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A Hierarchy of Automatic ω-Words having a Decidable MSO Theory

Published online by Cambridge University Press:  03 June 2008

Vince Bárány
Vince Bárány*
Affiliation:
Mathematische Grundlagen der Informatik, RWTH Aachen, Aachen, Germany; vbarany@logic.rwth-aachen.de
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Abstract

We investigate automatic presentations of ω-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 ω-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.

Keywords

Morphic wordsmonadic second-order logicautomatic structuresautomatic sequences.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 3: JM'06 , July 2008 , pp. 417 - 450
Copyright
© EDP Sciences, 2008

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