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INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Another Good Idea, A Little Less Wrong We reduce the propositional satisfiability problem (SAT) to MC(MSO,C). Given: w := (X 1 ∨ X 2 ∨¬X 3 )∧(X 4 ∨¬X 5 ∨ X 6 )... Problem: Decide if w is satisfiable. Reduction. Compute G w |w| 28 < tw(G w ) I(P,Q) G m,m in I(P,Q) Comp. GM (P,Q) MSO-definable. There is MSO 2 -formula ϕ(P, Q) saying (P, Q) are grid-like minor. Catch. We cannot delete edges in I(P,Q)! This means deleting vertices in G destroying the grid-like minors. STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 72/81
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 73/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Labelled Tree-Ordered Webs Encoding Words Definition. Labelled Tree-Ordered Webs encoding w Theorem. (K., Tazari 10) Let w ∈ {0, 1} ∗ be a word of length l and let d be a constant. There is a constant c and a polynomial time algorithm which, given a graph G of tree-width ≥ cl 14d computes a sub-graph G w ⊆ G which is a labelled tree-ordered web encoding w (with power d).
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