## Bounded Quantification with Bottom (1997)

Venue: | Computer Science Department, Indiana University |

Citations: | 5 - 3 self |

### BibTeX

@TECHREPORT{Pierce97boundedquantification,

author = {Benjamin C. Pierce},

title = {Bounded Quantification with Bottom},

institution = {Computer Science Department, Indiana University},

year = {1997}

}

### OpenURL

### Abstract

While numerous extensions of Cardelli and Wegner's calculus of polymorphism and subtyping, Kernel Fun, have been studied during the past decade, one quite simple one has received little attention: the addition of a minimal type Bot, dual to the familiar maximal type Top. We develop basic meta-theory for this extension. Although most of the usual properties of Kernel Fun (existence of meets and joins, decidability of subtyping and typing, subject reduction, etc.) also hold for the extended system, the presence of Bot introduces some surprising intricacies. In particular, a type variable bounded by Bot is actually a synonym for Bot; such "bottom variables" must be treated specially at several points. 1 Introduction The typed lambda-calculus Kernel Fun [CW85] and its wilder sibling, F [CG92, Ghe90, CMMS94] have become standard tools for the foundational study of programming languages combining subtyping and impredicative polymorphism. Their syntax and semantics have been investigated in ...

### Citations

759 | On Understanding Types, Data Abstraction, and Polymorphism
- Cardelli, Wegner
(Show Context)
Citation Context .... In particular, a type variable bounded by Bot is actually a synonym for Bot; such "bottom variables" must be treated specially at several points. 1 Introduction The typed lambda-calculus K=-=ernel Fun [CW85]-=- and its wilder sibling, F [CG92, Ghe90, CMMS94] have become standard tools for the foundational study of programming languages combining subtyping and impredicative polymorphism. Their syntax and sem... |

305 | Subtyping recursive types
- Amadio, Cardelli
- 1993
(Show Context)
Citation Context ... detail, and they have formed the basis for a number of experimental language designs. Moreover, many ways of enriching these pure systems have been studied, including extensions with recursive types =-=[AC93]-=-, extensible records [CM91], existential types [GP97], and type operators [Car90, CL91, PT94, HP95, PS94, Com94]. However, one quite obvious extension has apparently never been considered in detail. M... |

292 | Abstract types have existential types - Mitchell, Plotkin - 1985 |

166 | Local type inference - Pierce, Turner - 1998 |

158 | Simple type-theoretic foundations for object-oriented programming - Pierce, Turner - 1994 |

142 | Operations on records
- Cardelli, Mitchell
- 1989
(Show Context)
Citation Context ...ed the basis for a number of experimental language designs. Moreover, many ways of enriching these pure systems have been studied, including extensions with recursive types [AC93], extensible records =-=[CM91]-=-, existential types [GP97], and type operators [Car90, CL91, PT94, HP95, PS94, Com94]. However, one quite obvious extension has apparently never been considered in detail. Most presentations of Kernel... |

112 | An extension of system F with subtyping - Cardelli, Martini, et al. - 1994 |

102 | Bounded quantification is undecidable
- Pierce
- 1994
(Show Context)
Citation Context ...ubtype relation is contravariant in the bounds. Although the additional power offered by Full F is occasionally useful (e.g. [ACV96]), the system lacks many desirable properties, such as decidability =-=[Pie94]-=-. 3. Completeness of subtyping and typechecking algorithms. 4. Subject reduction. 1 Strictly speaking, Java's null value has all object types, but not types like integer or array. The languge studied ... |

64 | 1991] "A semantic basis for Quest - Cardelli, Longo - 1991 |

51 | An Interpretation of Objects and Object Types
- Abadi, Cardelli, et al.
- 1996
(Show Context)
Citation Context ...Kernel Fun, the bounds of the two types must be the same; in Full F , the subtype relation is contravariant in the bounds. Although the additional power offered by Full F is occasionally useful (e.g. =-=[ACV96]-=-), the system lacks many desirable properties, such as decidability [Pie94]. 3. Completeness of subtyping and typechecking algorithms. 4. Subject reduction. 1 Strictly speaking, Java's null value has ... |

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

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

37 |
Proof Theoretic Studies about a Minimal Type System Integrating Inclusion and Parametric Polymorphism
- Ghelli
- 1990
(Show Context)
Citation Context ...table variant of Curien and Ghelli [CG92, CMMS94]. The principal reason for this restriction is that it allows us to define meets and joins of all pairs of types, which may fail to exist in "Full=-= F " [Ghe90]-=-. Second, we give an algorithmic presentation of subtyping, in which the rules of transitivity and general reflexivity are omitted, and recover these rules (in the next subsection) as properties of th... |

37 | A unifying type-theoretic framework for objects - Hofmann, Pierce - 1994 |

25 | Bounded existentials and minimal typing
- Ghelli, Pierce
- 1998
(Show Context)
Citation Context ...of experimental language designs. Moreover, many ways of enriching these pure systems have been studied, including extensions with recursive types [AC93], extensible records [CM91], existential types =-=[GP97]-=-, and type operators [Car90, CL91, PT94, HP95, PS94, Com94]. However, one quite obvious extension has apparently never been considered in detail. Most presentations of Kernel Fun (and F ) include a ty... |

21 | Notes about F ! <:. Unpublished manuscript - Cardelli - 1990 |

9 | Bounded quanti cation is undecidable - Pierce - 1992 |

5 | Local type argument synthesis with bounded quantification
- Pierce, Turner
- 1997
(Show Context)
Citation Context ...ty of transitivity) for subtyping. 2. Existence of meets and joins in the subtype relation. Because the existence of meets and joins in the subtype relation is crucial for the development here and in =-=[PT97], the syte-=-m studied here extends Cardelli and Wegner's original Kernel Fun [CW85] rather than the "Full F " of Curien and Ghelli [CG92]. The only difference between these systems lies in the subtyping... |

1 | Ste en. Higher-order subtyping - Pierce, Martin - 1994 |