Order-Invariant MSO is Stronger than Counting MSO in the Finite

Authors Tobias Ganzow, Sasha Rubin

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Tobias Ganzow
Sasha Rubin

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Tobias Ganzow and Sasha Rubin. Order-Invariant MSO is Stronger than Counting MSO in the Finite. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 313-324, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


We compare the expressiveness of two extensions of monadic second-order logic (MSO) over the class of finite structures. The first, counting monadic second-order logic (CMSO), extends MSO with first-order modulo-counting quantifiers, allowing the expression of queries like ``the number of elements in the structure is even''. The second extension allows the use of an additional binary predicate, not contained in the signature of the queried structure, that must be interpreted as an arbitrary linear order on its universe, obtaining order-invariant MSO. While it is straightforward that every CMSO formula can be translated into an equivalent order-invariant MSO formula, the converse had not yet been settled. Courcelle showed that for restricted classes of structures both order-invariant MSO and CMSO are equally expressive, but conjectured that, in general, order-invariant MSO is stronger than CMSO. We affirm this conjecture by presenting a class of structures that is order-invariantly definable in MSO but not definable in CMSO.
  • MSO
  • Counting MSO
  • order-invariance
  • expressiveness
  • Ehrenfeucht-Fraissé game


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