1. Finite Model Theory Lecture 7
Complexity of FO (cont’d)

2. Outline
Complexity of conjunctive queries

3. Conjunctive Queries
The FO fragment consisting of: R(x,y,…) atomic formulas x=y equality f1 Æ phi conjunction 9 x.f existential quantifiers
Canonical form: 9 x1.9 x2… 9 xk.(G1 Æ … Æ Gm) or, simply: G1, …, Gm

4. Complexity of Conjunctive Queries
Theorem The query complexity of CQ is NP-hard. The combined complexity of CQ is NP-complete.
Proof NP membership. Let: f = 9 x1 … 9 xk G1 Æ … Æ Gm A = (A, R1A, …, RpA) Step 1: guess k values a1, …, ak 2 A Step 2: check if G1 Æ … Æ Gm is true after substituting x1 with a1, …, xk with ak

5. Proof (cont’d)
Hardness: will design a structure A s.t. the set {f | A ² f} is NP-hard.
By reduction from 3 colorability
A = ({0,1,2}, N), where N = {(i,j) | i ¹ j}
Let G = (V, E) be a graph, |V| = k
Define: f = 9 x1 … 9 xk (Æ(xi, xj) 2 E N(xi, xj))

6. Tree Decomposition Note: a conjunctive query = a hypergraph
A tree decomposition of a conjunctive query (or hypergraph) with variables (nodes) V is a tree T, and a set Bt µ V for each node t in T such that:
For every x 2 V, the set {t | x 2 Bt} is connected
Every hyperedge of the query (hypergraph) is contained in some Bt

7. Tree Decomposition Examples [in class]: f = R(x,y,z), R(z,u,v), S(v,w)
f = R(x,y,z), R(z,u,v), R(v,w,x)

8. Tree Decomposition
Definition A conjunctive query (or a hypergraph) is acyclic if there exists a tree decomposition such that 8 t, Bt is an hyperedge.
I.e. there are no redundant variables on the tree nodes.

9. Tree Decomposition
Theorem If f is an acyclic conjunctive query and A is a structure, then checking whether A ² f can be done in timme O(|f| |A|)
Note: |A| denotes the size of the entire structure, i.e. is more of the form n + n3 + n2 + n5 if the arities of the relations in A are 3, 2, 5.
Proof [in class]