Tractable Query Answering Under Probabilistic Constraints

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Tractable Query Answering Under Probabilistic Constraints

Antoine Amarilli¹, Pierre Bourhis², Pierre Senellart^1,3

1Télécom ParisTech

2CNRS-LIFL

3National University of Singapore

September 4th, 2014

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Tractable Query Evaluation On Probabilistic Instances

Antoine Amarilli¹, Pierre Bourhis², Pierre Senellart^1,3

1Télécom ParisTech

2CNRS-LIFL

3National University of Singapore

September 4th, 2014

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Background Ideas Results Consequences

Instances and queries

Given a relational instance(= set of facts, hypergraph) I={R(a,b),R(b,c),S(c)}

Given a conjunctive query(CQ) (existentially quantified) q:∃xy R(x,y)∧S(y)

→ Query evaluation(model checking) of q onI

→ Data complexity: qis fixed

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Instances and queries

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Instances and queries

Given a conjunctive query(CQ) (existentially quantified) q:∃xyR(x,y)∧S(y)

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Instances and queries

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Instances and queries

→ Data complexity: q is fixed

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Uncertain and probabilistic instances

Set of uncertain events

e_flight CDG → VIE flight AF1756 takes place e_bus Vienna → Bratislava buses are running

Annotate instance factswith formulaeon the events IsIn(AA, Paris) ¬e_flight

IsIn(AA, Vienna) e_flight∧ ¬e_bus IsIn(AA, Bratislava) e_flight∧e_bus

→ Semantics: aset of instances (possible worlds). Add aprobability distribution on each event

each event hasprobability0<p<1 of being true all events are assumed to beindependent

→ Semantics: aprobability distribution on instances.

→ Query evaluation: determine theprobability of q onbI.

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Uncertain and probabilistic instances

e_flight CDG → VIE flight AF1756 takes place e_bus Vienna → Bratislava buses are running Annotate instance factswith formulaeon the events

IsIn(AA, Paris) ¬e_flight IsIn(AA, Vienna) e_flight∧ ¬e_bus IsIn(AA, Bratislava) e_flight∧e_bus

→ Semantics: aset of instances (possible worlds). Add aprobability distribution on each event

each event hasprobability0<p<1 of being true all events are assumed to beindependent

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Uncertain and probabilistic instances

→ Semantics: aset of instances (possible worlds).

Add aprobability distribution on each event each event hasprobability0<p<1 of being true all events are assumed to beindependent

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Uncertain and probabilistic instances

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Uncertain and probabilistic instances

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Uncertain and probabilistic instances

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Hardness and tractability

Witharbitrary annotations

→ Query evaluation is#P-hardeven with a single fact (Immediate reduction from #SAT)

Withsimple annotations (one unique event per tuple)

→ Query evaluation is#P-hardon arbitrary instances (Use the instance to do the reduction)

Existing work:

→ Fix asimpleannotation scheme

→ Showdichotomybetween #P-hard and PTIME queries Our approach:

→ Find arestrictionon the instance and annotations

→ Show thatmany queriesare tractable in this case

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Hardness and tractability

Existing work:

→ Showdichotomybetween #P-hard and PTIME queries

Our approach:

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Hardness and tractability

Existing work:

→ Showdichotomybetween #P-hard and PTIME queries Our approach:

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Bounded treewidth

An idea from instances without probabilities...

If an instance has low treewidth then it is almost a tree Assume that the instance treewidth isconstant...

→ Linear time data complexity

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c) tree encoding T_I tree decomposition

O(|I|) for ﬁxed width instance I

R(a, b) R(b, c) S(c)

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

query q

∃xy R(x, y) ∧ S(y)

tree encoding T_I tree decomposition

O(|I|) for ﬁxed width

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

query q

∃xy R(x, y) ∧ S(y)

deterministic tree automaton Aq

rewriting O(1) data complexity

query q

∃xy R(x, y) ∧ S(y)

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

query q

∃xy R(x, y) ∧ S(y)

query q

∃xy R(x, y) ∧ S(y) deterministic tree automaton Aq

evaluation linear time

query answer

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

query q

∃xy R(x, y) ∧ S(y)

query q

query answer evaluation

linear time

query answer

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Bounded treewidth

instance I

R(a, b) R(b, c) S(c)

query q

∃xy R(x, y) ∧ S(y)

query q

query answer evaluation

linear time

query answer

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Tractable inference

An idea from probabilities without instances...

