## Decidability of Higher-Order Subtyping via Logical Relations (1997)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Compagnoni97decidabilityof,

author = {Adriana Compagnoni and Healfdene Goguen},

title = {Decidability of Higher-Order Subtyping via Logical Relations},

year = {1997}

}

### OpenURL

### Abstract

This paper uses logical relations for the first time to study the decidability of typechecking and subtyping. The result is proved for F ! , a language with higher-order subtyping and bounded operator abstraction not previously known to be decidable. The proof is via an intermediate system called a typed operational semantics, leading to a powerful and uniform technique for showing metatheoretic results of type systems with subtyping, such as strong normalization, subject reduction and decidability of subtyping. 1 Introduction During the last decade, object-oriented programming languages such as Smalltalk, C++, Modula 3, and Java have become popular because they encourage and facilitate software reuse and abstract design. One attempt to give a theoretical understanding of these object-oriented programming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [5, 7], for example bounded quantification [15] and recur...

### Citations

891 | A Theory of Objects
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- 1996
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Citation Context ...Int ! Bool ] BinTree = Y Comparable Int : ? : [ find : Y ! Bool; insert : Y ! BinTree Y; isEmpty : Bool ] Our framework for studying object-oriented programming is Abadi and Cardelli's object calculi =-=[1]-=-. They have demonstrated that typed object calculi are well-suited to giving semantics for many features of object-oriented programming languages, such as class hierarchies, inheritance, self types, a... |

758 | On understanding types, data abstraction, and polymorphism
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- 1985
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Citation Context ...except for those dealing with bounded type abstraction and type application and the rule for subtyping the quantifier. We chose Cardelli and Wegner's kernel Fun rule for quantifiers with equal bounds =-=[8]-=-, because the contravariant rule for quantifiers renders the system undecidable [20]. Furthermore, transitivity elimination in the presence of such a rule in the higher-order case remains an open prob... |

485 | A semantics of multiple inheritance
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- 1988
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Citation Context ...empt to give a theoretical understanding of these object-oriented programming languages has been to introduce type systems with features to model constructs from object-oriented programming languages =-=[5, 7]-=-, for example bounded quantification [15] and recursive types [2]. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property... |

305 | Subtyping recursive types
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- 1993
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Citation Context ...ogramming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [5, 7], for example bounded quantification [15] and recursive types =-=[2]-=-. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property of a type system is subject reduction or type preservation, whic... |

107 |
An algorithm for testing conversion in type theory
- Coquand
- 1991
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Citation Context ... theory, including partial interpretations [21], Kripkestyle models for strong normalization [14], typed operational semantics [16, 17], and binary logical relations to interpret judgemental equality =-=[13]-=-. Our approach gives a proof of decidability of subtyping that follows directly and naturally from the development of the other metatheoretic results for the type theory. We have modified one rule of ... |

68 | Subtyping dependent types
- Aspinall, Compagnoni
- 1996
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Citation Context ...her-order subtyping with bounded quantification. Other proofs have used reduction relations unrelated to the actual notion of computation of the type theory, for example the +-reduction of Compagnoni =-=[11, 12, 3]-=- or the \Gamma-reduction of Pierce and Steffen [19]. Treating these auxiliary reduction relations leads to syntactic complications unrelated to the basic problem of decidability. Furthermore, the proo... |

63 | A semantic basis for Quest
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- 1991
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Citation Context ...empt to give a theoretical understanding of these object-oriented programming languages has been to introduce type systems with features to model constructs from object-oriented programming languages =-=[5, 7]-=-, for example bounded quantification [15] and recursive types [2]. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property... |

61 |
Coherence of subsumption, minimum typing and the type checking
- Curien, Ghelli
- 1992
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Citation Context ... these object-oriented programming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [5, 7], for example bounded quantification =-=[15]-=- and recursive types [2]. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property of a type system is subject reduction or... |

52 |
A Typed Operational Semantics for Type Theory
- Goguen
- 1994
(Show Context)
Citation Context ...in Section 4.2. This is based on well-established ideas in dependent type theory, including partial interpretations [21], Kripkestyle models for strong normalization [14], typed operational semantics =-=[16, 17]-=-, and binary logical relations to interpret judgemental equality [13]. Our approach gives a proof of decidability of subtyping that follows directly and naturally from the development of the other met... |

40 | Decidability of higher-order subtyping with intersection types - Compagnoni - 1994 |

29 | Typing in object-oriented languages: Achieving expressiveness and safety
- Bruce
- 1996
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Citation Context ...erator abstraction. Cardelli and Harper are in favor of including this in ML2000 (private communication), the successor of Standard ML. The constructor is amply motivated by examples due to Kim Bruce =-=[4]-=-, including the following, which shows how bounded abstraction on types can be used to define the type of binary search trees with comparable elements: Comparable Int = [ eq : Int ! Int ! Bool; lt : I... |

21 |
Notes about F ! <:. Unpublished manuscript
- Cardelli
- 1990
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Citation Context ...ally with our existing development of the metatheory of F !s. The decidability of type-checking follows straightforwardly from that of subtyping. Because the term structure of our language and F ! !: =-=[6, 7, 18]-=- is the same (as opposed to the type structure), the proof is the same as for that system. We shall therefore only treat decidability of subtyping in this paper. As discussed extensively in our earlie... |

20 | A proof of strong normalization for the theory of constructions using a Kripke-like interpretation
- Coquand, Gallier
- 1990
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Citation Context ...s the model construction outlined in Section 4.2. This is based on well-established ideas in dependent type theory, including partial interpretations [21], Kripkestyle models for strong normalization =-=[14]-=-, typed operational semantics [16, 17], and binary logical relations to interpret judgemental equality [13]. Our approach gives a proof of decidability of subtyping that follows directly and naturally... |

12 | Typed operational semantics for higher order subtyping
- Compagnoni, Goguen
- 1997
(Show Context)
Citation Context ...ne important property of a type system is subject reduction or type preservation, which states that evaluation of programs preserves their type. This is one of the central results of an earlier paper =-=[10]-=-, which also showed the correctness of the algorithms for type-formation and subtyping. Another important property for type systems is the decidability of type-checking and subtyping: a compiler shoul... |

11 |
Coercive subtyping for the calculus of constructions
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Citation Context ...derivable in the semantics are also derivable in the original system. reduction under binders will no longer hold. Chen's proof of termination using \Gamma-reduction for the Calculus of Constructions =-=[9]-=- is not for bounded quantification, but the same problems would apply to any attempt to extend the technique. Compagnoni's proof of termination using +-reduction probably extends to bounded operator a... |

4 |
The metatheory of UTT
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Citation Context ...in Section 4.2. This is based on well-established ideas in dependent type theory, including partial interpretations [21], Kripkestyle models for strong normalization [14], typed operational semantics =-=[16, 17]-=-, and binary logical relations to interpret judgemental equality [13]. Our approach gives a proof of decidability of subtyping that follows directly and naturally from the development of the other met... |