## On sequential functionals of type 3 (2006)

Venue: | Math. Structures Comput. Sci |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Normann06onsequential,

author = {Dag Normann},

title = {On sequential functionals of type 3},

journal = {Math. Structures Comput. Sci},

year = {2006},

volume = {16},

pages = {2006}

}

### OpenURL

### Abstract

We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpo-enriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1

### Citations

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(Show Context)
Citation Context ... and the reader will not need much background from domain theory or computability theory in general. Further background in domain theory can be obtained from many sources, e.g. from Abramsky and Jung =-=[1]-=-, Stoltenberg-Hansen & al. [16] or Amadio and Curien [3]. The latter also provides a more than sufficient background on P CF . Acknowledgement Achim Jung gave valuable comments to a first draft of thi... |

397 |
LCF considered as a programming language
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Citation Context ...eterministic OR, and equivalently, the nondeterministic conditional ⊃nd are not captured by this calculus. These objects will be present in Scott’s model, but undefinable in his language for LCF . In =-=[12]-=- Plotkin viewed LCF as a programming language P CF . P CF is essentially typed λ-calculus with constants for fixed point operators for each type. P CF is given with its λ-calculus operational semantic... |

57 | Recursive functionals and quantifiers of finite types - Kleene - 1959 |

48 |
A type-theoretical alternative to ISWIM
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- 1993
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Citation Context ...al functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpo-enriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF . 1 Introduction In =-=[14]-=-, Scott defined the logic LCF , a logic for computable typed functionals. He constructed a“mathematical” model for LCF , a construction that led to the discovery of domain theory. By the Scott model w... |

42 |
Countable functionals
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Citation Context ...ous, unpublished manuscript for many years, but was later published as an historical document [15]. LCF has its root in Platek’s thesis [11] and the work on continuous functionals initiated by Kleene =-=[6]-=- and Kreisel [7]. Platek observed that nondeterministic OR, and equivalently, the nondeterministic conditional ⊃nd are not captured by this calculus. These objects will be present in Scott’s model, bu... |

38 |
A type theoretical alternative to
- Scott
- 1969
(Show Context)
Citation Context ...y the Scott model we will mean the hierarchy of algebraic domains constructed in [14]. [14] remained a famous, unpublished manuscript for many years, but was later published as an historical document =-=[15]-=-. LCF has its root in Platek’s thesis [11] and the work on continuous functionals initiated by Kleene [6] and Kreisel [7]. Platek observed that nondeterministic OR, and equivalently, the nondeterminis... |

30 |
Foundations of Recursion Theory
- Platek
(Show Context)
Citation Context ...hy of algebraic domains constructed in [14]. [14] remained a famous, unpublished manuscript for many years, but was later published as an historical document [15]. LCF has its root in Platek’s thesis =-=[11]-=- and the work on continuous functionals initiated by Kleene [6] and Kreisel [7]. Platek observed that nondeterministic OR, and equivalently, the nondeterministic conditional ⊃nd are not captured by th... |

29 |
Mathematical Theory of Domains,'' Cambridge Tracts
- Stoltenberg-Hansen, Lindstrom, et al.
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(Show Context)
Citation Context ...much background from domain theory or computability theory in general. Further background in domain theory can be obtained from many sources, e.g. from Abramsky and Jung [1], Stoltenberg-Hansen & al. =-=[16]-=- or Amadio and Curien [3]. The latter also provides a more than sufficient background on P CF . Acknowledgement Achim Jung gave valuable comments to a first draft of this paper. Later an anonymous ref... |

27 |
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- Milner
- 1977
(Show Context)
Citation Context ... We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpo-enriched. This shows that this model does not equal Milner’s =-=[9]-=- fully abstract model for P CF . 1 Introduction In [14], Scott defined the logic LCF , a logic for computable typed functionals. He constructed a“mathematical” model for LCF , a construction that led ... |

24 | On Full Abstraction for PCF: I. Models, observables and the full abstraction problem, II. Dialogue games and innocent strategies, III. A fully abstract and universal game model
- Hyland, Ong
(Show Context)
Citation Context ...m. Several attempts have been made to solve this problem. Most relevant for this paper are the game-semantical approaches in Abramsky, Jagadeesan and Malacaria [2] and independently in Hyland and Ong =-=[4]-=-. In both papers, the sequential functionals are constructed in two guises, as intensional and as extensional objects. The finitary and intensional sequential functionals are the functionals realized ... |

22 |
Interpretation of analysis by means of functionals of finite type
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- 1959
(Show Context)
Citation Context ... manuscript for many years, but was later published as an historical document [15]. LCF has its root in Platek’s thesis [11] and the work on continuous functionals initiated by Kleene [6] and Kreisel =-=[7]-=-. Platek observed that nondeterministic OR, and equivalently, the nondeterministic conditional ⊃nd are not captured by this calculus. These objects will be present in Scott’s model, but undefinable in... |

8 |
Full abstraction, totality and
- Plotkin
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(Show Context)
Citation Context ...s model. These are characterized via the extensional ordering of the intensional finitary functionals. The remaining problem, first mentioned in Hyland and Ong [4], but also pointed out by Plotkin in =-=[13]-=-, was if the infinitary objects in Milner’s model could be obtained directly from one of the infinitary sequential functionals, or if the extensional collapse of the sequential functionals from [2] or... |

4 |
Computations in Higher Types
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- 1977
(Show Context)
Citation Context ...ct that Kleene’s S1−S9 are restricted to the pure types, interpreting S1 − S9 over the Scott model is equivalent to P CF . The argument is originally due to Platek [11], and may be found in Moldestad =-=[10]-=-. Berger showed that the argument from [11] and [10] is valid in the continuous case as well. Thus an alternative approach to our main result would be to prove that the set of Scott-functionals of pur... |

2 |
Full abstraction for P CF (Extended abstract
- Abramsky, Jagadeesan, et al.
- 1994
(Show Context)
Citation Context ...as known as the fully abstraction problem. Several attempts have been made to solve this problem. Most relevant for this paper are the game-semantical approaches in Abramsky, Jagadeesan and Malacaria =-=[2]-=- and independently in Hyland and Ong [4]. In both papers, the sequential functionals are constructed in two guises, as intensional and as extensional objects. The finitary and intensional sequential f... |

1 | Finitary PCF is not decidable, Theoretical Computer Science - Loader - 2001 |