Tree-Automatic Well-Founded Trees

@inproceedings{Kartzow2012TreeAutomaticWT,
  title={Tree-Automatic Well-Founded Trees},
  author={Alexander Kartzow and J. Liu and Markus Lohrey},
  booktitle={CiE},
  year={2012}
}
We investigate tree-automatic well-founded trees. For this, we introduce a new ordinal measure for well-founded trees, called ∞-rank. The ∞-rankof a well-founded tree is always bounded from above by the ordinary (ordinal) rank of a tree. We also show that the ordinal rank of a well-founded tree of ∞-rankα is smaller than ω·(α+1). For string-automatic well-founded trees, it follows from [16] that the ∞-rankis always finite. Here, using Delhomme's decomposition technique for tree-automatic… Expand
Tree-Automatic Well-Founded Trees
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References

SHOWING 1-10 OF 24 REFERENCES
Tree-Automatic Well-Founded Trees
TLDR
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
Model-theoretic complexity of automatic structures
TLDR
The following results are proved: The ordinal height of any automatic well-founded partial order is bounded by ωω, and the ordinal heights of automaticWell-founded relations are unbounded below (ω1CK). Expand
Automatic linear orders and trees
TLDR
It is shown that every infinite path in an automatic tree with countably many infinite paths is a regular language. Expand
The isomorphism problem on classes of automatic structures with transitive relations
Automatic structures are finitely presented structures where the universe and all relations can be recognized by finite automata. It is known that the isomorphism problem for automatic structures isExpand
Automatic structures
  • Achim Blumensath, E. Grädel
  • Computer Science
  • Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)
  • 2000
TLDR
This work determines the complexity of model checking and query evaluation on automatic structures for fragments of first-order logic and gives model-theoretic characterisations for automatic structures via interpretations. Expand
Finite Presentations of Infinite Structures: Automata and Interpretations
TLDR
The model checking problem for FO(∃ω), first-order logic extended by the quantifier “there are infinitely many”, is proved to be decidable for automatic and ω-automatic structures and appropriate expansions of the real ordered group. Expand
Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable
TLDR
It is proved that the problem of whether a given tree automatic structure is already word automatic is decidable for tree automatic scattered linear orderings and in case of a positive answer a word automatic presentation is computable from the tree automatic presentation. Expand
Automatic structures of bounded degree revisited
Abstract The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove thatExpand
First-Order Model Checking on Generalisations of Pushdown Graphs
TLDR
It is proved that first-order logic extended by Ramsey quantifiers is decidable on all tree-automatic structures. Expand
Hausdorff ’s theorem for posets that satisfy the finite antichain property
Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulateExpand
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