Depth, Highness and DNR degrees

We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-[deep.sub.K], O(1)-[deep.sub.C], order-[deep.sub.K] and order-[deep.sub.C] sequences. Our main results are that Martin-Lof random sets are not order-[deep.sub.C], that every many-one...

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Bibliographic Details
Published in:Discrete Mathematics and Theoretical Computer Science Vol. 19; no. 4; p. 1
Main Authors: Moser, Philippe, Stephan, Frank
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.12.2017
Subjects:
Algorithms
Computation
Fuzzy sets
Logic
Mathematical research
Numbers, Random
Sequences
Set theory
Singapore
ISSN:1462-7264, 1365-8050
Online Access:Get full text
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Summary:We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-[deep.sub.K], O(1)-[deep.sub.C], order-[deep.sub.K] and order-[deep.sub.C] sequences. Our main results are that Martin-Lof random sets are not order-[deep.sub.C], that every many-one degree contains a set which is not O(1)-[deep.sub.C], that O(1)-[deep.sub.C] sets and order-[deep.sub.K] sets have high or DNR Turing degree and that no K-trival set is O(1)-[deep.sub.K]. Keywords: Bennett logical depth, Kolmogorov complexity, algorithmic randomness theory, computability and randomness.
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ISSN:1462-7264
1365-8050