[PPT] - Approximate DNF-Counting Kuldeep S. Meel 1 , Aditya A. Shrotri 2 , PowerPoint Presentation

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On Hashing-Based Approaches to Approximate DNF-Counting

Kuldeep S. Meel1, Aditya A. Shrotri2, Moshe Y. Vardi2

1 School of Computing, 2 Department of Computer Science,

National University of Singapore Rice University

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 1

SLIDE 2

Boolean formulas

Variables take values T/F
Logical Operations AND, OR, NOT, XOR
…

Formula encodings:

Conjunctive Normal Form (CNF)
(X
1 ∨ ¬X2 ∨ ¬X3 ) ∧ (¬X1 ∨ X4) ∧..

Disjunctive Normal Form (DNF)

(X
1 ∧ ¬X2 ∧ ¬X3 ) ∨ (¬X1 ∧ X4) ∨..

At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 2

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Counting Problem
number of satisfying assignments?
DNF-Counting Applications
Probabilistic Databases
Network Reliability
CNF-Counting Applications
Probabilistic Inference
Information Leakage

At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 3

SLIDE 4

Counting problem is hard [Valiant,’79]
“#P-Complete”
Relaxation: Approximate Counting
Challenge: Efficiently find an approximate answer that is guaranteed to

be close

At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 4

SLIDE 5

Hash functions show promise!

Hashing
based Approximate CNF-Counting

Improvements over

5+ years

[Chakraborty et al.

’13,’14,’16]

Scales to large formulas

Hashing
based Approximate DNF-Counting [Chakraborty et al. ‘16]

At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 5

SLIDE 6

Hash functions show promise!
Hashing-based Approximate CNF-Counting
Improvements over 5+ years
[Chakraborty et al. ’13,’14,’16]
Scales to large formulas
Hashing-based Approximate DNF-Counting [Chakraborty et al. ‘16]
Poor time complexity

At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 6

SLIDE 7

Problem

:
Approximate DNF-Counting

Our Solution Strategy

:

Design

efficient hashing techniques

Contributions

3
algorithmic improvements to the hashing framework

Result

Reduction in complexity from cubic to linear
Significance
Power and versatility of hashing
At a glance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 7

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9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 8

Boolean Formulas and the Counting problem

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

# variables n =

3

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 10

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Cubes: C1 C2 C3

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 11

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Cubes: C1 C2 C3 # cubes m =

3

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 12

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Assignment: X1 = F , X2 = T , X3 = T

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 13

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Assignment: X1 = F , X2 = T , X3 = T

(¬F ∧ T ) ∨ (T ∧T ) ∨ (F ∧ T ) = T ∨ T ∨ F = T

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 14

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Assignment: X

1 = F , X2 = T , X3 = T

σ = <

0, 1, 1>

σ satisfies C

2 ⇒ σ satisfies 

σ

⊨ C2

σ

⊨ 

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 15

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Assignment: X1 = F , X2 = T , X3 = T
σ = <0, 1, 1>
σ satisfies C2 ⇒ σ satisfies 
σ ⊨ C2
σ ⊨ 
Checking σ ⊨  takes linear time

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 16

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Universe of assignments U = {

0,1}n

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 17

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Universe of assignments U = {

0,1}n

Set of solutions S

 = { σ ∈ U : σ ⊨  }

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 18

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

Universe of assignments U = {

0,1}n

Set of solutions S

 = { σ ∈ U : σ ⊨  }

DNF

Counting: Determine |S| “Count of ”

Disjunctive Normal Form

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 19

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 = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∨ (𝑌1 ∧ 𝑌3)

|U| = 23 = 8
S = { <0, 1, 0> , <0, 1, 1> , <1,0,1>, <1, 1, 1> }
|S| = 4

DNF Counting Problem

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 20

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Checking satisfiability
UNSAT: ..∨ ( .. ∧ Xi ∧ ¬Xi ∧ .. ) ∨ ..

