Probabilities and Provenance on Trees and Treelike Instances

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Probabilities and Provenance on Trees and Treelike Instances

Antoine Amarilli¹, Pierre Bourhis², Pierre Senellart¹^,³^,⁴ September 7th, 2016

1Télécom ParisTech

2CNRS CRIStAL

3National University of Singapore

4École normale supérieure de Paris _1/7

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How to travel to Highlights from Paris?

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(Metro|RER)|(Bus|Tram)

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50%

(Metro|RER)|(Bus|Tram)

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50%

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50%

90%

(Metro|RER)|(Bus|Tram)

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50%

42%

37%

78% 90%

72%

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(Metro|RER)|(Bus|Tram)

50%

90%

42%

37%

90%

83%

78%

72%

What is the probability

that I can attend Highlights 2016?

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Problem statement

Input:

? QueryQ (Metro|RER)*|(Bus|Tram)*

DatabaseDor graph

% Probabilitieson facts or edges ^50%

Output: theprobabilitythat the query is true under the distribution (assuming independence of all probabilistic events)

Complexity: already#P-hardin the input database! (from #MONOTONE-SAT)

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Input:

DatabaseDor graph

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Input:

DatabaseDor graph

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Input:

DatabaseDor graph

Output:theprobabilitythat the query is true under the distribution (assuming independence of all probabilistic events)

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Input:

DatabaseDor graph

Output:theprobabilitythat the query is true under the distribution (assuming independence of all probabilistic events)

Complexity: already#P-hardin the input database!

(from #MONOTONE-SAT)

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Using treewidth to make the problem tractable

Treewidth by example:

• Treeshave treewidth1

• Cycleshave treewidth2

• k-cliquesand(k−1)-gridshave treewidthk−1

→ Treelike: thetreewidthis bounded by aconstant

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Tractability on treelike instances

(RER|metro)*

|(bus|tram)*

MSO query Treelike data

Theorem

For anyfixedBoolean MSO queryqandk∈N,

given a databaseDoftreewidth≤kwithindependent probabilities, we can compute inlinear timethe probability thatDsatisfiesq

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Tree automaton (RER|metro)*

|(bus|tram)*

Theorem

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Tree automaton Tree encoding

(RER|metro)*

|(bus|tram)*

Theorem

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(RER|metro)*

|(bus|tram)*

MSO query

linear [Courcelle]

Treelike data

Query answer

TRUE

Theorem

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(RER|metro)*

|(bus|tram)*

Theorem

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(RER|metro)*

|(bus|tram)*

MSO query

Provenance circuit

∧

linear Treelike data

Theorem

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(RER|metro)*

|(bus|tram)*

MSO query

∧

Probability

95%

linear

Theorem

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(RER|metro)*

|(bus|tram)*

MSO query

∧

Probability

95%

linear

Theorem

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Lower bound

What can we do for unbounded-treewidth instances?

... not much. Theorem

For any graph signatureσ, there is afirst-orderqueryqsuch that for any constructibleunbounded-treewidthclassI,

probability evaluation ofqonI is#P-hardunder RP reductions

3 1

2

4 3 4

2 1

maps vertices to vertices maps edges to vertex-disjoint paths

Proof idea:extract instances of a hard problem astopological minors using recentpolynomial bounds[Chekuri and Chuzhoy, 2014]

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Lower bound

What can we do for unbounded-treewidth instances?... not much.

Theorem

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Lower bound

Theorem

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Lower bound

Theorem

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Lower bound

Theorem

3 1

2

4 3 4

2 1

maps vertices to vertices maps edges to vertex-disjoint paths

Proof idea:extract instances of a hard problem astopological minors

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Future and ongoing work

• Improving thelower bound:

• Fromgraphstoarbitrary aritydatabases

• FromFOdown tounions of conjunctive querieswith6=

• Complexity inqueryanddatabase— currentlyΩ

2²^{. .}^.²

|Q|

× |D|

→ Which queries canefficientlybe compiled to automata?

• Non-Booleanqueries: efficientenumerationof query results?

• Other tasks: probabilisticconditioning

“Knowing that I’m here, what’s the probability that RER B is up?” Thanks for your attention!

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2²^{. .}^.²

|Q|

× |D|

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2²^{. .}^.²

|Q|

× |D|

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2²^{. .}^.²

|Q|

× |D|

“Knowing that I’m here, what’s the probability that RER B is up?”

Thanks for your attention!

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2²^{. .}^.²

|Q|

× |D|

“Knowing that I’m here, what’s the probability that RER B is up?”

Thanks for your attention!

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References I

Chekuri, C. and Chuzhoy, J. (2014).

Polynomial bounds for the grid-minor theorem.

InSTOC.

Courcelle, B. (1990).

The monadic second-order logic of graphs. I. Recognizable sets of finite graphs.

Inf. Comput., 85(1).

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Image credits

• Slide 2:

• https://commons.wikimedia.org/wiki/File:

Paris_Metro_map.svg(cropped), user Umx on Wikimedia Commons, public domain

• http://www.parisvoyage.com/images/cartoon18.jpg, ParisVoyage, fair use

• http://www.vianavigo.com/fileadmin/galerie/pdf/CGU_t_.pdf (cropped), RATP, fair use

• Slides 4 and 5: https://commons.wikimedia.org/wiki/File:

Carte_Transilien_RER_sch%C3%A9matique.svg(modified), user Benjamin Smith on Wikimedia Commons, license CC BY-SA 4.0 international.