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## Using dependent types to express modular structure (1986)

Venue: | In Thirteenth ACM Symposium on Principles of Programming Languages |

Citations: | 132 - 5 self |

### Citations

845 | On understanding types, data abstraction, and polymorphism.
- Cardelli, Wegner
- 1985
(Show Context)
Citation Context ...ssions of the form rep $t . s(t) t P : $t . s(t) where P is an expression of type s(t). These values are intended to model abstract data types, and were called data algebras in [MP85] and packages in =-=[CW85]-=-; we will use the term structure to agree with the terminology of [Mac85] that we will be adopting in later sections. The type component t will be called the witness or representation type of the stru... |

560 | The calculus of constructions. - Coquand, Huet - 1988 |

517 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...er them. Historically, these type systems are based on fundamental insights in proof theory, particularly the "formulas as types" notion that evolved through the work of Curry and Feys [CF58=-=], Howard [How80], de -=-Bruijn [deB80] and Scott [Sco70]. This notion provided the basis for Martin-Lo"f's formalizations of constructive logic as Intuitionistic Type Theory (ITT) [M-L71, M-L74, M-L82], and was utilized... |

321 |
Abstract types have existential type,".
- Mitchell, Plotkin
- 1988
(Show Context)
Citation Context ...nted a language similar to that used by Girard, and his version has come to be called the second-order lambda calculus. An extended form of this language, called SOL, was used by Mitchell and Plotkin =-=[MP85] to give a-=-n explanation of abstract data types. The programming languages ML [GMW78, Mil78] and Russell [BDD80, Hoo84, DD85] represent two distinctly different ways of realizing "polymorphism" by abst... |

316 | Constructive mathematics and computer programming. - Martin-Löf - 1979 |

208 | Modules for Standard ML. In:
- MacQueen
- 1984
(Show Context)
Citation Context ... sketch a ramified (i.e. stratified) system of dependent types from which we derive a small language called DL, which is a generalized and "desugared" version of the extended ML language pre=-=sented in [Mac85]-=-. The final section uses DL to illustrate some of the stylistic differences between ML and Pebble. 1. Shortcomings of SOL's existential types The SOL language [MP85] provides existential types of the ... |

207 | An intuitionistic theory of types: predicative part, - Martin-Lof - 1973 |

121 | Proofs as programs. - Bates, Constable - 1982 |

101 | A Theory of Type Polymorphism - Milner - 1978 |

60 |
A Kernel Language for Abstract Data Types and Modules
- Burstall, Lampson
- 1984
(Show Context)
Citation Context ...ursion. 7 __________________ 4 General product types are also called "indexed products", "Cartesian products," or "dependent function spaces." Other notations include x:A=-=sB(x) [CZ84], x:A --- >> B(x) [BL84], and "x: -=-A.B(x) (from the formulas as types isomorphism). 5 A "closed" version of the general sum, analogous to SOL's existential type, can be derived from the general product [Pra65], but the open v... |

60 |
de Bruijn. A survey of the project Automath
- G
- 1980
(Show Context)
Citation Context ...ly, these type systems are based on fundamental insights in proof theory, particularly the "formulas as types" notion that evolved through the work of Curry and Feys [CF58], Howard [How80], =-=de Bruijn [deB80] and -=-Scott [Sco70]. This notion provided the basis for Martin-Lo"f's formalizations of constructive logic as Intuitionistic Type Theory (ITT) [M-L71, M-L74, M-L82], and was utilized by Girard [Gir71],... |

31 |
Representation independence and data abstraction
- Mitchell
- 1986
(Show Context)
Citation Context ...relative strengths of predicative vs impredicative type systems, and reflexive vs irreflexive systems. It would be desirable to have a representation independence result analogous to that of Mitchell =-=[Mit86]-=- for the stratified system used here. Finally, it appears that the basic polymorphic type system of ML [Mil78] is in fact a ramified system, and that the system described in 2, rather than the second ... |

28 | An Investigation of a Programming Language with a Polymorphic Type Structure - McCracken - 1979 |

27 |
Constructive validity
- Scott
- 1970
(Show Context)
Citation Context ...tems are based on fundamental insights in proof theory, particularly the "formulas as types" notion that evolved through the work of Curry and Feys [CF58], Howard [How80], de Bruijn [deB80] =-=and Scott [Sco70]. Thi-=-s notion provided the basis for Martin-Lo"f's formalizations of constructive logic as Intuitionistic Type Theory (ITT) [M-L71, M-L74, M-L82], and was utilized by Girard [Gir71], who introduced a ... |

