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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TL;DR: The simulation technique introduced in Di Cosmo and Kesner (1997) is refined to show strong normalisation of $\l$-calculi with explicit substitutions via termination of cut elimination in proof nets.
Abstract: We refine the simulation technique introduced in Di Cosmo and Kesner (1997) to show strong normalisation of $\l$-calculi with explicit substitutions via termination of cut elimination in proof nets (Girard 1987). We first propose a notion of equivalence relation for proof nets that extends the one in Di Cosmo and Guerrini (1999), and show that cut elimination modulo this equivalence relation is terminating. We then show strong normalisation of the typed version of the $\ll$-calculus with de Bruijn indices (a calculus with full composition defined in David and Guillaume (1999)) using a translation from typed $\ll$ to proof nets. Finally, we propose a version of typed $\ll$ with named variables, which helps to give a better understanding of the complex mechanism of the explicit weakening notation introduced in the $\ll$-calculus with de Bruijn indices (David and Guillaume 1999).
36 citations
Cites background from "Proof Nets and Explicit Substitutio..."
...We are grateful to José Espı́rito Santo for suggesting a simpler termination proof for RE and to the anonymous referees of the current paper and of Di Cosmo et al. (2000) for their contributions to improve the presentation of this document....
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TL;DR: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets, and identifies a subcalculus that is shown to be as expressive as the full calculus.
Abstract: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize an isomorphism between the two systems: every single rewriting step on the calculus maps to a single step on proof nets, and vice-versa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way, and identify a subcalculus that is shown to be as expressive as the full calculus.
34 citations
01 Sep 2014
TL;DR: Two non-idempotent intersection type systems for the linear substitution calculus, a calculus with partial substitutions acting at a distance that is a computational interpretation of linear logic proof-nets, are defined.
Abstract: We define two non-idempotent intersection type systems for the linear substitution calculus, a calculus with partial substitutions acting at a distance that is a computational interpretation of linear logic proof-nets. The calculus naturally express linear-head reduction, a notion of evaluation of proof nets that is strongly related to abstract machines. We show that our first (resp. second) quantitave type system characterizes linear-head, head and weak (resp. strong) normalizing sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which decreases with respect to each reduction relation considered in the paper.
32 citations
Cites background from "Proof Nets and Explicit Substitutio..."
...Relationship with Linear Logic [24] and Relevant Logic [23, 18] provides an insight on the information refinement aspect of non-idempotent intersection types....
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...But calculi with ES can also be interpreted in Linear Logic [22, 28, 26, 5] by implementing another kind of operational semantics: their dynamics is defined using contexts (i.e. terms with holes) that allows the ES to act directly at a distance on single variable occurrences, with no need to commute with any other constructor in between....
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...But calculi with ES can also be interpreted in Linear Logic [22, 28, 26, 5] by implementing another kind of operational semantics: their dynamics is defined using contexts (i....
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19 Apr 2005
TL;DR: A simple term language with explicit operators for erasure, duplication and substitution enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's Proof Nets is presented.
Abstract: We present a simple term language with explicit operators for erasure, duplication and substitution enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's Proof Nets. We establish the good operational behaviour of the language by means of some fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of well-typed terms and step by step simulation. This formalism is the first term calculus with explicit substitutions having full composition and preserving strong normalisation.
31 citations
Cites background from "Proof Nets and Explicit Substitutio..."
...Also, it can be shown [18] that there is a simple translation from λws into the Proof...
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...1 In contrast to λws with names [17,18], where terms affected by substitutions have a complex format t[x, u, Γ, ∆]...
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TL;DR: The paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL) by showing how standardization for sps allows to prove SN of LL, using as usual Girard's reducibility candidates.
Abstract: The paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard's original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard's proof-nets, and we apply to sps Gandy's method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard's reducibility candidates.
30 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.
Abstract: The collection of TCS issues is about 1 meter high, 17,000 pages long and it contains 1100 papers. When in 1974 Einar Fredriksson and myself started talking about the creation of a journal dedicated to Theoretical Computer Science we were very far from even dreaming that it could take such an extension within twelve years. We were also a bit shy: what could such a journal, very theoretical indeed and hard to read, be useful to, and who would read it? Fortunately, some people encouraged us and indeed helped us a lot, Mike Paterson who was at that time President of EATCS and who accepted to become Associate Editor, Albert Meyer who was a very active editor at the beginning, Arto Salomaa, who was to become President of EATCS shortly afterwards. Indeed, I should mention all the first members of the Editorial Board, for TCS would never have come to existence without them. Theoretical Computer Science is not a clearly defined discipline with neat borderlines: it is more a state of mind, the conviction that the observed computation phenomena can be formally described and analysed as any physical phenomenon; the conviction that such a formal description helps to understand these phenomena and to master them in order to design better algorithms, better computers, better systems. Our fundamental activity is not to prove theorems in strange mathematical theories, it is to model a complicated reality and in this respect it has to be compared with theoretical physics or what we call in French “Mecanique rationnelle”. This comparison can be pursued rather far, for we also use all possible mathematical concepts and methods and when we do not find appropriate ones in traditional mathematics we create them. The aim is quite clear: using the compact and unambiguous language of mathematics brings to life concepts and methods which will be useful to all designers, builders and users of computer systems, exactly in the same way as matrix calculus or Fourier series and transforms are useful to all engineers and technicians in the electric and electronic industry. And when one thinks about the amount of time it took to build the mathematical theory of matrices and to polish and simplify it up to the state in which it could be taught to all future engineers and become a tool in daily use, we can be extremely satisfied by the development of Theoretical Computer Science. It is true that concepts and methods which were still vague and unclear when TCS was created became essential tools for all industrial designers and manufacturers, in algorithmics, in semantics, in automata theory and control, etc. . . . Certainly, TCS can be proud to have contributed to this development. Coming back to what I was saying a few minutes ago, 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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