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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 ..."
Abstract

Cited by 6 (1 self)
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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 3 (2 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.
Labelled calculi of resources
"... Lévy’s labelled λcalculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λevaluators: labels relate to the multiplicative information of paths. In this paper, we generalise the structure of labels, and the underlying t ..."
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Lévy’s labelled λcalculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λevaluators: labels relate to the multiplicative information of paths. In this paper, we generalise the structure of labels, and the underlying term structure, in order to keep track of exponential information too. We first define two labelled calculi with explicit substitutions and resource management, where labels are in close correspondence with paths in callbyvalue and callbyname translations of the λcalculus into linear logic proof nets, respectively. We observe a tight relationship between labels and the dynamics of substitutions; this will then guide us through the design of a third calculus that combines the advantages of the previous two, where labels fully reflect the dynamics of substitutions. 1
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
A Simplified Suspension Calculus and its Relationship to Other Explicit Substitution Calculi
"... This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this calculus provides a cumbersome encoding of substitution compos ..."
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This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this calculus provides a cumbersome encoding of substitution composition, an operation that is important to the efficient realization of reduction. This encoding is simplified here, resulting in a treatment that is easy to use directly in applications. The rationalization consists of the elimination of a practically inconsequential flexibility in the unravelling of substitutions that has the inadvertent side effect of losing contextual information in terms; the modified calculus now has a structure that naturally supports logical analyses, such as ones related to the assignment of types, over lambda terms. The overall calculus is shown to have pleasing theoretical properties such as a strongly terminating subcalculus for substitution and confluence even in the presence of term meta variables that are accorded a grafting interpretation. Another contribution of the paper is the identification of a broad set of properties that are desirable for explicit substitution calculi to support and a classification of a variety of proposed systems based on these. The suspension calculus is used as a tool in this study. In particular, mappings are described between it and the other calculi towards understanding the characteristics of the latter.
Compressing Polarized Boxes
"... Abstract—The sequential nature of sequent calculus provides a simple definition of cutelimination rules that duplicate or erase subproofs. The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graph ..."
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Abstract—The sequential nature of sequent calculus provides a simple definition of cutelimination rules that duplicate or erase subproofs. 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 cutelimination dynamics induced by implicit boxes, proving confluence and strong normalization.