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Comment by hakuseki | Hacker News Reader

The article contains a misleading description of NP-hardness:

> Computing the Kolmogorov complexity for a sequence X of N bits length is an NP-hard problem, since the search space of possible programs producing any sequence of length N is growing exponentially with respect to the input size N

NP-hardness would actually mean that any NP problem, e.g. a traveling salesman problem, could be "reduced" within polynomial time to calculating the Kolmogorov Complexity of a given string.

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I think NP-hard is correct? You can reduce the halting problem and Kolmogorov complexity computation to each other [1], so the question is whether TSP can be reduced to the halting problem. Well, you can reduce SAT to the halting problem (they explain how here [2]) and you can reduce TSP to SAT given they're both NP-complete so... we're done?

[1] https://www.nearly42.org/cstheory/halting_to_kolmogorov/

[2] https://en.wikipedia.org/wiki/NP-hardness#Examples

The reductions in your linked paper [1] are not polynomial-time reductions.

Construct a TM which enumerates all possible variable assignments and checks if the SAT problem is satisfied then halts if so. You can construct this TM in polynomial time, and it halts exactly if the SAT problem is satisfiable. So this is a polynomial reduction from SAT to the halting problem.

I do not dispute this. My comment was about the linked paper [1] regarding equivalence of the halting problem and Kolmogorov complexity, not the SAT problem.

Yes I am not aware of any proof on that one. Do you think it will suffice to generalize the statement to just "NP problem" instead?

I'm pretty sure determining Kolmogorov complexity is not in class NP, and in fact isn't computable.

(You can't just try all possible programs, because some of them never terminate and some others run for a very long time and you can't tell in advance which are which.)

I think you are right, since it entails the Halting problem it is not computable and thus not even in NP.

No, it is uncomputable, not an NP problem.

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