Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

  • Kord Eickmeyer
  • Michael Elberfeld
  • Frederik Harwath
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8634)


We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and order-invariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a fine-grained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth.


Expressivity succinctness first-order logic monadic second-order logic order-invariance tree-depth 


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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2014

Authors and Affiliations

  • Kord Eickmeyer
    • 1
  • Michael Elberfeld
    • 2
  • Frederik Harwath
    • 3
  1. 1.TU DarmstadtDarmstadtGermany
  2. 2.RWTH Aachen UniversityAachenGermany
  3. 3.Goethe-UniversitätFrankfurt am MainGermany

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