The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function $\mathrm{PPL}_{\mathbf{u}}(n)$ has been precisely computed is the Thue-Morse word $\mathbf{t}$. This word is $2$-automatic and, predictably, its function $\mathrm{PPL}_{\mathbf{t}}(n)$ is $2$-regular, but is this the case for all automatic words? In this paper, we prove that this function is $k$-regular for every $k$-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the period-doubling word, the prefix palindromic length does not look $2$-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.

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关于自动词前缀回文长度

无限单词 $\mathbf{u}$ 的前缀回文长度 $\mathrm{PPL}_{\mathbf{u}}(n)$ 是表达长度 $n$ 前缀所需的最小串联回文数$\mathbf{u}$。自 2013 年以来，$\mathrm{PPL}_{\mathbf{u}}(n)$ 是否对每个非周期性无限词 $\mathbf{u}$ 都是无界的仍然未知，尽管这已被证明几乎所有非周期性词。同时，唯一一个众所周知的非平凡无限词是函数 $\mathrm{PPL}_{\mathbf{u}}(n)$ 被精确计算的 Thue-Morse 词 $\mathbf{t }$。这个词是 $2$-automatic，可以预见的是，它的函数 $\mathrm{PPL}_{\mathbf{t}}(n)$ 是 $2$-regular，但是所有自动词都是这样吗？在本文中，我们证明这个函数对于每个只包含有限数量回文的 $k$-automatic 词是 $k$-regular。对于两个这样的词，即折纸词和 Rudin-Shapiro 词，我们推导出该函数的公式。我们的计算实验表明，通常情况并非如此：对于周期倍增词，前缀回文长度看起来不是 $2$-regular，而对于 Fibonacci 词，它看起来也不是 Fibonacci-regular。如果得到证实，这些结果将给出不规则的自动词的自然功能的罕见（如果不是第一个）示例。前缀回文长度看起来不是 $2$-regular，对于 Fibonacci 词，它看起来也不是 Fibonacci-regular。如果得到证实，这些结果将给出不规则的自动词的自然功能的罕见（如果不是第一个）示例。前缀回文长度看起来不是 $2$-regular，对于 Fibonacci 词，它看起来也不是 Fibonacci-regular。如果得到证实，这些结果将给出不规则的自动词的自然功能的罕见（如果不是第一次）示例。