Proof Nets and Explicit Substitutions
Roberto Di Cosmo,Delia Kesner,Emmanuel Polonovski +2 more
- pp 63-81
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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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Preservation of termination for explicit substitution
TL;DR: The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.
Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection
C.J. Bloo,Kristoffer H. Rose +1 more
TL;DR: It is shown that xgc is a conservative extension which preserves strong normalisation (PSN) of the untyped-calculus, which has two distinguishing features: rst, it retains the use of traditional variable names, specifying terms modulo renaming; this simpliies the reduction system.
Proceedings ArticleDOI
New foundations for the geometry of interaction
Samson Abramsky,Radha Jagadeesan +1 more
TL;DR: A new formal embodiment of J.-Y.
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.
Proceedings ArticleDOI
Strong normalization of explicit substitutions via cut elimination in proof nets
R. Di Cosmo,D. Kesner +1 more
TL;DR: In this paper, the authors show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for linear logic, via proof nets, and prove that a typed version of the lambda/x-calculus is strongly normalizing, as well as all the calculi that can be translated to it keeping normalization properties such as /spl lambda//sub v/, etc.