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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 10/81<br />
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
Complexity of First-Order Model-Checking<br />
Naïve algorithm gives running time and space:<br />
time: O(l · n m ) l: length of ϕ m: quantifier rank of ϕ<br />
space: O(m·log n) n: size of A<br />
Theorem: First-Order Model-Checking MC(FO) is PSPACE-complete.<br />
This is even true for MC(FO,A) for a fixed two element structure A.<br />
Proof. Reduction from satisfiability for Quantified Boolean Formulae<br />
Theorem. For any fixed ϕ,<br />
(data complexity)<br />
MC(ϕ, Str) ∈ AC 0 ⊆ LOGSPACE ⊆ PTIME<br />
However: Running time O(l · n m )
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