Nondeterministic Automatic Complexity of Overlap-Free and Almost Square-Free Words

Abstract

Shallit and Wang studied deterministic automatic complexity of words.  They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 1/2$.   We improve that result by showing that $I(\mathbf t)= 1/2$.  We prove that the nondeterministic automatic complexity $A_N(x)$ of a word $x$ of length $n$ is bounded by $b(n):=\lfloor n/2\rfloor + 1$.  This enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$.  If $x$ is square-free then $D(x)=0$. If $x$ is almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a overlap-free binary word such as the infinite Thue--Morse word, then $D(x)\le 1$.  On the other hand, there is no constant upper bound on $D$ for overlap-free words over a ternary alphabet, nor for cube-free words over a binary alphabet.The decision problem whether $D(x)\ge d$ for given $x$, $d$ belongs to $\mathrm{NP}\cap \mathrm{E}$.