## Normalization by evaluation for typed lambda calculus with coproducts (2001)

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Venue: | In LICS |

Citations: | 39 - 5 self |

### BibTeX

@INPROCEEDINGS{Altenkirch01normalizationby,

author = {T. Altenkirch and P. Dybjer},

title = {Normalization by evaluation for typed lambda calculus with coproducts},

booktitle = {In LICS},

year = {2001},

pages = {303--310},

publisher = {IEEE Computer Society Press}

}

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### Abstract

We solve the decision problem for simply typed lambda calculus with strong binary sums, equivalently the word problem for free cartesian closed categories with binary coproducts. Our method is based on the semantical technique known as “normalization by evaluation ” and involves inverting the interpretation of the syntax into a suitable sheaf model and from this extracting appropriate unique normal forms. There is no rewriting theory involved, and the proof is completely constructive, allowing program extraction from the proof. 1

### Citations

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...terminology, such lambda theories correspond exactly to almost bicartesian closed categories, that is, cartesian closed categories with nonempty finite coproducts (generated by a set of atomic types) =-=[LS86]-=-. As proved by Dougherty and Subrahmanyam � [DS95], a Friedman completeness theorem in Set holds for cartesian closed categories with binary coproducts. Therefore, the equality we decide is the natura... |

329 | Foundations for Programming Languages - Mitchell - 1996 |

66 | An Intuitionstic Theory of Types
- Martin-Löf
- 1995
(Show Context)
Citation Context ...e, the decision problem is a long-standing thorny issue in the subject. Our solution is based on normalization by evaluation (NBE) (also called “reductionfree normalisation”) introduced by Martin-Löf =-=[ML75]-=- for weak typed lambda calculus, and by Berger and Schwichtenberg [BS91] for typed lambda calculus ¤¦¥ with conversion. The technique has been further refined by the authors and coworkers using catego... |

36 | Eta-expansions in F - Ghani - 1997 |

34 | Isomorphisms of types: from *-calculus to in-formation retrieval and language design. Birkhauser - Cosmo - 1995 |

26 | equality for coproducts
- Ghani
(Show Context)
Citation Context ...e of coproducts uses difficult and involved syntactical arguments [DS95].sA decision procedure for cartesian closed categories with binary coproducts has been presented in Ghani’s thesis [Gh95a] (see =-=[Gh95b]-=- for a summary) although the proof involves intricate rewriting techniques whose details are daunting. Our method described here is quite different and we believe conceptually simpler. An algorithm fo... |

25 | Proof theory at work: Program development in the Minlog system
- Benl, Berger, et al.
- 1998
(Show Context)
Citation Context ...other systems, such as System F [AHS96]. As shown by Berger, Schwichtenberg, and Danvy [BS91, Da96], NBE techniques yield fast normalization algorithms, with applications in interactive proof systems =-=[BBSSZ98]-=- and type-directed partial evaluation [Da96, Da98, Fil01]. Here we show how to considerably extend the NBE techniques to take into account type systems with strong sums. The NBE method involves constr... |

25 | Adjoint Rewriting
- Ghani
- 1995
(Show Context)
Citation Context ...m for the case of coproducts uses difficult and involved syntactical arguments [DS95].sA decision procedure for cartesian closed categories with binary coproducts has been presented in Ghani’s thesis =-=[Gh95a]-=- (see [Gh95b] for a summary) although the proof involves intricate rewriting techniques whose details are daunting. Our method described here is quite different and we believe conceptually simpler. An... |

21 | Categorical reconstruction of a reductionfree normalization proof - Altenkirch, Hofmann, et al. - 1995 |

18 | Normalization by evaluation for the computational lambda-calculus
- Filinski
(Show Context)
Citation Context ... different and we believe conceptually simpler. An algorithm for type-directed partial evaluation for a call-by-value typed lambda calculus with sums has been given by Danvy [Da96, Da98] and Filinski =-=[Fil01]-=-. This algorithm uses continuations and is therefore also quite different from ours. In particular, it does not decide equality in cartesian closed categories with binary coproducts. Like Ghani and Do... |

4 | H.: Normalization by evaluation. Prospects for Hardware Foundations - Berger, Eberl, et al. - 1998 |

2 |
Reduction- free normalisation for a polymorphic system
- Altenkirch, Hofmann, et al.
- 1996
(Show Context)
Citation Context ...ith conversion. The technique has been further refined by the authors and coworkers using category-theoretic methods [CD97, AHS95, CDS97]. It has also been extended to other systems, such as System F =-=[AHS96]-=-. As shown by Berger, Schwichtenberg, and Danvy [BS91, Da96], NBE techniques yield fast normalization algorithms, with applications in interactive proof systems [BBSSZ98] and type-directed partial eva... |

2 | Type-directed partial evaluation, Partial evaluation, Practice and Theory - Danvy |

1 | Selected Domains and Lambda Calculi, Camb - Amadio, Curien - 1998 |

1 |
An inverse to the evaluation functional for typed
- Berger, Schwichtenberg
- 1991
(Show Context)
Citation Context ...Our solution is based on normalization by evaluation (NBE) (also called “reductionfree normalisation”) introduced by Martin-Löf [ML75] for weak typed lambda calculus, and by Berger and Schwichtenberg =-=[BS91]-=- for typed lambda calculus ¤¦¥ with conversion. The technique has been further refined by the authors and coworkers using category-theoretic methods [CD97, AHS95, CDS97]. It has also been extended to ... |

1 | Type-directed partial evaluation, POPL’96 - Danvy |

1 | Some lambda calculi with catgorical sums and products - Dougherty - 1993 |

1 |
Equality between Functionals
- Dougherty, Subrahmanyam
- 1995
(Show Context)
Citation Context ...ly to almost bicartesian closed categories, that is, cartesian closed categories with nonempty finite coproducts (generated by a set of atomic types) [LS86]. As proved by Dougherty and Subrahmanyam � =-=[DS95]-=-, a Friedman completeness theorem in Set holds for cartesian closed categories with binary coproducts. Therefore, the equality we decide is the natural extensional equality on proofs in intuitionistic... |

1 |
Lambda Definability with Sums via
- Fiore, Simpson
(Show Context)
Citation Context ...express which branch a certain � guard takes. This is the idea behind our Grothendieck topology � on : a “covering” expresses case-splitting. This use of Grothendieck topologies is closely related to =-=[FS99]-=- where they were used for proving a definability result for a language with coproducts. Definition 4.1 A neutral constrained environment, environment for short, is a pair -form. ����� where � is a typ... |

1 |
fij-equality for coproducts. TLCA '95
- Ghani
(Show Context)
Citation Context ...e of coproducts uses difficult and involved syntactical arguments [DS95]. A decision procedure for cartesian closed categories with binary coproducts has been presented in Ghani's thesis [Gh95a] (see =-=[Gh95b]-=- for a summary) although the proof involves intricate rewriting techniques whose details are daunting. Our method described here is quite different and we believe conceptually simpler. An algorithm fo... |