## Density and Choice for Total Continuous Functionals (1996)

Venue: | About and Around Georg Kreisel |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Schwichtenberg96densityand,

author = {Helmut Schwichtenberg},

title = {Density and Choice for Total Continuous Functionals},

booktitle = {About and Around Georg Kreisel},

year = {1996},

pages = {335--362}

}

### OpenURL

### Abstract

this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.

### Citations

398 | LCF considered as a programming language - Plotkin - 1977 |

259 |
Semantic Domains and Denotational Semantics. In
- Scott, Gunther
- 1990
(Show Context)
Citation Context ... may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott =-=[Sco82]-=- (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats inf... |

192 | Lectures on a mathematical theory of Computation - Scott - 1981 |

78 |
Interpretation of analysis by means of constructive functionals of finite types
- Kreisel
- 1959
(Show Context)
Citation Context ...e now prove the density theorem, which says that any finite partial continuous functional (i.e. any X with X 2 Con % ) can be extended to a total functional. This result is essentially due to Kreisel =-=[Kre59]-=-. The proof we give here has been obtained by specialization of a proof for a more general density theorem (for arbitrary f 0 --spaces) given by Berger in [Ber90, Ber93] (this proof also appears in [S... |

54 |
Using information systems to solve recursive domain equations
- Larsen, Winskel
- 1984
(Show Context)
Citation Context ...owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel =-=[LW84]-=- and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with ... |

49 |
Tω as universal domain
- Plotkin
(Show Context)
Citation Context ...tion systems C % have a rather pleasant property, which amounts to the possibility to locate inconsistencies in two--element sets of data objects; this property has been called coherent by Plotkin in =-=[Plo78]-=-, p.210. It is defined as follows. An information system A = (A; Con; `) is called coherent if for any set X ` fin A we have X 2 Con () (8a; b 2 X)fa; bg 2 Con: Clearly C nat = N is coherent, and furt... |

42 |
Countable functionals
- Kleene
- 1959
(Show Context)
Citation Context ...can be realized by a total functional f (which can be found uniformly in R). Both of these theorems also have effective versions, in a natural sense of the word. Kreisel's paper and the one by Kleene =-=[Kle59]-=- on Countable functionals in the same volume, which independently introduced equivalent concepts, have given rise to a lot of activity aimed at understanding and developing the mathematical notions in... |

36 |
Total sets and objects in domain theory
- Berger
- 1993
(Show Context)
Citation Context ... sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger =-=[Ber93]-=-. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Se... |

32 |
Foundations of Recursion Theory
- Platek
- 1966
(Show Context)
Citation Context ...ed. First of all there is the unpublished work of Tait in the so--called Stanford Report, which is based directly on Kreisel's work. Another early contribution of essential impact was Platek's thesis =-=[Pla66]-=-, who had started his work with Kreisel and later also worked with Dana Scott. Particularly Scott, in a number of attempts [Sco70, Sco82], contributed a lot to the subject. His work influenced Yuri Er... |

29 |
Model C of partial continuous functionals
- Ershov
- 1977
(Show Context)
Citation Context ...aim in this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals =-=[Ers77]-=-, essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental... |

29 |
Mathematical Theory of Domains,'' Cambridge Tracts
- Stoltenberg-Hansen, Lindstrom, et al.
- 1994
(Show Context)
Citation Context ...9]. The proof we give here has been obtained by specialization of a proof for a more general density theorem (for arbitrary f 0 --spaces) given by Berger in [Ber90, Ber93] (this proof also appears in =-=[SHGL94]-=-). Berger's proof in turn has been obtained by abstracting some general features from a proof of a density theorem given by Ershov in [Ers74, Ers75]. 5.7 Density Theorem. For any X 2 Con % we can find... |

22 | Totale Objekte und Mengen in der Bereichstheorie - Berger - 1990 |

11 |
Theorie der Numerierungen II, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 21
- ERSHOV
- 1975
(Show Context)
Citation Context ...ation. We obviously want to know that �� % is compatible with application. The only nontrivial part of this argument is to show that x �� % y =) zx �� oe zy. This has first been proved by =-=Yuri Ershov [Ers75]-=-, as an application of the density theorem. However, a simpler proof has later been found by Longo and Moggi [LM84], and we present that one. First we need some Lemmata. The first one notes a rather o... |

9 |
The hereditary partial effective functionals and recursion theory in higher types
- Longo, Moggi
- 1984
(Show Context)
Citation Context ... is to show that x �� % y =) zx �� oe zy. This has first been proved by Yuri Ershov [Ers75], as an application of the density theorem. However, a simpler proof has later been found by Longo an=-=d Moggi [LM84]-=-, and we present that one. First we need some Lemmata. The first one notes a rather obvious fact on total functionals. 5.3 Lemma. If z 2 G % , z 0 2 Cont % and z ` z 0 , then z 0 2 G % . Proof. By ind... |

8 |
Closing the gap between the continuous functionals and recursion
- Normann
- 1997
(Show Context)
Citation Context ...Ers75, Ers77]. Many other people have also made essential contributions, among them Feferman, Gandy, Hyland, Normann and Berger, mostly again in a rather general context. In particular Normann's work =-=[Nor93]-=- can be viewed as establishing the density theorem in a more general setting. My aim in this paper is to give complete proofs of the density theorem and the choice principle for total continuous funct... |

7 | Recursion on the partial continuous functionals - Schwichtenberg - 2006 |

2 | On the model G of the theory BR - Ershov - 1974 |

2 |
Lecture notes on domain theory
- Roscoe
- 1986
(Show Context)
Citation Context ...ted in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe =-=[Ros87]-=-, Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactl... |