Model Theoretic Complexity of Automatic Structures

TL;DR

This paper explores the complexity of automatic structures using concepts from logic and model theory, establishing bounds on ordinal heights, Scott ranks, and Cantor-Bendixson ranks for various classes of automatic structures.

Contribution

It provides new bounds and existence results for ordinal heights, Scott ranks, and Cantor-Bendixson ranks in automatic structures, linking automata theory with advanced model-theoretic concepts.

Findings

Ordinal height of automatic well-founded partial orders is bounded by ω^ω

Automatic structures can have Scott ranks reaching any computable ordinal

Existence of automatic structures with Scott rank at the first non-computable ordinal

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 \omega^\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.