## Explicit Substitutions and Reducibility (2001)

Venue: | Journal of Logic and Computation |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Herbelin01explicitsubstitutions,

author = {Hugo Herbelin},

title = {Explicit Substitutions and Reducibility},

journal = {Journal of Logic and Computation},

year = {2001},

volume = {11},

pages = {2001}

}

### OpenURL

### Abstract

. We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simply-typed call-by-name ~-calculus enriched with operators of explicit unary substitutions. The ~-calculus, dened by Curien & Herbelin, is a variant of -calculus with a let operator that exhibits symmetries such as terms/contexts and call-byname /call-by-value reduction. The ~-calculus embeds various standard -calculi (and Gentzen's style sequent calculi too) and as an application we derive the strong normalization of Parigot's simply-typed -calculus with explicit substitution. Introduction Explicit substitution in -calculus The traditional theory of -calculus relies on -reduction, that is the capture by a function of its argument followed by the process of substituting this argument to the places where it is used. The ...

### Citations

355 |
λµ-Calculus: an algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...lso provide for strong normalization for, say x-calculus. This is the purpose of this paper. The ~-calculus The ~-calculus dened by Curien & Herbelin [7] is a variant of Parigot's -calculus [23] with a let operator that provides with a term notation for proofs of LK. It embeds also various -calculi in their call-by-name (CBN) or call-by-value (CBV) variants, such as -calculus and usual -... |

186 | The duality of computation
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- 2000
(Show Context)
Citation Context ...agalin [12]. The reduction system he considers is a non con uent rewriting system but without overlapping between distinct redexes. Reading it along the lines of the correspondence with -calculus in [=-=16, 7-=-], it lacks the full generality of the propagation rule (tu)[x v] ! (t[x v])(u[x v]), a rule necessary to simulates-reduction. Therefore, strong normalization of, say x-calculus, cannot be inferred (a... |

183 | Call-by-name, call-by-value, and the lambda-calculus. Theoret - Plotkin - 1975 |

172 | M.: Reasoning About Programs in Continuation-Passing Style - Sabry, Felleisen - 1992 |

112 | A new deconstructive logic: Linear logic
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- 1997
(Show Context)
Citation Context ... directly) from Dragalin's proof. Another approach of strong normalization for sequent calculus is inherited from Girard's strong normalization of proof nets [14]. It has allowed for various con uent =-=[8]-=- and non con uent [29] strong normalization results for LK, all based on a global reduction system related tos-reduction. What about strong normalization of a reduction system includingsne-grained Dra... |

90 | A -calculus a la de Bruijn with explicit substitution
- Kamareddine, Ros
- 1995
(Show Context)
Citation Context ...alled de Bruijn levels (see Lescanne and Rouyer-Degli ? E-mail: Hugo.Herbelin@inria.fr -calculus [21]) and the second uses so-called de Bruijn indices (see s- and t-calculi by Kamareddine & Ros [18] and by Lescanne [20] 1 ). A good survey on the current state of the study of -calculus with substitution made explicit can be found in the introduction of [10]. Cut rule and explicit substitution... |

84 |
Proofs of strong normalization for second order classical natural deduction
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- 1997
(Show Context)
Citation Context ...ethod comes from Tait [28] who used it to prove the normalization of Godel's system T. It has been extended to system F by Girard [15] and to second-order -calculus (classical system F) by Parigot [24=-=]-=-. Strong normalization with explicit substitution raises several problems. In a calculus without operator of explicit substitution, a term can only interact against its arguments. In the presence of e... |

81 |
Natural deduction. A
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- 1965
(Show Context)
Citation Context ...t substitution Along the lines of the Curry-Howard correspondence, the proofs of implicational Gentzen's natural deduction [13] are -terms and Prawitz' process of normalization for natural deduction [=-=26]-=- iss-reduction. Explicitation of substitution can be transfered to natural deduction and it happens substitution is nothing but a cut rule. The cut rule is central in this other kind of formal systems... |

74 |
A Symmetric Lambda-Calculus for Classical Program Extraction
- Barbanera, Berardi
- 1996
(Show Context)
Citation Context ...lary we would get strong normalization for a formulation of call-by-value simply-typed -calculus. The proof pattern used for the strong normalization of Barbanera & Berardi's symmetrical -calculus [1] can without any doubt be adapted for ~-calculus as it has been by Urban & Bierman to get strong nondeterministic (big step) cut-elimination of LK [29]. Then we would get strong normalization for ... |

