## The Simply Typed Rewriting Calculus (2000)

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Venue: | In 3rd International Workshop on Rewriting Logic and its Applications |

Citations: | 10 - 5 self |

### BibTeX

@INPROCEEDINGS{Cirstea00thesimply,

author = {Horatiu Cirstea and Claude Kirchner},

title = {The Simply Typed Rewriting Calculus},

booktitle = {In 3rd International Workshop on Rewriting Logic and its Applications},

year = {2000},

publisher = {Electronic}

}

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### Abstract

The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambda-calculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly. We present a simply typed version of the rewriting calculus. With a good choice of the type system, we show that the calculus is type preserving and terminating, i.e. veries the subject reduction and strong normalization properties. 1 Introduction The rewriting calculus [CK99a,CK99b] is a general framework handling explicitly the three notions of rule formation, rule application and rule application result. The rule formation constructor is denoted ! and given two terms like x+ s(y) and s(x +y) it allows us to built the rewrite rule x+ s(y) ! s(x +y). Applying the previous rewrite rule to the top position of the sum 3 + 2 is performed using the application operator denoted [ ]( ) (where 3 is an abbreviation for s(s(s(0)))...

### Citations

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Citation Context ...K99a,Cir00] can be used to give a semantics to ELAN [KKV93,BKK + 98] 1 rules and strategies. This could be applied to many other frameworks, including rewrite based languages like ASF+SDF [Deu96], ML =-=[Mil84-=-], Maude [CELM96], CafeOBJ [FN97] or Stratego [Vis99] but also production systems and non-deterministic transition systems. In the context of rewriting logic [Mes92], proof terms are a subset of -term... |

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Citation Context ...lled internalization and wassrst used by Gandy [Gan80]. Another one is usually called reducibility and is based on the notions introduced by Tait [Tai67]. The latter technique has been generalized in =-=[Gir72-=-] and [JO97]. In what follows we use the notations, denitions and the proof line from [HS86] which is a variation of the Tait's method. Comparing to this method, in our approach we should handle prope... |

213 | Intensional Interpretations of Functionals of Finite Type - Tait - 1967 |

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124 | Principles of Maude
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Citation Context ... be used to give a semantics to ELAN [KKV93,BKK + 98] 1 rules and strategies. This could be applied to many other frameworks, including rewrite based languages like ASF+SDF [Deu96], ML [Mil84], Maude =-=[CELM96-=-], CafeOBJ [FN97] or Stratego [Vis99] but also production systems and non-deterministic transition systems. In the context of rewriting logic [Mes92], proof terms are a subset of -terms but there exis... |

105 |
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Citation Context ...the corresponding -redex reduces to ffx=ugtg where fx=ugt represents in both cases a higher order substitution application involving possible -conversions. The -calculus with patterns presented in [PJ87] can be given a direct representation in the -calculus. Let us consider, for example, the -term (PAIR x y):x that selects thesrst element of a pair and the applications(PAIR x y):x (PAIR a b) that... |

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Citation Context ...sm in the description of tactics and tacticals. 3 The typed -calculus The classical notations and denitions from this paper are inspired from those used for the typed -calculus used in [Hin97] and [HS=-=86-=-]. 1 A detailed description of ELAN can be found at http://elan.loria.fr/. 4 3.1 Syntax of the typed -calculus Let us consider a set K of atomic types K 1 ; K 2 ; : : : and the set of types inductivel... |

64 | The rewriting calculus
- Cirstea, Kirchner
- 2001
(Show Context)
Citation Context ...us and rewriting are combined. We are here in a dierent situation where rewriting (and not the normalization process) is treated at the same level as -abstraction. We have introduced and studied in [C=-=K9-=-9b] the untyped rewriting calculus and we present here a simply typed version of it. We brie y present in the next section the -calculus and some of its properties. In Section 3 we dene the typed -cal... |

