Proof structure

Basic idea: finding a tight pattern

The challenging part is to show:

Theorem

For any queryQclosed under homomorphisms andunbounded,PQE(Q)is#P-hard

Idea: find atight pattern, i.e., a graph with three distinguished edges such that:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

Theorem

Any unbounded query closed under homomorphisms has a tight pattern

Basic idea: finding a tight pattern

The challenging part is to show:

Theorem

For any queryQclosed under homomorphisms andunbounded,PQE(Q)is#P-hard Idea: find atight pattern, i.e., a graph with three distinguished edges such that:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

Theorem

Any unbounded query closed under homomorphisms has a tight pattern

8/12

Basic idea: finding a tight pattern

The challenging part is to show:

Theorem

For any queryQclosed under homomorphisms andunbounded,PQE(Q)is#P-hard Idea: find atight pattern, i.e., a graph with three distinguished edges such that:

•

• •

• satisfiesQ

but

•

•

•

•

•

• violatesQ

Theorem

Any unbounded query closed under homomorphisms has a tight pattern

Basic idea: finding a tight pattern

The challenging part is to show:

Theorem

For any queryQclosed under homomorphisms andunbounded,PQE(Q)is#P-hard Idea: find atight pattern, i.e., a graph with three distinguished edges such that:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

Theorem

Any unbounded query closed under homomorphisms has a tight pattern

8/12

Basic idea: finding a tight pattern

The challenging part is to show:

Theorem

For any queryQclosed under homomorphisms andunbounded,PQE(Q)is#P-hard Idea: find atight pattern, i.e., a graph with three distinguished edges such that:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

Theorem

Any unbounded query closed under homomorphisms has a tight pattern

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ... ... except we needmorefrom the tight pattern!

9/12

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ... ... except we needmorefrom the tight pattern!

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ... ... except we needmorefrom the tight pattern!

9/12

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ...

... except we needmorefrom the tight pattern!

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ...

... except we needmorefrom the tight pattern!

9/12

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ...

... except we needmorefrom the tight pattern!

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ...

... except we needmorefrom the tight pattern! 9/12

Using tight patterns to show hardness of PQE

•

Fix the queryQand thetight pattern:

•

is coded as

•

Idea:possible worlds at thelefthave a path that matchesQ₀

iff the corresponding possible world of the TID at therightsatisfies the queryQ...

... except we needmorefrom the tight pattern! 9/12

Saving the proof

We know that we have atight pattern:

•

Consider itsiterates

for eachn∈N:

•

• • • •

•

n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

• Consider itsiterates

for eachn∈N:

•

• • • •

•

n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained

Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

• Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

→ Reduce fromPQE(Q₀)as we explained Case 2:all iteratessatisfythe query:

•

→ Call this aniterable pattern

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

•

Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right

11/12

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected?

s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right

11/12

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right

11/12

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right 11/12

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right 11/12

Using iterable patterns to show hardness of PQE

We have aniterable pattern:

• Idea:reduce from the#P-hardproblemsource-to-target connectivity:

•

^Input:undirected graphwith asourcesandtargett, all edges have probability1/2

•

Output: what is theprobabilitythat the source and target areconnected? s

u

t

1/2 1/2

1/2

is coded as

•

Idea:There is apath connectingsandtin a possible world of the graph at the left iff the queryQissatisfiedin the corresponding possible world of the TID at the right 11/12

Dans le document Une dichotomie sur l’évaluation de requêtes closes sous homomorphismes sur les graphes probabilistes (Page 45-75)