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Weakly Computable Real Numbers and Total Computable Real Functions

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Computing and Combinatorics (COCOON 2001)

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Abstract

Let C sc and C wc be classes of the semi-computable and weakly computable real numbers, respectively, which are discussed by Weihrauch and Zheng 12. In this paper we show that both C sc and C wc are not closed under the total computable real functions of finite length on some closed interval, although such functions map always a semi-computable real numbers to a weakly computable one. On the other hand, their closures under general total computable real functions are the same and are in fact an algebraic field. This field can also be characterized by the limits of computable sequences of rational numbers with some special converging properties.

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Author information

Authors and Affiliations

  1. FernUniversität Hagen, 58084, Hagen, Germany

    Robert Rettinger1

  2. BTU Cottbus, 03044, Cottbus, Germany

    Xizhong Zheng, Romain Gengler & Burchard von Braunmühl

Authors
  1. Robert Rettinger1
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  2. Xizhong Zheng
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  3. Romain Gengler
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  4. Burchard von Braunmühl
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Massachusetts, One University Avenue, Lowell, MA, 01854, USA

    Jie Wang

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© 2001 Springer-Verlag Berlin Heidelberg

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Rettinger1, R., Zheng, X., Gengler, R., von Braunmühl, B. (2001). Weakly Computable Real Numbers and Total Computable Real Functions. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_65

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  • DOI: https://doi.org/10.1007/3-540-44679-6_65

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42494-9

  • Online ISBN: 978-3-540-44679-8

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