## Anti-Symmetry of Higher-Order Subtyping (1999)

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Venue: | In Proceedings of the 8th Annual Conference on Computer Science Logic (CSL’99), J. Flum and M. RodríguezArtalejo, Eds. Springer-Verlag LNCS 1683 |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Compagnoni99anti-symmetryof,

author = {Adriana Compagnoni and Healfdene Goguen},

title = {Anti-Symmetry of Higher-Order Subtyping},

booktitle = {In Proceedings of the 8th Annual Conference on Computer Science Logic (CSL’99), J. Flum and M. RodríguezArtalejo, Eds. Springer-Verlag LNCS 1683},

year = {1999},

pages = {420--438}

}

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### Abstract

. This paper shows that the subtyping relation of a higherorder lambda calculus, F ! , is anti-symmetric. It exhibits the rst such proof, establishing in the process that the subtyping relation is a partial order|reexive, transitive, and anti-symmetric up to -equality. While a subtyping relation is reexive and transitive by denition, anti-symmetry is a derived property. The result, which may seem obvious to the nonexpert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping oers a powerful new technology to solve the problem: of particular importance is our extended rule for the well-formedness of types with head variables. The paper also gives a presentation of F ! without a relation for -equality, apparently the rst such, and shows its equivalence with the traditional presentation. 1 Introduction Object-oriented programming languages such as Smalltalk, C++, Modula 3, and ...

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Citation Context ... University of Edinburgh, JCMB, The King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages =-=[1, 6, 21, 28]-=-, proof assistants [3, 25, 29], and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relat... |

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Citation Context ..., except for those dealing with bounded type abstraction and type application and the rule for subtyping the quantier. We chose Cardelli and Wegner's kernel Fun rule for quantiers with equal bounds [1=-=2-=-], because the contravariant rule for quantiers renders the system undecidable [31]. Furthermore, transitivity elimination in the presence of such a rule in the higher-order case remains an open probl... |

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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 =-=[8, 10-=-], for example bounded quantication [20] and recursive types [2]. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property ... |

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Citation Context ...2 A and B are equivalent in the subtyping relation if A B and B A. In our current work on F ! , the equivalence relation is the usual notion ofs-equality at the type level. However, Martin-Lof [26]=-=-=- has demonstrated how to capture more sophisticated equivalence relations in intensional type theory. He allows the equality on elements of a type to include an arbitrary decidable equivalence relatio... |

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Citation Context ...ogramming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [8, 10], for example bounded quantication [20] 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... |

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Citation Context ... University of Edinburgh, JCMB, The King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages =-=[1, 6, 21, 28]-=-, proof assistants [3, 25, 29], and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relat... |

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Citation Context ...n 2 , even though they are not syntactically identical." A similar situation appears in intersection types disciplines, where > = A!> and also A!> > and > A!>. Asnal example is extensible recor=-=ds [11]-=-, where the extension operator is associative and commutative in the subtyping relation. These equivalence relations have models in existing frameworks, for example game semantics [13] or PER models [... |

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Citation Context ...s-reduction for types. ; XA1 :K1 ` A2 : K2 ` C A1 : K1 ` (XA1 :K1:A2) C =sA2 [X C] : K2 [X C] (T-Eq-Beta) The type equality rules appear in Appendix B. The subtyping rules are again those of F !s[9, =-=10, 2-=-7], except for those dealing with bounded type abstraction and type application and the rule for subtyping the quantier. We chose Cardelli and Wegner's kernel Fun rule for quantiers with equal bounds ... |

87 | A modest model of records, inheritance, and bounded quantification - Bruce, Longo |

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Citation Context ...he King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages [1, 6, 21, 28], proof assistants =-=[3, 25, 29]-=-, and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relation for a higher-order calculu... |

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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 =-=[8, 10-=-], for example bounded quantication [20] and recursive types [2]. Metatheoretic properties of the type systems are important to justify the programming languages being modeled. One important property ... |

61 |
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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 [8, 10], for example bounded quantication [=-=20]-=- 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... |

51 | Higher-order subtyping - Pierce, Steffen - 1994 |

42 | Refinement types for logical frameworks
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Citation Context ...he King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages [1, 6, 21, 28], proof assistants =-=[3, 25, 29]-=-, and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relation for a higher-order calculu... |

40 | Decidability of higher-order subtyping with intersection types
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Citation Context ...S B w XA:K1:D : XA 0 :K 0 1 :K2 `S K2 [X C] n K `S D[X C] w E wn F : K `S C A : K 0 1 `S B C w E wn F : K (ST-Beta) The weak-head subtyping rules are motivated by the algorithmic rules in [16]. The rules SWS-Arrow, SWS-All, and SWS-TAbs are structural. The rule SWS-TApp implicitly uses transitivity, reducing the problem of a variable being less than another type to the problem of the bound... |

29 | Typing in object-oriented languages: Achieving expressiveness and safety
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Citation Context ... University of Edinburgh, JCMB, The King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages =-=[1, 6, 21, 28]-=-, proof assistants [3, 25, 29], and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relat... |

24 |
An operational semantics for
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Citation Context ...tics. Typed Operational Semantics Our proof of anti-symmetry of higher-order subtyping relies on our understanding of subtyping built up using typed operational semantics. Typed operational semantics =-=[22-=-] gives an alternative induction principle for type theories, by presenting type theory operationally rather than declaratively. The typed operational semantics for F ! has judgements for reduction t... |

