lower bounds for exact model counting and applications in probabilistic databases

Lower Bounds for Exact Model Counting and Applications in Probabilistic Databases

Paul Beame Jerry Li Sudeepa Roy Dan Suciu

University of Washington

2

Model Counting• Model Counting Problem:

Given a Boolean formula F, compute #F = #Models (satisfying assignments) of F

e.g. F = (x y) (x u w) (x u w z) #Assignments on x, y, u, z, w which make F = true

• Probability Computation Problem:Given F, and independent Pr(x), Pr(y), Pr(z), …,

compute Pr(F)

3

Model Counting• #P-hard

▫ Even for formulas where satisfiability is easy to check

• Applications in probabilistic inference ▫ e.g. Bayesian net learning

• There are many practical model counters that can compute both #F and Pr(F)

4

•CDP•Relsat•Cachet•SharpSAT•c2d•Dsharp•…

Exact Model Counters

Search-based/DPLL-based(explore the assignment-space and count the satisfying ones)

Knowledge Compilation-based(compile F into a “computation-friendly” form)

[Survey by Gomes et. al. ’09]

Both techniques explicitly or implicitly • use DPLL-based algorithms • produce FBDD or Decision-DNNF compiled forms (output or trace)

[Huang-Darwiche’05, ’07]

[Birnbaum et. al.’99]

[Bayardo Jr. et. al. ’97, ’00]

[Sang et. al. ’05]

[Thurley ’06]

[Darwiche ’04]

[Muise et. al. ’12]

5

Model Counters Use Extensions to DPLL

• Caching Subformulas▫ Cachet, SharpSAT, c2d, Dsharp

• Component Analysis▫ Relsat, c2d, Cachet , SharpSAT, Dsharp

• Conflict Directed Clause Learning▫ Cachet, SharpSAT, c2d, Dsharp

• DPLL + caching + (clause learning) FBDD• DPLL + caching + component + (clause learning) Decision-DNNF

How much more does component analysis add?i.e. how much more powerful are decision-DNNFs than FBDDs?

6

Theorem:

• Decision-DNNF of size N FBDD of size Nlog N + 1

• If the formula is k-DNF, then FBDD of size Nk

• Algorithm runs in linear time in the size of its output

Main Result

7

Consequence: Running Time Lower Bounds

Model counting algorithm running time ≥ compiled form size

Lower bound on compiled form size Lower bound on running time

▫Note: Running time may be much larger than the size▫e.g. an unsatisfiable CNF formula has a trivial compiled form

8

Our quasipolynomial conversion+ Known exponential lower bounds on FBDDs

[Bollig-Wegener ’00, Wegener’02]

Exponential lower bounds on decision-DNNF size

Exponential lower bounds on running time of exact model counters

Consequence: Running Time Lower Bounds

9

Outline

•Review of DPLL-based algorithms▫Extensions (Caching & Component Analysis)▫Knowledge Compilation (FBDD & Decision-DNNF)

•Our Contributions▫Decision-DNNF to FBDD conversion▫ Implications of the conversion▫Applications to Probabilistic Databases

•Conclusions

DPLL Algorithms

Davis, Putnam, Logemann, Loveland [Davis et. al. ’60, ’62]

10

x

z

0

y

1

u 01

1

0

w

1

0

0

1

1 0

u11

1

0

w

1

0

0

1

1 0

1 0 1 0

01

11

F: (xy) (xuw) (xuwz)

uwz

uw

w

uw

½

¾ ¾

y(uw)3/87/8

5/8

w½

Assume uniform distribution for simplicity

// basic DPLL:Function Pr(F):

if F = false then return 0if F = true then return 1select a variable x, return

½ Pr(FX=0) + ½ Pr(FX=1)

DPLL Algorithms

11

x

z

0

y

1

u 01

1

0

w

1

0

0

1

1 0

u11

1

0

w

1

0

0

1

1 0

1 0 1 0

01

11


uwz

uw

w

uw

½

¾ ¾

y(uw)3/87/8

5/8

w½

The trace is a Decision-Tree for F

12

Extensions to DPLL

• Caching Subformulas

• Component Analysis

• Conflict Directed Clause Learning▫ Affects the efficiency of the algorithm, but not the final “form” of the trace

Traces of• DPLL + caching + (clause learning) FBDD• DPLL + caching + component + (clause learning) Decision-DNNF

Caching

13



½ Pr(FX=0) + ½ Pr(FX=1)

x

z

0

y

1

u 01

1

0

w

1

0

0

1

1 0

u11

1

0

w

1

0

0

1

1 0


uwz

uw

w

uw

y(uw)

w

// DPLL with caching:Cache F and Pr(F);look it up before computing

Caching & FBDDs

14

x

z

0

y

1

0

1 0

u11

1

0

w

1

0

0

1

1 0


uwz

uw

w

y(uw)The trace is a decision-DAG for F

FBDD (Free Binary Decision Diagram)or

ROBP (Read Once Branching Program)

• Every variable is tested at most once on any path

• All internal nodes are decision-nodes

Decision-Node

Component Analysis

15

x

z

0

y

1

0

1 0

u11

1

0

w

1

0

0

1

1 0


uwz

uw

w

y (uw)



½ Pr(FX=0) + ½ Pr(FX=1)

// DPLL with component analysis (and caching):

if F = G Hwhere G and H have disjoint set of variablesPr(F) = Pr(G) × Pr(H)

Components & Decision-DNNF

16

x

z

1u1

1

1

0

w

1

0

0

1

1 0

uwz

w

y (uw)

0

y

1

0


The trace is a Decision-DNNF [Huang-Darwiche ’05, ’07]

FBDD + “Decomposable” AND-nodes

(Two sub-DAGs do not share variables)

Decision Node

y

01AND Node

uw

How much power do they add?

