An incompressibility theorem for automatic complexity

Abstract

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.