On prefix palindromic length of automatic words
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The prefix palindromic length PPL_๐ฎ(n) of an infinite word ๐ฎ is the minimal number of concatenated palindromes needed to express the prefix of length n of ๐ฎ. Since 2013, it is still unknown if PPL_๐ฎ(n) is unbounded for every aperiodic infinite word ๐ฎ, 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 PPL_๐ฎ(n) has been precisely computed is the Thue-Morse word ๐ญ. This word is 2-automatic and, predictably, its function PPL_๐ญ(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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