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### Citations

436 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1992
(Show Context)
Citation Context ...esign of indexed versions of known named calculi. 1 Introduction This article is devoted to explicit substitutions (ES, for short), a formalism that has attracted attention since the appearance of λσ =-=[1]-=- and, later, of Melliès’ counterexample [17], showing the lack of the preservation of β -strong normalization property (PSN, for short) in λσ . One of the main motivations behind the field of ES is s... |

129 | 1996], `Confluence properties of weak and strong calculi of explicit substitutions
- Curien, Hardin, et al.
(Show Context)
Citation Context ... such a task hard. In this direction, a mix of ideas from λ rex and calculi with n-ary substitutions (i.e., λσ -styled calculi) may lead to the solution of both issues. Particularly, a swap-based λσ⇑ =-=[7]-=- could be an option. This comes from the following observation: in λσ⇑, the (Lamb) rule is: (Lamb) (λa)[s]→ λa[⇑(s)] where the intuitive semantics of ⇑(s) is: 1 ·(s ◦↑). We observe here that this is n... |

108 |
a calculus of explicit substitutions which preserves strong normalisation
- Benaissa, Briaud, et al.
- 1995
(Show Context)
Citation Context ...pendently discovered, the swapping mechanism introduced in this article was first depicted by de Bruijn for his ES calculus Cλξφ [6], and, later, updated w.r.t. notation – λξφ – and compared to λυ in =-=[2]-=-. We will now briefly discuss the main differences between these calculi and our swapping-based approach. Firstly, neither Cλξφ nor λξφ have composition of substitutions nor Garbage Collection, two ke... |

89 | A λ-calculus a la de Bruijn with Explicit Substitutions.
- Kamareddine, Rıos
- 1995
(Show Context)
Citation Context ...less notion makes sense in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s =-=[11]-=-, λ se [12] or λ t [13]. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free... |

77 | Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection
- Bloo, Rose
- 1995
(Show Context)
Citation Context ... first devised λ r, and then made substitutions explicit orienting the definition for λ r’s meta-substitution. At that point, we got a calculus we called λ re, which turned out to be isomorphic to λx =-=[4, 3]-=-. Encouraged by this result, we added Garbage Collection to λ re, obtaining a calculus isomorphic to λxgc [4]: λ regc. Finally, we added composition of substitutions in the style of λex to λ regc, obt... |

35 | Explicit Substitution: on the Edge of Strong Normalisation
- Bloo, Geuvers
- 1998
(Show Context)
Citation Context ... first devised λ r, and then made substitutions explicit orienting the definition for λ r’s meta-substitution. At that point, we got a calculus we called λ re, which turned out to be isomorphic to λx =-=[4, 3]-=-. Encouraged by this result, we added Garbage Collection to λ re, obtaining a calculus isomorphic to λxgc [4]: λ regc. Finally, we added composition of substitutions in the style of λex to λ regc, obt... |

33 | Nominal rewriting.
- Fernández, Gabbay
- 2007
(Show Context)
Citation Context ...le to λ rex should be fairly simple using the decrement meta-operator. Finally, studying the possible relation between these swapping-based formalisms and nominal logic or nominal rewriting (see e.g. =-=[10, 9]-=-) could be an interesting approach in gathering a deeper understanding of λ r’s underlying logic. 10 Swapping: a natural bridge between named and indexed explicit substitution calculi Acknowledgements... |

25 |
Typed lambda-calculi with explicit substitutions may not terminate
- Melliès
- 1995
(Show Context)
Citation Context ...culi. 1 Introduction This article is devoted to explicit substitutions (ES, for short), a formalism that has attracted attention since the appearance of λσ [1] and, later, of Melliès’ counterexample =-=[17]-=-, showing the lack of the preservation of β -strong normalization property (PSN, for short) in λσ . One of the main motivations behind the field of ES is studying how substitution behaves when interna... |

23 | From λσ to λυ, a journey through calculi of explicit substitutions - Lescanne - 1994 |

19 |
Extending a λ-calculus with explicit substitution which preserves strong normalisationinto a confluentcalculus on open terms
- Kamareddine, Ríos
- 1997
(Show Context)
Citation Context ... makes sense in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s [11], λ se =-=[12]-=- or λ t [13]. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free variable s... |

11 |
2000): Higher order unification via explicit substitutions
- Hardin, Kirchner
(Show Context)
Citation Context ... a priority, because the main merit of λ rex is evidencing the accessory nature of index updates. From a different perspective, an attempt to use λ rex in proof assistants or higher order unification =-=[8]-=- implementations may be taken into account. In such a case, a typed version of λ rex should be developed as well. Also, adding an η rule to λ rex should be fairly simple using the decrement meta-opera... |

10 | Bridging de Bruijn indices and variable names in explicit substitutions calculi
- Kamareddine, R'ios
- 1996
(Show Context)
Citation Context ... in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s [11], λ se [12] or λ t =-=[13]-=-. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free variable should be sub... |

8 |
Pitts: 2002, ‘A New Approach to Abstract Syntax with Variable Binding’. Formal Aspects of Computing 13
- Gabbay, M
(Show Context)
Citation Context ...le to λ rex should be fairly simple using the decrement meta-operator. Finally, studying the possible relation between these swapping-based formalisms and nominal logic or nominal rewriting (see e.g. =-=[10, 9]-=-) could be an interesting approach in gathering a deeper understanding of λ r’s underlying logic. 10 Swapping: a natural bridge between named and indexed explicit substitution calculi Acknowledgements... |

6 | A theory of explicit substitutions with safe and full composition
- Kesner
- 2009
(Show Context)
Citation Context ...en shown to have a whole set of desirable properties: simulation of β -reduction, PSN, meta-confluence, full composition, etc. For a detailed introduction to the ES field, we refer the reader to e.g. =-=[16, 15, 20]-=-. In 2008, D. Kesner proposed λex [14, 15], a formalism with variable names that has the entire set of properties expected from an ES calculus. As Kesner points in [15], for implementation purposes a ... |

5 |
de Bruijn (1972): Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem
- G
(Show Context)
Citation Context ...d variable renaming (i.e., working modulo α-equivalence) is known to be error-prone and computationally expensive. Among others, one of the ways this problem is tackled is by using de Bruijn notation =-=[5]-=-, which is a technique that simply avoids the need of working modulo α-equivalence. As far as we know, no ES calculus with de Bruijn indexes and the whole set of properties enjoyed by λex exists to da... |

4 |
2008): Perpetuality for Full and Safe Composition (in a Constructive Setting
- Kesner
(Show Context)
Citation Context ...erties: simulation of β -reduction, PSN, meta-confluence, full composition, etc. For a detailed introduction to the ES field, we refer the reader to e.g. [16, 15, 20]. In 2008, D. Kesner proposed λex =-=[14, 15]-=-, a formalism with variable names that has the entire set of properties expected from an ES calculus. As Kesner points in [15], for implementation purposes a different approach to variable names shoul... |

1 |
de Bruijn (1978): A namefree λ calculus with facilities for internal definition of expressions and segments
- G
(Show Context)
Citation Context ...SNλex. 5 Related work It is important to mention that, even though independently discovered, the swapping mechanism introduced in this article was first depicted by de Bruijn for his ES calculus Cλξφ =-=[6]-=-, and, later, updated w.r.t. notation – λξφ – and compared to λυ in [2]. We will now briefly discuss the main differences between these calculi and our swapping-based approach. Firstly, neither Cλξφ n... |