![The SAT Problem
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 3 / 12
■ Can we satisfy k literals in c clauses (CNF)?
▲ If yes, what are the assignments?
▲ If no, exhaustive search needed? (P = NP)
■ Problem is NP-complete (Cook ’71)[1]
■ Example:
(¬A ∨ B ∨ ¬C)∧
(¬A ∨ ¬B ∨ D) ∧ (¬D)](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-5-320.jpg)
![The SAT Problem
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 3 / 12
■ Can we satisfy k literals in c clauses (CNF)?
▲ If yes, what are the assignments?
▲ If no, exhaustive search needed? (P = NP)
■ Problem is NP-complete (Cook ’71)[1]
■ Example:
(¬A ∨ B ∨ ¬C)∧
(¬A ∨ ¬B ∨ D) ∧ (¬D)
Some solutions:
A=F, B=T, C=F, D=F
A=F, B=T, C=T, D=F
A=T, B=F, C=F, D=F](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-6-320.jpg)
![DP (DLL)
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 6 / 12
Algorithm 1 DP Algorithm[2] [3]
1: procedure DP(c) ⊲ Derive ∅ or F from CNF(c)
2: while c = {F} and c = ∅ do
3: δ ← SetAtomicLiterals(c)
4: if |δ| ≤ 1 return c
5: δ ← SetPureLiterals(c)
6: if δ = c then loop ⊲ Propagate pure literals
7: c ← SetV ar(x ∈ δ, δ)
8: end while
9: return c
10: end procedure](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-11-320.jpg)
![Visualizations
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 9 / 12
Figure 1: Literal-Based Renderings[4]](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-14-320.jpg)
![Visualizations (2)
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 10 / 12
Figure 2: Literal-Based Renderings[4]](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-15-320.jpg)
![Visualizations (3)
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 11 / 12
Figure 3: Clause-Based Renderings[5]](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-16-320.jpg)
![References
Some Terminology
The SAT Problem
Satisfiable?
More Difficult?
DP (DLL)
Stochastic Search
State Space
Visualizations
Visualizations (2)
Visualizations (3)
References
Visualization November 13, 2013 Dunn – 12 / 12
[1] Cook, S. A. (1971). “The complexity of theorem-proving
procedures”. Proceedings of the 3rd Annual ACM
Symposium on Theory of Computing: 151-158.
[2] Davis, Martin and Putnam, Hilary (1960). “A Computing
Procedure for Quantification Theory”. Journal of the
ACM 7 (3): 201-215.
[3] Davis, M.; Logemann, G.; Loveland, D. (1962). “A machine
program for theorem-proving”. Communications of the
ACM 5 (7): 394-397.
[4] Sinz, Carsten. “Visualizing the internal structure of SAT
instances (preliminary report)”. SAT. Citeseer, 2004.
[5] Sinz, Carsten. “Visualizing SAT instances and runs of the
DPLL algorithm”. Journal of Automated Reasoning (Vol.
39 Num. 2): 219-243. Springer, 2007.](https://image.slidesharecdn.com/satvis-150402193828-conversion-gate01/85/SAT-Visualization-17-320.jpg)
SAT Visualization
- 1. The SAT Problem Structure through Visualization Stephen Dunn November 13, 2013
- 2. Some Terminology Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 2 / 12 ■ A variable is a Boolean variable which can be assigned either T or F ■ A literal is a variable which may be negated (¬x vs x)
- 3. Some Terminology Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 2 / 12 ■ A variable is a Boolean variable which can be assigned either T or F ■ A literal is a variable which may be negated (¬x vs x) ■ We write OR as ∨, AND as ∧ ■ Example: If we let x = T, then ¬x = F ■ Example: (A ∨ ¬B ∨ C) ∧ (A ∨ B ∨ ¬C) ∧ (¬A ∨ ¬B ∨ C)
- 4. The SAT Problem Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 3 / 12 ■ Can we satisfy k literals in c clauses (CNF)?
