Model Theoretic Complexity of Automatic Structures (Extended Abstract)

  • Bakhadyr Khoussainov
  • Mia Minnes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4978)


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 ωω; 2) The ordinal heights of automatic well-founded relations are unbounded below \(\omega_{1}^{CK}\); 3) For any infinite computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank \(\omega_1^{CK}, \omega_1^{CK}+1\); 4) For any ordinal \(\alpha<\omega_1^{CK}\), there is an automatic successor tree of Cantor-Bendixson rank α.


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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bakhadyr Khoussainov
    • 1
  • Mia Minnes
    • 2
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  2. 2.Mathematics DepartmentCornell University, IthacaNew York

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