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 CNF𝒍(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: 𝑌1 ∨ ¬𝑌5 ∨ 𝑌6 ⋀ ¬𝑌1 ∨ 𝑌3 ∨ 𝑌5 is one possible value for CNF3(6,1/3).
• Prob
• blem: Fixing 𝑙 and 𝑠, what is the asymptotic probability that CNF𝑙(𝑜, 𝑠) is

satisfiableas 𝑜 goes to infinity?

2 Combining the k-CNF and XOR Phase-Transitions

k-CNF F Ph Phase e Transit ition

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 Probability of Satisfiability r: 3-CNF Clause Density (#clauses / #variables)

Probability that CNF3 400,𝑠 is satisfiable

3 Combining the k-CNF and XOR Phase-Transitions

k-CNF F Ph Phase e Transit ition

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 Probability of Satisfiability r: 3-CNF Clause Density (#clauses / #variables)

Probability that CNF3 400,𝑠 is satisfiable

4

k-CN CNF Phase-Transition

• n Con
• nject

cture: For every 𝑙 ≥ 2, there is a constant 𝑠

𝑙 > 0 such that:

𝑠

3

Combining the k-CNF and XOR Phase-Transitions

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: 𝑌2⨁𝑌4, 1⨁𝑌1⨁𝑌2⨁𝑌7

• 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.

5

Prob

• blem: 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

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: 𝑌2⨁𝑌4, 1⨁𝑌1⨁𝑌2⨁𝑌7

• 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.

6

Prob

• blem: Fixing 𝑡, what is the asymptotic probability that XOR(𝑜, 𝑡) is satisfiable

as 𝑜 goes to infinity?

lim

𝑜→∞Pr XOR(𝑜, 𝑡) is sat. = ቊ1 if 𝑡 < 1

0 if 𝑡 > 1

Combining the k-CNF and XOR Phase-Transitions

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.

7 Combining the k-CNF and XOR Phase-Transitions

Random k-CNF-XOR Satisfia iabili ility

• Definitio

ion:Let 𝝎𝒍(𝒐,𝒔,𝒕) be a random variable denoting CNF𝑙(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

• blem: Fixing 𝑙, 𝑠, and 𝑡, what is the asymptotic probability that 𝜔𝑙(𝑜, 𝑠, 𝑡) is

satisfiable as 𝑜 goes to infinity?

8 Combining the k-CNF and XOR Phase-Transitions

k-CNF-XOR: Wh What Do Do We Ex Expec ect to See? e?

Probability that 𝜔5 𝑜, 𝑠, 𝑡 = CNF5(𝑜, 𝑠) ∧ XOR(𝑜,𝑡) is satisfiable

9

r: 5-CNF Clause Density s: XOR Clause Density

?

Probability that 𝜔5 100,𝑠, 𝑡 = CNF5(100,𝑠) ∧ XOR(100,𝑡) is satisfiable

10

r: 5-CNF Clause Density s: XOR Clause Density

Theorem 1: 1: The k-CNF-XOR Ph Phas ase-Tran ansition Ex Exists

𝜔𝑙 𝑜, 𝑠,𝑡 = CNF𝑙(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 𝜚𝑙 and constants 𝛽𝑙 ≥ 1 such that random k-CNF-XOR formulas have a phase-transition located at 𝑡 = 𝜚𝑙(𝑠) when r < 𝛽𝑙. For all 𝑡 ≥ 0, and 0 ≤ 𝑠 ≤ 𝛽𝑙 (except for at most countably many 𝑠): What can we say about 𝜚𝑙?

11 Combining the k-CNF and XOR Phase-Transitions

Theo eorem em 2: 2: Loca catin ing the e Ph Phase-Transit itio ion

lim

𝑜→∞ Pr 𝜔𝑙(𝑜, 𝑠, 𝑡) is sat. = 1

What can we say about 𝜚𝑙, the location of the k-CNF-XOR phase-transition?

Thm2: For 𝑙 ≥ 3, we have linear

upper and lower bounds on 𝜚𝑙(𝑠).

12

lim

𝑜→∞ Pr 𝜔𝑙(𝑜, 𝑠, 𝑡) is sat. = 0

r: 5-CNF Clause Density s: XOR Clause Density

Combining the k-CNF and XOR Phase-Transitions

Conclu clusio ion

• There is a phase-transition in the satisfiability of random k-CNF-XOR

formulas at k-CNF clause densities below 𝛽𝑙.

• We have some explicit bounds on the location.

Future Work:

• Conjecture: There is a phase-transition in k-CNF-XOR formulas at all k-CNF

clause densities.

• Conjecture: 𝜚𝑙(𝑠) is linear for k-CNF clause densities below some 𝛽𝑙

∗ > 0.

• How does the runtime of SAT solvers on k-CNF-XOR equations behave near

the phase-transition?

13 Combining the k-CNF and XOR Phase-Transitions

Thanks!

14

r: 5-CNF Clause Density s: XOR 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 Mathematics, 5(1):77–87, 1983.

• [Chakraborty et al. 2013] S. Chakraborty, K. S. Meel, and M. Y. Vardi. A scalable and nearly uniform generator
• f 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.

15 Combining the k-CNF and XOR Phase-Transitions

Runtime Behavio ior at the Transit itio ion

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 Clause Density r (#Clauses/#Variables) Probability of Satisfiability Solve Time

Average satisfiability and solve time of 𝐺

3 200, 200𝑠

16 Combining the k-CNF and XOR Phase-Transitions