Hardness of DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 21

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Enumerating satisfying assignments

 = (X1 ∧¬X2 ) ∨ (X3 ∧ X4) ∨ …
<1 , 0, 0, 0>
<1 , 0, 0, 1>
<1 , 0, 1, 0>
<1 , 0, 1, 1>
𝑇𝑑1 = {<1 , 0, 0, 0>, <1 , 0, 0, 1>, <1 , 0, 1, 0>, <1 , 0, 1, 1>}

Hardness of DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 22

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Enumerating satisfying assignments
 = (X1 ∧¬X2 ) ∨ (X3 ∧ X4) ∨ …
<1 , 0, 0, 0>
<1 , 0, 0, 1>
<1 , 0, 1, 0>
<1 , 0, 1, 1>
𝑇𝑑1 = {<1 , 0, 0, 0>, <1 , 0, 0, 1>, <1 , 0, 1, 0>, <1 , 0, 1, 1>}
𝑇𝑑1 ∩ 𝑇𝑑2 is not empty

Hardness of DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 23

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Enumerating satisfying assignments

 = (X1 ∧¬X2 ) ∨ (X3 ∧ X4) ∨ …
<1 , 0, 0, 0>
<1 , 0, 0, 1>
<1 , 0, 1, 0>
<1 , 0, 1, 1>
𝑇𝑑1 = {<1 , 0, 0, 0>, <1 , 0, 0, 1>, <1 , 0, 1, 0>, <1 , 0, 1, 1>}
𝑇𝑑1 ∩ 𝑇𝑑2 is not empty ⇒

|S| = |𝑇𝑑1 ∪ 𝑇𝑑2 ∪ 𝑇𝑑3 … ∪ 𝑇𝑑𝑛|

Hardness of DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 24

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DNF

Counting is #P-Complete [Valiant,’79]

Class contains entire Polynomial Hierarchy!

Need to Approximate!
Hardness of DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 25

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Input: DNF Formula  Tolerance ε 0 < ε < 1 Confidence δ 0 < δ < 1 Output: Approximate Count C s.t. Pr [|S|⋅(1-ε) < C < |S|⋅(1+ε) ] > 1-δ

Approximate DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 26

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Input: DNF Formula  Tolerance ε 0 < ε < 1 Confidence δ 0 < δ < 1 Output: Approximate Count C s.t. Pr [|S|⋅(1-ε) < C < |S|⋅(1+ε) ] > 1-δ Challenge: Design a poly(m, n, 1

𝜗 , log( 1 𝜀 )) time algorithm “FPRAS”

Approximate DNF-Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 27

where m = #cubes n = #vars

SLIDE 28

Monte Carlo Sampling for Approximate DNF

1.

Counting

[Karp,

1.

Luby ‘83] [Karp,

2.

Luby, Madras ‘89]

3.

[Dagum, Karp, Luby, Ross ‘00]

Hashing

2.

Based Algorithms for Approximate DNF-Counting

Originated with [

1.

Sipser, ‘83], [Bellare et al. ‘98 ], [Gomes et al. ‘06] DNF

2.

Counting FPRAS: [Chakraborty et al., ‘16]

Previous Work

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 28

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Randomized algorithms based on drawing independent samples

Method 1: Monte Carlo Sampling

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 29

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Randomized algorithms based on drawing independent samples

Method 1: Monte Carlo Sampling

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 30

U S

Randomly sample from U

“Throwing darts”

SLIDE 31

Randomized algorithms based on drawing independent samples

Method 1: Monte Carlo Sampling

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 31

U S

Randomly sample from U

“Throwing darts”

Count darts on target

|S
| ≈

#• #•+#• * |U|

SLIDE 32

Randomized algorithms based on drawing independent samples

Method 1: Monte Carlo Sampling

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 32

U S

Randomly sample from U

“Throwing darts”

Count darts on target

|S
| ≈

#• #•+#• * |U|

𝑃

(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2) [Karp et al., ‘89] where m = #cubes n = #vars

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Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 33

U S “Cutting slices of a pie”

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Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 34

