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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 20/81<br />
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
Feferman-Vaught Style Theorems<br />
Theorem. Let G, H be graphs, v ∈ V(G)<br />
u ∈ V(G) such that u = V(G)∩V(H)<br />
w ∈ V(H)<br />
For all q ≥ 0, tp q (G∪H, uvw) is determined by tp q (G, uv) and tp q (uw)<br />
Furthermore, there is an algorithm that computes tp q (G∪H, uvw) from<br />
tp q (G, uv) and tp q (uw).<br />
This suggests a model-checking algorithm on graphs which can<br />
recursively be decomposed into sub-graphs of constant size.<br />
graphs of bounded tree-width