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Probabilistic Query Evaluation: Towards Tractable Combined Complexity Mikal Monet 1 , 2 , supervised by Pierre Senellart 2 , 3 and Antoine Amarilli 1 May 31th, 2017 1 LTCI, Tlcom ParisTech, Universit Paris-Saclay; Paris, France 2 Inria

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Probabilistic Query Evaluation: Towards Tractable Combined Complexity

Mikaël Monet1,2, supervised by Pierre Senellart2,3 and Antoine Amarilli1

May 31th, 2017

1LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France 2Inria Paris; Paris, France 3École normale supérieure, PSL Research University; Paris, France

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Introduction

  • Uncertainty in data

→ Untrustworthy sources, automated information extraction, imperfect sensor precision in experimental sciences, etc.

  • Need framework to model this uncertainty and reason about it

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Introduction

  • Uncertainty in data

→ Untrustworthy sources, automated information extraction, imperfect sensor precision in experimental sciences, etc.

  • Need framework to model this uncertainty and reason about it

→ Probabilistic Databases!

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Plan

1) Define TID model and probabilistic query evaluation (PQE)

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Plan

1) Define TID model and probabilistic query evaluation (PQE) 2) Existing approaches (efficient PQE in the data)

2/20

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Plan

1) Define TID model and probabilistic query evaluation (PQE) 2) Existing approaches (efficient PQE in the data) 3) Efficient PQE in the query and the data

2/20

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Plan

1) Define TID model and probabilistic query evaluation (PQE) 2) Existing approaches (efficient PQE in the data) 3) Efficient PQE in the query and the data 4) Efficient PQE in the data, reasonable complexity in the query

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Tuple-independent databases (TID)

  • Probabilistic databases: model uncertainty about data
  • Simplest model: tuple-independent databases (TID)
  • A relational database I
  • A probability valuation π mapping each fact of I to [0, 1]
  • Semantics of a TID (I, π): a probability distribution on I′ ⊆ I:
  • Each fact F ∈ I is either present or absent with probability π(F)
  • Assume independence across facts

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Example: TID

S a b .5 a c .2

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution:

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c .5 × (1 − .2) S a b

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c .5 × (1 − .2) S a b (1 − .5) × .2 S a c

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c .5 × (1 − .2) S a b (1 − .5) × .2 S a c (1 − .5) × (1 − .2) S

4/20

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Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

5/20

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Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

Probabilistic query evaluation (PQE) problem for Q and I:

  • Given a query q ∈ Q
  • Given an instance I ∈ I and a probability valuation π
  • Compute the probability that (I, π) satisfies q

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Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

Probabilistic query evaluation (PQE) problem for Q and I:

  • Given a query q ∈ Q
  • Given an instance I ∈ I and a probability valuation π
  • Compute the probability that (I, π) satisfies q

→ Pr((I, π) | = q) =

J⊆I, J| =q Pr(J)

5/20

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Complexity of probabilistic query evaluation (PQE)

Question: what is the (data, combined) complexity of PQE depending on the class Q of queries and class I of instances?

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ UCQs \S

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ UCQs \S

  • Existing data dichotomy result on instances

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ UCQs \S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ UCQs \S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015] → There is an FO query for which PQE is #P-hard on any unbounded-treewidth graph family I (under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

7/20

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Data complexity results: related work

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • I is all instances
  • There is a class S ⊆ UCQs of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ UCQs \S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015] → There is an FO query for which PQE is #P-hard on any unbounded-treewidth graph family I (under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

What about combined complexity?

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Wish list

We want:

  • PQE tractable in combined complexity

OR

  • PQE tractable in the data, reasonable in the query

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Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7

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Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) → x y z t R S S R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7

9/20

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Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) → x y z t R S S R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7 → d c b a 1. R .1 R R .1 R .05 S a .7 R a .8

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Restrict instances to trees

Q = one-way paths (1WP), I = polytrees (PT)

10/20

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Restrict instances to trees

Q = one-way paths (1WP), I = polytrees (PT) Q: T S S S T

10/20

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Restrict instances to trees

Q = one-way paths (1WP), I = polytrees (PT) Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

10/20

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Restrict instances to trees

Q = one-way paths (1WP), I = polytrees (PT) Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T Proposition PQE of 1WP on PT is #P-hard

10/20

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Our graph classes

1WP 2WP R S S T R S S T R DWT PT 1WP 2WP DWT PT Connected All ⊆ ⊆ ⊆ ⊆ ⊆ ⊆

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Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected 2 labels

