## A Predicative Type-Theoretic Interpretation of Objects (1997)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Hickey97apredicative,

author = {Jason J. Hickey},

title = {A Predicative Type-Theoretic Interpretation of Objects},

year = {1997}

}

### OpenURL

### Abstract

Predicative type theories are powerful tools for giving foundational interpretations of programming languages. Due to their explicit inductive construction, predicative type theories have multiple mathematical models that provide precise definitions of programming language features. However, not all features have predicative interpretations, and current interpretations of objects rely on impredicative type theories, such as Girard's System F, because of the difficulty in specifying a type for objects in the presence of self-application. In this paper we show that objects have a predicative interpretation. We show that predicativity is associated with method monotonicity, and that binary methods prevent the inductive type construction. Our interpretation differs from impredicative accounts by replacing the use of recursive types for objects with conditions for method polymorphism over the self type. We further give a propositional meaning to objects in the type theory, providing a calc...

### Citations

950 | A Theory of Objects
- Abadi, Cardelli
- 1996
(Show Context)
Citation Context ... presence of method override. Self application occurs during method selection when the entire object is passed to the method as an implicit "self " argument. Recent type-theoretic accounts o=-=f objects [1, 21, 5, 10]-=- have been encoded in impredicative type theory (mainly variants of Girard's System F [9]), which allow self-application through an impredicative interpretation of the self parameter. In this paper, w... |

305 |
Interpretation fonctionelle et elimination des coupures de l'arithmetique d'ordre superieur
- Girard
- 1972
(Show Context)
Citation Context ... is passed to the method as an implicit "self " argument. Recent type-theoretic accounts of objects [1, 21, 5, 10] have been encoded in impredicative type theory (mainly variants of Girard's=-= System F [9]-=-), which allow self-application through an impredicative interpretation of the self parameter. In this paper, we show that the construction of objects has an interpretation in predicative type theory.... |

181 |
An intuitionistic theory of types: Predicative part. Logic Colloquium ‘73
- Martin-Löf
- 1975
(Show Context)
Citation Context ... a modular and object-oriented logical framework [12]. The object interpretation is more delicate but we get more information. For example, predicative theories have a strictly inductive construction =-=[19, 4]-=-, so the account of objects is inductive. An interesting feature is that impredicative type recursion is replaced by method polymorphism. The explicit inductive construction is also valid in impredica... |

167 | A lambda calculus of objects and method specialization - Fisher, Honsell, et al. |

165 | Simple type-theoretic foundations for object-oriented programming. Journal of Functional Programming. To appear. A preliminary version appeared
- Pierce, Turner
- 1993
(Show Context)
Citation Context ... presence of method override. Self application occurs during method selection when the entire object is passed to the method as an implicit "self " argument. Recent type-theoretic accounts o=-=f objects [1, 21, 5, 10]-=- have been encoded in impredicative type theory (mainly variants of Girard's System F [9]), which allow self-application through an impredicative interpretation of the self parameter. In this paper, w... |

120 | Paradigmatic object-oriented programming language: Design, static typing and semantics
- Bruce
- 1994
(Show Context)
Citation Context ... presence of method override. Self application occurs during method selection when the entire object is passed to the method as an implicit "self " argument. Recent type-theoretic accounts o=-=f objects [1, 21, 5, 10]-=- have been encoded in impredicative type theory (mainly variants of Girard's System F [9]), which allow self-application through an impredicative interpretation of the self parameter. In this paper, w... |

69 |
Recursive definitions in type theory
- Constable, Mendler
- 1985
(Show Context)
Citation Context ...ough grant 93-11-271, and from AASERT through grant N00014-95-1-0985. ticular, the Nuprl type theory that we use in this paper has multiple mathematical models, notably set theoretic [17], PER models =-=[3, 20]-=-, denotational models [23], and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11], a critical step toward our goal of providing a modular... |

56 |
A Non-Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ...ough grant 93-11-271, and from AASERT through grant N00014-95-1-0985. ticular, the Nuprl type theory that we use in this paper has multiple mathematical models, notably set theoretic [17], PER models =-=[3, 20]-=-, denotational models [23], and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11], a critical step toward our goal of providing a modular... |

54 | Objective ML: a simple object-oriented extension of ML - Rémy, Vouillon - 1997 |

50 | An interpretation of objects and object types
- Abadi, Cardelli, et al.
- 1996
(Show Context)
Citation Context ...es have propositional meaning. We further extend the calculus with dependent method types to provide a basis for formal program analysis. We develop our account on the object calculus of Abadi et al. =-=[1, 2]-=-, which gives a precise characterization of self application and method override. We show that the type system can be developed without the use of recursive types or weak sums, and without sacrificing... |

46 | Positive subtyping
- Hofmann, Pierce
- 1995
(Show Context)
Citation Context ... type theory. Although the existential encoding restricts method update, the interpretation of objects does not require the use of recursive types, allowing the development of expressive type systems =-=[6, 14]-=-. The interpretation of object relies on a restricted subtyping relation (we use notation ) to describe valid subobjects. This is important because the general subtype relation would disallow updates.... |

