Co Combining the k-CN CNF and XOR Phase se-Transitions Jeffrey M. - PowerPoint PPT Presentation

Co Combining the k-CN CNF and XOR Phase se-Transitions Jeffrey M. Dudek , Kuldeep S. Meel, & Moshe Y. Vardi Rice University

Random k-CNF F Satis isfia iabil ilit ity [Franco and Paull, 1983] • Definitio ion:Let CNF 𝒍 (𝒐,𝒔) be a random variable denoting a uniformly chosen k- CNF formula with 𝑜 variables and 𝑜𝑠 k-CNF clauses. • 𝒐 : The number of variables. • 𝒍 : The width of every CNF clause. • r : CNF clause density = Ratio of # of CNF clauses to # of variables. • Ex: 𝑌 1 ∨ ¬𝑌 5 ∨ 𝑌 6 ⋀ ¬𝑌 1 ∨ 𝑌 3 ∨ 𝑌 5 is one possible value for CNF 3 (6,1/3) . • Prob oblem: Fixing 𝑙 and 𝑠, what is the asymptotic probability that CNF 𝑙 (𝑜, 𝑠) is satisfiableas 𝑜 goes to infinity? Combining the k-CNF and XOR Phase-Transitions 2

k-CNF F Ph Phase e Transit ition Probability that CNF 3 400,𝑠 is satisfiable 1 Probability of Satisfiability 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 r : 3-CNF Clause Density (#clauses / #variables) Combining the k-CNF and XOR Phase-Transitions 3

k-CNF F Ph Phase e Transit ition Probability that CNF 3 400,𝑠 is satisfiable k-CN CNF Phase-Transition on Con onject cture: 1 For every 𝑙 ≥ 2 , there is a constant Probability of Satisfiability 0.8 𝑠 𝑙 > 0 such that: 0.6 0.4 0.2 𝑠 3 0 0 1 2 3 4 5 6 7 8 r : 3-CNF Clause Density (#clauses / #variables) Combining the k-CNF and XOR Phase-Transitions 4

XOR Ph Phase-Transit itio ion [Creignou and Daudé, 1999] • Definitio ion: An XOR R clause is the exclusive-or of a set of variables, possibly including 1 as well. Ex Ex: 𝑌 2 ⨁𝑌 4 , 1⨁𝑌 1 ⨁𝑌 2 ⨁𝑌 7 • Definitio ion: Let XOR(𝒐,𝒕) be a random variable denoting a uniformly chosen XOR formula with 𝑜 variables and 𝑜𝑡 XOR clauses. • n : The number of variables. • s : XOR clause density = Ratio of # of XOR clauses to # of variables. Prob oblem: Fixing 𝑡 , what is the asymptotic probability that XOR(𝑜, 𝑡) is satisfiable as 𝑜 goes to infinity? Combining the k-CNF and XOR Phase-Transitions 5

XOR Ph Phase-Transit itio ion [Creignou and Daudé, 1999] • Definitio ion: An XOR R clause is the exclusive-or of a set of variables, possibly including 1 as well. Ex Ex: 𝑌 2 ⨁𝑌 4 , 1⨁𝑌 1 ⨁𝑌 2 ⨁𝑌 7 • Definitio ion: Let XOR(𝒐,𝒕) be a random variable denoting a uniformly chosen XOR formula with 𝑜 variables and 𝑜𝑡 XOR clauses. • n : The number of variables. • s : XOR clause density = Ratio of # of XOR clauses to # of variables. Prob oblem: Fixing 𝑡 , what is the asymptotic probability that XOR(𝑜, 𝑡) is satisfiable as 𝑜 goes to infinity? 𝑜→∞ Pr XOR(𝑜, 𝑡) is sat. = ቊ1 if 𝑡 < 1 lim 0 if 𝑡 > 1 Combining the k-CNF and XOR Phase-Transitions 6

Combin ining k-CNF and XOR Together • Motivation: Hashing-based sampling and counting algorithms use formulas with both k-CNF and XOR clauses. [Gomes et al. 2007], [Chakraborty et al. , 2013], [Ermon et al. 2013] • • Definition: A k-CNF-XOR formula is the conjunction of k-CNF and XOR clauses. • Goal : Analyze the “behavior” of k -CNF-XOR formulas. • In this work we analyze the asymptotic satisfiability of random k-CNF-XOR formulas. Combining the k-CNF and XOR Phase-Transitions 7

