# The Rank of Tree-Automatic Linear Orderings

@inproceedings{Huschenbett2013TheRO,
title={The Rank of Tree-Automatic Linear Orderings},
author={Martin Huschenbett},
booktitle={STACS},
year={2013}
}
A tree-automatic structure is a structure whose domain can be encoded by a regular tree language such that each relation is recognisable by a finite automaton processing tuples of trees synchronously. The finite condensation rank (FC-rank) of a linear ordering measures how far it is away from being dense. We prove that the FC-rank of every tree-automatic linear ordering is below omega^omega. This generalises Delhomme's result that each tree-automatic ordinal is less than omega^omega^omega… Expand
8 Citations

#### Figures and Topics from this paper

Tree-automatic scattered linear orders
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 2016
It is shown that there is no tree-automatic scattered linear order, and therefore no tree's automatic well-order, on the set of all finite labeled trees, and that a regular tree language admits a tree- automatic scattered linear orders if and only if for some n, no binary tree of height n can be embedded into the union of the domains of its trees. Expand
The model-theoretic complexity of automatic linear orders
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
The Field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic
This work lifts Delhomm\'e's relative-growth-technique from the automatic and tree-automatic setting to the ordinal- automatic setting, which implies that the random graph is not Ordinal-automatic and infinite integral domains are not ordinals below $\omega_1+\omega^ \omega$ where $\omegas_1$ is the first uncountable ordinal. Expand
Pumping for ordinal-automatic structures
• Mathematics, Computer Science
• Comput.
• 2017
A pumping lemma for alpha-automata (processing finite alpha-words, i.e., words of length alpha that have one fixed letter at all but finitely many positions) is developed and a sharp bound on the height of the finite word alpha-automatic well-founded order forests is provided. Expand
Isomorphisms of scattered automatic linear orders
• D. Kuske
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 2014
Abstract We prove that the isomorphism of scattered tree-automatic linear orders as well as the existence of automorphisms of scattered word-automatic linear orders are undecidable. For the existenceExpand
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
L O ] 2 5 A pr 2 01 2 Isomorphisms of scattered automatic linear orders
We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable. For the existence ofExpand
Isomorphisms of scattered automatic linear orders
• D. Kuske
• Mathematics, Computer Science
• CSL
• 2012
We prove the undecidability of the existence of an isomorphism between scattered tree-automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders.Expand

#### References

SHOWING 1-10 OF 14 REFERENCES
Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable
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 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
Tree-Automatic Well-Founded Trees
• Mathematics, Computer Science
• Log. Methods Comput. Sci.
• 2013
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
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
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 isExpand
Automata Presenting Structures: A Survey of the Finite String Case
• S. Rubin
• Mathematics, Computer Science
• Bulletin of Symbolic Logic
• 2008
The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability. Expand
LINEAR ORDERINGS
We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable, such that for any countable linear ordering B, L does not embed into BExpand
Rabin's Uniformization Problem
• Mathematics, Computer Science
• J. Symb. Log.
• 1983
It is proved that there is no monadic second-order formula 0*(X, y) such that for every nonempty subset X of T there is a uniquey e X that satisfies 0q*( X, y), and it is proposed that this language allows quantification over elements of T and over arbitrary subsets of T. Expand
Automata-based presentations of infinite structures
• Mathematics, Computer Science
• Finite and Algorithmic Model Theory
• 2011
Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones. Expand
Automaticité des ordinaux et des graphes homogènes
Resume Les structures automatiques (resp. arbre-automatiques) sont les structures relationnelles dont le domaine est un ensemble regulier de mots (resp. de termes) finis et dont chaque relationExpand