# Tree-Automatic Well-Founded Trees

@article{Huschenbett2013TreeAutomaticWT, title={Tree-Automatic Well-Founded Trees}, author={Martin Huschenbett and Alexander Kartzow and J. Liu and Markus Lohrey}, journal={Log. Methods Comput. Sci.}, year={2013}, volume={9} }

We investigate tree-automatic well-founded trees. Using Delhomme's
decomposition technique for tree-automatic structures, we show that the
(ordinal) rank of a tree-automatic well-founded tree is strictly below
omega^omega. Moreover, we make a step towards proving that the ranks of
tree-automatic well-founded partial orders are bounded by omega^omega^omega: we
prove this bound for what we call upwards linear partial orders. As an
application of our result, we show that the isomorphism problem… Expand

#### 3 Citations

Tree-Automatic Well-Founded Trees

- Mathematics, Computer Science
- CiE
- 2012

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 (under Turing-reductions). Expand

The Rank of Tree-Automatic Linear Orderings

- Mathematics, Computer Science
- STACS
- 2013

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

Structures without Scattered-Automatic Presentation

- Mathematics, Computer Science
- CiE
- 2013

This paper proves the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a fixed \(\ mathfrak {L}\) of finite condensation rank 1 + α. Expand

#### References

SHOWING 1-10 OF 37 REFERENCES

Tree-Automatic Well-Founded Trees

- Mathematics, Computer Science
- CiE
- 2012

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 (under Turing-reductions). Expand

Automatic linear orders and trees

- Mathematics, Computer Science
- TOCL
- 2005

It is shown that every infinite path in an automatic tree with countably many infinite paths is a regular language. Expand

Model-theoretic complexity of automatic structures

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 2009

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 structures

- Computer Science
- Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)
- 2000

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

Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable

- Computer Science, Mathematics
- CiE
- 2012

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

The isomorphism problem on classes of automatic structures with transitive relations

- Mathematics
- 2013

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 is… Expand

Finite Presentations of Infinite Structures:
Automata and Interpretations

- Mathematics, Computer Science
- Theory of Computing Systems
- 2004

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

First-Order Model Checking on Generalisations of Pushdown Graphs

- Mathematics, Computer Science
- ArXiv
- 2012

It is proved that first-order logic extended by Ramsey quantifiers is decidable on all tree-automatic structures. Expand

Computable structures and the hyperarithmetical hierarchy

- Mathematics
- 2000

Preface. Computability. The arithmetical hierarchy. Languages and structures. Ordinals. The hyperarithmetical hierarchy. Infinitary formulas. Computable infinitary formulas. The Barwise-Kreisel… Expand

Automatic structures: richness and limitations

- Computer Science, Mathematics
- Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.
- 2004

It is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations, and the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete. Expand