Precise Expression for the Algorithmic Information Distance
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We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitányi, 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 ⊆{0,1}^n given any of its elements, has length at most max_w ∈ S K(S|w) + O(1), provided this maximum is at least logarithmic in n.
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