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June 2010 Effectivity and effective continuity of multifunctions
Dieter Spreen
J. Symbolic Logic 75(2): 602-640 (June 2010). DOI: 10.2178/jsl/1268917496

Abstract

If one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed into better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) into (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest. In this paper the setting is extended to the case of multifunctions. Various ways of coding (indexing) sets are discussed and their relationship is investigated. Moreover, effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Mostly, in addition to being effective the multifunctions need also possess certain witnessing functions. Important special cases are discussed where such witnessing functions always exist.

Citation

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Dieter Spreen. "Effectivity and effective continuity of multifunctions." J. Symbolic Logic 75 (2) 602 - 640, June 2010. https://doi.org/10.2178/jsl/1268917496

Information

Published: June 2010
First available in Project Euclid: 18 March 2010

zbMATH: 1209.03034
MathSciNet: MR2648157
Digital Object Identifier: 10.2178/jsl/1268917496

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 2 • June 2010
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