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MASS PROBLEMS AND INITIAL SEGMENT COMPLEXITY

Published online by Cambridge University Press:  17 April 2014

W. M. PHILLIP HUDELSON
W. M. PHILLIP HUDELSON*
Affiliation:
301 GRANT STREET, SUITE 2600, PITTSBURGH, PA 15219, USAE-mail:[email protected]
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Abstract

By the complexity of a finite sequence of 0’s and 1’s we mean the Kolmogorov complexity, that is the length of the shortest input to a universal recursive function which returns the given sequence as output. By initial segment complexity of an infinite sequence of 0’s and 1’s we mean the asymptotic behavior of the complexity of its finite initial segments. In this paper, we construct infinite sequences of 0’s and 1’s with given recursive lower bounds on initial segment complexity which do not compute any infinite sequences of 0’s and 1’s with a significantly larger recursive lower bound on initial segment complexity. This improves several known results about randomness extraction and separates many natural degrees in the lattice of Muchnik degrees.

Keywords

Partial randomnessalgorithmic randomnessKolmogorov complexityMartin-Löf randomnesseffective Hausdorff dimension
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 20 - 44
Copyright
Copyright © Association for Symbolic Logic 2014 

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