On the Parameterised Intractability of Monadic Second-Order Logic

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.