Random k-CNF-XOR Satisfia iabili ility • Definitio ion:Let 𝝎 𝒍 (𝒐,𝒔,𝒕) be a random variable denoting CNF 𝑙 (𝑜, 𝑠) ∧ XOR(𝑜, 𝑡) • i.e. the conjunction of 𝑜𝑠 random k-CNF clauses and 𝑜𝑡 random XOR clauses. • n : The number of variables. • k : The width of every CNF clause. • r : k-CNF clause density. • s : XOR clause density. Prob oblem: Fixing 𝑙 , 𝑠 , and 𝑡, what is the asymptotic probability that 𝜔 𝑙 (𝑜, 𝑠, 𝑡) is satisfiable as 𝑜 goes to infinity? Combining the k-CNF and XOR Phase-Transitions 8

k-CNF-XOR: Wh What Do Do We Ex Expec ect to See? e? Probability that 𝜔 5 𝑜, 𝑠, 𝑡 = CNF 5 (𝑜, 𝑠) ∧ XOR(𝑜,𝑡) is satisfiable s: XOR Clause Density ? 9 r: 5-CNF Clause Density

Probability that 𝜔 5 100,𝑠, 𝑡 = CNF 5 (100,𝑠) ∧ XOR(100,𝑡) is satisfiable s: XOR Clause Density r: 5-CNF Clause Density 10

Theorem 1: 1: The k-CNF-XOR Ph Phas ase-Tran ansition Ex Exists 𝜔 𝑙 𝑜, 𝑠,𝑡 = CNF 𝑙 (𝑜, 𝑠) ∧ XOR 𝑜,𝑡 is a random variable denoting a uniformly chosen k -CNF-XOR formula over n variables with CNF-density r and XOR-density s . Thm 1: For all 𝑙 ≥ 2 , there are functions 𝜚 𝑙 and constants 𝛽 𝑙 ≥ 1 such that random k-CNF-XOR formulas have a phase-transition located at 𝑡 = 𝜚 𝑙 (𝑠) when r < 𝛽 𝑙 . For all 𝑡 ≥ 0 , and 0 ≤ 𝑠 ≤ 𝛽 𝑙 (except for at most countably many 𝑠 ): What can we say about 𝜚 𝑙 ? Combining the k-CNF and XOR Phase-Transitions 11

Theo eorem em 2: 2: Loca catin ing the e Ph Phase-Transit itio ion 𝑜→∞ Pr 𝜔 𝑙 (𝑜, 𝑠, 𝑡) is sat. = 0 lim What can we say about 𝜚 𝑙 , the location of the k-CNF-XOR phase-transition? s: XOR Clause Density Thm2 : For 𝑙 ≥ 3 , we have linear upper and lower bounds on 𝜚 𝑙 (𝑠) . 𝑜→∞ Pr 𝜔 𝑙 (𝑜, 𝑠, 𝑡) is sat. = 1 lim r: 5-CNF Clause Density Combining the k-CNF and XOR Phase-Transitions 12

Conclu clusio ion formulas at k-CNF clause densities below 𝛽 𝑙 . • There is a phase-transition in the satisfiability of random k-CNF-XOR • We have some explicit bounds on the location. Future Work: ∗ > 0 . • Conjecture: There is a phase-transition in k-CNF-XOR formulas at all k-CNF • Conjecture: 𝜚 𝑙 (𝑠) is linear for k-CNF clause densities below some 𝛽 𝑙 clause densities. • How does the runtime of SAT solvers on k-CNF-XOR equations behave near the phase-transition? Combining the k-CNF and XOR Phase-Transitions 13

Thanks! s: XOR Clause Density 14 r: 5-CNF Clause Density

Citatio ions • [Ermon et al. 2013] S. Ermon, C. P. Gomes, A. Sabharwal, and B. Selman. Taming the curse of dimensionality: Discrete integration by hashing and optimization. In Proc. of ICML , pages 334 – 342, 2013. • [Franco and Paull, 1983] J. Franco and M. Paull. Probabilistic analysis of the Davis – Putnam procedure for solving the satisfiability problem. Discrete Applied Math ematics, 5(1):77 – 87, 1983. • [Chakraborty et al. 2013] S. Chakraborty, K. S. Meel, and M. Y. Vardi. A scalable and nearly uniform generator of SAT witnesses. In Proc. of CAV , pages 608 – 623, 2013. • [Creignou and Daudé, 1999] N. Creignou and H. Daudé. Satisfiability threshold for random xor-cnf formulas. Discrete Applied Mathematics , 9697:41 – 53, 1999. • [Gomes et al. 2007] C.P. Gomes, A. Sabharwal, and B. Selman. Near-Uniform sampling of combinatorial spaces using XOR constraints. In Proc. of NIPS , pages 670 – 676, 2007 • [Goerdt, 1996] A. Goerdt. A threshold for unsatisfiability . Journal of Computer and System Sciences, 53(3):469 – 486, 1996. Combining the k-CNF and XOR Phase-Transitions 15

Runtime Behavio ior at the Transit itio ion Average satisfiability and solve time of 𝐺 3 200, 200𝑠 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 Clause Density r (#Clauses/#Variables) Probability of Satisfiability Solve Time Combining the k-CNF and XOR Phase-Transitions 16