Structures without Scattered-Automatic Presentation

@article{Kartzow2013StructuresWS,
  title={Structures without Scattered-Automatic Presentation},
  author={Alexander Kartzow and Philipp Schlicht},
  journal={ArXiv},
  year={2013},
  volume={abs/1304.0912}
}
Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order \(\mathfrak{L}\). Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order \(\mathfrak{L}\) there is the class of \(\mathfrak{L}\)-automatic structures. In this paper we prove the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a… Expand
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References

SHOWING 1-10 OF 25 REFERENCES
Complementation of rational sets on countable scattered linear orderings
TLDR
It is proved that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach. Expand
Automata-based presentations of infinite structures
TLDR
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
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
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
Automata on ordinals and automaticity of linear orders
TLDR
A method for proving non-automaticity is described and this is applied to determine the optimal bounds for the ranks of linear orders recognized by finite state automata. Expand
Automatic structures: richness and limitations
TLDR
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
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
The Rank of Tree-Automatic Linear Orderings
TLDR
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
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
Automata on linear orderings
TLDR
It is proved that for countable scattered linear orderings, the two notions of finite automata and rational expressions are equivalent, which extends Kleene's theorem. Expand
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