## Definability and reducibility in higher types over the reals

Venue: | the proceedings of Logic Colloquium ’03 |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Normann_definabilityand,

author = {Dag Normann},

title = {Definability and reducibility in higher types over the reals},

booktitle = {the proceedings of Logic Colloquium ’03},

year = {}

}

### OpenURL

### Abstract

We consider sets CtR(σ) of total, continuous functionals of type σ over the reals. A subset A ⊆ CtR(σ) is reducible if A can be reduced to totality in one of the other spaces. We show that all Polish spaces are homeomorphic to a reducible subset of R → R and that the class of reducible sets is closed under the formation of function spaces and some comprehension. 1

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(Show Context)
Citation Context ...ed how complete metric spaces in general can be represented as domains, and how effective metric spaces can be represented as effective domains. Polish spaces are characterised as the Gδ-subspaces of =-=[0, 1]-=- N . In Section 4 we will use this to show that Polish spaces are homeomorphic to reducible subsets of R → R. Moreover we will show that certain definable subsets of reducible sets will be reducible, ... |

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Citation Context ... but we have made no deep exploration of this connection. 4 Polish spaces A Polish Space is a separable topological space that admits a complete metric. We reccomend Hoffmann-Jørgensen [7] or Kechris =-=[8]-=- as standard references on Polish spaces. A useful characterisation is that the Polish spaces are exactly the topological spaces homeomorphic to Gδ subsets of [0, 1] N . Moreover, a Gδ subspace of a P... |

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Citation Context ...of forming function spaces. This hierarchy also have numerous characterisations. It is equivalent to the one studied by Weihrauch and his group in his project on Type Two Enumerability, see Weihrauch =-=[15]-=-. The equivalence is proved combining the characterisations due to Schröder [13] of the Weihrauch hierarchy and to Normann [10] of the hierarchy in this paper as the hierarchy obtained in the category... |

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(Show Context)
Citation Context ...ll call CtR(σ) for some finite type σ. The point is that when a structure A is represented as a reducible set in this way, we have access to internal notions of computability, like Escardó’s Real PCF =-=[5]-=-, for computing on elements in A. Blanck [3] showed how complete metric spaces in general can be represented as domains, and how effective metric spaces can be represented as effective domains. Polish... |

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Citation Context ...oint is that when a structure A is represented as a reducible set in this way, we have access to internal notions of computability, like Escardó’s Real PCF [5], for computing on elements in A. Blanck =-=[3]-=- showed how complete metric spaces in general can be represented as domains, and how effective metric spaces can be represented as effective domains. Polish spaces are characterised as the Gδ-subspace... |

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Citation Context ...ch and his group in his project on Type Two Enumerability, see Weihrauch [15]. The equivalence is proved combining the characterisations due to Schröder [13] of the Weihrauch hierarchy and to Normann =-=[10]-=- of the hierarchy in this paper as the hierarchy obtained in the category of Kuratowski limit spaces. In his thesis [4] De Jaeger gave a characterisation in the category of filter spaces. In this pape... |

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Citation Context ...σ as follows: • ¯ R(0) is the set of ideals α ⊆ R0(0) such that ∩α consists of one real number. • ¯ R(σ → τ) = {x ∈ R(σ → τ) | ∀y ∈ ¯ R(σ)(x(y) ∈ ¯ R(τ))}. The following was proved by Longo and Moggi =-=[9]-=- for the natural numbers. The proof works for this hierarchy as well, see e.g. Normann [10] Proposition 1 When x1 and x2 are elements of ¯ R(σ), let x1 ≈ x2 when x1 ⊓ x2 ∈ ¯ R(σ). Then ≈ is an equival... |

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Citation Context ... It is equivalent to the one studied by Weihrauch and his group in his project on Type Two Enumerability, see Weihrauch [15]. The equivalence is proved combining the characterisations due to Schröder =-=[13]-=- of the Weihrauch hierarchy and to Normann [10] of the hierarchy in this paper as the hierarchy obtained in the category of Kuratowski limit spaces. In his thesis [4] De Jaeger gave a characterisation... |

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Citation Context ...k). The approximation lemma may be used instead of an extension theorem in many situations. Our main application will be in Section 3.2. Further applications will be found in the forthcomming Normann =-=[12]-=-. We need a more accurate description of the finitary elements of R(k + 1) in the proof of the approximation lemma. If σ is a finitary element in R(k) and [p, q] ∈ R0(0), then the pair (σ, [p, q]) def... |

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Citation Context ...hisms be the continuous functions, the algebraic domains then form a cartesian closed category. We will assume some familiarity with the theory for algebraic domains, see e.g Stoltenberg-Hansen & al. =-=[14]-=-, Abramsky and Jung [1] or Gierz & al. [6]. We will consider a typed hierarchy over the reals: Definition 2 a) We define the algebraic domain R(σ) for each type term σ as follows: • R0(0) = {R} ∪ {[p,... |

2 |
Calculabilité sur les réels, Thesis Paris VII
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(Show Context)
Citation Context ...haracterisations due to Schröder [13] of the Weihrauch hierarchy and to Normann [10] of the hierarchy in this paper as the hierarchy obtained in the category of Kuratowski limit spaces. In his thesis =-=[4]-=- De Jaeger gave a characterisation in the category of filter spaces. In this paper we will define the concept of a reducible set, which technically will be a subset of a special kind of what we will c... |

1 |
The theory of analytic spaces, Various Publication Series No
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(Show Context)
Citation Context ...s of realizers, but we have made no deep exploration of this connection. 4 Polish spaces A Polish Space is a separable topological space that admits a complete metric. We reccomend Hoffmann-Jørgensen =-=[7]-=- or Kechris [8] as standard references on Polish spaces. A useful characterisation is that the Polish spaces are exactly the topological spaces homeomorphic to Gδ subsets of [0, 1] N . Moreover, a Gδ ... |