Model-theoretic complexity of automatic structures

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.