Partition universe into

‘cells’

U S

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Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 35

Partition universe into

‘cells’

Ensure

‘roughly’ equal solutions to  in each cell

U S

SLIDE 36

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 36

Partition universe into

‘cells’

Ensure

‘roughly’ equal solutions to  in each cell

Pick a cell at random

U

S

SLIDE 37

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 37

U S

Partition universe into

‘cells’

Ensure

‘roughly’ equal solutions to  in each cell

Pick a cell at random

Count solutions only in the picked cell

SLIDE 38

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 38

U S

Partition universe into

‘cells’

Ensure

‘roughly’ equal solutions to  in each cell

Pick a cell at random

Count solutions only in the picked cell
Estimate count as #(solutions to
 in cell) x #cells

SLIDE 39

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 39

U S

Partition universe into

‘cells’

Hash function

: Conjunction of Hash Constraints

Hash Constraint:

Boolean formula randomly chosen

from special set (Hash family)

SLIDE 40

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 40

U S Q1

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

SLIDE 41

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 41

U S Q1

Partition universe into

‘cells’

Hash function

: Conjunction of Hash Constraints

Hash Constraint:

Boolean formula randomly chosen

from special set (Hash family)

Q1 partitions U into 2 ‘cells’
Q1 = 0
Q1 = 1

SLIDE 42

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 42

U S Q1 Q2

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 partitions U into 4 ‘cells’
Q1 , Q2 ∈ {00 , 01, 10, 11}

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Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 43

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’

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Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 44

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’
Ensure ‘roughly’ equal solutions to  in each cell

SLIDE 45

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 45

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’
Ensure ‘roughly’ equal solutions to  in each cell
Use 2-universal Hash Families

SLIDE 46

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 46

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’
Ensure ‘roughly’ equal solutions to  in each cell
Use 2-universal Hash Families
Pick cell at random
Q1Q2Q3 …. QL = 0100…1

SLIDE 47

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 47

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’
Ensure ‘roughly’ equal solutions to  in each cell
Use 2-universal Hash Families
Pick cell at random
Q1Q2Q3 …. QL = 0100…1
Count solutions in the picked cell
’ =  ∧ Q
Q =(Q1 ⇔ 0) ∧ (Q2 ⇔ 1) ∧ (Q3 ⇔ 0) .. (QL ⇔ 1)
Calculate |S’|

SLIDE 48

Method 2: Hashing - Based

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 48

U S

Partition universe into ‘cells’
Hash function: Conjunction of Hash Constraints
Hash Constraint: Boolean formula randomly chosen

from special set (Hash family)

Q1 , Q2 , Q3 …. , QL partitions U into 2L ‘cells’
Ensure ‘roughly’ equal solutions to  in each cell
Use 2-universal Hash Families
Pick cell at random
Q1Q2Q3 …. QL = 0100…1
Count solutions in the picked cell
’ =  ∧ Q
Q =(Q1 ⇔ 0) ∧ (Q2 ⇔ 1) ∧ (Q3 ⇔ 0) .. (QL ⇔ 1)
Calculate |S’|
Estimate count as #(solutions to  in cell) x #cells
|S| ≈ |S’| x (# cells)

SLIDE 49

Monte Carlo Sampling

“Dart Throwing”

DNF

Counting

Complexity:

𝑃(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

[Karp et al.,

’89]

Hashing Based

“Pie Slicing”
DNF-Counting
Complexity: 𝑃(𝑛 ⋅ 𝑜3 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

‘ApproxMC’ [Chakraborty et al., ’16]

Comparison of Approaches

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 49

where m = #cubes n = #vars

SLIDE 50

Monte Carlo Sampling

“Dart Throwing”
DNF-Counting
Complexity: 𝑃(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

[Karp et al., ’89]
CNF-Counting
Does not scale well

Hashing Based

“Pie Slicing”
DNF-Counting
Complexity: 𝑃(𝑛 ⋅ 𝑜3 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

‘ApproxMC’ [Chakraborty et al., ’16]
CNF-Counting
Very successful in practice

Comparison of Approaches

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 50

where m = #cubes n = #vars

SLIDE 51

Power of Hashing?

