Information Distance Revisited

TL;DR

This paper revisits the concept of information distance, clarifies misconceptions about its prefix version, and establishes conditions under which certain equalities hold, refining the theoretical understanding of this measure.

Contribution

It corrects a previous claim about the prefix version of information distance and specifies when the known equality is valid, enhancing theoretical clarity.

Findings

The claim that prefix information distance equals max(K(x|y), K(y|x)) is false in general.

The equality holds if the information distance is at least super logarithmic.

Provides a refined understanding of the conditions for the equality to hold.

Abstract

We consider the notion of information distance between two objects x and y introduced by Bennett, G\'acs, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. It was claimed by Mahmud [11] that the prefix version of information distance equals max(K(x|y), K(y|) + O(1) (this equality with logarithmic precision was one of the main results of the paper by Bennett, G\'acs, Li, Vitanyi, and Zurek). We show that this claim is false, but does hold if the information distance is at least super logarithmic.