Model theoretic complexity of automatic structures

Authors:
Bakhadyr Khoussainov;Mia Minnes
Affiliations:
Department of Computer Science, University of Auckland, Auckland, New Zealand;Mathematics Department, Cornell University, Ithaca, New York
Venue:
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Year:
2008

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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 ωω; 2) The ordinal heights of automatic well-founded relations are unbounded below (ω1CK ; 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 (ω1CK,ω1CK + 1; 4) For any ordinal α 1CK, there is an automatic successor tree of Cantor-Bendixson rank α.