56 |
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Citation Context ...e when T is empty, the syntactic matching substitution from t to t 0 , when it exists, is unique and can be computed, in the untyped case, by a simple recursive algorithm given for example by G. Huet =-=[Hue76]-=-. It can also be computed by the following set of rules SyntacticMatching where the symbol ^ is assumed to be associative and commutative. We denote Match(t; t 0 ; E; E 0 ) the substitution obtained b... |

52 |
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(Show Context)
Citation Context ...alculus in order, in particular, to insure termination of the evaluation of well-typed -terms. When dealing with the combination of term rewriting and -calculus, thesrst such result was obtained in [G=-=BT8-=-9] and [Oka89]. One of the originalities of the -calculus with respect to the previous approaches is that we consider only one calculus by opposition to the situation where the two frameworks of -calc... |

49 |
Conditional Rewriting Logic as a Uni Model of Concurrency
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- 1992
(Show Context)
Citation Context ... languages like ASF+SDF [Deu96], ML [Mil84], Maude [CELM96], CafeOBJ [FN97] or Stratego [Vis99] but also production systems and non-deterministic transition systems. In the context of rewriting logic =-=[Mes-=-92], proof terms are a subset of -terms but there exist -terms that do not correspond to any proof in rewriting logic. This extends the classical representation of proof terms by -terms used in logica... |

44 | Abstract Data Type Systems
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- 1997
(Show Context)
Citation Context ...lization and wassrst used by Gandy [Gan80]. Another one is usually called reducibility and is based on the notions introduced by Tait [Tai67]. The latter technique has been generalized in [Gir72] and =-=[JO97-=-]. In what follows we use the notations, denitions and the proof line from [HS86] which is a variation of the Tait's method. Comparing to this method, in our approach we should handle properly the ter... |

34 | Strategic Pattern Matching
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(Show Context)
Citation Context ...[KKV93,BKK + 98] 1 rules and strategies. This could be applied to many other frameworks, including rewrite based languages like ASF+SDF [Deu96], ML [Mil84], Maude [CELM96], CafeOBJ [FN97] or Stratego =-=[Vis9-=-9] but also production systems and non-deterministic transition systems. In the context of rewriting logic [Mes92], proof terms are a subset of -terms but there exist -terms that do not correspond to ... |

25 | An overview of CAFE specification environment – an algebraic approach for creating, verifying, and maintaining formal specifications over networks - Futatsugi, Nakagawa - 1997 |

24 |
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(Show Context)
Citation Context ...rder, in particular, to insure termination of the evaluation of well-typed -terms. When dealing with the combination of term rewriting and -calculus, thesrst such result was obtained in [GBT89] and [O=-=ka8-=-9]. One of the originalities of the -calculus with respect to the previous approaches is that we consider only one calculus by opposition to the situation where the two frameworks of -calculus and rew... |

22 |
Higher-order Uni via Explicit Substitutions
- Dowek, Hardin, et al.
- 2000
(Show Context)
Citation Context ...described at the meta-level of the calculus, except for explicit substitution frameworks. For the description of the -calculus that we give here, we use (higher-order) substitutions and not grafting [=-=DHK00-=-], i.e. the application takes care of variable bindings and therefore uses -conversion. In the general case, we consider a higher-order matching, but in practical cases it will be higher-order-pattern... |

20 | Combining higher-order and first-order computation using ρ-calculus: Towards a semantics of ELAN - Cirstea, Kirchner - 1999 |

13 |
contrôle de la réécriture: étude et implantation d’un formalisme de stratégies. Thèse de Doctorat d’Université, Université Henri Poincaré – Nancy 1
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- 1998
(Show Context)
Citation Context ...m features. The untyped -calculus has been used to give a semantics to the language ELAN and this approach can be smoothly extended to a representation of the ELAN typed rewrite rules and strategies [=-=Bor9-=-8] in the typed -calculus. We have recently proposed a new presentation of the -calculus [CKL00] allowing us to encode two major object-calculi [AC96,FHM94] in a very natural and simple way. This appr... |