Can
ApproxMC be made competitive with state-of-the-art?

Motivating Questions

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 51

SLIDE 52

Better Hashing techniques for Approximate DNF-Counting

Row Echelon Hash Family

Symbolic Hashing
Stochastic Cell
Counting

Contributions

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 52

SLIDE 53

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 53

Row Echelon Hash Family

SLIDE 54

Boolean Formulas with only XOR

⊕
2-Universal

Sampling procedure:

Pick each variable with probability
0.5

XOR picked variables

XOR Family of Hash Functions

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 54

SLIDE 55

Sample each constraint at random (n=

7)
Q1 = X1 ⊕ X4 ⊕ X5
Q2 = X3 ⊕ X4 ⊕ X6 ⊕ X7
…
QL = X2 ⊕ X4 ⊕ X6

XOR Family: Hash Matrix

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 55

SLIDE 56

Sample each constraint at random (n=

7)
Q1 = X1 ⊕ X4 ⊕ X5
Q2 = X3 ⊕ X4 ⊕ X6 ⊕ X7
…
QL = X2 ⊕ X4 ⊕ X6

Equivalent to sampling a

0/1 matrix of dimension L x 7

X1 ⊕ X4 ⊕ X5 1 0 0 1 1 0 0 ∧ X3 ⊕ X4 ⊕ X6 ⊕ X7 ⇒ 0 0 1 1 0 1 1 ∧ .. … ∧ X2 ⊕ X4 ⊕ X6 0 1 0 1 0 1 0

XOR Family: Hash Matrix

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 56

L N=7

SLIDE 57

Sample a hash matrix

1.

Q1 = X1 ⊕ X4 ⊕ X5
Q2 = X3 ⊕ X4 ⊕ X6 ⊕ X7
…
QL = X2 ⊕ X4 ⊕ X6

Pick a cell (L bits) at random

2.

10

…1

Count solutions to

3.

 in cell

’ =  ∧ Q where Q = (Q1 ⇔ 1) ∧ (Q2 ⇔ 0) ∧ .. (QL ⇔ 1)

Need to find |S

’|

XOR Family: Counting using XORs

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 57

SLIDE 58

1.

Sample a hash matrix

Q1 = X1 ⊕ X4 ⊕ X5
Q2 = X3 ⊕ X4 ⊕ X6 ⊕ X7
…
QL = X2 ⊕ X4 ⊕ X6

2.

Pick a cell (L bits) at random

10…1

3.

Count solutions to  in cell

’ =  ∧ Q where Q = (Q1 ⇔ 1) ∧ (Q2 ⇔ 0) ∧ .. (QL ⇔ 1)
Need to find |S’| : Enumerate solutions to Q and check if satisfy 

XOR Family: Counting using XORs

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 58

SLIDE 59

Constraints Cell Hash Matrix Cell Vector X1 ⊕ X4 ⊕ X5 = 1 1 0 0 1 1 0 0 X1 1 ∧ X3 ⊕ X4 ⊕ X6 ⊕ X7 = 0 ⇒ 0 0 1 1 0 1 1 X2 = 0 ∧ … .. … .. .. ∧ X2 ⊕ X4 ⊕ X6 = 1 0 1 0 1 0 1 0 X7 1

Q is a system of linear equations (mod

2)

Simplify using Gaussian Elimination!

XOR Family: Enumerating Solutions

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 59

SLIDE 60

1

1 . . 1 1 1 . . 1 1 . . 1 . . . . . . . . . . 1 1 𝑌1 𝑌2 . . 𝑌𝑜′ = 1 1 . . . . 1

𝐻𝑏𝑣𝑡𝑡𝑗𝑏𝑜 𝐹𝑚𝑗𝑛𝑗.

