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Automatic complexity of Fibonacci and Tribonacci words
Discrete Applied Mathematics ( IF 1.1 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.dam.2020.10.014
Bjørn Kjos-Hanssen

For a complexity function $C$, the lower and upper $C$-complexity rates of an infinite word $\mathbf{x}$ are \[ \underline{C}(\mathbf x)=\liminf_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n,\quad \overline{C}(\mathbf x)=\limsup_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n \] respectively. Here $\mathbf{x}\upharpoonright n$ is the prefix of $x$ of length $n$. We consider the case $C=\mathrm{A_N}$, the nondeterministic automatic complexity. If these rates are strictly between 0 and $1/2$, we call them intermediate. Our main result is that words having intermediate $\mathrm{A_N}$-rates exist, viz. the infinite Fibonacci and Tribonacci words.

中文翻译:

Fibonacci 和 Tribonacci 词的自动复杂度

对于复杂度函数 $C$,无限词 $\mathbf{x}$ 的下限和上限 $C$-复杂度率为 \[ \underline{C}(\mathbf x)=\liminf_{n\to\ infty} \frac{C(\mathbf{x}\upharpoonright n)}n,\quad \overline{C}(\mathbf x)=\limsup_{n\to\infty} \frac{C(\mathbf{x }\upharpoonright n)}n \] 分别。这里 $\mathbf{x}\upharpoonright n$ 是长度为 $n$ 的 $x$ 的前缀。我们考虑 $C=\mathrm{A_N}$ 的情况,即非确定性自动复杂度。如果这些费率严格介于 0 和 $1/2$ 之间,我们称它们为中级。我们的主要结果是存在具有中间 $\mathrm{A_N}$-rates 的单词,即。无限的斐波那契和 Tribonacci 词。
更新日期:2021-01-01
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