# Automatic linear orders and trees

@article{Khoussainov2005AutomaticLO, title={Automatic linear orders and trees}, author={Bakhadyr Khoussainov and Sasha Rubin and Frank Stephan}, journal={ACM Trans. Comput. Log.}, year={2005}, volume={6}, pages={675-700} }

We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor--Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and… Expand

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#### 68 Citations

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It is proved that the FC-rank of every tree-automatic linear ordering is below omega^omega, and an analogue for tree- automatic linear orderings where the branching complexity of the trees involved is bounded is shown. Expand

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It is shown that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions. Expand

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This thesis studies the model-theoretic complexity of automatic linear orders in terms of two complexity measures: the finite-condensation rank and the Ramsey degree. Expand

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- Mathematics, Computer Science
- CiE
- 2011

This work investigates structures recognizable by α-automata with running time a limit ordinal α and determines the suprema of the α-automatic ordinals and the ranks of α- automatic linear orders. Expand

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- Mathematics, Computer Science
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An algorithm is described that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic. Expand

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- CAI
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