Slides.

STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 62/81
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS
Intractability on Grids
Theorem. Let G be the class of coloured grids. Then MC(MSO,G) is not
fixed-parameter tractable unless P=NP.
Proof. We reduce SAT to MC(MSO,G) as follows.
1. Given SAT instance w of length n, construct an n 2 × n 2 -grid G w and
colour its bottom row by w.
2. Construct a formula ϕ M ∈ MSO which guesses a colouring of the
grid and checks that this encodes a successful run of M on input w.
Then w ∈ SAT if, and only if, G w |= ϕ M .
∃X 0 ∃X 1 ∃X □ ∃X q0 ...X qk ψ... ∈ MSO 1