## Retractions of Types with Many Atoms (2001)

Citations: | 3 - 0 self |

### BibTeX

@TECHREPORT{Regnier01retractionsof,

author = {Laurent Regnier and Pawel Urzyczyn and Uniwersytet Warszawski and Instytut Informatyki},

title = {Retractions of Types with Many Atoms},

institution = {},

year = {2001}

}

### OpenURL

### Abstract

We de ne a sound and complete proof system for ane -retractions in simple types (built over many atoms), and we state a necessary condition for arbitrary -retractions in simple types. We also show a simple necessary condition for polymorphic -retractability and we disprove an earlier conjecture about a stronger necessary condition.

### Citations

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(Show Context)
Citation Context ...orphisms, respectively. The main results concerning right and/or left invertibility were obtained mostly in the 70's (see e.g. [2, 4, 9]) and an exposition of the theory can be found in Chapter 21 of =-=[1]-=-. But not everything has been understood in full, and there is still a progress in this line of research, see [6]. Thesrst paper, to our best knowledge, that provides a characterization of isomorphic ... |

52 | Inductive and coinductive types with iteration and recursion
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(Show Context)
Citation Context ...se, see Example 5.1. The necessary condition we can show, namely: If E then FV () FV ( ), 1 Note the dierence between \recursive" and \inductive". Inductive types are representable, see e=-=.g. [7, -=-13]. 3 is too weak to generalize the result of [13]. Thus, proving that recursive types cannot be dened in system F vias-reductions requires a dierent approach. In particular, we conjecture the follow... |

18 |
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(Show Context)
Citation Context ...morphic types (simple types with products and an inhabited type constant), is Soloviev's work [12] from 1981, published in English in 1983. Bruce and Longo have obtained a similar characterization in =-=[3-=-], and they also introduced the notion of a type retraction. In fact, they only considers-retractions. It occurs, however, that the more general case ofs-retractions is more dicult. We say that a type... |

18 |
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(Show Context)
Citation Context ...1 + maxfrank( i ) j i = 1; : : : ; kg. The above condition wassrst shown by Padovani [11] for types over a single atom. A comprehensive study of type isomorphisms in typed lambda calculi is the book [=-=5]-=- of Di Cosmo. He gives complete proof rules for isomorphisms of simple as well as polymorphic types. However, there is no discussion of embeddings nor retractions for polymorphic types. The notion of ... |

8 |
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- 2001
(Show Context)
Citation Context ...8] is a complete proof system for \linear", or more adequately, ane retractability, that is retractability by means of BCK-terms (at most one occurrence of each variable). Padovani, in a recent p=-=ape-=-r [11], proves that the retractability relation is decidable, still under the single atom assumption. For this, he shows that if E then the coder-decoder pair F : ! and G : ! can be chosen in a ... |

6 |
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Citation Context ... notions of right and left invertibility, corresponding to epi- and monomorphisms, respectively. The main results concerning right and/or left invertibility were obtained mostly in the 70's (see e.g. =-=[2, 4, 9]-=-) and an exposition of the theory can be found in Chapter 21 of [1]. But not everything has been understood in full, and there is still a progress in this line of research, see [6]. Thesrst paper, to ... |

5 |
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(Show Context)
Citation Context ... All the above denitions are generic in that they apply to every reasonable type system. The notions ofs-retractability ands-embeddability in simple types can be easily characterized as follows (see [8, 13]): Es i = 1 ! ! k ! , for some 1 ; : : : ; k ; s i = 1 ! ! k ! , for some 1 ; : : : ; k , such that ` i , for all i. In the above, the notation ` i... |

4 |
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(Show Context)
Citation Context ...iated the following tree: a # # # # # # # # # # # # # # # # b c d a a We do not know about the origin of this representation (far less common than the \ordinary" one). We learned it from Hanno Nickau [10]. This idea can be extended to the polymorphic case, provided we agree to identify types 8a( ! ) and ! 8a, whenever a is not free in . Then a type of the form 8~a 1 : 1 ! 8~a 2 : 2 ! ! ... |

3 |
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Citation Context ...s line of research, see [6]. Thesrst paper, to our best knowledge, that provides a characterization of isomorphic types (simple types with products and an inhabited type constant), is Soloviev's work =-=[12]-=- from 1981, published in English in 1983. Bruce and Longo have obtained a similar characterization in [3], and they also introduced the notion of a type retraction. In fact, they only considers-retrac... |

2 |
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(Show Context)
Citation Context ... notions of right and left invertibility, corresponding to epi- and monomorphisms, respectively. The main results concerning right and/or left invertibility were obtained mostly in the 70's (see e.g. =-=[2, 4, 9]-=-) and an exposition of the theory can be found in Chapter 21 of [1]. But not everything has been understood in full, and there is still a progress in this line of research, see [6]. Thesrst paper, to ... |

2 | Completeness, invariance, and lambda-de - Statman - 1982 |

1 |
Retracts in simply typed
- Liguoro, Piperno, et al.
- 1992
(Show Context)
Citation Context ... All the above denitions are generic in that they apply to every reasonable type system. The notions ofs-retractability ands-embeddability in simple types can be easily characterized as follows (see [8, 13]): Es i = 1 ! ! k ! , for some 1 ; : : : ; k ; s i = 1 ! ! k ! , for some 1 ; : : : ; k , such that ` i , for all i. In the above, the notation ` i... |

1 | and left invertibility in - Margaria, Zacchi, et al. - 1983 |