## Reducibility of Domain Representations and Cantor-Weihrauch Domain Representations (2006)

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@TECHREPORT{Blanck06reducibilityof,

author = {Jens Blanck},

title = {Reducibility of Domain Representations and Cantor-Weihrauch Domain Representations},

institution = {},

year = {2006}

}

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### Abstract

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.

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Citation Context ...n and Fredrik Dahlgren for many invaluable discussions on this paper. 2 Preliminaries 2.1 Domains We will briefly give some background to domain theory. For a fuller background on domains we refer to =-=[1, 25]-=-. Let D = (D, ⊑) be a partially ordered set. A subset A ⊆ D is an upper set if x ∈ A and x ⊑ y implies y ∈ A. Let ↑A = {y ∈ D : ∃x ∈ A(x ⊑ y)}. We will abbreviate ↑{x} by ↑x. A subset A ⊆ D is directe... |

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Citation Context ...R ,δ) is a Cantor domain representation of D. By composition, we have that (C,δ −1 [D R ],ρδ) is a Cantor domain representation of X. TTE also comes with a notion of reduction between naming systems (=-=Weihrauch 2000-=-, Definition 2.3.2), which are continuous partial functions. We will see that these reductions correspond to (total) continuous reductions in our sense. Definition 4.7. Let δ :Σ ω ⇀Xand ɛ :Σ ω ⇀Xbe TT... |

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Citation Context ...(as used in Computable Algebra). Reducibility between numberings (when one numbering factors through another) is one of the basic tools in studying numberings (Mal’cev 1961; Ershov 1973; 1975; 1977a; =-=Stoltenberg-Hansen and Tucker 1995-=-; 1999a). We will generalise reducibilityJ. Blanck 1032 to a very general class of representations of topological spaces and study basic properties of reducibility, in particular, for domain represen... |

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Citation Context ...rge class of topological spaces is to use domain representations. Representations of topological spaces by domains or embeddings of topological spaces into domains have been studied by several people =-=[2, 3, 4, 10, 11, 12, 13, 14, 18, 21, 24, 26, 27, 32, 35]-=-. Domain representations are also closely related to Type-2 Theory of Effectivity (TTE) [29, 33, 34] introduced by Weihrauch. Any T0 space can be given domain representations [6]. Some of these have n... |

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Citation Context ...representations within our framework. Weihrauch’s representation theory has been an inspiration for this work. Domain representations have been reinvented with the introduction of equilogical spaces (=-=Scott 1996-=-). The category Equ of equilogical spaces was shown in Bauer et al. (2004) to be equivalent to the category DPER(Dom) of dense partial equivalence relations over domains, which is just another view of... |

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Citation Context ...entation. 1.1. Related work The use of ordered structures to study effectivity on uncountable structures has appeared from time to time, for example, in higher types (Ershov 1977b) and weighted cpos (=-=Weihrauch and Schreiber 1981-=-). Weihrauch developed the idea of representations further in his successful Type-2 Theory of Effectivity (TTE), where he uses representations from Baire or Cantor space (Weihrauch 1987; 2000). Both o... |

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Citation Context ...to the proofs above. It is known that W-admissibility implies S-admissibility within CRep, and, trivially, we have that H-admissibility implies κ-admissibility for any κ. By characterisation results (=-=Schröder 2002-=-, Theorem 13; Hamrin 2005, Theorem 6.8), we have that spaces have S-admissible representations if and only if they have ω-admissible representations. We would like to relate S-admissibility and ω-admi... |

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Citation Context ...reductions for numberings (as used in Computable Algebra). Reducibility between numberings (when one numbering factors through another) is one of the basic tools in studying numberings (Mal’cev 1961; =-=Ershov 1973-=-; 1975; 1977a; Stoltenberg-Hansen and Tucker 1995; 1999a). We will generalise reducibilityJ. Blanck 1032 to a very general class of representations of topological spaces and study basic properties of... |

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Citation Context ... Other lifting results where density of D is not required can be found in [7, 19, 22] 2.3 Partial Domain Functions We will use the notion of continuous partial domain functions introduced by Dahlgren =-=[9]-=-. Definition 2.14. Let D and E be domains. A continuous partial function from D to E is a pair (S,f) where S ⊆ D is a non-empty closed subset of D, and f is a strict continuous function from S to E. N... |

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Citation Context ...e top element of E. Hence, the above result could be stated for arbitrary D and a retract representation E with a top element. Other lifting results where density of D is not required can be found in =-=[7, 19, 22]-=- 2.3 Partial Domain Functions We will use the notion of continuous partial domain functions introduced by Dahlgren [9]. Definition 2.14. Let D and E be domains. A continuous partial function from D to... |

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Citation Context ...arallel to the use of reductions for numberings (used in Computable Algebra). Reductions between numberings (when a numbering factors through another) is one of the basic tools in studying numberings =-=[15, 16, 17, 20, 27, 28]-=-. We generalise reducibility to a very general class of representations of topological spaces and study basic properties of reducibility, in particular for domain representations. Our aim is to study ... |

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Citation Context ...e top element of E. Hence, the above result could be stated for arbitrary D and a retract representation E with a top element. Other lifting results where density of D is not required can be found in =-=[7, 19, 22]-=- 2.3 Partial Domain Functions We will use the notion of continuous partial domain functions introduced by Dahlgren [9]. Definition 2.14. Let D and E be domains. A continuous partial function from D to... |

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Citation Context ...a,b] : a < x < b, a,b ∈ Q}. Lemma 6.1. (R, R R ,ρR,ηR) is an admissible representation of the reals. Proof. A standard proof shows that the representation is a dense retract domain representation. In =-=[8]-=- centred dyadic approximations are considered for efficient implementations of exact real arithmetic. These form an interesting substructure of the interval domain. Definition 6.2. A centred dyadic in... |

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Citation Context ...arallel to the use of reductions for numberings (used in Computable Algebra). Reductions between numberings (when a numbering factors through another) is one of the basic tools in studying numberings =-=[15, 16, 17, 20, 27, 28]-=-. We generalise reducibility to a very general class of representations of topological spaces and study basic properties of reducibility, in particular for domain representations. Our aim is to study ... |

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Theorie der Numerierungen II
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Citation Context ...arallel to the use of reductions for numberings (used in Computable Algebra). Reductions between numberings (when a numbering factors through another) is one of the basic tools in studying numberings =-=[15, 16, 17, 20, 27, 28]-=-. We generalise reducibility to a very general class of representations of topological spaces and study basic properties of reducibility, in particular for domain representations. Our aim is to study ... |