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].
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14 Dec 2013
TL;DR: Strong normalization of the typed atomic lambda-calculus is proved using Tait’s reducibility method to prove fully lazy sharing to be reproduced in a typed setting.
Abstract: The atomic lambda-calculus is a typed lambda-calculus with explicit sharing, which originates in a Curry-Howard interpretation of a deep-inference system for intuitionistic logic. It has been shown that it allows fully lazy sharing to be reproduced in a typed setting. In this paper we prove strong normalization of the typed atomic lambda-calculus using Tait’s reducibility method.
5 citations
Cites background from "Proof Nets and Explicit Substitutio..."
...It is a leading principle behind, among others, explicit substitution calculi [1, 18, 8, 9, 15, 2], term calculi with strategies or higher-order transformations [14, 3], and sharing graphs in the style of Lamping [17, 4, 21]....
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25 Jun 2013
TL;DR: It is shown that logical polarity can be exploited to obtain an implicit, compact, and natural representation of boxes: in an expressive polarized dialect of linear logic, boxes may be represented by simply recording some of the polarity changes occurring in the box at level 0.
Abstract: The sequential nature of sequent calculus provides a simple definition of cut-elimination rules that duplicate or erase sub-proofs. The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graphs. We show that logical polarity can be exploited to obtain an implicit, compact, and natural representation of boxes: in an expressive polarized dialect of linear logic, boxes may be represented by simply recording some of the polarity changes occurring in the box at level 0. The content of the box can then be recovered locally and unambiguously. Moreover, implicit boxes are more parallel than explicit boxes, as they realize a larger quotient. We provide a correctness criterion and study the novel and subtle cut-elimination dynamics induced by implicit boxes, proving confluence and strong normalization.
4 citations
Cites background or methods from "Proof Nets and Explicit Substitutio..."
...Let us point out that via the Curry-Howard correspondence between MELLP and .µ-calculus this work provides implicit boxes for classical logic....
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...Therefore, non-linear cut-elimination rules can easily be de.ned by duplicating or erasing sub-trees, as in Figure 1.a. Switching to proof nets, the situation radically changes....
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25 Aug 2003
TL;DR: The λ ws -calculus is a λ-calculus with explicit substitutions with desired properties: step by step simulation of β, confluence on terms with meta-variables and preservation of the strong normalization.
Abstract: The λ ws -calculus is a λ-calculus with explicit substitutions introduced in [4]. It satisfies the desired properties of such a calculus: step by step simulation of β, confluence on terms with meta-variables and preservation of the strong normalization. It was conjectured in [4] that simply typed terms of λ ws are strongly normalizable. This was proved in [7] by Di Cosmo & al. by using a translation of λ ws into the proof nets of linear logic. We give here a direct and elementary proof of this result. The strong normalization is also proved for terms typable with second order types (the extension of Girard’s system F). This is a new result.
3 citations
Cites background or methods from "Proof Nets and Explicit Substitutio..."
...Di Cosmo, Kesner and Polonovsky [ 7 ] understood the relation between ¸ws and linear logic and, by using a translation of ¸ws into proof nets, they proved this conjecture....
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...This was proved in [ 7 ] by Di Cosmo & al. by using a translation of ¸ws into the proof nets of linear logic....
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...Theorem 5 below has first been proved in [ 7 ] by Di Cosmo & al. It is of course a trivial consequence of theorem 7 of section 5. However, the proof presented below is interesting in itself because it is purely arithmetical whereas the one of section 5 is not....
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...Note that permutation of rules is also the main ingredient in the proof of [ 7 ]....
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Posted Content•
TL;DR: This dissertation proves lower bounds on the inherent difficulty of deciding flow analysis problems in higher-order programming languages by giving exact characterizations of the computational complexity of 0CFA, the kCFA hierarchy, and related analyses.
Abstract: This dissertation proves lower bounds on the inherent difficulty of deciding flow analysis problems in higher-order programming languages. We give exact characterizations of the computational complexity of 0CFA, the $k$CFA hierarchy, and related analyses. In each case, we precisely capture both the expressiveness and feasibility of the analysis, identifying the elements responsible for the trade-off.
0CFA is complete for polynomial time. This result relies on the insight that when a program is linear (each bound variable occurs exactly once), the analysis makes no approximation; abstract and concrete interpretation coincide, and therefore pro- gram analysis becomes evaluation under another guise. Moreover, this is true not only for 0CFA, but for a number of further approximations to 0CFA. In each case, we derive polynomial time completeness results.
For any $k > 0$, $k$CFA is complete for exponential time. Even when $k = 1$, the distinction in binding contexts results in a limited form of closures, which do not occur in 0CFA. This theorem validates empirical observations that $k$CFA is intractably slow for any $k > 0$. There is, in the worst case---and plausibly, in practice---no way to tame the cost of the analysis. Exponential time is required. The empirically observed intractability of this analysis can be understood as being inherent in the approximation problem being solved, rather than reflecting unfortunate gaps in our programming abilities.
3 citations
Dissertation•
01 Jan 2019
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3 citations
References
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TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .
2,304 citations
"Proof Nets and Explicit Substitutio..." refers background or methods in this paper
...In this paper 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]....
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...While we refer the interested reader to [13] for more details on linear logic in general, we give here a one-sided presentation of the sequent calculus for MELL:...
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TL;DR: This contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science.
1,480 citations
01 Dec 1989
TL;DR: The λ&sgr;-calculus is a refinement of the λ-Calculus where substitutions are manipulated explicitly, and provides a setting for studying the theory of substitutions, with pleasant mathematical properties.
Abstract: The ls-calculus is a refinement of the l-calculus where substitutions are manipulated explicitly. The ls-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical l-calculus and concrete implementations.
577 citations
"Proof Nets and Explicit Substitutio..." refers background in this paper
...The pioneer calculus with explicit substitutions, λσ, was introduced in [1] as a bridge between the classical λ-calculus and concrete implementations of functional programming languages....
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TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.
Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.
321 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]....
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