- 5. The SAT Problem Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 3 / 12 ■ Can we satisfy k literals in c clauses (CNF)? ▲ If yes, what are the assignments? ▲ If no, exhaustive search needed? (P = NP) ■ Problem is NP-complete (Cook ’71)[1] ■ Example: (¬A ∨ B ∨ ¬C)∧ (¬A ∨ ¬B ∨ D) ∧ (¬D)
- 6. The SAT Problem Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 3 / 12 ■ Can we satisfy k literals in c clauses (CNF)? ▲ If yes, what are the assignments? ▲ If no, exhaustive search needed? (P = NP) ■ Problem is NP-complete (Cook ’71)[1] ■ Example: (¬A ∨ B ∨ ¬C)∧ (¬A ∨ ¬B ∨ D) ∧ (¬D) Some solutions: A=F, B=T, C=F, D=F A=F, B=T, C=T, D=F A=T, B=F, C=F, D=F
- 7. Satisfiable? Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 4 / 12 (A ∨ B ∨ C)∧ (¬A ∨ B ∨ C)∧ (A ∨ ¬B ∨ C)∧ (¬A ∨ ¬B ∨ C)∧ (A ∨ B ∨ ¬C)∧ (¬A ∨ B ∨ ¬C)∧ (A ∨ ¬B ∨ ¬C)∧ (¬A ∨ B ∨ D)∧ (¬A ∨ ¬B ∨ ¬D)
- 8. Satisfiable? Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 4 / 12 (A ∨ B ∨ C)∧ (¬A ∨ B ∨ C)∧ (A ∨ ¬B ∨ C)∧ (¬A ∨ ¬B ∨ C)∧ (A ∨ B ∨ ¬C)∧ (¬A ∨ B ∨ ¬C)∧ (A ∨ ¬B ∨ ¬C)∧ (¬A ∨ B ∨ D)∧ (¬A ∨ ¬B ∨ ¬D) Yes! A = T, B = T, C = T, D = F
- 9. More Difficult? Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 5 / 12 (A ∨ B ∨ C)∧ (¬A ∨ B ∨ C)∧ (A ∨ ¬B ∨ C)∧ (¬A ∨ ¬B ∨ C)∧ (A ∨ B ∨ ¬C)∧ (¬A ∨ B ∨ ¬C)∧ (A ∨ ¬B ∨ ¬C)∧ (¬A ∨ ¬B ∨ D)∧ ← B is now negated (¬A ∨ ¬B ∨ ¬D)
- 10. More Difficult? Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 5 / 12 (A ∨ B ∨ C)∧ (¬A ∨ B ∨ C)∧ (A ∨ ¬B ∨ C)∧ (¬A ∨ ¬B ∨ C)∧ (A ∨ B ∨ ¬C)∧ (¬A ∨ B ∨ ¬C)∧ (A ∨ ¬B ∨ ¬C)∧ (¬A ∨ ¬B ∨ D)∧ ← B is now negated (¬A ∨ ¬B ∨ ¬D) No solution!
- 11. DP (DLL) Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 6 / 12 Algorithm 1 DP Algorithm[2] [3] 1: procedure DP(c) ⊲ Derive ∅ or F from CNF(c) 2: while c = {F} and c = ∅ do 3: δ ← SetAtomicLiterals(c) 4: if |δ| ≤ 1 return c 5: δ ← SetPureLiterals(c) 6: if δ = c then loop ⊲ Propagate pure literals 7: c ← SetV ar(x ∈ δ, δ) 8: end while 9: return c 10: end procedure
- 12. Stochastic Search Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 7 / 12 ■ Randomly assign variables ■ Improve working solution ▲ Escape local minima ▲ Avoid full state space ▲ Determine unsat. without exhaustive search ∅ x1 x2 x3 ¬x3 ¬x2 . . . ¬x1 . . .
- 13. State Space Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 8 / 12
- 14. Visualizations Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 9 / 12 Figure 1: Literal-Based Renderings[4]
- 15. Visualizations (2) Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 10 / 12 Figure 2: Literal-Based Renderings[4]
- 16. Visualizations (3) Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 11 / 12 Figure 3: Clause-Based Renderings[5]
- 17. References Some Terminology The SAT Problem Satisfiable? More Difficult? DP (DLL) Stochastic Search State Space Visualizations Visualizations (2) Visualizations (3) References Visualization November 13, 2013 Dunn – 12 / 12 [1] Cook, S. A. (1971). “The complexity of theorem-proving procedures”. Proceedings of the 3rd Annual ACM Symposium on Theory of Computing: 151-158. [2] Davis, Martin and Putnam, Hilary (1960). “A Computing Procedure for Quantification Theory”. Journal of the ACM 7 (3): 201-215. [3] Davis, M.; Logemann, G.; Loveland, D. (1962). “A machine program for theorem-proving”. Communications of the ACM 5 (7): 394-397. [4] Sinz, Carsten. “Visualizing the internal structure of SAT instances (preliminary report)”. SAT. Citeseer, 2004. [5] Sinz, Carsten. “Visualizing SAT instances and runs of the DPLL algorithm”. Journal of Automated Reasoning (Vol. 39 Num. 2): 219-243. Springer, 2007.