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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 29/81<br />
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
Courcelle’s Theorem<br />
Theorem: (Courcelle 1990)<br />
For any class C of bounded tree-width<br />
MC(MSO 2 , C)<br />
Input: Graph G ∈ C, ϕ ∈ MSO<br />
Parameter: |ϕ|<br />
Problem: Decide G |= ϕ<br />
is fixed-parameter tractable (linear time for each fixed ϕ).<br />
What about the parameter dependence?<br />
Theorem: (Frick, Grohe, 01)<br />
1. Unless P=NP, there is no fpt-algorithm for MSO model checking on<br />
trees with elementary parameter dependence.<br />
2. Unless FPT=W[1], there is no fpt-algorithm for FO model checking<br />
on trees with elementary parameter dependence.
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