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The theory of calculi with explicit substitutions revisited
 CSL 2007
, 2007
"... 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 metalevel substitutions) they were implementing. In this paper we fi ..."
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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 metalevel 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 lambdacalculus which enjoys fundamental properties such as simulation of onestep betareduction, confluence on metaterms, preservation of betastrong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proofnets.
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... 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 proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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Cited by 6 (3 self)
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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 proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
Investigations into the duality of computation
"... The work presented here is an extension of a previous work realised jointly with PierreLouis Curien [CH00]. The current work focuses on the pure calculus of variables and binders that operates at the core of the duality between callbyname and callbyvalue evaluations. A CurryHowardde Bruijn co ..."
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Cited by 1 (0 self)
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The work presented here is an extension of a previous work realised jointly with PierreLouis Curien [CH00]. The current work focuses on the pure calculus of variables and binders that operates at the core of the duality between callbyname and callbyvalue evaluations. A CurryHowardde Bruijn correspondence is given that shed light on some aspects of Gentzen’s sequent calculus. This includes a sequentfree presentation of it.
A Typed Calculus Supporting Shallow Embeddings of Abstract Machines
, 2005
"... The goal of this work is to draw a formal connection between steps taken by abstract machines and reductions in a system of proof terms for a version the sequent calculus. We believe that by doing so we shed light on some essential characteristics of abstract machines, proofs in sequent calculus sys ..."
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The goal of this work is to draw a formal connection between steps taken by abstract machines and reductions in a system of proof terms for a version the sequent calculus. We believe that by doing so we shed light on some essential characteristics of abstract machines, proofs in sequent calculus systems, and weak normalization of λterms. The machines that we consider are the (callbyname) Krivine machine and a callbyvalue machine that may be called a “righttoleft CEK machine ” but with some modifications can be seen as a protoZINC machine. The formal connection we exhibit is, in fact, an embedding of the machines into the term calculus. We embed runtime data structures, such as the control stack and environment, in such a way that the operational semantics of the machine corresponds to reduction steps in the calculus. The abstract machine machine transitions are captured as term reductions. This is in contrast to specifying the operational semantics on top of the
The duality of computation (notes for the 3rd International Workshop on HigherOrder Rewriting)
"... Abstract. The work presented here is an extension of a previous work realised jointly with PierreLouis Curien [1]. The title has been kept unchanged. The current work focuses on the pure calculus of variables and binders that operates at the core of the duality between callbyname and callbyvalu ..."
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Abstract. The work presented here is an extension of a previous work realised jointly with PierreLouis Curien [1]. The title has been kept unchanged. The current work focuses on the pure calculus of variables and binders that operates at the core of the duality between callbyname and callbyvalue evaluations. A CurryHowardde Bruijn correspondence is given that shed light on some aspects of Gentzen’s sequent calculus. This includes a sequentfree presentation of it.
Deriving SN from PSN: a general proof technique
, 909
"... In the framework of explicit substitutions there is two termination properties: preservation of strong normalization (PSN), and strong normalization (SN). Since there are not easily proved, only one of them is usually established (and sometimes none). We propose here a connection between them which ..."
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In the framework of explicit substitutions there is two termination properties: preservation of strong normalization (PSN), and strong normalization (SN). Since there are not easily proved, only one of them is usually established (and sometimes none). We propose here a connection between them which helps to get SN when one already has PSN. For this purpose, we formalize a general proof technique of SN which consists in expanding substitutions into “pure ” λterms and to inherit SN of the whole calculus by SN of the “pure ” calculus and by PSN. We apply it successfully to a large set of calculi with explicit substitutions, allowing us to establish SN, or, at least, to trace back the failure of SN to that of PSN. Contents
Spécialite ́ : Mathématiques et Informatique
, 2009
"... properties of symmetric logical calculi ..."
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