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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 5/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Model-Checking Monadic Second-Order Logic. First-Order Logic by quantification over sets of elements. Formula building rules. • ∃Xϕ, ∀Xϕ: there is a/for all sets of elements ϕ holds • ∃xϕ, ∀xϕ: there is an/for all elements ϕ holds • Boolean connectives and atomic formulas Example. In the language σ := {E} of graphs G := (V, E) we can write ( ∃C 1 ∃C 2 ∃C } {{ } 3 there are sets C 1 , C 2 , C 3 ∀x 3∨ C i (x) i=1 } {{ } ev. node has a col. ∧ ∀x∀y(E(x, y) → 3∧ ) ¬(C i (x)∧C i (y))) i=1 } {{ } endpoints of edges have different colours to say that a graph is 3-colourable.
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 6/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Monadic Second-Order Logic on Graphs There is a subtlety in how we encode graphs as logical structures. Standard encoding. Signature σ g := {E}. A graph G := (V, E) is encoded as σ g -structure G := (V, E) Incidence encoding. Signature σ i := {V, E, inc}. A graph G := (V, E) is encoded as σ i -structure G := (V ∪ E,σ) with • V G := V(G), E G := E(G) and • (x, e) ∈ inc G if the vertex x ∈ V(G) is incident to edge e ∈ E(G) Over the incidence encoding we can say in MSO that a graph has a Hamiltonian cycle, which we cannot say in the standard encoding. ∃P ⊆ E ( P forms a path and every vertex occurs exactly once on P ) We will refer to MSO as MSO 2 whenever we mean the incidence encoding and to MSO if we use the standard encoding.
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