17

Main Technical Result

Decision-DNNF FBDDEfficient construction

Size N Size Nlog N+1

(quasipolynomial)

Size Nk

(polynomial)k-DNFe.g. 3-DNF: (x y z) (w y z)

18

Outline

•Review of DPLL algorithms▫Extensions (Caching & Component Analysis)▫Knowledge Compilation (FBDDs & Decision-DNNF)


•Conclusions

19

Need to convertall AND-nodes to Decision-nodeswhile evaluating the same formula F

Decision-DNNF FBDD

A Simple Idea

20

G H

0 1 0 1

G

H0

0 1Decision-DNNF FBDD

G and H do not share variables, so every variable is still tested at most once on any path

1

FBDD

But, what if sub-DAGs are shared?

21

G H

0 10 1

Decision-DNNF

Conflict!

g’

hG

H0

0 1

H

G

0 1

0g’

h

22

G H

010 1

g’

h

Obvious Solution: Replicate Nodes

G H

No conflictApply the simple idea

But, may need recursive replicationCan have exponential blowup!

Main Idea: Replicate Smaller Sub-DAG

23

Edges coming from other nodes in the decision-DNNF

Smaller sub-DAG

Larger sub-DAG

Each AND-node creates a private copy of its smaller sub-DAG

Light and Heavy Edges

24

Smaller sub-DAG

Larger sub-DAG

Light Edge Heavy Edge

Each AND-node creates a private copy of its smaller sub-DAG

Þ Recursively each node u is replicated #times in a smaller sub-DAG

Þ #Copies of u = #sequences of light edges leading to u

Quasipolynomial Conversion

25

L = Max #light edges on any path

L ≤ log N

N = Nsmall + Nbig ≥ 2 Nsmall ≥ ... ≥ 2L

#Copies of each node ≤ NL ≤ Nlog N

We also show that our analysis is tight

#Nodes in FBDD ≤ N. Nlog N

26

Polynomial Conversion for k-DNFs

•L = #Max light edges on any path ≤ k – 1

•#Nodes in FBDD ≤ N. NL = Nk

27

Outline



•Conclusions

Separation Results

AND-FBDDDecision-DNNF

FBDDd-DNNF

• FBDD: Decision-DAG, each variable is tested once along any path

• Decision-DNNF: FBDD + decomposable AND-nodes (disjoint sub-DAGs)

Exponential Separation

Poly-size AND-FBDD or d-DNNF exists

Exponential lower bound on decision-DNNF size

• AND-FBDD: FBDD + AND-nodes (not necessarily decomposable) [Wegener’00]

• d-DNNF: Decomposable AND nodes + OR-nodes with sub-DAGs not simultaneously satisfiable [Darwiche ’01, Darwiche-Marquis ’02]

29

Outline



•Conclusions

Probabilistic Databases AsthmaPatien

t

Ann

Bob

Friend

Ann Joe

Ann Tom

Bob Tom

Smoker

Joe

Tom

Boolean query Q: x y AsthmaPatient(x) Friend (x, y) Smoker(y)

• Tuples are probabilistic (and independent)▫ “Ann” is present with probability 0.3

• What is the probability that Q is true on D?▫ Assign unique variables to tuples

• Boolean formula FQ,D = (x1y1z1) (x1y2z2) (x2y3z2)▫ Q is true on D FQ,D is true

x1

x2

z1

z2

y1

y2

y3

0.30.1

0.51.0

0.90.5

0.7

Pr(x1) = 0.3

Probabilistic Databases

• FQ,D = (x1y1z1) (x1y2z2) (x2y3z2)

• Probability Computation Problem: Compute Pr(FQ,D) given Pr(x1), Pr(x2), ….

• FQ,D can be written as a k-DNF ▫ for fixed, monotone queries Q

For an important class of queries Q, we get exponential lower bounds on decision-DNNFs and model counting algorithms

32

Outline



•Conclusions

33

Summary

• Quasi-polynomial conversion of any decision-DNNF into an FBDD (polynomial for k-DNF)

• Exponential lower bounds on model counting algorithms • d-DNNFs and AND-FBDDs are exponentially more

powerful than decision-DNNFs

• Applications in probabilistic databases

34

Open Problems

• A polynomial conversion of decision-DNNFs to FBDDs?

• A more powerful syntactic subclass of d-DNNFs than decision-DNNFs?▫ d-DNNF is a semantic concept▫ No efficient algorithm to test if two sub-DAGs of an OR-node are

simultaneously satisfiable

• Approximate model counting?

35

Thank You

Questions?

lower bounds for exact model counting and applications in probabilistic databases

Documents