Kolmogorov complexity and the Recursion Theorem

Bjørn Kjos-Hanssen; Wolfgang Merkle; Frank Stephan

doi:10.1090/s0002-9947-2011-05306-7

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arXiv · 0901.3933

Kolmogorov complexity and the Recursion Theorem

Bjørn Kjos-Hanssen·Wolfgang Merkle·Frank Stephan

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Abstract

Several classes of 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 segements 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 solves the halting problem 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 do no longer permit the usage of the Recursion Theorem.

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BibTeX RIS

Bjørn Kjos-Hanssen, Wolfgang Merkle, Frank Stephan. 2009-01-26. Kolmogorov complexity and the Recursion Theorem. https://doi.org/10.1090/s0002-9947-2011-05306-7

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