Slides.

STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 32/81
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS
Locality of First-Order Logic
Notation: Let G be a graph
dist G (u, v) :
e.g. the Gaifman graph of a structure
length of the shortest path between u and v
N G r (v) := {u ∈ V(G) : dist G (u, v) ≤ r}
N G r (v): r-neighbourhood of v in G.
Definition:
A formula ϕ(x) ∈ FO is r-local if for all graphs G and all v ∈ V(G)
G |= ϕ(v) ⇐⇒ G [ N r (v) ] |= ϕ(v).
Hence, truth at v only depends on the vertices around v.