1 . . 1 1 1 . . 1 1 1 . . 1 . . . . . . . . . . . . . . 𝑌1 𝑌2 . . 𝑌𝑜′ = 1 . . . . 1

Row Echelon Form:

First block is the identity matrix
Zero rows after non
zero rows

XOR Hash: Gaussian Elimination

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 60

Row Echelon Form

SLIDE 61

1

1 . . 1 1 1 . . 1 1 . . 1 . . . . . . . . . . 1 1 𝑌1 𝑌2 . . 𝑌𝑜′ = 1 1 . . . . 1

𝐻𝑏𝑣𝑡𝑡𝑗𝑏𝑜 𝐹𝑚𝑗𝑛𝑗.

1 . . 1 1 1 . . 1 1 1 . . 1 . . . . . . . . . . . . . . 𝑌1 𝑌2 . . 𝑌𝑜′ = 1 . . . . 1

Row Echelon Form:

First block is the identity matrix
Zero rows after non
zero rows

Drawback: Gaussian Elimination is O(n

3)

ApproxMC

Complexity 𝑃(𝑛 ⋅ 𝑜3 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

XOR Hash: Gaussian Elimination

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 61

Row Echelon Form

SLIDE 62

1

0. .

1 1 1

0. .

1 1

1. .

. . . . . . . . . . . .

0. .

1 1 1 𝑌1 𝑌2 . . 𝑌𝑜 = 1 . . 1

Sample a hash matrix in Row-Echelon form directly!

Row Echelon Hash Family

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 62

L n

SLIDE 63

1

0. .

𝟐 𝟐 1

0. .

𝟐 𝟐

1. .

𝟏 𝟏 . . . . . . . . . . . .

0. .

1 𝟐 𝟐 𝑌1 𝑌2 . . 𝑌𝑜 = 1 . . 1

Sample a hash matrix in Row-Echelon form directly!

Only sample 𝑀 × (𝑜 − 𝑀) non-identity part

Row Echelon Hash Family

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 63

L n

SLIDE 64

1

0. .

𝟐 𝟐 1

0. .

𝟐 𝟐

1. .

𝟏 𝟏 . . . . . . . . . . . .

0. .

1 𝟐 𝟐 𝑌1 𝑌2 . . 𝑌𝑜 = 1 . . 1

Sample a hash matrix in Row-Echelon form directly!

Only sample 𝑀 × (𝑜 − 𝑀) non-identity part
Eliminates expensive Gaussian Elimination step

Row Echelon Hash Family

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 64

L n

SLIDE 65

1

0. .

𝟐 𝟐 1

0. .

𝟐 𝟐

1. .

𝟏 𝟏 . . . . . . . . . . . .

0. .

1 𝟐 𝟐 𝑌1 𝑌2 . . 𝑌𝑜 = 1 . . 1

Sample a hash matrix in Row-Echelon form directly!

Only sample 𝑀 × (𝑜 − 𝑀) non-identity part
Eliminates expensive Gaussian Elimination step
Is it 2-Universal?

Row Echelon Hash Family

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 65

L n

SLIDE 66

1

0. .

𝟐 𝟐 1

0. .

𝟐 𝟐

1. .

𝟏 𝟏 . . . . . . . . . . . .

0. .

1 𝟐 𝟐 𝑌1 𝑌2 . . 𝑌𝑜 = 1 . . 1

Sample a hash matrix in Row-Echelon form directly!

Only sample 𝑀 × (𝑜 − 𝑀) non-identity part
Eliminates expensive Gaussian Elimination step
Is it 2-Universal? Yes!
Theorem: Row-Echelon Family is 2-Universal

Row Echelon Hash Family

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 66

L n

SLIDE 67

XOR Hash Family O(n

2) space

O(n

3) time to generate (Gaussian Elim.)

O(n

2) time to enumerate one solution

Row Echelon Hash Family O(n

2) space

O(n

2) time (Sampling the matrix)

O(n) time to enumerate one solution

Hash Family Comparison

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 67

where m = #cubes n = #vars

SLIDE 68

XOR Hash Family O(n

2) space

O(n

3) time to generate (Gaussian Elim.)

