## A Set-Theoretic Model for a Typed Polymorphic Lambda Calculus - A Contribution to MetaSoft (1988)

Venue: | Proceedings VDM'88 Symposium, Lecture Notes in Computer Science 328 |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Borzyszkowski88aset-theoretic,

author = {Andrzej Borzyszkowski and Ryszard Kubiak and Stefan Sokolowski},

title = {A Set-Theoretic Model for a Typed Polymorphic Lambda Calculus - A Contribution to MetaSoft},

journal = {Proceedings VDM'88 Symposium, Lecture Notes in Computer Science 328},

year = {1988},

volume = {328},

pages = {267--298}

}

### OpenURL

### Abstract

this paper is to define precisely the set-theoretic interpretation of the ML-like polymorphic system of types. The aim of this Introduction is to convince the Reader that this is worthwhile to try.

### Citations

364 |
Towards a theory of type structure
- Reynolds
- 1974
(Show Context)
Citation Context ...erly define the Types . In particular, it is important to realize whether polymorphic types, such as T above, belong themselves to Types . In Girard's and Reynold's second order polymorphic -calculus =-=[9, 15]-=- a polymorphic type may be instantiated by itself. This causes non-existence of set-theoretic model for it (cf. [16]) similarly to the case of the untyped -calculus. The construction of a model by Pit... |

285 |
Interpre'tation fonctionelle et e'limination des coupures duns l'arithme'tique d'ordre supe'rieure
- Girard
- 1972
(Show Context)
Citation Context ...erly define the Types . In particular, it is important to realize whether polymorphic types, such as T above, belong themselves to Types . In Girard's and Reynold's second order polymorphic -calculus =-=[9, 15]-=- a polymorphic type may be instantiated by itself. This causes non-existence of set-theoretic model for it (cf. [16]) similarly to the case of the untyped -calculus. The construction of a model by Pit... |

3 | The semantics of second-order polymorphic lambda calculus - Bruce, Meyer - 1984 |

3 |
Introduction to combinators and -calculus. Cambridge Unversity
- Hindley, Seldin
- 1985
(Show Context)
Citation Context ...of the set-theory. In [6] Scott's denotational model is given, this model, however, is highly unintuitive and not "naive" in our sense. The typed -calculus does have a simple set-theoretic m=-=odel (cf. [12]). We have-=- chosen the same remedy, i.e. strong typing, to enable a set-theoretic model for polymorphism; we have thus "typed the types". This gives rise to an infinite hierarchy of objects, types, sup... |

2 |
Cohn: Universal Algebra
- M
- 1981
(Show Context)
Citation Context ...allowness for simplicity. In the sequel we show how to construct the domain Types that is sufficiently universal for our needs (though smaller than the one discussed in a similar context by [4] after =-=[7]-=-). We start from a certain collection of sets Prim considered primitive. This collection is one of the parameters of the whole approach and it should correspond to user's needs. For instance, for nume... |

2 | A Theory of Polymorphism - Milner - 1978 |

1 |
Andrzej Tarlecki Naive Denotational Semantics in "Information Processing 83
- Blikle
- 1983
(Show Context)
Citation Context ...its typed version without self-application. We cannot solve any type equations but we are happy with the useful ones that do not require reflexive domains. This "naive" approach has been pre=-=sented in [4]-=-. In this paper we give a set-theoretic model for polymorphism. We hope the model is both consistent and intuitive. We believe our approach is adequate for discussing polymorphism and also for explain... |

1 |
Sokolowski Set-Theoretic Type Theory in preparation
- Borzyszkowski, Kubiak, et al.
(Show Context)
Citation Context ... the types". This gives rise to an infinite hierarchy of objects, types, supertypes,: : : etc., all of which may be polymorphic, superpolymorphic,: : : etc., and still with a set-theoretic model =-=(see [5]). This re-=-port presents a more conventional version. The polymorphism studied here is the shallow explicit one. "Shallow" means that polymorphic types may only depend on monomorphic ones, like in ML (... |

1 |
O'Donnell The Expressiveness of Simple and Second-Order Type
- Fortune, Leivant, et al.
- 1983
(Show Context)
Citation Context ...ng. Let sen:var = tv 1 then the types of the following two expressions (POLY tv 1 )var and (POLY tv 2 )var that are intended to be ff-convertible to each other, are not ff-convertible themselves (cf. =-=[8]-=-). The polymorphism under the operator LAMBDA of functional abstraction is prohibited. But this does not preclude our capability to deal with polymorphic functions; for example the polymorphic functio... |

1 |
Polymorphism is Set Theoretic, Constructively in "Category Theory
- Pitts
- 1987
(Show Context)
Citation Context ...polymorphic type may be instantiated by itself. This causes non-existence of set-theoretic model for it (cf. [16]) similarly to the case of the untyped -calculus. The construction of a model by Pitts =-=[14] does not -=-falsify this result since it is done under a non-standard view of the set-theory. In [6] Scott's denotational model is given, this model, however, is highly unintuitive and not "naive" in ou... |

1 |
Polymorphism Is Not Set-Theoretic in "Semantics of Data Types
- Reynolds
(Show Context)
Citation Context ...Calculus A Contribution to MetaSoft Andrzej Borzyszkowski Ryszard Kubiak Stefan Soko/lowski Project MetaSoft 1 May 1988 1 Introduction The Reader who knows that polymorphism is not set-theoretic (cf. =-=[16]-=-) is nevertheless implored not to quit reading out of hand. We are aware that the complexity of full polymorphism escapes a set-theoretic description. The situation, however, is not as hopeless with m... |

1 |
Data types as lattices Programming Research Group
- Scott
- 1974
(Show Context)
Citation Context ...peration of application as the genuine application of a function to an argument. However, there are numerous models of -calculus, for instance P!, the powerset of natural numbers, proposed by D.Scott =-=[17]. Unfortunately, all these mod-=-els are highly unintuitive. For example, to explain the meaning of "((x)x+1):2" in P! a subset of Nat is assigned to "(x)x+ 1" and another subset to "2", the both subsets... |