Normalized information distance and the oscillation hierarchy
Klaus Ambos‐Spies, Wolfgang Merkle, Sebastiaan A. Terwijn · Journal of Computer and System Sciences · 2021
We study the complexity of computing the normalized information distance. We introduce a hierarchy of limit-computable functions by considering the number of oscillations. This is a function version of the difference hierarchy for sets. We show that the normalized information distance is not in any level of this hierarchy, strengthening previous nonapproximability results. As an ingredient to the proof, we demonstrate a conditional undecidability result about the independence of pairs of random strings.