Kolmogorov complexity and the recursion theorem

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 ≥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. © 2011 American Mathematical Society.