Nondeterministic Automatic Complexity of Almost Square-Free and Strongly Cube-Free Words

Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension I(t) of the infinite Thue word satisfies 1/3 ≤ I(t) ≤ 2/3. We improve that result by showing that I(t) ≥ 1/2. For nondeterministic automatic complexity we show I(t) = 1/2. We prove that such complexity AN of a word x of length n satisfies AN (x) ≤ b(n) := bn/2c+ 1. This enables us to define the complexity deficiency D(x) = b(n)−AN (x). If x is squarefree 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. The decision problem whether D(x) ≥ d for given x, d belongs to NP∩E.