Exact Expression For Information Distance

TL;DR

This paper provides an exact formula for information distance among multiple strings using Kolmogorov complexity, extending the concept beyond pairs to multisets with a rigorous proof.

Contribution

It introduces an elementary proof for expressing multiset information distance exactly in terms of Kolmogorov complexity, generalizing previous pairwise definitions.

Findings

Exact expression for multiset information distance

Proof of bounds equality up to a constant

Extension of information distance to multisets

Abstract

Information distance can be defined not only between two strings but also in a finite multiset of strings of cardinality greater than two. We give an elementary proof for expressing the information distance in terms of plain Kolmogorov complexity. It is exact since for each cardinality of the multiset the lower bound for some multiset equals the upper bound for all multisets up to a constant additive term.