Abstract
We study the complexity of computing the normalized information distance. We introduce a hierarchy of limit-computable functions 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 demonstrate a conditional undecidability result about the independence of pairs of random strings.
Highlights
Normalized information distance The normalized information distance NID is a distance measure for binary strings that is based on prefix-free Kolmogorov complexity K
The normalized information distance is defined as NID(x, y) =
Torenvliet, and Vitányi [9] have shown that NID can neither be computably approximated from below nor from above, i.e., such a computable approximation f of NID can neither be increasing nor decreasing in s
Summary
Normalized information distance The normalized information distance NID is a distance measure for binary strings that is based on prefix-free Kolmogorov complexity K. The value K(x) is the minimum length of a string p that describes x in the sense that U (p) = x for some fixed additively optimal Turing machine with prefix-free domain. Observe that such a machine cannot be defined on the empty string, all values of K are nonzero. We improve on these nonapproximability results by confirming their conjecture [9, Section 5] that for any computable approximation of NID, the number of oscillations is not bounded by a constant, or, equivalently, that NID is not in the oscillation hierarchy. The stronger result can be viewed as a conditional immunity statement and is used in the proof of our main result
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