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Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected 2 labels ↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected No labels

12/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE

13/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures

13/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures
  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

13/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures
  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

  • Established the complexity for all combinations of the graph

classes we considered

13/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures
  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

  • Established the complexity for all combinations of the graph

classes we considered Drawbacks and future work:

  • Our graph classes may seem “arbitrary”

13/20

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Led to a publication in PODS’2017

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures
  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

  • Established the complexity for all combinations of the graph

classes we considered Drawbacks and future work:

  • Our graph classes may seem “arbitrary”
  • Not yet a dichotomy, just starting to understand the problem
  • Practical applications?

13/20

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Lowering our expectations

What if we want the complexity to be:

  • Tractable in the data
  • Not too horrible in the query

Can we then support a more expressive query language (e.g., disjunctions, negations, recursion)?

14/20

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Starting point

  • Existing data dichotomy result on instances

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Starting point

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

15/20

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Starting point

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

  • Problem: nonelementary in the query 22

. . . |Q|

15/20

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Starting point

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

  • Problem: nonelementary in the query 22

. . . |Q|

The instance class is parameterized

15/20

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Starting point

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

  • Problem: nonelementary in the query 22

. . . |Q|

The instance class is parameterized Idea: one parameter for the instances and one parameter for the queries

15/20

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Parameterized Complexity

Idea: one parameter kI for the instance (treewidth) AND one parameter kQ for the query

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Parameterized Complexity

Idea: one parameter kI for the instance (treewidth) AND one parameter kQ for the query

  • Instance classes I1, I2, · · ·

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Parameterized Complexity

Idea: one parameter kI for the instance (treewidth) AND one parameter kQ for the query

  • Instance classes I1, I2, · · ·
  • Query classes Q1, Q2, · · ·

16/20

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Parameterized Complexity

Idea: one parameter kI for the instance (treewidth) AND one parameter kQ for the query

  • Instance classes I1, I2, · · ·
  • Query classes Q1, Q2, · · ·

Definition The problem is fixed-parameter tractable (FPT) linear if there exists a computable function f such that it can be solved in time f(kI, kQ) × |Q| × |I|

16/20

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Publication in ICDT’2017

1) A new language...

  • We introduce the language of intentional-clique-guarded

Datalog (ICG-Datalog), parameterized by body-size kP

17/20

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Publication in ICDT’2017

1) A new language...

  • We introduce the language of intentional-clique-guarded

Datalog (ICG-Datalog), parameterized by body-size kP 2) ... with FPT-linear (combined) evaluation...

  • Given an ICG-Datalog program P with body-size kP and a

relational instance I of treewidth kI, checking if I | = P can be done in time f(kP, kI) × |P| × |I|

17/20

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Publication in ICDT’2017

1) A new language...

  • We introduce the language of intentional-clique-guarded

Datalog (ICG-Datalog), parameterized by body-size kP 2) ... with FPT-linear (combined) evaluation...

  • Given an ICG-Datalog program P with body-size kP and a

relational instance I of treewidth kI, checking if I | = P can be done in time f(kP, kI) × |P| × |I| 3) ... and also FPT-linear (combined) computation of provenance

  • We design a new concise provenance representation based on

cyclic Boolean circuits: cycluits

17/20

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Tree encoding E Two-way Alternating Tree Automaton A Database I

  • f treewidth ≤ kI

C(x) ← Subway("Corvisart",x) “Under which conditions is it impossible to go from station Corvisart to station Châtelet with the subway?" Provenance Cycluit

1 2

(Paris Metro map)

C(x) ← C(y) Subway(y,x) ∧ ICG-Datalog program P

  • f body-size ≤ kP

O( g(kP, kI) |P| ) O( g'(kI) |I| ) O( |A| • |E| )

Goal() ← ¬ C("Châtelet")

18/20

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Application to PQE

Theorem Having fixed kP and kI, we can solve PQE in O(22|P|α |I| |P|).

  • 2EXP, but still better than previous nonelementary bounds

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Conclusion

Up to now:

  • Study of the combined complexity of PQE
  • Tractable cases quite restricted
  • If we lower our expectations then we can capture more

expressive query languages Ongoing and future work:

  • Lots of open technical questions
  • Started a collaboration with Dan Olteanu (Univ. of Oxford) on

mixed probabilistic models

  • Practical applications?

Thanks for your attention!

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