1. On the Power of Clause-Learning SAT Solvers with Restarts
Knot Pipatsrisawat and Adnan Darwiche
Computer Science Department UCLA
CP-09

2. Boolean Satisfiability (SAT)
Conjunctive Normal Form (CNF)
(AvBvC) (~AvDvE) (~DvF)
(~CvDvGvH) …
clauses

3. CL SAT Solvers and Resolution
Clause-learning SAT solvers
Resolution Refutation proof of 
UNSAT CNF 
 = C1, C2, C3, …, Ci, …, CN-1, false
Clauses from CNF 
Empty clause

4. CL SAT Solvers and Resolution
Clause-learning SAT solvers
Resolution Refutation proof of 
UNSAT problem 
resolution
(YvZvX)
(~XvWvV)
 = C1, C2, C3, …, Ci, …, CN-1, false
(YvZvWvV)
Resolvents of earlier clauses
resolvent
Size of  = || = # of clauses

5. CL SAT Solvers and Resolution
Clause-learning SAT solvers
Resolution Refutation proof
UNSAT problem
Asserting clause learning
Non-chronological backtracking
Restarting
E.g., Chaff, MiniSat, PicoSat, Rsat, …

6. CL SAT Solvers and Resolution
Decision heuristic 2
Decision heuristic 3
Decision heuristic …
Decision heuristic 1
Does CLR contain short proofs for every UNSAT (compared to RES) ?
Clause-learning SAT solvers
UNSAT problems
Restart policy 2
Restart policy …
Restart policy 3
Restart policy 1
RES Proofs
CLR Proofs …
1 2 3 …
1 2 3 …
1 2 3 …

7. Previous Results
[Beame et al, JAIR 04] Yes, if you allow the solver to branch on assigned variables
[Hertel et al, AAAI 08] Yes, if you first transform the problem
[Buss et al, LMCS 08] Yes, if you first transform the problem

8. Our Results
Yes.
For any solvers that learn asserting clauses and restart

9. Clause-Learning SAT Solvers
Decision:
A=true
Implications:
B=true
C=false
D=false
Decision:
E=false
Implications:
F=true
Decision:
G=true
Implications:
H=false
SATISFIABLE
I=false

10. Clause-Learning SAT Solvers
Decision:
A=true
Implications:
B=true
C=false
D=false
Decision:
E=false
Implications:
F=true
Decision:
G=true
Implications:
H=false
CONFLICT

11. Clause-Learning SAT Solvers
Decision:
A=true
Implications:
B=true
C=false
D=false
Some of these are wrong!
(~A v E v ~G)
Decision:
E=false
(~A v D v H)
Implications:
F=true
Decision:
G=true
Implications:
H=false
CONFLICT

12. Clause-Learning SAT Solvers
Decision:
A=true
(~A v D v H)
Implications:
B=true
C=false
D=false
Decision:
E=false
Implications:
F=true
Decision:
G=true
Implications:
H=false
CONFLICT

13. Clause-Learning SAT Solvers
Decision:
A=true
(~A v D v H)
Implications:
B=true
free literal
C=false
D=false
Implications:
H=true
Decision:
E=false
Implications:
F=true

14. Clause-Learning SAT Solvers
Decision:
A=true
Restart = undo all decisions & consequent implications
Implications:
B=true
C=false
D=false
Decision:
E=false
Implications:
F=true
Decision:
G=true
Implications:
H=false

15. Clause-Learning SAT Solvers
Decision:
B=false
Restart = undo all decisions & consequent implications
Implications:
D=true
G=false

16. Clause-Learning Algorithm
Falsified clause:
(AvDvG)
Unit clauses:
(AvDvF)
(Fv~G)
(AvDvE)
(Ev~F)
(AvCvD)
(AvCv~E)
(AvBvC)
(BvCv~D)
Each derivation has at most n intermediate clauses!
Intermediate clauses
Conflict clause

17. What We Will Show Resolution refutation  of CNF 
Given
We will construct
Resolution refutation  of CNF 
CLR refutation ’ of CNF  s.t. |’| = poly(||, ||)

18. What We Will Show Resolution refutation  of CNF 
Given
We will construct
Resolution refutation  of CNF 
CLR refutation ’ of CNF  s.t. |’|  poly(||, ||)
n||

19. Proof Ideas  = C1, C2, C3, …, Ci, …, CN-1, false
Past work simulated the solver to learn every clause in 

20. Proof Ideas  = C1, C2, C3, …, Ci, …, CN-1, false
We will learn only clauses in  with certain properties

21. Clauses CL Solvers Can Learn
CNF 
(XvYvZvW)
X=false
All literals in the clause are falsified …
Y=false …
These assignments clauses an immediate conflict
Z=false …
W=false
CONFLICT

22. Proof Ideas  = C1, C2, C3, …, Ci, …, CN-1, false Ci Cj Ck Cm false
We will learn only clauses in  with certain properties

23. Proof Ideas
 = C1, C2, C3, …, Ci, …, CN-1, false
Ci Cj Ck Cm … false

24. Proof Ideas  = C1, C2, C3, …, Ci, …, CN-1, false
1 , Ci , 2 , Cj , 3 , Ck , 4 , Cm , 5 , …, false
Intermediate clauses
Each i has size at most n

25. Proof Ideas  = C1, C2, C3, …, Ci, …, CN-1, false
’ = 1 , Ci , 2 , Cj , 3 , Ck , 4 , Cm , 5 , …, false
Each i has size at most n
|’|  n ||

26. Clauses We Are After C = ( v l) is 1-empowering C is 1-provable
Unit resolution cannot derive l when ~ is asserted
Asserting ~C results in a conflict
C allows unit resolution to see a new implication
Easy to show that C is implied by the CNF
Solvers can make decisions in order to falsify C
Solvers will run into a conflict after falsifying C

27. How We Derive Them … CNF  C = ( v l) : 1-provable, 1-empowering
Set ~ as decisions
No conflict!
Because C is 1-empowering
l, ~l are not implied
Set ~l as the next decision
CONFLICT
Because C is 1-provable

28. How We Derive Them … CNF  C = ( v l) : 1-provable, 1-empowering
Set ~ as decisions
No conflict!
Because C is 1-empowering
l, ~l are not implied
Set ~l as the next decision
CONFLICT
Because C is 1-provable
C can be derived by the “Decision” learning scheme

29. Putting It All Together
UNSAT CNF 
 = C1, C2, C3, …, Ci, …, CN-1, false Ci
1-provable & 1-empowering

30. Putting It All Together
UNSAT CNF 
 = C1, C2, C3, …, Ci, …, CN-1, false 1 Ci

31. Putting It All Together
UNSAT CNF , Ci
 = C1, C2, C3, …, Ci, …, CN-1, false
1, Ci
Restart before proceeding

32. Putting It All Together
UNSAT CNF , Ci
 = C1, C2, C3, …, Ci, …, CN-1, false
1, Ci Cj

33. Putting It All Together
UNSAT CNF , Ci, Cj
 = C1, C2, C3, …, Ci, …, CN-1, false
1, Ci , 2 , Cj

34. Putting It All Together
UNSAT CNF , Ci, Cj, …
 = C1, C2, C3, …, Ci, …, CN-1, false
’ = 1, Ci ,2,Cj, … , false
CLR proof

35. Conclusions CL SAT solvers have the full power of general resolution!
Appropriate decision & restart heuristics required

36. Open Question & Future Work
Are restarts necessary?
Can we achieve the same result with no or bounded # of restarts?

37. Open Question & Future Work
More attention on heuristics
For guiding resolution as opposed to search
Beyond resolution

38. Thank you!