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 ω; 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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Khoussainov, B., Minnes, M. (2008). Model Theoretic Complexity of Automatic Structures (Extended Abstract). In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_45
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DOI: https://doi.org/10.1007/978-3-540-79228-4_45
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