Theory of representations

Kreitz, Christoph; Weihrauch, Klaus

doi:10.1016/0304-3975(85)90208-7

Cited by 128 publications

(56 citation statements)

References 9 publications

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“…An answer can be given only relatively to computability concepts on the measures and functions under consideration, which must be defined in advance. For studying computability on general spaces we use the representation approach for computable analysis (TTE, Type Two theory of Effectivity) [KW85,Wei00,BHW08]. In TTE computability on finite words w ∈ Σ * and infinite sequences p ∈ Σ N is defined explicitly, for example by Turing machines, and then such finite or infinite sequences are used as names of abstract objects.…”

Section: Introductionmentioning

confidence: 99%

Computability of the Radon-Nikodym Derivative

Hoyrup

Rojas

Weihrauch

2011

Models of Computation in Context

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We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f, g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN.

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Section: Introductionmentioning

confidence: 99%

Computability of the Radon-Nikodym Derivative

Hoyrup

Rojas

Weihrauch

2011

Models of Computation in Context

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“…(in terminology of Section 5), principal continuous representations. This notion was introduced in [KW85] for countably based spaces and it was extensively studied by many authors. In [BH02] a close relation of admissible representations of countably based spaces to open continuous representations was established.…”

Section: Admissible Total Representationsmentioning

confidence: 99%

Total Representations

Selivanov¹

2013

Logical Methods in Computer Science

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Abstract. Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.

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“…Turing associates the real number (2c 0 − 1)n + ∞ r=1 (2c r − 1) (2/3) r to the infinite binary sequence c 0 1 n 0c 1 c 2 · · · ∈ {0, 1} ω . 1 Our choice of representation differs from Turing's, but it is equivalent in the sense of Kreitz and Weihrauch [15]; in particular, they both induce the same computable structure on R. See Weihrauch [43] for a thorough introduction to the theory of representations.…”

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confidence: 99%

“…Let β = i∈N k∈A (i) ,k≤n−i 2 −k−1 . For 6 The reverse is not true, so b-representation is not equivalent-again in the sense of Kreitz and Weihrauch [15,43] …”

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confidence: 99%

Degrees of unsolvability of continuous functions

Miller¹

2004

J. symb. log.

112

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Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0,1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ C[0, 1] computes a non-computable subset of N; there is a non-total degree between Turing degrees a < T b iff b is a PA degree relative to a; S ⊆ 2 N is a Scott set iff it is

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Theory of representations

Cited by 128 publications

References 9 publications

Computability of the Radon-Nikodym Derivative

Computability of the Radon-Nikodym Derivative

Total Representations

Degrees of unsolvability of continuous functions

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