Model Theoretic Complexity of Automatic Structures (Extended Abstract)

We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well-founded partial order is bounded by ω; 2) The ordinal heights of automatic well-founded relations are unbounded below ω 1 ; 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 ω 1 , ω CK 1 + 1; 4) For any ordinal α < ω CK 1 , there is an automatic successor tree of Cantor-Bendixson rank α.