On Extended Boundary Sequences of Morphic and Sturmian Words

Authors Michel Rigo , Manon Stipulanti , Markus A. Whiteland

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Michel Rigo
  • Department of Mathematics, University of Liège, Belgium
Manon Stipulanti
  • Department of Mathematics, University of Liège, Belgium
Markus A. Whiteland
  • Department of Mathematics, University of Liège, Belgium


We thank Jean-Paul Allouche for references [Ethan M. Coven, 1974; Parvaix, 1997; Michael E. Paul, 1974], and Jeffrey Shallit for discussions about the "logical approach". The anonymous referees are warmly thanked for providing useful feedback improving the quality of the text.

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Michel Rigo, Manon Stipulanti, and Markus A. Whiteland. On Extended Boundary Sequences of Morphic and Sturmian Words. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 79:1-79:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Generalizing the notion of the boundary sequence introduced by Chen and Wen, the nth term of the 𝓁-boundary sequence of an infinite word is the finite set of pairs (u,v) of prefixes and suffixes of length 𝓁 appearing in factors uyv of length n+𝓁 (n ≥ 𝓁 ≥ 1). Otherwise stated, for increasing values of n, one looks for all pairs of factors of length 𝓁 separated by n-𝓁 symbols. For the large class of addable numeration systems U, we show that if an infinite word is U-automatic, then the same holds for its 𝓁-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). We also provide examples of numeration systems and U-automatic words with a boundary sequence that is not U-automatic. In the second part of the paper, we study the 𝓁-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Mathematics of computing → Combinatorics on words
  • Boundary sequences
  • Sturmian words
  • Numeration systems
  • Automata
  • Graph of addition


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