An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Kolmogorov's Structure Functions with an Application to the Foundations of Model Selection
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
IEEE Transactions on Information Theory
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Let x be a binary string of length n. Consider the set Px of all pairs of integers (a,b) such that the randomness deficiency of x in a finite set S of Kolmogorov complexity at most a is at most b. The paper [4] proves that there is no algorithm that for every given x upper semicomputes the minimal deficiency function βx(a)=min{b | (a,b)∈px}with precision n/log4n. We strengthen this result in two respects. First, we improve the precision to n/4. Second, we show that there is no algorithm that for every given x enumerates a set at distance at most n/4 from Px, which is easier than to upper semicompute the minimal deficiency function of x with the same accuracy.