# Automatic structures

@article{Blumensath2000AutomaticS,
title={Automatic structures},
author={Achim Blumensath and Erich Gr{\"a}del},
journal={Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)},
year={2000},
pages={51-62}
}
• Published 26 June 2000
• Computer Science
• Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)
We study definability and complexity issues for automatic and /spl omega/-automatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover they admit effective (in fact automatic) evaluation of all first-order queries. Therefore, automatic structures provide an interesting framework for extending many algorithmic and logical methods from finite structures to infinite ones. We explain the notion of (/spl omega/-)automatic…
334 Citations

## Topics from this paper

Advice Automatic Structures and Uniformly Automatic Classes
• Computer Science
CSL
• 2017
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.
Finite Presentations of Infinite Structures: Automata and Interpretations
• Mathematics, Computer Science
Theory of Computing Systems
• 2004
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.
Climbing up the Elementary Complexity Classes with Theories of Automatic Structures
• Computer Science
CSL
• 2018
A positive answer to the question whether there are automatic structures of arbitrary high elementary complexity is given and it is shown that for every h ≥ 0 the forest of all finite trees of height at most h+ 2 is automatic and it’s theory is complete for STA(∗, exph(n, poly(n), poly( n)), an alternating complexity class between h-fold exponential time and space.
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.
Invariants of Automatic Presentations and Semi-synchronous Transductions
The main result is that a one-to-one function on words preserves regularity as well as non-regularity of all relations iff it is a semi-synchronous transduction.
AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES
• Computer Science, Mathematics
The Journal of Symbolic Logic
• 2019
This paper shows that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1$-complete and uses similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.
Computability and complexity properties of automatic structures and their applications
• Mathematics
• 2008
Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Continuing a tradition of
On automatic partial orders
• Mathematics, Computer Science
18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings.
• 2003
It is shown that every infinite path in an automatic tree with countably many infinite paths is a regular language.
First-order and counting theories of ω-automatic structures
• Mathematics
Journal of Symbolic Logic
• 2008
Abstract The logic extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with
Algorithmic Solutions via Model Theoretic Interpretations
• Mathematics
• 2017
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,

## References

SHOWING 1-10 OF 46 REFERENCES
Towards a Theory of Recursive Structures
This paper summarizes the recent work on recursive structures and data bases, including the high undecidability of many problems on recursive graphs and structures, a method for deducing results on the descriptive complexity of nitary NP optimization problems from results onthe computational complexity of their innnitary analogues.
Word problems requiring exponential time(Preliminary Report)
• Computer Science
STOC
• 1973
A number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines are considered.
On the Equivalence, Containment, and Covering Problems for the Regular and Context-Free Languages
• Computer Science, Mathematics
J. Comput. Syst. Sci.
• 1976
We consider the complexity of the equivalence and containment problems for regular expressions and context-free grammars, concentrating on the relationship between complexity and various language
On Relations Defined by Generalized Finite Automata
• Mathematics, Computer Science
IBM J. Res. Dev.
• 1965
A transduction, in the sense of this paper, is a n-ary word relation (which may be a function) describable by a finite directed labeled graph that constitutes the equilibrium (potential) behavior of 1-dimensional, bilateral iterative networks.
On Finite Model Theory
The subject of this paper is the part of finite model theory intimately related to the classical model theory. In the very beginning of our career in computer science, we attended a few lectures on
More about recursive structures: descriptive complexity and zero-one laws
• Mathematics, Computer Science
Proceedings 11th Annual IEEE Symposium on Logic in Computer Science
• 1996
This paper investigates the descriptive complexity of several logics over recursive structures, including first-order, second- order, and fixpoint logic, and proposes a version that applies to recursive structures and uses it to prove several non-expressibility results.
The theory of functions and sets of natural numbers
Recursiveness and Computability. Induction. Systems of Equations. Arithmetical Formal Systems. Turing Machines. Flowcharts. Functions as Rules. Arithmetization. Church's Thesis. Basic Recursion
SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC
• Mathematics
• 1974
Lower bounds are established on the computational complexity of the decision problem and on the inherent lengths of proofs for two classical decidable theories of logic: the first-order theory of the
The complexity of relational query languages (Extended Abstract)
The pattern which will be shown is that the expression complexity of the investigated languages is one exponential higher then their data complexity, and for both types of complexity the authors show completeness in some complexity class.
Ehrenfeucht Games, the Composition Method, and the Monadic Theory of Ordinal Words
• W. Thomas
• Mathematics, Computer Science
Structures in Logic and Computer Science
• 1997
Shelah's extension of the method, the “composition of monadic theories”, is reviewed, explained in the example of the monadic theory of the ordinal ordering (ω, <), and compared with the automata theoretic approach due to Buchi.