Structures without Scattered-Automatic Presentation

  title={Structures without Scattered-Automatic Presentation},
  author={Alexander Kartzow and Philipp Schlicht},
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
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
Second-Order Finite Automata
It is shown that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation, and emptiness of intersection and inclusion are decidable. Expand
Advice Automatic Structures and Uniformly Automatic Classes
It is proved that the class of all torsion-free Abelian groups of rank one is uniformly omega-automatic and that there is a uniform omega-tree-automatic presentation of the classof all Abelian Groups up to elementary equivalence and of theclass of all countable divisible Abelian Group groups. Expand
Pumping for ordinal-automatic structures
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
Algorithmic Solutions via Model Theoretic Interpretations
Model theoretic interpretations are an important tool in algorithmic model theory. Their applications range from reductions between logical theories to the construction of algorithms for problems,Expand
Automatic Structures: Twenty Years Later
This tutorial presents an introduction into the history and basic definitions of automatic structures, and surveys the achievements in the study of different variants ofautomatic structures. Expand


Complementation of rational sets on countable scattered linear orderings
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
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
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
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
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
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
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
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
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