21 |
Notes about F ! <:. Unpublished manuscript
- Cardelli
- 1990
(Show Context)
Citation Context ...s-reduction for types. ; XA1 :K1 ` A2 : K2 ` C A1 : K1 ` (XA1 :K1:A2) C =sA2 [X C] : K2 [X C] (T-Eq-Beta) The type equality rules appear in Appendix B. The subtyping rules are again those of F !s[9, =-=10, 2-=-7], except for those dealing with bounded type abstraction and type application and the rule for subtyping the quantier. We chose Cardelli and Wegner's kernel Fun rule for quantiers with equal bounds ... |

17 | B.C.: Higher-order intersection types and multiple inheritance
- Compagnoni, Pierce
(Show Context)
Citation Context ...], where the extension operator is associative and commutative in the subtyping relation. These equivalence relations have models in existing frameworks, for example game semantics [13] or PER models =-=[17]-=-. Such equivalences also arise in the context of programming languages. For example, consider object types with a private section and a public interface, where two object types O 1 and O 2 may satisfy... |

17 | A new type-assignment for -terms - Coppo, Dezani-Ciancaglini - 1978 |

17 | editors. Theoretical Aspects of ObjectOriented - Gunter, Mitchell - 1994 |

12 | Typed operational semantics for higher order subtyping
- Compagnoni, Goguen
- 1997
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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 =-=[15-=-] about F ! , 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 c... |

11 | Game Semantics and Subtyping
- Chroboczek
- 2003
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Citation Context ...ensible records [11], where the extension operator is associative and commutative in the subtyping relation. These equivalence relations have models in existing frameworks, for example game semantics =-=[13]-=- or PER models [17]. Such equivalences also arise in the context of programming languages. For example, consider object types with a private section and a public interface, where two object types O 1 ... |

10 | Subtyping for object type constructors
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- 1999
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Citation Context |

10 | Higher-order subtyping. Theoretical Computer Science - Pierce, Steffen - 1997 |

8 | Equational axiomatization of bicoercibility for polymorphic types
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Citation Context ...alculus a la Curry with subtyping, a completely dierent style of subtyping from the one we consider here, anti-symmetry has been studied under the name of equational axiomatization of bicoercibility [34]. There, A and B are called bicoercible if A B and B A, and the paper proves that if A and B are bicoercible then A B, for an appropriate equivalence relation . However, the problem of bicoerci... |

6 | Some Algorithmic and Proof-Theoretical Aspects of Coercive Subtyping
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Citation Context ...he King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of its importance in applications to programming languages [1, 6, 21, 28], proof assistants =-=[3, 25, 29]-=-, and metatheoretical studies [4, 5, 7, 10, 14-16, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relation for a higher-order calculu... |

6 |
Subtyping + extensionality: Confluence of fij-reductions
- Curien, Ghelli
(Show Context)
Citation Context ...4]. The subtyping relation has been extensively researched because of its importance in applications to programming languages [27, 1, 6, 21], proof assistants [28, 3, 24], and metatheoretical studies =-=[18, 4, 7, 10, 5, 19, 16, 23, 29, 15, 14]-=-, to name a few. However, none of these studies has established the anti-symmetry of the subtyping relation for a higher-order calculus. In some cases it has been conjectured, as in [34]. In other cas... |

2 | Subtyping + Extensionality: Con of Reduction - Curien, Ghelli - 1991 |

2 |
Bounded quanti is undecidable
- Pierce
- 1992
(Show Context)
Citation Context ...he rule for subtyping the quantier. We chose Cardelli and Wegner's kernel Fun rule for quantiers with equal bounds [12], because the contravariant rule for quantiers renders the system undecidable [31=-=-=-]. Furthermore, transitivity elimination in the presence of such a rule in the higher-order case remains an open problem. Type equality is included in subtyping. ` A =sB : K ` A B : K (S-Conv) The su... |

2 |
An Object Oriented Programming Logic Based on Type Theory
- Zwanenburg
- 1999
(Show Context)
Citation Context ...6, 18, 19, 24, 30], to name a few. However, none of these studies has established the anti-symmetry of the subtyping relation for a higher-order calculus. In some cases it has been conjectured, as in [35]. In other cases, the problem is avoided by taking an equality that satises anti-symmetry by denition: A = B is dened as A B and B A. Steen [33] has showed the simpler property, appropriate to... |

2 |
n K implies
- `S
(Show Context)
Citation Context |

1 | Decidability of higher-order subtyping via logical relations
- Compagnoni, Goguen
- 1997
(Show Context)
Citation Context ...rogramming errors. Decidability of type-checking ensures that this will always be possible, and decidability of subtyping is a crucial step to proving this. This result was proved for our calculus in =-=[14]-=-. ? Most of this author's work was carried out at LFCS, University of Edinburgh, JCMB, The King's Buildings, Edinburgh, EH9 3JZ, UK. The subtyping relation has been extensively researched because of i... |

1 |
Bounded quanti with bottom
- Pierce
- 1997
(Show Context)
Citation Context ...ons The equivalence relation induced by A B and B A may be stronger than the usual intensional equality associated with type theory, syntactic equivalence on normal forms. One such case occurs in [3=-=2], whe-=-re the types 8(XsBot)X!X and 8(XsBot)Bot!Bot are \equivalent in the subtype relation 2 , even though they are not syntactically identical." A similar situation appears in intersection types disci... |

1 |
n K and `S K n K then K K . Lemma 14 (Subject Reduction). `S A w B wn C : K and A ! A imply there is a B such that
- K
(Show Context)
Citation Context ...ogramming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [8, 10], for example bounded quantication [20] 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... |

1 |
wn C : K then `S A
- B, B
(Show Context)
Citation Context ...ogramming languages has been to introduce type systems with features to model constructs from object-oriented programming languages [8, 10], for example bounded quantication [20] 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... |