60 |
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- 1988
(Show Context)
Citation Context ... precisely relies on the progressive elimination of cuts through application of local rewriting steps. To our knowledge, thesrst strong normalization proof for sequent calculus LK comes from Dragalin =-=[12-=-]. The reduction system he considers is a non con uent rewriting system but without overlapping between distinct redexes. Reading it along the lines of the correspondence with -calculus in [16, 7], it... |

45 | Preservation of Termination for Explicit Substitutions
- Bloo
- 1997
(Show Context)
Citation Context ...of weakening if any) is turned into asne-grain reduction process allowing for possible interleaving with other reductions. For simply typed terms, SN has been inferred from PSN in x-calculus by Bloo [2] and in s-calculus by Kamareddine & Ros [19]. Some proofs have also been proposed using an embedding into proof nets (assuming reduction in proof nets is proved strongly normalizing). This works fo... |

34 |
A lambda-calculus structure isomorphic to sequent calculus structure
- Herbelin
- 1994
(Show Context)
Citation Context ...agalin [12]. The reduction system he considers is a non con uent rewriting system but without overlapping between distinct redexes. Reading it along the lines of the correspondence with -calculus in [=-=16, 7-=-], it lacks the full generality of the propagation rule (tu)[x v] ! (t[x v])(u[x v]), a rule necessary to simulates-reduction. Therefore, strong normalization of, say x-calculus, cannot be inferred (a... |

32 | Explicit Substitutions with de Bruijn’s Levels
- Lescanne, Rouyer-Degli
- 1995
(Show Context)
Citation Context ... (notations as above), two strategies based on numbers have specially been studied. Thesrst uses so-called de Bruijn levels (see Lescanne and Rouyer-Degli ? E-mail: Hugo.Herbelin@inria.fr -calculus [21]) and the second uses so-called de Bruijn indices (see s- and t-calculi by Kamareddine & Ros [18] and by Lescanne [20] 1 ). A good survey on the current state of the study of -calculus with su... |

25 | Strong normalization of explicit substitutions via cut elimination in proof nets
- Cosmo, Kesner
- 1997
(Show Context)
Citation Context ...d t-calculi by Kamareddine & Ros [18] and by Lescanne [20] 1 ). A good survey on the current state of the study of -calculus with substitution made explicit can be found in the introduction of [10]. Cut rule and explicit substitution Along the lines of the Curry-Howard correspondence, the proofs of implicational Gentzen's natural deduction [13] are -terms and Prawitz' process of normalization ... |

18 | Combinatory reduction systems with explicit substitutions that preserve strong normalisation
- Bloo, Rose
- 1996
(Show Context)
Citation Context ...On another side, Bloo & Rose proved that for a large subset of interesting Combinatory Reduction Systems (CRS), the extension with explicit substitution (up to -conversion) satises the PSN property [4]. This includes -calculus and ~-calculus which therefore are SN as soon as their version without explicit substitution is. 1 The ~-Calculus The syntax relies on three syntactic categories: term... |

18 |
Theoretical Computer Science 50
- Girard
- 1987
(Show Context)
Citation Context ... x-calculus, cannot be inferred (at least directly) from Dragalin's proof. Another approach of strong normalization for sequent calculus is inherited from Girard's strong normalization of proof nets [=-=14]-=-. It has allowed for various con uent [8] and non con uent [29] strong normalization results for LK, all based on a global reduction system related tos-reduction. What about strong normalization of a ... |

16 | The s-calculus: its typed and its extended versions
- Kamareddine, Ros
- 1995
(Show Context)
Citation Context ... reduction process allowing for possible interleaving with other reductions. For simply typed terms, SN has been inferred from PSN in x-calculus by Bloo [2] and in s-calculus by Kamareddine & Ros [19]. Some proofs have also been proposed using an embedding into proof nets (assuming reduction in proof nets is proved strongly normalizing). This works for David & Guillaume l - calculus (see Di Cosm... |

