Model-theoretic complexity of automatic structures
Name
Khoussainov-2009-Model-theoretic comp.pdf
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686 KB
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Author(s)
Khoussainov, Bakhadyr
Minnes, Mia
Date Issued
August 2009
Journal
Annals of Pure and Applied Logic
Publisher
Elsevier
Citation
Khoussainov, Bakhadyr, and Mia Minnes. “Model-Theoretic Complexity of Automatic Structures.” Annals of Pure and Applied Logic 161, no. 3 (December 2009): 416–426. © 2009 Elsevier B.V.
Version
Final published version
Abstract
We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor–Bendixson ranks (of trees). We prove the following results: (1) The ordinal height of any automatic well-founded partial order is bounded by ω[superscript ω]. (2) The ordinal heights of automatic well-founded relations are unbounded below ω[subscript 1 superscript CK], the first non-computable ordinal. (3) For any computable ordinal α, there is an automatic structure of Scott rank at least αα. Moreover, there are automatic structures of Scott rank ω[subscript 1 superscript CK],ω[subscript 1 superscript CK] +1. (4) For any computable ordinal α, there is an automatic successor tree of Cantor–Bendixson rank α.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1016/j.apal.2009.07.012