13 | Implementing computational systems with constraints - Kirchner, Kirchner, et al. - 1993 |

12 |
Basic Simple Type Theory, volume 42 of Cambridge Tracts in Theoretical
- Hindley
- 1997
(Show Context)
Citation Context ...on-determinism in the description of tactics and tacticals. 3 The typed -calculus The classical notations and denitions from this paper are inspired from those used for the typed -calculus used in [Hi=-=n97-=-] and [HS86]. 1 A detailed description of ELAN can be found at http://elan.loria.fr/. 4 3.1 Syntax of the typed -calculus Let us consider a set K of atomic types K 1 ; K 2 ; : : : and the set of types... |

11 |
Intensional interpretation of functionals of type I
- Tait
- 1967
(Show Context)
Citation Context ... the strong normalization of the -calculus. One is called internalization and wassrst used by Gandy [Gan80]. Another one is usually called reducibility and is based on the notions introduced by Tait [=-=Tai67-=-]. The latter technique has been generalized in [Gir72] and [JO97]. In what follows we use the notations, denitions and the proof line from [HS86] which is a variation of the Tait's method. Comparing ... |

7 |
Combining higher-order and computation using - calculus: Towards a semantics of ELAN
- Cirstea, Kirchner
- 1999
(Show Context)
Citation Context ...them and we abbreviate the function Solution(E ` t ? ; E 0 ` t 0 ) by Solution(t ? ; t 0 ). 3.6 The evaluation rules of the typed -calculus The evaluation rules of the untyped ; -calculus from [CK9=-=-=-9a] enriched with the typing information are presented in Figure 3. The rules that are modied are the ones handling rewrite rules: F ire and SwitchR . The typed T -calculus (the -calculus with a give... |

4 |
An overview of CAFE speci environment { an algebraic approach for creating, verifying, and maintaining formal speci over networks
- Futatsugi
- 1997
(Show Context)
Citation Context ... semantics to ELAN [KKV93,BKK + 98] 1 rules and strategies. This could be applied to many other frameworks, including rewrite based languages like ASF+SDF [Deu96], ML [Mil84], Maude [CELM96], CafeOBJ =-=[FN9-=-7] or Stratego [Vis99] but also production systems and non-deterministic transition systems. In the context of rewriting logic [Mes92], proof terms are a subset of -terms but there exist -terms that d... |

4 |
Proof of strong normalisation
- Gandy
- 1980
(Show Context)
Citation Context ...from the proof of the strong normalization of the -calculus. There are several methods for proving the strong normalization of the -calculus. One is called internalization and wassrst used by Gandy [G=-=an80]-=-. Another one is usually called reducibility and is based on the notions introduced by Tait [Tai67]. The latter technique has been generalized in [Gir72] and [JO97]. In what follows we use the notatio... |

3 |
An Overview of ASF+SDF
- Deursen
- 1996
(Show Context)
Citation Context ...-calculus [CK99a,Cir00] can be used to give a semantics to ELAN [KKV93,BKK + 98] 1 rules and strategies. This could be applied to many other frameworks, including rewrite based languages like ASF+SDF =-=[Deu96]-=-, ML [Mil84], Maude [CELM96], CafeOBJ [FN97] or Stratego [Vis99] but also production systems and non-deterministic transition systems. In the context of rewriting logic [Mes92], proof terms are a subs... |

2 |
Le Rho Calcul: Fondements et Applications
- Cirstea
- 2000
(Show Context)
Citation Context ...d the use of sets for representing the reduction results is the main cause of non-con uence. The con uence can be recovered if the evaluation rules of -calculus are guided by an appropriate strategy [=-=Cir00-=-] handling properly the propagation of the failure (i.e. ;) and the sets with more than one element. The -calculus is both conceptually simple as well as very expressive. This allowed us to represent ... |