## Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection (1995)

Venue: | IN CSN-95: COMPUTER SCIENCE IN THE NETHERLANDS |

Citations: | 65 - 7 self |

### BibTeX

@INPROCEEDINGS{Bloo95preservationof,

author = {Roel Bloo and Kristoffer H. Rose},

title = {Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection},

booktitle = {IN CSN-95: COMPUTER SCIENCE IN THE NETHERLANDS},

year = {1995},

pages = {62--72},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we introduce and study a new lambda-calculus with explicit substitution, lambda-xgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rules for explicit garbage collection; this simplifies several proofs. We show that lambda-xgc is a conservative extension which preserves strong normalisation (PSN) of the untyped lambda-calculus. The result is obtained in a modular way by first proving it for garbage-free reduction and then extending to `reductions in garbage'. This provides insight into the counterexample to PSN for lambda-sigma of Melliès (1995); we exploit the abstract nature of lambda-xgc to show how PSN is in conflict with any reasonable substitution composition rule (except for trivial composition rules of which we mention one). Key words: lambda calculus, explicit substitution, strong normalisation, garbage collection.

### Citations

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Citation Context ... Kamareddine and Nederpelt 1993, R'ios 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990, =-=Abadi et al. 1991-=-, Benaissa, Briaud, Lescanne and Rouyer-Degli 1995). Too many reductions! However, it turns out that there are too many possible ways to reduce with the traditional explicit substitution calculi in a ... |

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Citation Context .... Finally we give an overview of the following sections. The reader is expected to be familiar with the untyped -calculus (Barendregt 1984) and we make free use of abstract reduction system concepts (=-=Huet 1980-=-, Klop 1987), in particular the `diagram stencils' of Rosen (1973). Due to space limitations, most proofs are omitted; the full paper includes detailed proofs and will be available as a technical repo... |

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Citation Context ...uired to compute something does not correspond well to the number of computation steps needed by any reasonable `real' computer. This led to the study of abstract machines , notably the SECD machine (=-=Landin 1964-=-), and later to the study of combinator reduction (Turner 1979) that both satisfy the need for an elementary reduction that reflects a realistic measure of complexity. The price, however, is that the ... |

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Citation Context ...of strong normalisation of -terms) property and why it is difficult to obtain. Finally we give an overview of the following sections. The reader is expected to be familiar with the untyped -calculus (=-=Barendregt 1984-=-) and we make free use of abstract reduction system concepts (Huet 1980, Klop 1987), in particular the `diagram stencils' of Rosen (1973). Due to space limitations, most proofs are omitted; the full p... |

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Citation Context ...ed infinite reduction. Recovering preservation of strong normalisation (PSN). Since the announcement of Melli`es's result several similar explicit substitution calculi have been shown to possess PSN (=-=Benaissa et al. 1995-=-, Bloo 1995, Lescanne and Rouyer-Degli 1995, Kamareddine and R'ios 1995). The calculus described in this paper was developed independently and is shown to have the PSN property. Our proof is construct... |

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Citation Context ...ation (PSN). Since the announcement of Melli`es's result several similar explicit substitution calculi have been shown to possess PSN (Benaissa et al. 1995, Bloo 1995, Lescanne and Rouyer-Degli 1995, =-=Kamareddine and R'ios 1995-=-). The calculus described in this paper was developed independently and is shown to have the PSN property. Our proof is constructive whereas the proofs in the aforecited papers are all highly nonconst... |

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Citation Context ...en reported (cf. Rose 1993, Kamareddine and Nederpelt 1993, R'ios 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. =-=Curien 1990-=-, Cr'egut 1990, Abadi et al. 1991, Benaissa, Briaud, Lescanne and Rouyer-Degli 1995). Too many reductions! However, it turns out that there are too many possible ways to reduce with the traditional ex... |

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Citation Context ...notably oe of Abadi, Cardelli, Curien and L'evy (1991) developed from Curien's (1986) Categorical Combinators. In recent years several explicit substitution calculi have been reported (cf. Rose 1993, =-=Kamareddine and Nederpelt 1993-=-, R'ios 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990, Abadi et al. 1991, Benaissa, Br... |

33 |
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Citation Context ...evy (1991) developed from Curien's (1986) Categorical Combinators. In recent years several explicit substitution calculi have been reported (cf. Rose 1993, Kamareddine and Nederpelt 1993, R'ios 1993, =-=Lescanne 1994-=-). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990, Abadi et al. 1991, Benaissa, Briaud, Lescanne and Rouyer-D... |

32 |
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Citation Context ...s 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990, Abadi et al. 1991, Benaissa, Briaud, =-=Lescanne and Rouyer-Degli 1995-=-). Too many reductions! However, it turns out that there are too many possible ways to reduce with the traditional explicit substitution calculi in a certain sense: Melli`es (1995) observed that oe is... |

22 |
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Citation Context ...urien and L'evy (1991) developed from Curien's (1986) Categorical Combinators. In recent years several explicit substitution calculi have been reported (cf. Rose 1993, Kamareddine and Nederpelt 1993, =-=R'ios 1993-=-, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990, Abadi et al. 1991, Benaissa, Briaud, Lescan... |

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Citation Context ... xgc with reductions that have such an effect breaks PSN. Further work. Extensions to this work proceed in several directions. First, we are generalising the results to combinatory reduction systems (=-=Rose 1995-=-). Second, we are working on how to model sharing of identical subterms within explicit substitution and deriving efficient abstract reduction machines from them, this includes generalising the result... |

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Citation Context ...e give an overview of the following sections. The reader is expected to be familiar with the untyped -calculus (Barendregt 1984) and we make free use of abstract reduction system concepts (Huet 1980, =-=Klop 1987-=-), in particular the `diagram stencils' of Rosen (1973). Due to space limitations, most proofs are omitted; the full paper includes detailed proofs and will be available as a technical report. Infinit... |

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3 |
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Citation Context ... invented, notably oe of Abadi, Cardelli, Curien and L'evy (1991) developed from Curien's (1986) Categorical Combinators. In recent years several explicit substitution calculi have been reported (cf. =-=Rose 1993-=-, Kamareddine and Nederpelt 1993, R'ios 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, Cr'egut 1990,... |

1 | Preservation of strong normalisation for some explicit substitution calculi - Bloo, Geuvers - 1995 |

1 |
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Citation Context ...cf. Rose 1993, Kamareddine and Nederpelt 1993, R'ios 1993, Lescanne 1994). For several of these, abstract machines have been constructed based on the reduction steps of the calculus (cf. Curien 1990, =-=Cr'egut 1990-=-, Abadi et al. 1991, Benaissa, Briaud, Lescanne and Rouyer-Degli 1995). Too many reductions! However, it turns out that there are too many possible ways to reduce with the traditional explicit substit... |