We investigate automatic presentations of -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 -words enjoying the above mentioned definability and decidability properties. We introduce -lexicographic presentations, and morphisms of level stacks and show that these are inter-translatable, thus giving rise to the same classes of -lexicographic or level morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every -lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mappings. The classes of -lexicographic words are shown to constitute an infinite hierarchy.
Keywords: morphic words, monadic second-order logic, automatic structures, automatic sequences
@article{ITA_2008__42_3_417_0,
author = {B\'ar\'any, Vince},
title = {A hierarchy of automatic $\omega $-words having a decidable {MSO} theory},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {417--450},
year = {2008},
publisher = {EDP Sciences},
volume = {42},
number = {3},
doi = {10.1051/ita:2008008},
mrnumber = {2434027},
zbl = {1152.03030},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ita:2008008/}
}
TY - JOUR AU - Bárány, Vince TI - A hierarchy of automatic $\omega $-words having a decidable MSO theory JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2008 SP - 417 EP - 450 VL - 42 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ita:2008008/ DO - 10.1051/ita:2008008 LA - en ID - ITA_2008__42_3_417_0 ER -
%0 Journal Article %A Bárány, Vince %T A hierarchy of automatic $\omega $-words having a decidable MSO theory %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2008 %P 417-450 %V 42 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ita:2008008/ %R 10.1051/ita:2008008 %G en %F ITA_2008__42_3_417_0
Bárány, Vince. A hierarchy of automatic $\omega $-words having a decidable MSO theory. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 3, pp. 417-450. doi: 10.1051/ita:2008008
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