Hello,


is the problem of pattern recognition (for a given sequence of n numbers, find the shortest Turing machine with an alphabet of 42 elements that will output these n numbers in, say, 5\*n^3 time) an NP-complete problem?


To me, it seems practically impossible that the problem is in P: If you had the level of understanding of algorithms required to show quickly that all algorithms smaller than this-or-that size will produce a different output or require more than 5\*n^3 time, what reasons could there be that prevent you from having the level of understanding required to solve the Halting problem? (in this case, actually, the problem of determining whether a program will run in less than 5\*n^3 steps) But I have no clue how you could possibly *prove* this. On the one hand, this seems to be for exactly the same reason: If you use the method "cleverly encode your SAT problem as a pattern; then prove that if the SAT instance is satisfiable, you could use a Turing machine of form A; if it is not satisfiable, you must use a longer Turing machine of form B" (or vice versa), how do you want to show that there exists no Turing machine C that is smaller than both of them? And: If you had the understanding to prove that, how could you possibly not be able to solve the Halting problem?


Even if you left that issue aside, I can't even think of approaches that could create such an A-B dichotomy.


Is the problem possibly NP-intermediate?



Is pattern recognition NP-complete?
The key phrase you are looking for is "**resource-bounded Kolmogorov complexity**". [This paper](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.9495) by Allender, et. al. may be a good starting point. Also, [this PhD thesis](http://www.research.rutgers.edu/~troyjlee/thesis.html) might provide some helpful background.


*Edited to add*:


According to the first article, [Ko, K.-I. "On the complexity of learning minimum time-bounded Turing machines", *SIAM Journal on Computing*, 20 (1991)](http://portal.acm.org/citation.cfm?id=92591) may be even more relevant. In that paper, Ko demonstrates that determining whether or not computing time-bounded Kolmogorov complexity is NP-hard requires non-relativizing techniques.


