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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Kolmogorov complexity and the Recursion Theorem


Authors: Bjørn Kjos-Hanssen, Wolfgang Merkle and Frank Stephan
Journal: Trans. Amer. Math. Soc. 363 (2011), 5465-5480
MSC (2010): Primary 03D28, 03D32, 68Q30
Published electronically: April 27, 2011
MathSciNet review: 2813422
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Abstract: Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers $ A$ can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of $ A$. Furthermore, $ A$ can Turing compute a DNR function iff there is a nontrivial $ A$-recursive lower bound on the Kolmogorov complexity of the initial segments of $ A$. $ A$ is PA-complete, that is, $ A$ can compute a $ \{0,1\}$-valued DNR function, iff $ A$ can compute a function $ F$ such that $ F(n)$ is a string of length $ n$ and maximal $ C$-complexity among the strings of length $ n$. $ A \geq_T K$ iff $ A$ can compute a function $ F$ such that $ F(n)$ is a string of length $ n$ and maximal $ H$-complexity among the strings of length $ n$. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which no longer permit the usage of the Recursion Theorem.


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Additional Information

Bjørn Kjos-Hanssen
Affiliation: Department of Mathematics, University of Hawaii at Mānoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822
Email: bjoern@math.hawaii.edu

Wolfgang Merkle
Affiliation: Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany
Email: merkle@math.uni-heidelberg.de

Frank Stephan
Affiliation: Departments of Computer Science and Mathematics, National University of Singapore, 3 Science Drive 2, Singapore 117543, Republic of Singapore
Email: fstephan@comp.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05306-7
PII: S 0002-9947(2011)05306-7
Received by editor(s): December 23, 2009
Received by editor(s) in revised form: February 2, 2010
Published electronically: April 27, 2011
Additional Notes: The first author was partially supported by NSF-USA grants DMS-0652669 and DMS-0901020.
Article copyright: © Copyright 2011 American Mathematical Society