Abstract
One of Courcelle’s celebrated results states that if \({\mathcal C}\) is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO 2) is fixed-parameter tractable (fpt) on \({\mathcal C}\) by linear time parameterised algorithms. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width.
In this paper we show that in terms of tree-width, the theorem can not be extended much further. More specifically, we show that if \({\mathcal C}\) is a class of graphs which is closed under colourings and satisfies certain constructibility conditions such that the tree-width of \({\mathcal C}\) is not bounded by log16 n then MSO 2-model checking is not fpt unless Sat can be solved in sub-exponential time. If the tree-width of \({\mathcal C}\) is not poly-log. bounded, then MSO 2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.
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Kreutzer, S. (2009). On the Parameterised Intractability of Monadic Second-Order Logic. In: Grädel, E., Kahle, R. (eds) Computer Science Logic. CSL 2009. Lecture Notes in Computer Science, vol 5771. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04027-6_26
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DOI: https://doi.org/10.1007/978-3-642-04027-6_26
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