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The theory of calculi with explicit substitutions revisited

Delia Kesner
- Vol. 4646, pp 238-252
TLDR
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.

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Citations
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Book ChapterDOI

From reduction-based to reduction-free normalization

Olivier Danvy
TL;DR: The overall method builds on previous work by the author and his students on a syntactic correspondence between reduction semantics and abstract Machines and on a functional correspondence between evaluators and abstract machines.
Proceedings ArticleDOI

An Abstract Factorization Theorem for Explicit Substitutions

Beniamino Accattoli
TL;DR: A simple form of standardization, here called factorization, for explicit substitutions calculi, i.e. lambda-calculi where beta-reduction is decomposed in various rules, is studied and an abstract theorem deducing factorization from some axioms on local diagrams is developed.
Journal ArticleDOI

Non-idempotent intersection types and strong normalisation

Alexis Bernadet, +1 more
- 06 Oct 2013 - 
TL;DR: A typing system with non-idempotent intersection types, typing a term syntax covering three different calculi, and the instance based on filters is shown to be better at proving strong normalisation results for {\ lambda}S and {\lambda}lxr.
Book ChapterDOI

The structural λ-calculus

Beniamino Accattoli, +1 more
TL;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.
Journal ArticleDOI

Non-idempotent intersection types and strong normalisation

Alexis Bernadet, +1 more
- 10 Oct 2013 - 
TL;DR: In this paper, a typing system with non-idempotent intersection types is presented, where a term is typable if and only if it is strongly normalising, as it is the case in (many) systems with idempotent intersections.
References
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Journal ArticleDOI

Linear logic

Jean-Yves Girard
Book

The Lambda Calculus. Its Syntax and Semantics

Henk Barendregt
TL;DR: In this article, the Lambda-Calculus has been studied as a theory of composition and reduction, and the theory of reduction has been used to construct models of Lambda Theories.
Book

Term rewriting and all that

Franz Baader, +1 more
TL;DR: This chapter discusses abstract reduction systems, universal algebra, and Grobner bases and Buchberger's algorithm, and a bluffer's guide to ML Bibliography Index.
Book

The Definition of Standard ML

Robin Milner, +2 more
TL;DR: This book provides a formal definition of Standard ML for the benefit of all concerned with the language, including users and implementers, and the authors have defined their semantic objects in mathematical notation that is completely independent of StandardML.
Journal ArticleDOI

Linear logic

Patrick Lincoln
- 01 May 1992 - 
TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "The theory of calculi with explicit substitutions revisited" ?

In this paper the authors first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. 

The authors leave this for future work. Note however that λes-reduction can be translated to the correspondent notion of reduction in this calculus: thus for example App1 can be obtained by App followed by Gc. The authors believe that simultaneous substitutions will be needed to avoid axiom C while some technology like de Bruijn notation will be needed to avoid axiom α ( as in the λσ⇑ -calculus ).