15 |
Preservation of strong normalization in named lambda calculi with explicit substitution and garbage collection
- Bloo, Rose
- 1995
(Show Context)
Citation Context ...s of the -calculus have been developed. The simplest of them, in the sense it is up to -conversion and uses the minimal apparatus for substitution, is the x-calculus deeply studied in Bloo & Rose [3] (but described by Lins [22] in an independent context). Various naming strategies are possible in order to make explicit the weakening rule in x-calculus. Besides the simple renaming of x into a var... |

15 |
a journey through calculi of explicit substitutions
- Lescanne, J
- 1994
(Show Context)
Citation Context ...see Lescanne and Rouyer-Degli ? E-mail: Hugo.Herbelin@inria.fr -calculus [21]) and the second uses so-called de Bruijn indices (see s- and t-calculi by Kamareddine & Ros [18] and by Lescanne [20] 1 ). A good survey on the current state of the study of -calculus with substitution made explicit can be found in the introduction of [10]. Cut rule and explicit substitution Along the lines of the ... |

12 |
Séquents qu’on calcule, Thèse de Doctorat, Université Paris 7, available from http://pauillac.inria.fr/˜herbelin
- Herbelin
- 1995
(Show Context)
Citation Context ...[]M) ! [](M) (if Dom() 6= ) Consider the following embedding from -calculus to -calculus (which is ~-calculus without ~ 5 ): 5 The resulting -calculus diers slightly from the one in [16, 17] by the treatment of axiom rules and contexts x n = x (x:M) n = x:M n (:c) n = :c n ([]M) n = hM n ji (MN) n = :hM n jN n i (for some 62 (MN)) (M) n = M n n [x N ] n = [x N n ] [ s... |

12 |
Intensional interpretation of functionals of type I
- Tait
- 1967
(Show Context)
Citation Context ... u, as a substituend, has traversed some x:v. In l -calculus the information is preserved and [x := t] is erased instead of propagated as in x-calculus. 3 Reducibility has been introduced by Tait [28=-=-=-] to prove the normalization of Godel's system T. It has been reused in various frameworks, most noticeable is probably its extension for system F by Girard [15]. A set X of term variables is given. T... |

8 |
A new formula for the execution of a categorical combinators
- Lins
- 1986
(Show Context)
Citation Context ...een developed. The simplest of them, in the sense it is up to -conversion and uses the minimal apparatus for substitution, is the x-calculus deeply studied in Bloo & Rose [3] (but described by Lins [2=-=2-=-] in an independent context). Various naming strategies are possible in order to make explicit the weakening rule in x-calculus. Besides the simple renaming of x into a variable not occurring in u (no... |

5 |
Perpetuality in a named lambda calculus with explicit substitutions
- Bonelli
(Show Context)
Citation Context ...esigned in order to forbid these interferences 2 . Proofs by reducibility Our own approach is to develop a direct proof by reducibility 3 . It turned out this approach was already followed by Bonelli =-=[5-=-] to prove strong normalization for Church-style system F with explicit substitution. Bonelli's proof is based on a standard denition of reducibility sets, with the extra condition of closure by rever... |

5 |
The l -calculus
- David, Guillaume
- 1999
(Show Context)
Citation Context ...alculus (see Di Cosmo et al [11]) but fails to manage some pathological interferences between weakening and substitution for the x-calculus style of reduction. Notice David & Guillaume l -calculus [9=-=]-=- is precisely designed in order to forbid these interferences 2 . Proofs by reducibility Our own approach is to develop a direct proof by reducibility 3 . It turned out this approach was already follo... |

5 |
Strong Normalization of Cut-Elimination in Classical Logic
- Urban, Bierman
(Show Context)
Citation Context ...lin's proof. Another approach of strong normalization for sequent calculus is inherited from Girard's strong normalization of proof nets [14]. It has allowed for various con uent [8] and non con uent =-=[29]-=- strong normalization results for LK, all based on a global reduction system related tos-reduction. What about strong normalization of a reduction system includingsne-grained Dragalin's cut-eliminatio... |

2 |
Investigations into logical deduction (1935), e.g. in Gentzen collected
- Gentzen
- 1969
(Show Context)
Citation Context ... made explicit can be found in the introduction of [10]. Cut rule and explicit substitution Along the lines of the Curry-Howard correspondence, the proofs of implicational Gentzen's natural deduction =-=[13-=-] are -terms and Prawitz' process of normalization for natural deduction [26] iss-reduction. Explicitation of substitution can be transfered to natural deduction and it happens substitution is nothing... |