Une dichotomie sur l’évaluation de requêtes closes sous homomorphismes sur les graphes probabilistes

Une dichotomie sur l’évaluation de requêtes closes sous homomorphismes sur les graphes probabilistes

Antoine Amarilli¹and İsmail İlkan Ceylan² 29 octobre 2020

1Télécom Paris

2University of Oxford

1/12

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/12

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/12

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/12

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud

Benny Technion

İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data ^2/12

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

80% 10%

40% 80%

100%

90% 90% 50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/12

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/12

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

80% 10%

40% 80%

100%

90% 90% 50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/12

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

80% 10%

40% 80%

100%

90% 90% 50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?

0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?

0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/12

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

What is theprobabilityof this possible world?0.03%

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/12

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

4/12

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

4/12

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

4/12

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

4/12

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime”

Queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

→ We want ahomomorphismfrom the pattern to the graph (not necessarilyinjective)

•

Union of conjunctive queries(UCQ): can I find a match ofsome pattern?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

Intuition about homomorphism-closed queries:

•

^{GeneralizeCQsandUCQs, but also}regular path queries(RPQs),Datalog, etc.

•

Do not allow forinequalitiesornegation

•

A homomorphism-closed query can be seen as aninfinite union of CQs:

→ The query isboundedif the union is finite (it is a UCQ),unboundedotherwise

•

Allows pretty wild things, e.g., “There is a path whose length is prime” ^4/12

Problem statement: Probabilistic query evaluation (PQE)

•

^{Wefixa queryQ}, for instance the CQ: ^{x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40% 80%

100%

90% 90% 50%

90%

100%

•

^{Theoutputis the}total probabilityof the worlds which satisfy the query:

• Formally:P

W⊆D,W|=QPr(W)

→ Intuition:theprobabilitythat the query is true

→ What is thecomplexityof the problemPQE(Q), depending on the queryQ?

Problem statement: Probabilistic query evaluation (PQE)

•

^{Wefixa queryQ}, for instance the CQ: ^{x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}total probabilityof the worlds which satisfy the query:

• Formally:P

W⊆D,W|=QPr(W)

→ Intuition:theprobabilitythat the query is true

→ What is thecomplexityof the problemPQE(Q), depending on the queryQ?

5/12

Problem statement: Probabilistic query evaluation (PQE)

•

^{Wefixa queryQ}, for instance the CQ: ^{x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}total probabilityof the worlds which satisfy the query:

• Formally:P

W⊆D,W|=QPr(W)

→ Intuition:theprobabilitythat the query is true

→ What is thecomplexityof the problemPQE(Q), depending on the queryQ?

Problem statement: Probabilistic query evaluation (PQE)

•

^{Wefixa queryQ}, for instance the CQ: ^{x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}total probabilityof the worlds which satisfy the query:

• Formally:P

W⊆D,W|=QPr(W)

→ Intuition:theprobabilitythat the query is true

→ What is thecomplexityof the problemPQE(Q), depending on the queryQ?

5/12

Problem statement: Probabilistic query evaluation (PQE)

•

^{Wefixa queryQ}, for instance the CQ: ^{x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}total probabilityof the worlds which satisfy the query:

• Formally:P

W⊆D,W|=QPr(W)

→ Intuition:theprobabilitythat the query is true

→ What is thecomplexityof the problemPQE(Q), depending on the queryQ?

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

6/12

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe

What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

6/12

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

6/12

Existing results on PQE

Dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

•

^{The CQ x y z issafe}, but the CQ ^{x y z w} isunsafe What aboutmore expressive queries?

•

Work by [Fink and Olteanu, 2016] aboutnegation

•

No work aboutrecursive queries(but no works about RPQs, Datalog, etc.)

•

Only exception: work onontology-mediated query answering[Jung and Lutz, 2012]

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

7/12

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

7/12

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

7/12

Our result

We study PQE forhomomorphism-closed queriesand show:

Theorem

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQ(hencebounded) andPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

•

This extends the result of [Jung and Lutz, 2012] and covers RPQs, Datalog, etc.

•

Example: theRPQQ: ^∗

• It isnot equivalent to a UCQ: infinite disjunction ⁱ

for alli∈N

• Hence,PQE(Q)is#P-hard

•

We do not study thecomplexity of deciding which case applies

• Depends on how queries arerepresented

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂ 1/2

1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

• 1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w

a⁰₁ a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂ 1/2

1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

• 1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂ 1/2

1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

• 1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂ 1/2

1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

• 1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂

1/2 1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

•

1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂

1/2 1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

•

1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂

1/2 1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

•

1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

AND •

•

•

•

•

• violatesQ

•

We reduce from PQE for theunsafeCQ:Q₀: x y z w a⁰₁

a⁰₂ a⁰₃

a1

a2

a3

b1

b2

b⁰₁

b⁰₂

1/2 1/2 1/2

1/2

1/2

is coded as

•

•

•

•

•

•

•

•

•

1/21/21/2

1/2 1/2

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:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ

Consider itsiterates

for eachn∈N:

•

• • • •

•

n

Case 1:some iterateviolatesthe query:

•

• • • •

•

i

satisfiesQ

but •

• • • •

•

i+1

violatesQ

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

•

• • • •

•

n

satisfiesQfor alln ∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ Consider itsiterates

for eachn∈N:

•

• • • •

•

n

Case 1:some iterateviolatesthe query:

•

• • • •

• i

satisfiesQ

but •

• • • •

• i+1

violatesQ

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

•

• • • •

• n

satisfiesQfor alln∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern

Saving the proof

We know that we have atight pattern:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

• • • •

• i

satisfiesQ

but •

• • • •

• i+1

violatesQ

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

•

• • • •

• n

satisfiesQfor alln∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

• • • •

• i

satisfiesQ

but •

• • • •

• i+1

violatesQ

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

•

• • • •

• n

satisfiesQfor alln∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern

Saving the proof

We know that we have atight pattern:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

• • • •

• i

satisfiesQ

but •

• • • •

• i+1

violatesQ

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

•

• • • •

• n

satisfiesQfor alln∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern

10/12

Saving the proof

We know that we have atight pattern:

•

• •

• satisfiesQ

but •

•

•

•

•

• violatesQ Consider itsiteratesfor eachn∈N:

•

• • • •

• n

Case 1:some iterateviolatesthe query:

•

• • • •

• i

satisfiesQ

but •

• • • •

• i+1

violatesQ

→ Reduce fromPQE(Q₀)as we explained

Case 2:all iteratessatisfythe query:

•

• • • •

• n

satisfiesQfor alln∈N

but •

•

•

•

•

• violatesQ

→ Call this aniterable pattern