Proof Nets and Explicit Substitutions
25 Mar 2000-pp 63-81
TL;DR: 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].
Citations
More filters
19 Aug 2014
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.
Abstract: It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. 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. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.
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])....
[...]
11 Sep 2007
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.
Abstract: Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing.
In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology 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. The calculus also admits a natural translation into Linear Logic's proof-nets.
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....
[...]
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.
Abstract: Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. 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. A typed version is also presented.
49 citations
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 ?
Abstract: We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier?s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol?s untyped ?-calculus with resources extended with a linear?nonlinear reduction a la Ehrhard and Regnier?s differential ?-calculus. Such reduction comes in two flavours: baby-step and giant-step s-reduction. The translation, based on Girard?s encoding A?B~!A?B and as such extending the usual one for ?-calculus into proof nets, enjoys bisimulation for giant-step s-reduction. From this result we also derive confluence of both reductions.
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....
[...]
Proceedings Article•
23 Aug 2010
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.
Abstract: Inspired by a recent graphical formalism for λ-calculus based on Linear Logic technology, we introduce an untyped structural λ-calculus, called λj, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction. Firstly, we prove fundamental properties such as confluence and preservation of β-strong normalisation. Secondly, we use λj to describe known notions of developments and superdevelopments, and introduce a more general one called XL-development. Then we show how to reformulate Regnier's s-equivalence in λj so that it becomes a strong bisimulation. Finally, we prove that explicit composition or de-composition of substitutions can be added to λj while still preserving β-strong normalisation.
37 citations
References
More filters
DOI•
01 Jan 1997
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.
Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
129 citations
"Proof Nets and Explicit Substitutio..." refers background in this paper
...As customary in explicit substitutions calculi with names [3], we work modulo α-conversion, so that we can suppose that in the rule Weak the set ∆ does not contain variables that are bound in M ....
[...]
01 Jan 1995
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.
Abstract: In this paper we introduce and study a new-calculus with explicit substitution, xgc, which has two distinguishing features: rst, it retains the use of traditional variable names, specifying terms modulo renaming; this simpliies the reduction system. Second, it includes reduction rules for explicit garbage collection; this simpliies several proofs. We show that xgc is a conservative extension which preserves strong normalisation (PSN) of the untyped-calculus. The result is obtained in a modular way by rst proving it for garbage-free reduction and then extending tòreductions in garbage'. This provides insight into the counterexample to PSN for of Melli es (1995); we exploit the abstract nature of xgc to show how PSN is in connict with any reasonable substitution composition rule (except for trivial composition rules of which we mention one).
111 citations
"Proof Nets and Explicit Substitutio..." refers background in this paper
...Finally, in [10], the first two authors of this work showed for the first time that explicit substitutions could be tightly related to linear logic’s proof nets, by providing a translation into a variant of proof nets from λx [19, 4], a simple calculus with explicit substitutions and named variables, but no composition....
[...]
22 Jun 1992
TL;DR: A new formal embodiment of J.-Y.
Abstract: A new formal embodiment of J.-Y. Girard's (1989) geometry of interaction program is given. The geometry of interaction interpretation considered is defined, and the computational interpretation is sketched in terms of dataflow nets. Some examples that illustrate the key ideas underlying the interpretation are given. The results, which include the semantic analogue of cut-elimination, stated in terms of a finite convergence property, are outlined. >
98 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]....
[...]
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.
Abstract: Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. 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. A typed version is also presented.
49 citations
29 Jun 1997
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.
Abstract: In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for linear logic, via proof nets. This correspondence allows us to prove that a typed version of the /spl lambda/x-calculus is strongly normalizing, as well as of all the calculi that can be translated to it keeping normalization properties such as /spl lambda//sub v/, /spl lambda//sub s/, /spl lambda//sub d/ and /spl lambda//sub f/. In order to achieve this result, we introduce a new notion of reduction in proof nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it corresponds to more identifications of proofs in linear logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.
47 citations