46 | Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra
- Jackson
- 1995
(Show Context)
Citation Context ..." part of S with a new element of T . In their account, Hofmann and Pierce use the subtyping to define an existential encoding of objects in F !s:. Another existential system was developed by Jac=-=kson [18]-=- to formalize a significant portion of constructive algebra in Nuprl. While Jackson's framework includes many of the properties of objects, the subtyping requirements prohibit the use of self-applicat... |

40 | A unifying type-theoretic framework for objects
- Hofmann, Pierce
- 1995
(Show Context)
Citation Context ...ve three parts: an object state, the object methods, and the proofs of method correctness (dependent objects with exactly one dependency). Their interpretation uses an existential encoding of objects =-=[21, 16]-=- in an impredicative type theory. Although the existential encoding restricts method update, the interpretation of objects does not require the use of recursive types, allowing the development of expr... |

33 |
A Non-type-theoretic Definition of Martin-Löf’s Types
- Allen
- 1987
(Show Context)
Citation Context ... a modular and object-oriented logical framework [12]. The object interpretation is more delicate but we get more information. For example, predicative theories have a strictly inductive construction =-=[19, 4]-=-, so the account of objects is inductive. An interesting feature is that impredicative type recursion is replaced by method polymorphism. The explicit inductive construction is also valid in impredica... |

30 | Formal objects in type theory using very dependent types
- Hickey
- 1996
(Show Context)
Citation Context ...hematical models, notably set theoretic [17], PER models [3, 20], denotational models [23], and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs =-=[15, 11]-=-, a critical step toward our goal of providing a modular and object-oriented logical framework [12]. The object interpretation is more delicate but we get more information. For example, predicative th... |

19 | Partial Objects in Type Theory
- Smith
- 1989
(Show Context)
Citation Context ...troduction of recursive methods, a formal account of totality would require a specification of all the intermediate computations. Our account of partial objects is based on the partial types of Smith =-=[25]-=-, which have been simplified by Crary [7]. A partial type, denotedsT , contains a term x if the evaluation of x diverges, or if x is an element of T . The induction principle for partial types relies ... |

17 | Higher-order intersection types and multiple inheritance
- Compagnoni, Pierce
- 1996
(Show Context)
Citation Context ... type theory. Although the existential encoding restricts method update, the interpretation of objects does not require the use of recursive types, allowing the development of expressive type systems =-=[6, 14]-=-. The interpretation of object relies on a restricted subtyping relation (we use notation ) to describe valid subobjects. This is important because the general subtype relation would disallow updates.... |

13 | Nuprl-light: An implementation framework for higherorder logics
- Hickey
- 1997
(Show Context)
Citation Context ... In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11], a critical step toward our goal of providing a modular and object-oriented logical framework =-=[12]-=-. The object interpretation is more delicate but we get more information. For example, predicative theories have a strictly inductive construction [19, 4], so the account of objects is inductive. An i... |

6 | Inductive De nition in Type Theory - Mendler - 1987 |

5 |
Semantic Foundations for Embedding HOL
- Howe
- 1996
(Show Context)
Citation Context ...9, from DARPA through grant 93-11-271, and from AASERT through grant N00014-95-1-0985. ticular, the Nuprl type theory that we use in this paper has multiple mathematical models, notably set theoretic =-=[17]-=-, PER models [3, 20], denotational models [23], and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11], a critical step toward our goal of... |

5 | A non-type-theoretic de nition of Martin-Lof's types - Allen - 1987 |

4 | Inheritance of proofs
- Hofmann, Narashewski, et al.
- 1996
(Show Context)
Citation Context ...lus. The functionality of the type theory guarantees progress, preservation, and substitution properties. 5 Related Work Another framework for expressive object calculi is developed by Hofmann et el. =-=[13]-=-, who have implemented a verification calculus based on the existential interpretation of objects. In their system, objects have three parts: an object state, the object methods, and the proofs of met... |

4 |
Bounded quantification and record-update problems. Message to Types electronic mail list
- Robinson, Tennent
- 1988
(Show Context)
Citation Context ...for any T that is a subtype of fx: fg ! Zg in the predicative type universe Uk . Unfortunately, this typing also fails to produce the expected behavior because it contains only the identity functions =-=[24]-=-. Essentially, we don't want polymorphism over subtypes of fx: fg ! Zg, but over record extensionssof fx: fg ! Z g. Since the move method is to modify the value of the method x, the type for x must re... |

3 |
Semantics of constructive type theory
- Rezus
- 1985
(Show Context)
Citation Context ...m AASERT through grant N00014-95-1-0985. ticular, the Nuprl type theory that we use in this paper has multiple mathematical models, notably set theoretic [17], PER models [3, 20], denotational models =-=[23]-=-, and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11], a critical step toward our goal of providing a modular and object-oriented logic... |

2 |
Recursive computation in foundational type theory
- Crary
(Show Context)
Citation Context ...account of totality would require a specification of all the intermediate computations. Our account of partial objects is based on the partial types of Smith [25], which have been simplified by Crary =-=[7]-=-. A partial type, denotedsT , contains a term x if the evaluation of x diverges, or if x is an element of T . The induction principle for partial types relies on an admissiblity condition, which holds... |

1 | Bounded quanti cation and record-update problems. Message to types email list - Robinson, Tennent - 1988 |