26 | A.Demers: Data types are values - Donahue - 1985 |

20 |
Towards a theory of type structure, in: Colloq. sur la
- Reynolds
- 1974
(Show Context)
Citation Context ...AUTOMATH system and Martin-Lo"f's ITT, is also central to the "programming logics", PL/CV3 and nu-PRL developed by Constable and his coworkers [CZ84, BC85]. In the programming language =-=area, Reynolds [Rey74]-=- independently invented a language similar to that used by Girard, and his version has come to be called the second-order lambda calculus. An extended form of this language, called SOL, was used by Mi... |

17 |
The type theory of PL/CV3
- Constable, Zlatin
- 1984
(Show Context)
Citation Context ..."sum" when we mean "general product" and "general sum." 2.2. Small and large types The stratified type system we will be working with is basically a simplified version of=-= the type system described in [CZ84]-=-. It has several (in fact infinitely many) levels, though only the first two or three will be mentioned here. At the bottom of the hierarchy are the small types, contained in the level 1 type universe... |

16 | Programming with modules as typed functional programming - Burstall - 1985 |

14 | Persistence and type abstraction
- Cardelli, MacQueen
- 1985
(Show Context)
Citation Context ...cognizable type within the scope of its definition. This indefiniteness seems to be the price paid for being able to treat the abstract type structure as an ordinary value rather than as a type. (See =-=[CM85], where we use the terms "-=-;virtual witness," "abstract witness", and "transparent witness" to describe three possible treatments of the witness type in an existential structure.) Hierarchies of structu... |

12 |
Type' is not a type
- Meyer, Reinhold
- 1986
(Show Context)
Citation Context ...s, the paradox implies at least the existence of divergent expressions, but it is not yet clear whether more serious pathologies might follow from it (see Meyer and Reinhold's paper, this proceedings =-=[MR86]-=-). Since types are simply values belonging to the type Type, reflexive type systems tend to obscure the distinction between types and the values they are meant to describe, and this in turn tends to c... |

10 | An informal description of Russell. - Boehm, Demers, et al. - 1980 |

6 |
A theory of types. Unpublished manuscript
- Martin-Löf
- 1971
(Show Context)
Citation Context ...uages, Russell and Pebble are distinguished by having "reflexive" type systems, meaning that there is a "type of all types" that is a member of itself (TypeType). Martin-Lo"f'=-=s initial version of ITT [M-L71] was also -=-reflexive in this sense, but he abandoned this version in favor of a "ramified " 2 system with a hierarchy of type universes when Girard's Paradox [Gir71] showed that the reflexive system wa... |

2 |
Une extension de l'interpre'tation de Go"del a` l'analyse, et son application a` l'e'limination des coupures dans l'analyse et la te'orie des types
- Girard
- 1971
(Show Context)
Citation Context ... [deB80] and Scott [Sco70]. This notion provided the basis for Martin-Lo"f's formalizations of constructive logic as Intuitionistic Type Theory (ITT) [M-L71, M-L74, M-L82], and was utilized by Gi=-=rard [Gir71], who introduce-=-d a form of second-order typed lambda calculus as a tool in his proof-theoretic work. The "formulas as types" notion, as developed in de Bruijn's AUTOMATH system and Martin-Lo"f's ITT, ... |

2 | Understanding Russell--a first attempt, in Semantics of Data Types - Hook - 1984 |

1 | The impredicative typed l-calculus, unpublished manuscript - Cardelli - 1985 |

1 | The impredicative typed λ-calculus, unpublished manuscript - Cardelli - 1985 |

1 |
Une extension de l’interpretation de Go del " a `l’analyse, et son application a ` l’e limination ´ des coupures dans l’analyse et la the orie ´ des types
- Girard
- 1971
(Show Context)
Citation Context ...deB80] and Scott [Sco70]. This notion provided the basis for Martin-Lo f’s " formalizations of constructive logic as Intuitionistic Type Theory (ITT) [M-L71, M-L74, M-L82], and was utilized by Girard =-=[Gir71]-=-, who introduced a form of second-order typed lambda calculus as a tool in his proof-theoretic work. The "formulas as types" notion, as developed in de Bruijn’s AUTOMATH system and Martin-Lo f’s " ITT... |

1 |
f, " A theory of types, unpublished manuscript
- Martin-Lo
- 1971
(Show Context)
Citation Context ...Russell and Pebble are distinguished by having ‘‘reflexive’’ type systems, meaning that there is a ‘‘type of all types’’ that is a member of itself (Type∈Type). Martin-Lo f’s " initial version of ITT =-=[M-L71]-=- was also reflexive in this sense, but he abandoned this version in favor of a ‘‘ramified’’ 2 system with a hierarchy of type universes when Girard’s Paradox [Gir71] showed that the reflexive system w... |

1 | An intuitionistic theory of types: predicative part, Logic Colloquium 73 - f - 1974 |

1 | f, " Constructive mathematics and computer programming - Martin-Lo - 1982 |