## Explicit substitutions (1996)

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Citations: | 410 - 11 self |

### BibTeX

@MISC{Abadi96explicitsubstitutions,

author = {M. Abadi and L. Cardelli and P.-L. Curien and J.-J. Lévy},

title = {Explicit substitutions},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

The λσ-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λσ-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.

### Citations

1183 |
The lambda calculus: Its syntax and semantics
- Barendregt
- 1984
(Show Context)
Citation Context ...he theory and its implementations becomes highly nontrivial, and the correctness of the implementations can be di cult to establish. In this paper we study the -calculus, a re nement of the -calculus =-=[1]-=- where substitutions are manipulated explicitly. Substitutions have syntactic representations, and if a is a term and s is a substitution then the term a[s] represents a with the substitution s. We ca... |

262 | Combinatory Reduction Systems - Klop - 1980 |

142 | Typeful Programming
- Cardelli
- 1989
(Show Context)
Citation Context ...erve in designing typechecking algorithms. In particular, their study has been of help for both soundness and e ciency in the design of the 2stypechecking algorithm for the Quest programming language =-=[3]-=-. We postpone discussion of the untyped calculi to section 3 and of the typed calculi to sections 4 and 5. We now proceed with a general technical overview. 2 Overview The technical details of the -ca... |

130 | Con properties of weak and strong calculi of explicit substitutions - Curien, Hardin, et al. - 1996 |

118 |
Semantics and Pragmatics of the Lambda-Calculus
- Wadsworth
(Show Context)
Citation Context ..., relevant in the implementation of both logics and programming languages. The term afb=xg may contain many copies of b (for instance, if a = xxxx); without sophisticated structure-sharing mechanisms =-=[18]-=-, performing substitutions immediately causes a size explosion. Therefore, in practice, substitutions are delayed and explicitly recorded; the application of substitutions is independent, and not coup... |

102 | Equations and rewrite rules: A survey
- Huet, Oppen
- 1980
(Show Context)
Citation Context ...he largest variant considered in 3.2). Hardin and Laville have established the termination of SUBST [12]. Noetherianity simpli es the proof of con uence. By a well-known lemma, local con uence su ces =-=[14]-=-; it can be checked by examining critical pairs, according to the Knuth-Bendix test. For example, for the critical pair (1[id])[s] ! 1[s] and (1[id])[s] ! 1[id s] local con uence is ensured through th... |

46 |
On laziness and optimality in lambda interpreters: Tools for specification and analysis
- Field
- 1990
(Show Context)
Citation Context ...f con uence properties and to the practicality of the -calculus. Simultaneously with our work, Field developed a system almost identical to our basic system, too, and claimed some of the same results =-=[10]-=-. Thus, we share a starting point. However, Field's paper is an investigation of optimality properties of reduction schemes, so for example Field went on to consider a labelled calculus. In contrast, ... |

42 |
Con Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL
- Hardin
- 1989
(Show Context)
Citation Context ...tract λσ machines. We describe one machine, which extends Krivine’s weak reduction machine [13] with strong reduction. In her study of categorical combinators, Hardin proposed systems similar to ours =-=[8]-=-. In particular, Hardin’s system E + (Beta) is the homomorphic image of our basic system. We rely on some of her techniques to prove our results, and not surprisingly we find confluence properties sim... |

41 |
A con calculus of substitutions
- Levy
- 1989
(Show Context)
Citation Context ...ly, Hardin and Lévy have succeeded in obtaining confluence with metavariables for both terms and substitutions, by slightly changing the syntax and the set of equations. Their results are reported in =-=[10]-=-. 3.3 The λσ-calculus with names Let us discuss a more traditional formulation of the calculus, with variable names x,y,z,... , as a small digression. Two ways seem viable. In one approach, we conside... |

19 |
Intuitionistic type theory. Notes by G. Sambin of a series of lectures given
- Martin-Löf
- 1980
(Show Context)
Citation Context ...nil The typing rules become more complex because types may contain type variables, which must be looked up in the appropriate environments. (The problem arises in full generality with dependent types =-=[17], and-=- some readers may nd it helpful to think about calculi of substitutions with dependent types.) In particular, the typing axiom for 1 shown above becomes the rule: E ` A :: Ty A; E ` 1 : A["] The ... |

9 |
Bruijn, Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, Indagationes Mathematicae
- de
- 1972
(Show Context)
Citation Context ...tion methods. We give reduction rules that extend those of the classical -calculus and investigate their con uence. We concentrate on a presentation that relies on De Bruijn's numbering for variables =-=[2]-=-, and brie y discuss presentations with more traditional variable names. Then we proceed to consider typed calculi of substitutions, in De Bruijn notation. We discuss typing rules for a rst-order syst... |