Represent a propositional formulaF as a Boolean circuit Assume the circuit has constant treewidth

→ Probability of F can be computed in linear time (usingjunction tree algorithm for Bayesian networks) (assuming constant-time arithmetic operations)

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cc-tables

Boolean circuit for the annotations

1/2 1/2

1/2

∧

R(a, b) R(b, c)

R(c, d) Circuitmust have low treewidth Instance must have low treewidth

→ Needsimultaneous decomposition

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cc-tables

Boolean circuit for the annotations 1/2

1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

Circuitmust have low treewidth Instance must have low treewidth

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cc-tables

1/2 1/2

∧

R(a, b) R(b, c)

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cc-tables

1/2 1/2

∧

R(a, b) R(b, c)

1/2 R(a, b) 1/2

R(b, c)

∧

1/2

R(c, d)

∧

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

tree decomposition O(|I|) for ﬁxed width

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

query q

∃xy R(x, y) ∧ S(y)

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

query q

∃xy R(x, y) ∧ S(y)

query q

∃xy R(x, y) ∧ S(y)

deterministic tree automaton A_q rewriting

O(1) data complexity

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

query q

∃xy R(x, y) ∧ S(y)

query q

∃xy R(x, y) ∧ S(y)

instrumentation linear time

bounded treewidth circuit C

1/2

∧

1/2

∧ A A

A

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

query q

∃xy R(x, y) ∧ S(y)

query q

∃xy R(x, y) ∧ S(y)

1/2

∧

1/2

∧ A A

A

1/2

∧

1/2

∧ A A

A

probability p probabilistic inference O(|C|) for ﬁxed width

0.42

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Main result

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

instance I

1/2 1/2 1/2

∧

R(a, b) R(b, c) R(c, d)

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

tree encoding T_I

1/2 R(a, b) 1/2

R(b, c)

∧ 1/2

R(c, d)

∧

query q

∃xy R(x, y) ∧ S(y)

query q

∃xy R(x, y) ∧ S(y)

1/2

∧

1/2

∧ A A

A

1/2

∧

1/2

∧ A A

A

0.42

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Consequences

For queries representable as deterministic automata...

→ CQs

→ Monadic second-order

→ Guarded second-order

... on various probabilistic models...

→ Tuple-independent tables

→ Block-independent disjoint tables

→ pc-tables (presented before)

→ Probabilistic XML

... assuming bounded treewidth(for reasonable definitions) ...

→ ... probability of fixed q can be computed in O(|bI|)!

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Consequences

→ CQs

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Consequences

→ CQs

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Consequences

→ CQs

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Conclusion

We can combinethe following techniques:

Computingtree decompositions

Encoding problems toautomataon tree encodings of instances Evaluatingprobabilitieson bounded-treewidth circuits

Applications:

Tractable probabilistic query evaluation inpractice? Reasoning underuncertain rules

(hence the bait-and-switch on the title...) Questions:

Othersemiringsthan Boolean AND/OR? Other tasks thanprobabilistic inference? 5

votes

0

answers

48

views

What are bounded-treewidth circuits good for?

circuit-complexity pr.probability treewidth arithmetic-circuits

modiﬁed aug 28 at 13:05a3nm1,432 http://cstheory.stackexchange.com/q/25624

Thanks for your attention!

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Conclusion

Applications:

Tractable probabilistic query evaluation inpractice?

Reasoning underuncertain rules

(hence the bait-and-switch on the title...)

Questions:

Othersemiringsthan Boolean AND/OR? Other tasks thanprobabilistic inference? 5

votes

0

answers

48

views

(43)

Conclusion

Applications:

Othersemiringsthan Boolean AND/OR?

Other tasks thanprobabilistic inference?

5

votes

0

answers

48

views

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Conclusion

Applications:

5

votes

0

answers

48

views

(45)

Conclusion

Applications:

5

votes

0

answers

48

views