Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice - PowerPoint PPT Presentation

Theory and Practice of Efficient Approximate Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice University Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 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 Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 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! Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 3

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 Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 4

Paradigms for #DNF FPRAS U • Monte Carlo Sampling [Karp et al., ‘89] F 𝟐 𝟐 • Complexity: 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉( 𝜺 ) ⋅ 𝜻 𝟑 ) U • Hashing – based [Chakraborty et al., ‘16] F • Complexity: O(𝒏 𝟒 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉( 𝟐 𝟐 𝜺 ) ⋅ 𝜻 𝟑 ) Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 5

Open Questions • What is the power of hashing? • Is hashing as powerful as Monte Carlo? • How do algorithms compare empirically? • No rigorous empirical comparison Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 6

Our Contributions New hashing-based algorithm SymbolicDNFApproxMC 1. 𝟐 𝟐 Complexity 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉( 𝜺 ) ⋅ 𝜻 𝟑 ) 1. • Significance: General technique of hashing is as powerful as specialized technique of Monte Carlo for DNF-Counting! Introduced generic new concepts applicable beyond DNF formulas 2. • Symbolic Hashing • A new highly space and time efficient 2-Universal Hash Family • Stochastic Cell Counting Improved complexity of older hashing algorithm DNFApproxMC by 𝑃(𝑜) 2. First large scale experimental comparison of 5 FPRASs 3. Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 7

Experimental Results • Worst-case complexity not the last word • DNFApproxMC is most robust and solves largest number of benchmarks! • Previous works used Vazirani Counter – not the best choice Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 8

Future Directions • Best complexity possible? • Extend to Weighted DNF-Counting • Hashing-based FPRAS for other problems • Counting Linear Extensions, Perfect Matchings, Knapsacks … • Practically efficient Exact Counting Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 9

Publications “Not All FPRASs are Equal: Demystifying FPRASs for DNF- Counting” Kuldeep S. Meel, Aditya A. Shrotri, and Moshe Y. Vardi Constraints 1-24 (2018) “On Hashing -Based Approaches to Approximate DNF- Counting” Kuldeep S. Meel, Aditya A. Shrotri, and Moshe Y. Vardi FST&TCS (2017) Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 10