Precise Expression for the Algorithmic Information Distance

We consider the notion of information distance between two objects $x$ and $y$ introduced by Bennett, Gacs, Li, Vitanyi, and Zurek in 1998 as the minimal length of a program that computes $x$ from $y$ as well as computing $y$ from $x$. In this paper, it was proven that the distance is equal to $\max (K(x|y),K(y|x))$ up to additive logarithmic terms, and it was conjectured that this could not be improved to $O(1)$ precision. We revisit subtle issues in the definition and prove this conjecture. We show that if the distance is at least logarithmic in the length, then this equality does hold with $O(1)$ precision for strings of equal length. Thus for such strings, both the triangle inequality and the characterization hold with optimal precision. Finally, we extend the result to sets $S$ of bounded size. We show that for each constant~$s$, the shortest program that prints an $s$-element set $S \subseteq \{0,1\}^n$ given any of its elements, has length at most $\max_{w \in S} K(S|w) + O(1)$, provided this maximum is at least logarithmic in~$n$.