP → P[z/x], if P is pure. Proof. 1. We start by showing, by induction on the structure of nets, that P - Then
1	xN � β·δ) � δ † �z〈z.α〉 →A (exp) �xN - Then
1	x.δ〉 � δ † �z〈z.α〉 →A (cap) 〈x.α〉 = ∆ 〈x.δ〉[α/δ - Then
1	P� δ † �z〈z.α〉 →A P� δ † �z〈z.α〉, and the result follows from the above. 2. Similar. Lemma 11. If N1 and N2 are natural numbers, 1. (N1�α [f] �x〈x.α〉)�α - Then
1	there exists B such that Γ ⊢λ P : B→A and Γ ⊢λ Q :B. By induction, both D1 :: ⌈P ⌋ γ · · · Γ ⊢ γ:B→A and D2 :: ⌈Q ⌋ β · · · Γ ⊢ β:B exist, and we can construct - Then