Author

# Emmanuel Polonovski

Bio: Emmanuel Polonovski is an academic researcher from University of Paris. The author has contributed to research in topics: Equivalence relation & De Bruijn sequence. The author has an hindex of 3, co-authored 3 publications receiving 85 citations.

##### Papers

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25 Mar 2000TL;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].

38 citations

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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

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29 Mar 2004

TL;DR: This work proves the strong normalization (SN) of simply typed \(\overline{\lambda}\mu\widetilde{\mu}\)-calculus with explicit substitutions by formalizing a proof technique of SN via PSN (preservation of strongnormalization), and proves PSN by the perpetuality technique, as formalized by Bonelli.

Abstract: The \(\overline{\lambda}\mu\widetilde{\mu}\)-calculus, defined by Curien and Herbelin [7], is a variant of the λμ-calculus that exhibits symmetries such as term/context and call-by-name/call-by-value. Since it is a symmetric, and hence a non-deterministic calculus, usual proof techniques of normalization needs some adjustments to be made to work in this setting. Here we prove the strong normalization (SN) of simply typed \(\overline{\lambda}\mu\widetilde{\mu}\)-calculus with explicit substitutions. For that purpose, we first prove SN of simply typed \(\overline{\lambda}\mu\widetilde{\mu}\)-calculus (by a variant of the reducibility technique from Barbanera and Berardi [2]), then we formalize a proof technique of SN via PSN (preservation of strong normalization), and we prove PSN by the perpetuality technique, as formalized by Bonelli [5].

13 citations

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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.

77 citations

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11 Sep 2007TL;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

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TL;DR: The operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strongnormalisation, strong normalisation of simply typed terms, step by step simulation of @b-reduction and full composition are shown.

Abstract: We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply typed terms, step by step simulation of @b-reduction and full composition.

51 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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23 Aug 2010TL;DR: An untyped structural λ-calculus, called λj, is introduced, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction, and fundamental properties such as confluence and preservation of β-strong normalisation are proved.

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.

46 citations