Proof Nets and Explicit Substitutions
Roberto Di Cosmo,Delia Kesner,Emmanuel Polonovski +2 more
- pp 63-81
TLDR
The simulation technique introduced in [10] is refined to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets and a version of typed λl with named variables is proposed which helps to better understand the complex mechanism of the explicit weakening notation.Abstract:
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]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λl-calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λl to proof nets. Finally, we propose a version of typed λl with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λl-calculus with de Bruijn indices [8].read more
Citations
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Proceedings ArticleDOI
Distilling abstract machines
TL;DR: The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic, and shows that the LSC is a complexity-preserving abstraction of abstract machines.
Book ChapterDOI
The theory of calculi with explicit substitutions revisited
TL;DR: Very simple technology is used to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition.
Journal ArticleDOI
A λ-calculus with explicit weakening and explicit substitution
René David,Bruno Guillaume +1 more
TL;DR: The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening.
Journal ArticleDOI
Intuitionistic differential nets and lambda-calculus
TL;DR: Normalization of the exponential reduction and confluence of the full one is proved and a translation of Boudol?s untyped ?-calculus with resources extended with a linear?nonlinear reduction a la Ehrhard and Regnier?s differential ?
Proceedings Article
The structural lambda-calculus
Beniamino Accattoli,Delia Kesner +1 more
TL;DR: In this article, an untyped structural λ-calculus, called λj, was introduced, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction.
References
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Book ChapterDOI
Explicit Cyclic Substitutions
TL;DR: It is demonstrated how this may be used to describe standard binding constructions (let and letrec)—directly using substitution and fixed point induction as well as using ‘small-step’ rewriting semantics where substitution is interleaved with the mechanics of the following β-reductions.
Book ChapterDOI
Strong Normalization of Proof Nets Modulo Structural Congruences
TL;DR: A notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links is proposed, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes.
Dissertation
Un calcul de substitution avec etiquettes
TL;DR: Kamareddine et rios as mentioned in this paper propose le lambda-se-calculus, which is a lambda-sigma-calculus with substitutions explicite.
Journal ArticleDOI
λCalculi with Explicit Substitutions Preserving Strong Normalization
TL;DR: In this article, the preservation of β-strong normalization by two different confluent λ-calculi with explicit substitutions defined in [96] was studied. But the particularity of these calculi, called λ d and λ n respectively, is that both have a (weak) composition operator for substitutions.