On prefix palindromic length of automatic words

Abstract

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