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A foundation for computable analysis

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Foundations of Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1337))

Abstract

While for countable sets there is a single well established computability theory (ordinary recursion theory), Computable Analysis is still underdeveloped. Several mutually non-equivalent theories have been proposed for it, none of which, however, has been accepted by the majority of mathematicians or computer scientists. In this contribution one of these theories, TTE (Type 2 Theorie of Effectivity), is presented, which at least in the author's opinion has important advantages over the others. TTE intends to characterize and study exactly those functions, operators etc. known from Analysis, which can be realized correctly by digital computers. The paper gives a short introduction to basic concepts of TTE and shows its general applicability by some selected examples.

First, Turing computability is generalized from finite to infinite sequences of symbols. Assuming that digital computers can handle (w.l.o.g.) only sequences of symbols, infinite sequences of symbols are used as names for “infinite objects” such as real numbers, open sets, compact sets or continuous functions. Naming systems are called representations. Since only very few representations are of interest in applications, a very fundamental principle for defining effective representations for T0-spaces with countable bases is introduced. The concepts are applied to real numbers, compact sets and continuous functions. The problem of zero-finding is considered. Computational complexity is discussed. The paper concludes with some remarks on other models for Computable Analysis.

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Authors and Affiliations

  1. FernUniversität Hagen, D-58084, Hagen

    Klaus Weihrauch

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  1. Klaus Weihrauch
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Christian Freksa Matthias Jantzen Rüdiger Valk

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© 1997 Springer-Verlag Berlin Heidelberg

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Weihrauch, K. (1997). A foundation for computable analysis. In: Freksa, C., Jantzen, M., Valk, R. (eds) Foundations of Computer Science. Lecture Notes in Computer Science, vol 1337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0052087

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  • DOI: https://doi.org/10.1007/BFb0052087

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  • Print ISBN: 978-3-540-63746-2

  • Online ISBN: 978-3-540-69640-7

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