Normalized Information Distance and the Oscillation Hierarchy

TL;DR

This paper investigates the complexity of approximating normalized information distance, introducing a hierarchy based on oscillations, and proves it cannot be approximated within this hierarchy, highlighting its noncomputability.

Contribution

It introduces a hierarchy of computable approximations for normalized information distance and proves its nonapproximability within this hierarchy, strengthening previous results.

Findings

Normalized information distance is not in any level of the oscillation hierarchy.

The hierarchy is a function version of the difference hierarchy for sets.

A conditional undecidability result about independence is also proved.

Abstract

We study the complexity of approximations to the normalized information distance. We introduce a hierarchy of computable approximations 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 also prove a conditional undecidability result about independence.