7 |
Con uence Properties of Weak and Strong Calculi of Explicit Substitutions
- Curien, Hardin, et al.
- 1991
(Show Context)
Citation Context ... works as a counterexample to con uence: Y (Y ( x[1[u (1 id)] (" (u ((21) id)))])) where Y is a xpoint combinator, x is a term metavariable, and u is a substitution metavariable. The proof appear=-=s in [6]-=-. Let us just summarize the informal argument. Call b = Y (c) the term above. It reduces to both x[u ((cb) id)] and c(x[u ((cb) id)]). To get a common reduct of these two terms, we need to apply SCons... |

6 |
Un resultat de Completude pour les substitutions explicites. Comptes Rendus de l'Academie des Sciences
- Curien, Ros
- 1991
(Show Context)
Citation Context ...losures, by undoing Beta steps. This reconstruction can be performed in a systematic way with a suitable rewriting system; soundness and completeness results follow. We refer the interested reader to =-=[8]-=-. 5 Second-order theories Type rules and typecheckers are also needed for second-order calculi. Unfortunately, the situation is more complex than at the rst order, because types include binding constr... |

6 |
Proof of Termination of the Rewriting System SUBST on CCL, Theoretical Computer Science 46
- Hardin, Laville
- 1986
(Show Context)
Citation Context ...to a one-step rewriting of a system SUBST of categorical rewriting rules (the exact translation of the largest variant considered in 3.2). Hardin and Laville have established the termination of SUBST =-=[12]-=-. Noetherianity simpli es the proof of con uence. By a well-known lemma, local con uence su ces [14]; it can be checked by examining critical pairs, according to the Knuth-Bendix test. For example, fo... |

4 |
The λρ-calculi : an abstract framework for closures
- Curien
- 1988
(Show Context)
Citation Context ...d is a conservative extension of the -calculus. Moreover, the -calculus is strongly connected with the categorical understanding of the -calculus, where a substitution is interpreted as a composition =-=[4]-=-. 1sWe propose the -calculus as a step in closing the gap between the classical -calculus and concrete implementations. The calculus is a vehicle for designing, understanding, verifying, and comparing... |

2 |
A Con uent Calculus of Substitutions
- Hardin, Levy
- 1989
(Show Context)
Citation Context ...tly, Hardin and Levy have succeeded in obtaining con uence with metavariables for both terms and substitutions, by slightly changing the syntax and the set of equations. These results are reported in =-=[13, 6]-=-. 3.3 The -calculus with names Let us discuss a more traditional formulation of the calculus, with variable names x; y; z; : :: , as a small digression. Two ways seem viable. In one approach, we consi... |

1 |
R os, Normalisation Forte des Substitutions
- Curien, Hardin, et al.
- 1991
(Show Context)
Citation Context ...t. For example, for the critical pair (1[id])[s] ! 1[s] and (1[id])[s] ! 1[id s] local con uence is ensured through the IdL rule. A di erent proof of termination for SUBST and has been found recently =-=[7]. 2 Since is no-=-etherian, let us examine the form of normal forms. A substitution in normal form is necessarily in the form a 1 (a 2 (:::(am U):::)) where U is either id or a composition " (:::(" "):::... |

1 |
Con uence Results for the Pure Strong Categorical Combinatory Logic CCL: -calculi as
- Hardin
- 1989
(Show Context)
Citation Context ...abstract machines. We describe one machine, which extends Krivine's weak reduction machine [16] with strong reduction. In her study of categorical combinators, Hardin proposed systems similar to ours =-=[11]-=-. In particular, Hardin's system E +(Beta) is a homomorphic image of our basic system. We rely on some of her techniques to prove our results, and not surprisingly we nd con uence properties similar, ... |

1 |
Proof of Termination of the Rewriting
- Hardin, Laville
- 1986
(Show Context)
Citation Context ...to a one-step rewriting of a system SUBST of categorical rewriting rules (the exact translation of the largest variant considered in 3.2). Hardin and Laville have established the termination of SUBST =-=[12]-=-. Noetherianity simpli es the proof of con uence. By a well-known lemma, local con uence su ces [14]; it can be checked by examining critical pairs, according to the Knuth-Bendix test. For example, fo... |

1 | The Lambda Calculus: Its Syntar and Semantics - Barendregt - 1985 |

1 | Bruijn, Lambda-calcuIus Notation with Nameless Dummies, a Tool for Automatic Formula X,Ianipulation - De - 1972 |