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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1353
URN: urn:nbn:de:0030-drops-13535
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Ganzow, Tobias ; Rubin, Sasha

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

22011.GanzowTobias.Paper.1353.pdf (0.2 MB)


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.

BibTeX - Entry

  author =	{Tobias Ganzow and Sasha Rubin},
  title =	{{Order-Invariant MSO is Stronger than Counting MSO in the Finite}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{313--324},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-13535},
  doi =		{10.4230/LIPIcs.STACS.2008.1353},
  annote =	{Keywords: MSO, Counting MSO, order-invariance, expressiveness, Ehrenfeucht-Fraiss{\'e} game}

Keywords: MSO, Counting MSO, order-invariance, expressiveness, Ehrenfeucht-Fraissé game
Collection: 25th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2008
Date of publication: 06.02.2008

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