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Algorithmic randomness of continuous functions

Algorithmic randomness of continuous functions We investigate notions of randomness in the space $${{\mathcal C}(2^{\mathbb N})}$$ of continuous functions on $${2^{\mathbb N}}$$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random $${\Delta^0_2}$$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any $${y \in 2^{\mathbb N}}$$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

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References (40)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Springer-Verlag
Subject
Mathematics; Algebra; Mathematics, general; Mathematical Logic and Foundations
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-007-0060-4
Publisher site
See Article on Publisher Site

Abstract

We investigate notions of randomness in the space $${{\mathcal C}(2^{\mathbb N})}$$ of continuous functions on $${2^{\mathbb N}}$$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random $${\Delta^0_2}$$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any $${y \in 2^{\mathbb N}}$$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Jan 4, 2008

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