Nondeterministic automatic complexity of overlap-free and almost square-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) <= 1/2. We improve that result by showing that I(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) := [n/2] + 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) <= 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) >= d for given x, d belongs to NP boolean AND E.