An incompressibility theorem for automatic complexity

Bjørn Kjos-Hanssen

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arXiv · 1908.10843

An incompressibility theorem for automatic complexity

Bjørn Kjos-Hanssen

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Abstract

Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in{\{\mathtt{0},\mathtt{1}\}}^n$. They also stated that Holger Petersen had informed them that the constant 13 can be reduced to 7. Here we show that it can be reduced to $2+\epsilon$ for any $\epsilon>0$. The result also applies to nondeterministic automatic complexity $A_N(x)$. In that setting the result is tight inasmuch as $A_N(x)\le n/2+1$ for all $x$.

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Bjørn Kjos-Hanssen. 2019-08-26. An incompressibility theorem for automatic complexity. https://arxiv.org/abs/1908.10843

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