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Model Theoretic Complexity of Automatic Structures
 PROC. TAMC ’08, LNCS 4978
, 2008
"... We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any autom ..."
Abstract

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We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic wellfounded partial order is bounded by ωω; 2) The ordinal heights of automatic wellfounded relations are unbounded below ωCK 1, the first noncomputable ordinal; 3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ωCK 1, ωCK 1 +1; 4) For any computable ordinal α, there is an automatic successor tree of CantorBendixson rank α.
COMPUTABILITY, TRACEABILITY AND BEYOND
"... This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability ..."
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This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is Π0 4complete. This shows that the problem of deciding if a c.e. set is strongly jump traceable, is as hard as it can be. We define a strengthening of strong jump traceability, called hyper jump traceability, and prove some interesting results about this new class. Despite the fact that the hyper jump traceable sets have their origins in algorithmic randomness, we are able to show that they are natural examples of several Turing degree theoretic properties. For instance, we show that the hyper jump traceable sets are the first example of a lowness class with no promptly simple members. We also study the dual highness notions obtained from strong jump traceability, and explore their degree theoretic properties.