First-Order and Counting Theories of ω-Automatic Structures

  • Dietrich Kuske
  • Markus Lohrey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3921)


The logic \({\mathcal L}({\mathcal Q}_u)\) 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 an injective ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [4]. It is shown that, as in the case of automatic structures [13], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are \(\varkappa\) many elements satisfying ...”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of \({\mathcal L}({\mathcal Q}_u)\) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.


Decidable Theory Turing Machine Relational Symbol Annual IEEE Symposium Bounded Degree 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dietrich Kuske
    • 1
  • Markus Lohrey
    • 2
  1. 1.Institut für InformatikUniversität LeipzigGermany
  2. 2.Universität Stuttgart, FMIGermany

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