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