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Shallit and Wang studied deterministic automatic complexity of words. Theyshowed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infiniteThue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result byshowing that $I(\mathbf t)\ge 1/2$. For nondeterministic automatic complexitywe show $I(\mathbf t)=1/2$. We prove that such complexity $A_N$ of a word $x$of length $n$ satisfies $A_N(x)\le b(n):=\lfloor n/2\rfloor + 1$. This enablesus to define the complexity deficiency $D(x)=b(n)-A_N(x)$. If $x$ issquare-free then $D(x)=0$. If $x$ almost square-free in the sense of Fraenkeland Simpson, or if $x$ is a strongly cube-free binary word such as the infiniteThue word, then $D(x)\le 1$. On the other hand, there is no constant upperbound on $D$ for strongly cube-free words in a ternary alphabet, nor forcube-free words in a binary alphabet. The decision problem whether $D(x)\ge d$for given $x$, $d$ belongs to $NP\cap E$.

Nondeterministic automatic complexity of overlap-free and almost square-free words