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Revision #1 to TR13-007 | 8th January 2013 15:39

Short lists with short programs in short time

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Revision #1
Authors: Bruno Bauwens, Anton Makhlin, Nikolay Vereshchagin, Marius Zimand
Accepted on: 8th January 2013 15:39
Downloads: 316
Keywords: 


Abstract:

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a $O(\log|x|)$-short program for $x$. We also show that there exist computable functions that map every $x$ to a list of size $O(|x|^2)$ containing a $O(1)$-short program for $x$ and this is essentially optimal because we prove that such a list must have size $\Omega(|x|^2)$. Finally we show that for some machines, computable lists containing a shortest program must have length $\Omega(2^{|x|})$.



Changes to previous version:

Corrected the year of Jason's Teutsch paper.


Paper:

TR13-007 | 8th January 2013 14:40

Short lists with short programs in short time


Abstract:

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a $O(\log|x|)$-short program for $x$. We also show that there exist computable functions that map every $x$ to a list of size $O(|x|^2)$ containing a $O(1)$-short program for $x$ and this is essentially optimal because we prove that such a list must have size $\Omega(|x|^2)$. Finally we show that for some machines, computable lists containing a shortest program must have length $\Omega(2^{|x|})$.



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