The normalized algorithmic information distance can not be approximated
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It is known that the normalized algorithmic information distance N is not computable and not semicomputable. We show that for all ϵ < 1/2, there exist no semicomputable functions that differ from N by at most ϵ. Moreover, for any computable function f such that |lim_t f(x,y,t) - N(x,y)| <ϵ and for all n, there exist strings x,y of length n such that ∑_t |f(x,y,t+1) - f(x,y,t)| >Ω(log n). This is optimal up to constant factors. We also show that the maximal number of oscillations of a limit approximation of N is Ω(n/log n). This strengthens the ω(1) lower bound from [K. Ambos-Spies, W. Merkle, and S.A. Terwijn, 2019, Normalized information distance and the oscillation hierarchy], see arXiv:1708.03583 .
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