A hierarchy of automatic $\omega$-words having a decidable MSO theory
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 3, pp. 417-450.

We investigate automatic presentations of $\omega$-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSO-interpretability in algebraic trees, and the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is in a certain sense canonical. We then generalize our techniques to a hierarchy of classes of $\omega$-words enjoying the above mentioned definability and decidability properties. We introduce $k$-lexicographic presentations, and morphisms of level $k$ stacks and show that these are inter-translatable, thus giving rise to the same classes of $k$-lexicographic or level $k$ morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every $k$-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mappings. The classes of $k$-lexicographic words are shown to constitute an infinite hierarchy.

DOI: 10.1051/ita:2008008
Classification: 03D05,  68Q42,  68Q45,  68R15
Keywords: morphic words, monadic second-order logic, automatic structures, automatic sequences
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Bárány, Vince. A hierarchy of automatic $\omega$-words having a decidable MSO theory. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 3, pp. 417-450. doi : 10.1051/ita:2008008. http://www.numdam.org/articles/10.1051/ita:2008008/

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