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 $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that $I(\mathbf t)\ge 1/2$. For nondeterministic automatic complexity we 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 enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$. If $x$ is square-free then $D(x)=0$. If $x$ almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a strongly cube-free binary word such as the infinite Thue word, then $D(x)\le 1$. On the other hand, there is no constant upper bound on $D$ for strongly cube-free words in a ternary alphabet, nor for cube-free words in a binary alphabet. The decision problem whether $D(x)\ge d$ for given $x$, $d$ belongs to $NP\cap E$.