The isomorphism problem on classes of automatic structures with transitive relations

@article{Kuske2013TheIP,
  title={The isomorphism problem on classes of automatic structures with transitive relations},
  author={Dietrich Kuske and J. Liu and Markus Lohrey},
  journal={Transactions of the American Mathematical Society},
  year={2013},
  volume={365},
  pages={5103-5151}
}
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 complete for Σ 1 , the first existential level of the analytical hierarchy. Positive results on ordinals and on Boolean algebras raised hope that the isomorphism problem is simpler for transitive relations. We prove that this is not the case. More precisely, this paper shows: (i) The isomorphism… 
The isomorphism problem for ω-automatic trees
Abstract The main result of this paper states that the isomorphism problem for ω -automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This
Isomorphism of regular trees and words
TLDR
The computational complexity of the isomorphism problem for regular trees, regular linear orders, and regular words is analyzed and techniques can be used to show that one can check in polynomial time whether a given regular linear order has a non-trivial automorphism.
Automatic Equivalence Structures of Polynomial Growth
TLDR
It is proved that the isomorphism problem for structures from the class EqP is undecidable and the isomorphic problem is decidable forstructures from EquP with domains of quadratic growth.
The model-theoretic complexity of automatic linear orders
TLDR
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.
Automatic Structures — Recent Results and Open Questions
Regular languages are languages recognised by finite automata; automatic structures are a generalisation of regular languages where one also uses automatic relations (which are relations recognised
Degrees of Categoricity and the Isomorphism Problem
In this thesis, we study notions of complexity related to computable structures. We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal α,
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 (under Turing-reductions).
Pumping for ordinal-automatic structures
TLDR
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.
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.
Tree-automatic scattered linear orders
TLDR
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.
...
1
2
3
...

References

SHOWING 1-10 OF 56 REFERENCES
The isomorphism problem for ω-automatic trees
Abstract The main result of this paper states that the isomorphism problem for ω -automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This
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.
Isomorphism of regular trees and words
TLDR
The computational complexity of the isomorphism problem for regular trees, regular linear orders, and regular words is analyzed and techniques can be used to show that one can check in polynomial time whether a given regular linear order has a non-trivial automorphism.
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.
Ehrenfeucht-Fraïssé goes elementarily automatic for structures of bounded degree
TLDR
This work proposes a general method based on Ehrenfeucht-Fraisse games to give upper bounds on the size of these automata and on the time required to build them and concludes that the very general and simple automata-based algorithm works well to decide the first-order theories over these structures.
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.
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 that
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).
Model Theoretic Complexity of Automatic Structures (Extended Abstract)
TLDR
The following results are proved: The ordinal height of any automatic well-founded partial order is bounded by ω ω ; the ordinal heights of automaticWell-founded relations are unbounded below \(\omega_{1}^{CK}\); and for any infinite computable ordinal α, there is an automatic structure of Scott rank at least α.
Where Automatic Structures Benefit from Weighted Automata
  • D. Kuske
  • Mathematics, Computer Science
    Algebraic Foundations in Computer Science
  • 2011
TLDR
It is proved that the extension of first-order logic by the infinity ∃∞, the modulo ∃(p,q), and the (new) boundedness quantifier is decidable.
...
1
2
3
4
5
...