FPRASs for DNF-Counting Kuldeep S. Meel 1 , Aditya A. Shrotri 2 , - PowerPoint PPT Presentation

Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel 1 , Aditya A. Shrotri 2 , Moshe Y. Vardi 2 1 School of Computing, 2 Department of Computer Science, National University of Singapore Rice University Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 1

Beyond SAT: #SAT • SAT: Does there exist a satisfying assignment? • #SAT: How many satisfying assignments ? • Complexity: #P-Complete (contains entire polynomial hierarchy) • In Practice: Harder than SAT Measuring Probabilistic Information Inference Leakage #SAT Network Probabilistic Reliability Databases Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 2

The Disjunctive Normal Form • F = ¬𝑌 1 ∧ 𝑌 2 ∨ 𝑌 2 ∧ 𝑌 3 ∧ 𝑌 4 ∨ (𝑌 1 ∧ 𝑌 3 ∧ ¬𝑌 5 ) • Disjunction of Cubes • DNF-SAT is in P • #DNF is #P- Complete [Valiant, ’79] • Need to Approximate! Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 3

Approximate DNF-Counting Input: DNF Formula F Tolerance ε 0 < ε < 1 Confidence δ 0 < δ < 1 Output: Approximate Count C s.t. Pr [# 𝐆 ⋅ (1- ε) < C < # 𝐆 ⋅ (1+ε) ] > 1 - δ Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 4

Fully Polynomial Randomized Approximation Scheme Input: DNF Formula F Tolerance ε 0 < ε < 1 Confidence δ 0 < δ < 1 Output: Approximate Count C s.t. Pr [# 𝐆 ⋅ (1- ε) < C < # 𝐆 ⋅ (1+ε) ] > 1 - δ Challenge: Design a poly(m, n, 1 𝜗 , log( 1 𝜀 ) ) time algorithm where m = #cubes n = #vars Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 5

Paradigms for #DNF FPRAS • Monte Carlo Sampling 𝟐 𝟐 • Best complexity: 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉( 𝜺 ) ⋅ 𝜻 𝟑 ) • Hashing – based 𝟐 𝟐 • Best complexity: ෩ 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉( 𝜺 ) ⋅ 𝜻 𝟑 ) Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 6

Motivation Can hashing – based approach have same complexity as Monte 1. Carlo? How do the various algorithms compare empirically? 2. Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 7

Our Contribution Can hashing – based approach have same complexity as Monte 1. Carlo? How do the various algorithms compare empirically? 2. In this work: • Improved complexity of hashing – based algorithm • Improves practical performance • First comprehensive empirical evaluation of FPRASs for #DNF • Much more nuanced than complexity analysis alone Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 8

Method 1: Monte Carlo Sampling U • Draw independent samples from U F Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 9

Method 1: Monte Carlo Sampling U U • Draw independent samples from U F F Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 10

Method 1: Monte Carlo Sampling U U U • Draw independent samples from U • Count samples that satisfy F F F F Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 11

Method 1: Monte Carlo Sampling U U U • Draw independent samples from U • Count samples that satisfy F F F F #• • #F ≈ #•+#• * |U| Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 12

Method 1: Monte Carlo Sampling U U U • Draw independent samples from U • Count samples that satisfy F F F F #• • #F ≈ #•+#• * |U| #𝐺 • Requirement: 𝑉 is large • Solution density plays crucial role Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 13

Monte Carlo FPRAS • KL Counter [Karp, Luby , ‘83] • Insight: Transform solution space 1 1 • KLM Counter [Karp et al., ‘89] 𝑃(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕( 𝜀 ) ⋅ 𝜁 2 ) • Insight: Sample using geometric distribution • Vazirani Counter [Vazirani , ’13] • Insight: Low variance ⟹ fewer samples Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 14

Method 2: Hashing-Based • Partition U into small cells U U • Count solutions in a random cell ‘ C cell ’ F F • #F ≈ C cell * no. of cells On Hashing-Based Approaches to Approximate DNF Counting 9/5/2018 15

Method 2: Hashing-Based • Partition U into small cells U U • Count solutions in a random cell ‘ C cell ‘ F F • #F ≈ C cell * no. of cells • Requirements: • Roughly equal solutions in each cell • Right size of cell On Hashing-Based Approaches to Approximate DNF Counting 9/5/2018 16

Method 2: Hashing-Based • Requirement: Roughly Equal Solutions in each cell • Solution: Hash Functions • Conjunction of random XOR formulas • h = 𝐼 1 ∧ 𝐼 2 ∧ … ∧ 𝐼 𝑗 • h: 2 𝑜 → 2 𝑗 Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 17

Method 2: Hashing-Based • Requirement: Right size cell • Solution: Count up to threshold 𝑈 • 𝑈 depends on 𝜁 and 𝜀 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 ⟹ cell too large • • Increase no. of constraints 𝑗 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 18

Forward Search • Requirement: Right size partition • Solution: Count up to threshold 𝑈 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 𝒋 0 𝑫 𝒅𝒇𝒎𝒎 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 Cell size Too large Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 19

Forward Search • Requirement: Right size partition • Solution: Count up to threshold 𝑈 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 𝒋 0 1 𝑫 𝒅𝒇𝒎𝒎 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 Cell size Too large Too large Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 20

Forward Search • Requirement: Right size partition • Solution: Count up to threshold 𝑈 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 𝒋 0 1 2 𝑫 𝒅𝒇𝒎𝒎 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 Cell size Too large Too large Too large Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 21

Forward Search • Requirement: Right size partition • Solution: Count up to threshold 𝑈 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 𝒋 0 1 2 … 𝑚 − 1 𝑚 𝑫 𝒅𝒇𝒎𝒎 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 … 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 Cell size Too large Too large Too large … Too large Perfect • #F ≈ C cell * 2 𝑚 Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 22

Hashing-Based FPRAS DNFApproxMC [Chakraborty et al., ‘16] 1. • Insight: Satisfiability of partitioned DNF formulas is polytime 𝜀 ) ⋅ 1 1 SymbolicDNFApproxMC [Meel et al., ‘17] ෨ 𝑃(𝑛 ⋅ 𝑜 ⋅ 𝑚𝑝𝑕( 𝜁 2 ) 2. • Insight: Symbolic hash constraints applied to transformed space Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 23

Forward Search • Requirement: Right size partition • Solution: Count up to threshold 𝑈 • Find 𝑗 such that 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 𝒋 0 1 2 … 𝑚 − 1 𝑚 𝑫 𝒅𝒇𝒎𝒎 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 … 𝐷 𝑑𝑓𝑚𝑚 > 𝑈 𝐷 𝑑𝑓𝑚𝑚 ≤ 𝑈 Cell size Too large Too large Too large … Too large Perfect • #F ≈ C cell * 2 𝑚 Drawback: Re-enumerates solutions multiple times Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting Kuldeep S. Meel, Aditya A. Shrotri, Moshe Y. Vardi 9/5/2018 24