Model Theoretic Complexity of Automatic Structures (Extended Abstract)

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.