# Automatic linear orders and trees

@article{Khoussainov2005AutomaticLO,
title={Automatic linear orders and trees},
author={Bakhadyr Khoussainov and Sasha Rubin and Frank Stephan},
journal={ACM Trans. Comput. Log.},
year={2005},
volume={6},
pages={675-700}
}
• Published 2005
• Mathematics, Computer Science
• ACM Trans. Comput. Log.
We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor--Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and… Expand

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#### References

SHOWING 1-10 OF 24 REFERENCES
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
Some results on automatic structures
• Mathematics, Computer Science
• Proceedings 17th Annual IEEE Symposium on Logic in Computer Science
• 2002
This work reduces the problem of existence of an automatic presentation of a structure to that for a graph, and exhibits a series of properties of automatic equivalence structures, linearly ordered sets and permutation structures. Expand
Automatic structures: richness and limitations
• Computer Science
• LICS 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
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
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
Is Cantor's Theorem Automatic?
• D. Kuske
• Mathematics, Computer Science
• LPAR
• 2003
It is shown that the canonical presentation from [5] is automatic-homogeneous, any tuple can be mapped to any other tuple by an automatic automorphism, and that there are automatic linear orders that are not isomorphic to any regular subset of the canonicalpresentation. Expand
Definability and Regularity in Automatic Structures
• Mathematics, Computer Science
• STACS
• 2004
It is said that a relation R is intrinsically regular in a structure $$\mathcal{A}$$ if it is FA recognisable in every automatic presentation of the structure. Expand
Theory of Recursive Functions and Effective Computability
Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular. A second group of topics has to do with generalizationsExpand
Sets recognized by n-tape automata
• Mathematics
• 1969
The n-tape automata studied here, bear a very close rcscmblancc to ordinary finite automata, since all n-tapes move simultaneously. This kind of tape action differs from that of the n-tape automataExpand
Automata theory and its applications
• Mathematics
• 2001
1. Basic Notions.- 1.1 Sets.- 1.2 Sequences and Tuples.- 1.3 Functions, Relations, Operations.- 1.4 Equivalence Relations.- 1.5 Linearly Ordered Sets.- 1.6 Partially Ordered Sets.- 1.7 Graphs.- 1.8Expand