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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05306-7

Published electronically:
April 27, 2011

MathSciNet review:
2813422

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Abstract | References | Similar Articles | Additional Information

Abstract: Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of . Furthermore, can Turing compute a DNR function iff there is a nontrivial -recursive lower bound on the Kolmogorov complexity of the initial segments of . is PA-complete, that is, can compute a -valued DNR function, iff can compute a function such that is a string of length and maximal -complexity among the strings of length . iff can compute a function such that is a string of length and maximal -complexity among the strings of length . 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:
https://doi.org/10.1090/S0002-9947-2011-05306-7

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