Kolmogorov complexity and the Recursion Theorem

Abstract

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.

Cone avoidance and randomness preservation

“…Remark 1.13. Recall from [18] the notions of autocomplexity and complexity for X ∈ {0, 1} N . By [13, §7] we have the following characterizations in terms of strong f -randomness.…”

Computability theory, algorithmic randomness and Turing's anticipation

“…For example, you can show that this ability corresponds to traceing, and the speed of growth of the initial segment complexity of a real. As an illustration, A is h-complex if C(A n) ≥ h(n) for all n. A is autocomplex if there is an Acomputable order h such that A is h-complex, where an order is a nondecreasing unbounded function with f (0) ≥ 1.. Theorem 14 (Kjos-Hanssen, Merkle, and Stephan [73]). A set is autocomplex iff it is of DNC degree.…”

Joining non-low C.E. sets with diagonally non-computable functions

“…Kjos-Hanssen, Merkle and Stephan [8] showed an elegant characterization of DNC degrees in terms of Kolmogorov complexity. Proposition 4.2 (Kjos-Hanssen, Merkle, Stephan).…”

“…Kjos-Hanssen, Merkle and Stephan actually show this result with plain Kolmogorov complexity C instead of prefix-free Kolmogorov complexity. The argument given in [8] applies to prefix-free complexity as well. For completeness of presentation we recall that argument.…”