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.
The decision problem whether D(x) ≥ d for given x, d belongs to NP ∩ E.
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References
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Hyde, K.K., Kjos-Hanssen, B. (2014). Nondeterministic Automatic Complexity of Almost Square-Free and Strongly Cube-Free Words. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_6
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DOI: https://doi.org/10.1007/978-3-319-08783-2_6
Publisher Name: Springer, Cham
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