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A simple and powerful approach for studying constructivity, computability, and complexity

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Constructivity in Computer Science (Constructivity in CS 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 613))

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Abstract

In this contribution a natural and simple as well as general and efficient frame for studying effectivity (Type 2 theory of effectivity, TTE) is presented.

TTE is a straightforward “logic free” extension of ordinary computability theory. Three basic kinds of effectivity for functions on ∑* and ∑ω are distinguished: continuity, computability, and easy computability (computational complexity).

As the most remarkable property, continuity in TTE can be very adequateley interpreted as a basic kind of constructivity. Effectivity is transferred from ∑* and ∑ω to other sets by notations, (where finite words serve as names) and by representations (where ω-words serve as names), respectively.

In this contribution the structure of TTE is explained and its applicability is demonstrated by simple examples mainly from analysis. Especially it is shown how the “effectivity gap” between Abstract Analysis and Numerical Analysis can be closed step by step by introducing stronger and stronger effectivity requirements.

It is suggested that the theory outlined in this paper is adequate to introduce effectivity in Analysis into Comupter Science curricula.

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Authors and Affiliations

  1. Theoretische Informatik, Fern-Universität, D-5800, Hagen, Germany

    Klaus Weihrauch

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  1. Klaus Weihrauch
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J. Paul Myers Jr. Michael J. O'Donnell

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© 1992 Springer-Verlag Berlin Heidelberg

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Weihrauch, K. (1992). A simple and powerful approach for studying constructivity, computability, and complexity. In: Myers, J.P., O'Donnell, M.J. (eds) Constructivity in Computer Science. Constructivity in CS 1991. Lecture Notes in Computer Science, vol 613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021094

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  • DOI: https://doi.org/10.1007/BFb0021094

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  • Print ISBN: 978-3-540-55631-2

  • Online ISBN: 978-3-540-47265-0

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