Slides.

STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 25/81
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS
Courcelle’s Theorem
Theorem: (Courcelle 1990)
For any class C of bounded tree-width
MC(MSO 2 , C)
Input: Graph G ∈ C, ϕ ∈ MSO 2
Parameter: |ϕ|
Problem: Decide G |= ϕ
is fixed-parameter tractable (linear time for each fixed ϕ).
MSO 2 : tree-width of a graph equals tree-width of its incidence encoding.
Example: 3-COLOURABILITY
( 3∨
∃C 1 ∃C 2 ∃C
} {{ } 3 ∀x C i (x)
there are sets
i=1
} {{ }
C 1 , C 2 , C 3 ev. node has a col.
∧ ∀x∀y(E(x, y) →
3∧ )
¬(C i (x)∧C i (y)))
i=1
} {{ }
endpoints of edges have different colours