O(n

2) time to enumerate one solution

Complexity of

ApproxMC (XOR hash)

[Chakraborty et al, ‘16]: 𝑃(𝑛 ⋅ 𝑜3 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

Row Echelon Hash Family O(n

2) space

O(n

2) time (Sampling the matrix)

O(n) time to enumerate one solution

Complexity of
ApproxMC with RE

hash: 𝑃(𝑛 ⋅ 𝑜2 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

Hash Family Comparison

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 68

where m = #cubes n = #vars

SLIDE 69

XOR Hash Family O(n

2) space

O(n

3) time to generate (Gaussian Elim.)

O(n

2) time to enumerate one solution

Complexity of

ApproxMC (XOR hash)

[Chakraborty et al, ‘16]: 𝑃(𝑛 ⋅ 𝑜3 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

Row Echelon Hash Family

O(n2) space
O(n2) time (Sampling the matrix)
O(n) time to enumerate one solution
Complexity of ApproxMC with RE

hash: 𝑃(𝑛 ⋅ 𝑜2 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

Hash Family Comparison

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 69

Karp et al. Algorithm: 𝑃(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕(

1 𝜀) ⋅ 1 𝜁2)

SLIDE 70

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 70

Symbolic Hashing and Stochastic Cell-Counting

SLIDE 71

Time to sample a matrix from RE family:

𝑀 × (𝑜 − 𝑀)

If L

≈ n then sample time = O(n)

Room for Improvement

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 71

SLIDE 72

Time to sample a matrix from RE family: 𝑀 × (𝑜 − 𝑀)
If L ≈ n then sample time = O(n)
L ≈ n when |S| ≈ |U|

Room for Improvement

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 72

SLIDE 73

Time to sample a matrix from RE family: 𝑀 × (𝑜 − 𝑀)
If L ≈ n then sample time = O(n)
L ≈ n when |S| ≈ |U|
But |S| can be much smaller than |U|
Exponentially smaller in worst case

Room for Improvement

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 73

SLIDE 74

Transform U to U

1.

’ [Karp et al., ‘83]

|S

| is polynomially smaller than |U’| in worst case

Hash over the transformed universe U

2.

’ (our contribution)

Symbolic Hashing: Outline

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 74

SLIDE 75

Transform U to U

1.

’ [Karp et al., ‘83]

U = {

0,1}n

Set of all assignments

σ

Includes satisfying and unsatisfying assignments to



Symbolic Hashing: Space Transform

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 75

SLIDE 76

Transform U to U

1.

’ [Karp et al., ‘83]

U’ = { (σ , Ci ) | σ ⊨ Ci }

Multiset of satisfying assignments ( Recall: σ

⊨Ci ⇒ σ ⊨  )

Each

σ occurs at most m = #cubes times

|U

’

| ⩽ m.|S|

Symbolic Hashing: Space Transform

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 76

SLIDE 77

 = (¬X1 ∧ X2 ) ∨ (X2 ∧X3 ) ∨ ( X1 ∧ X3 )

Symbolic Hashing: Space Transform Example

|U| = 8 |S| = 4 |U| ⩽ 2.|S|

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 77

<0,0,1> <1,0,0> <1,1,0> <0,0,0> U

<0,1,1> <1,0,1> <0,1,0> <1,1,1>

S

SLIDE 78

 = (¬X1 ∧ X2 ) ∨ (X2 ∧X3 ) ∨ ( X1 ∧ X3 )

Symbolic Hashing: Space Transform Example

|U| = 8 |S| = 4 |U| ⩽ 2.|S|

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 78

|U’| = 6 |S| = 4 |U’ | ⩽ 1.5 |S| <0,0,1> <1,0,0> <1,1,0> <0,0,0> U <0,1,1>, C2 <0,1,1>, C1 <1,1,1>, C3 <1,0,1>, C3 <1,1,1>, C2 <0,1,0>, C1 U’

<0,1,1> <1,0,1> <0,1,0> <1,1,1>

S

SLIDE 79

Partition U

2.

