Definability and Regularity in Automatic Structures

  • Bakhadyr Khoussainov
  • Sasha Rubin
  • Frank Stephan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)


An automatic structure \(\mathcal{A}\) is one whose domain A and atomic relations are finite automaton (FA) recognisable. A structure isomorphic to \(\mathcal{A}\) is called automatically presentable. Suppose R is an FA recognisable relation on A. This paper concerns questions of the following type. For which automatic presentations of \(\mathcal{A}\) is (the image of) R also FA recognisable? To this end we say that a relation R is intrinsically regular in a structure \(\mathcal{A}\) if it is FA recognisable in every automatic presentation of the structure. For example, in every automatic structure all relations definable in first order logic are intrinsically regular. We characterise the intrinsically regular relations of some automatic fragments of arithmetic in the first order logic extended with quantifiers ∃  ∞  interpreted as ‘there exists infinitely many’, and ∃ (i) interpreted as ‘there exists a multiple of i many’.


Turing Machine Order Logic Finite Automaton Unary Relation Reachability Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Bakhadyr Khoussainov
    • 1
  • Sasha Rubin
    • 2
  • Frank Stephan
    • 3
  1. 1.Computer Science DepartmentThe University of AucklandNew Zealand
  2. 2.Mathematics DepartmentThe University of AucklandNew Zealand
  3. 3.Sydney Research Laboratory at KensingtonNational ICT AustraliaAustralia

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