Nondeterministic Automatic Complexity of Almost Square-Free and Strongly Cube-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) ≤ 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.