An incompressibility theorem for automatic complexity

Abstract

Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in {\{0,1\}}^n$. 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 $x$.