Slides.

STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 8/81
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS
Complexity of Monadic Second-Order Model-Checking
Given:
Problem:
Finite structure A := (A,σ)
MSO-formula ϕ
Decide A |= ϕ
Naïve algorithm: Evaluation following the structure of the formula
• Existential second-order quantification: ϕ := ∃Xψ
for all U ⊆ A check whether (A, X ↦→ U) |= ψ
• Existential first-order quantification: ϕ := ∃xψ
for all a ∈ A check whether (A, x ↦→ a)ψ
• Boolean connectives ∧,∨,¬: easy
• Atomic formulae: direct look up in the structure
Running time and space:
Time: exponential in |ϕ| and |A|
Space:
linear in both |ϕ| and |A|