’ = { (σ , Ci ) | σ ⊨ Ci }

Use hash constraints with

n + log(m) variables
X1 … Xn
Y1 … Ylog(m) - auxiliary variables
“Bit-Blasting”

First n bits select

σ

Last log(m) bits select C

i

Binary representation of the index

i

Symbolic Hashing: Hashing over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 79

where m = #cubes n = #vars

SLIDE 80

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 80

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

SLIDE 81

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 81

1) Add constraints ..

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

SLIDE 82

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 82

1) Add constraints : Q1

SLIDE 83

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 83

1) Add constraints : Q1 , Q2

SLIDE 84

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 84

1) Add constraints : Q1 , Q2 , … , QL

SLIDE 85

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 85

1) Add constraints : Q1 , Q2 , … , QL 2) Pick a cell at random

SLIDE 86

(𝝉𝒌𝟑, 𝑫𝒋𝟑) (𝝉𝒌𝟐, 𝑫𝒋𝟐) (𝝉𝒌𝟓, 𝑫𝒋𝟓) (𝝉𝒌𝟔, 𝑫𝒋𝟔) (𝝉𝒌𝟖, 𝑫𝒋𝟖) (𝝉𝒌𝟔, 𝑫𝒋𝟔) U’ (𝝉𝒌𝟒, 𝑫𝒋𝟒) (𝝉𝒌𝟕, 𝑫𝒋𝟕)

Symbolic Hashing: Counting over U’

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 86

Add constraints : Q 1)

1 , Q2 , … , QL

Pick a cell at random 2) Estimate |S 3)

| ≈ |S’| x (# cells)

Details in paper

SLIDE 87

Theorem:

The estimate |S’| × 2L in U’ provides the required

tolerance 𝜗 and confidence 𝜀 [Chakraborty et al.

‘13, ‘16] imposed tight coupling between

input formula and hash function

Removed this restriction

U’ is never explicitly constructed

Adapted ideas from Monte Carlo to Hashing

Symbolic Hashing: Significance

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 87

SLIDE 88

Idea: Use Monte Carlo within cell!

Sample assignments uniformly from cell
No need to calculate
|S’| exactly

Probabilistic estimate

Y ≈ |S’| adapting [Karp et al., ‘89]

Theorem:

The estimate Y × 2L in U’ provides the required

tolerance 𝜗 and confidence 𝜀

Stochastic Cell Counting

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 88

SLIDE 89

DNF

ApproxMC

Row Echelon Hash Functions

Symbolic Hashing
Stochastic Cell
Counting

Theorem:

DNF-ApproxMC runs in time ෩

𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(

𝟐 𝜺) ⋅ 𝟐 𝜻𝟑)

Complexity

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 89

SLIDE 90

Problem:

Approximate DNF-Counting

Previous work

: Poor time complexity

Our contributions

: Improvements to the hashing framework

Row Echelon Hash Family, Symbolic Hashing, Stochastic Cell Counting

Result
: New complexity ෩

𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(𝟐

𝜺) ⋅ 𝟐 𝜻𝟑)

Significance:

General technique of hashing is as powerful as

specialized technique of Monte Carlo for DNF-Counting!

Summary

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 90

SLIDE 91

Extend techniques to Weighted DNF

Counting

Utilize techniques to improve CNF

Counting

Techniques do not depend on encoding of formula

Sparsity of Row Echelon
Hash functions

Future Work

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SLIDE 92

Theorem

1: Row-Echelon Hash Family is 2-Universal

Theorem

2: The estimate obtained from DNF-ApproxMC

provides the required tolerance 𝜗 and confidence 𝜀 Theorem

3: DNF-ApproxMC runs in time ෩

𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(

𝟐 𝜺) ⋅ 𝟐 𝜻𝟑)

Main Results

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SLIDE 93

Questions?

Thank you

9/5/2018 On Hashing-Based Approaches to Approximate DNF Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 93