Probabilities and Provenance on Trees and Treelike Instances Antoine - - PowerPoint PPT Presentation

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Probabilities and Provenance on Trees and Treelike Instances Antoine - - PowerPoint PPT Presentation

Feb 07, 2023 243 likes •823 views

Probabilities and Provenance on Trees and Treelike Instances Antoine Amarilli 1 , Pierre Bourhis 2 , Pierre Senellart 1 , 3 , 4 September 7th, 2016 1 Tlcom ParisTech 2 CNRS CRIStAL 3 National University of Singapore 4 cole normale

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Probabilities and Provenance

  • n 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 1/7

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How to travel to Highlights from Paris?

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How to travel to Highlights from Paris?

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How to travel to Highlights from Paris?

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How to travel to Highlights from Paris?

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How to travel to Highlights from Paris?

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

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How to travel to Highlights from Paris?

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How to travel to Highlights from Paris?

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

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How to travel to Highlights from Paris?

50%

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

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SLIDE 10

How to travel to Highlights from Paris?

50%

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How to travel to Highlights from Paris?

50% 90%

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

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How to travel to Highlights from Paris?

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

50% 90% 42% 37% 90% 83% 78% 72%

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How to travel to Highlights from Paris?

(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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SLIDE 14

Problem statement

Input:

? Query Q

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

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Problem statement

Input:

? Query Q

(Metro|RER)*|(Bus|Tram)* Database D or graph

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Problem statement

Input:

? Query Q

(Metro|RER)*|(Bus|Tram)* Database D or graph

% Probabilities on facts or edges

50%

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SLIDE 17

Problem statement

Input:

? Query Q

(Metro|RER)*|(Bus|Tram)* Database D or graph

% Probabilities on facts or edges

50%

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

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SLIDE 18

Problem statement

Input:

? Query Q

(Metro|RER)*|(Bus|Tram)* Database D or graph

% Probabilities on facts or edges

50%

Output: the probability that the query is true under the distribution (assuming independence of all probabilistic events) Complexity: already #P-hard in the input database! (from #MONOTONE-SAT)

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Using treewidth to make the problem tractable

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Using treewidth to make the problem tractable

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Using treewidth to make the problem tractable

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

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Using treewidth to make the problem tractable

Treewidth by example:

  • Trees have treewidth 1
  • Cycles have treewidth 2
  • k-cliques and (k − 1)-grids have treewidth k − 1

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Using treewidth to make the problem tractable

Treewidth by example:

  • Trees have treewidth 1
  • Cycles have treewidth 2
  • k-cliques and (k − 1)-grids have treewidth k − 1

→ Treelike: the treewidth is bounded by a constant

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Tractability on treelike instances (RER|metro)* |(bus|tram)* MSO query Treelike data

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Tractability on treelike instances Tree automaton (RER|metro)* |(bus|tram)* MSO query Treelike data

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query Treelike data

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query linear [Courcelle] Treelike data Query answer

TRUE

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query Treelike data

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query Provenance circuit

∧

linear Treelike data

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query Provenance circuit

∧

linear Treelike data Probability

42%

linear

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Tractability on treelike instances Tree automaton Tree encoding (RER|metro)* |(bus|tram)* MSO query Provenance circuit

∧

linear Treelike data Probability

42%

linear

Theorem For any fixed Boolean MSO query q and k ∈ N, given a database D of treewidth ≤ k with independent probabilities, we can compute in linear time the probability that D satisfies q

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Lower bound

What can we do for unbounded-treewidth instances?

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Lower bound

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

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Lower bound

Theorem For any graph signature σ, there is a first-order query q such that for any constructible unbounded-treewidth class I, probability evaluation of q on I is #P-hard under RP reductions

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Lower bound

Theorem For any graph signature σ, there is a first-order query q such that for any constructible unbounded-treewidth class I, probability evaluation of q on I is #P-hard under RP reductions Proof idea: extract instances of a hard problem as topological minors using recent polynomial bounds [Chekuri and Chuzhoy, 2014]

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Lower bound

Theorem For any graph signature σ, there is a first-order query q such that for any constructible unbounded-treewidth class I, probability evaluation of q on I is #P-hard under 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 as topological minors using recent polynomial bounds [Chekuri and Chuzhoy, 2014]

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Future and ongoing work

  • Improving the lower bound:
  • From graphs to arbitrary arity databases
  • From FO down to unions of conjunctive queries with =

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Future and ongoing work

  • Improving the lower bound:
  • From graphs to arbitrary arity databases
  • From FO down to unions of conjunctive queries with =
  • Complexity in query and database — currently Ω
  • 22...2|Q|

× |D|

  • → Which queries can efficiently be compiled to automata?

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SLIDE 53

Future and ongoing work

  • Improving the lower bound:
  • From graphs to arbitrary arity databases
  • From FO down to unions of conjunctive queries with =
  • Complexity in query and database — currently Ω
  • 22...2|Q|

× |D|

  • → Which queries can efficiently be compiled to automata?
  • Non-Boolean queries: efficient enumeration of query results?

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Future and ongoing work

  • Improving the lower bound:
  • From graphs to arbitrary arity databases
  • From FO down to unions of conjunctive queries with =
  • Complexity in query and database — currently Ω
  • 22...2|Q|

× |D|

  • → Which queries can efficiently be compiled to automata?
  • Non-Boolean queries: efficient enumeration of query results?
  • Other tasks: probabilistic conditioning

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

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SLIDE 55

Future and ongoing work

  • Improving the lower bound:
  • From graphs to arbitrary arity databases
  • From FO down to unions of conjunctive queries with =
  • Complexity in query and database — currently Ω
  • 22...2|Q|

× |D|

  • → Which queries can efficiently be compiled to automata?
  • Non-Boolean queries: efficient enumeration of query results?
  • Other tasks: probabilistic conditioning

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

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SLIDE 56

References I

Chekuri, C. and Chuzhoy, J. (2014). Polynomial bounds for the grid-minor theorem. In STOC. Courcelle, B. (1990). The monadic second-order logic of graphs. I. Recognizable sets

  • f 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.