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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 60/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS First Example: Coloured Grids Coloured Grid. A grid whose vertices may be coloured red or blue. Let G be the class of all finite coloured grids. Theorem. Let G be the class of coloured grids. Then MC(MSO,G) is not fixed-parameter tractable unless P=NP. (4×5)-grid
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 61/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Intractability on Grids Theorem. Let G be the class of coloured grids. Then MC(MSO,G) is not fixed-parameter tractable unless P=NP. Proof. Let SAT be the NP-complete propositional satisfiability problem. SAT can be solved in quadratic time by an NTM M. Given: SAT-instance w := (X 1 ∨¬X 3 ...) ˆ= 010... Look at the time-space diagram of an acc. run of M on w.
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