# 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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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.

71 citations

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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

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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.

48 citations

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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.

42 citations

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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

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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 .

2,199 citations

### "Proof Nets and Explicit Substitutio..." refers background or methods in this paper

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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

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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.

559 citations

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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.

305 citations

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