1. Width-Based Restart Policies for Clause-Learning Satisfiability Solvers
Knot Pipatsrisawat and Adnan Darwiche
Computer Science Department UCLA
SAT’09

2. Outline Modern SAT solvers & resolution proofs
Existing restart policies (size-based)
Width-based restart policies
Hybrid restart policies
Experimental results

3. Modern SAT solvers Decisions + unit propagation Conflict analysis
Derive conflict clauses by resolving existing clauses together
For UNSAT, solver can produce a refutation proof upon termination

4. Example (a+b+c) (b+c+d) (c+~d) (y+~w) (x+y+w) (x+~y+z) (~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)

5. Example a = f (a+b+c) (b+c+d) (c+~d) (y+~w) (x+y+w) (x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
a = f

6. Example a = f b = f c = t, d= t/f (conflict) (a+b+c) (b+c+d) (c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
a = f
b = f
c = t, d= t/f (conflict)

7. Example a = f b = f c = t, d= t/f (conflict) (a+b+c) (b+c+d) (c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(a+b)
a = f
b = f
c = t, d= t/f (conflict)

8. Example a = f b = t (a+b+c) (b+c+d) (c+~d) (y+~w) (x+y+w) (x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(a+b)
a = f
b = t

9. Example a = f b = t c = f d = f (a+b+c) (b+c+d) (c+~d) (y+~w) (x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(a+b)
a = f
b = t
c = f
d = f

10. Example a = f b = t c = f d = f x = f (a+b+c) (b+c+d) (c+~d) (y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(a+b)
a = f
b = t
c = f
d = f
x = f

11. Example a = f b = t c = f d = f x = f y = f w = t/f (conflict) (a+b+c)
(b+c+d)
(c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(a+b)
a = f
b = t
c = f
d = f
x = f
y = f
w = t/f (conflict)

12. Example a = f b = t c = f d = f x = f y = f w = t/f (conflict) (a+b+c)
(b+c+d)
(c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(x+y)
(a+b)
a = f
b = t
c = f
d = f
x = f
y = f
w = t/f (conflict)

13. Example a = f b = t c = f d = f x = f y = t, z = t, u = t/f (conflict)
(a+b+c)
(b+c+d)
(c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(b+c)
(x+y)
(a+b)
a = f
b = t
c = f
d = f
x = f
y = t, z = t, u = t/f (conflict)

14. Example a = f b = t c = f d = f x = f y = t, z = t, u = t/f (conflict)
(a+b+c)
(b+c+d)
(c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(x+~y+~z)
(b+c)
(x+y)
(x+~y)
(a+b)
a = f
b = t
c = f
d = f
x = f
y = t, z = t, u = t/f (conflict)
(x)

15. Example x = t, z = t, u = t, w = t/f (conflict) (a+b+c) (b+c+d) (c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(x+~y+~z)
(b+c)
(x+y)
(x+~y)
(a+b)
x = t, z = t, u = t,
w = t/f (conflict)
(x)

16. Example x = t, z = t, u = t, w = t/f (conflict) (a+b+c) (b+c+d) (c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(x+~y+~z)
(~z+~u)
(b+c)
(x+y)
(x+~y)
(~x+~z)
(a+b)
x = t, z = t, u = t,
w = t/f (conflict)
(x)
(~x)

17. Example x = t, z = t, u = t, w = t/f (conflict) (a+b+c) (b+c+d) (c+~d)
(y+~w)
(x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(x+~y+~z)
(~z+~u)
(b+c)
(x+y)
(x+~y)
(~x+~z)
(a+b)
x = t, z = t, u = t,
w = t/f (conflict)
(x)
(~x)
false

18. Example Refutation proof (a+b+c) (b+c+d) (c+~d) (y+~w) (x+y+w)
(x+~y+z)
(~y+~z+~u)
(x+~z+u)
(~x+z)
(~x+u)
(~z+w)
(~u+~w)
(x+~y+~z)
(~z+~u)
(b+c)
(x+y)
(x+~y)
(~x+~z)
(a+b)
(x)
(~x)
false
Refutation proof

19. Properties of Refutation Proofs
Resolution Proof:
Size = # of clauses in proof
Width = length of largest clause in proof

20. Existing Restart Policies
Arithmetic: threshold += C
zChaff, Berkmin, Siege, Eureka
Geometric: threshold *= C
MiniSat 1.14, 2.0
Inner-outer geometric: like Geo. But keep another threshold
PicoSat
Luby’s: threshold = Li L = x,x,2x,x,x,2x,4x,… where x is Luby’s unit
TiniSat, Rsat, MiniSat (SAT-Race’08)
(see Huang IJCAI-07)

21. Considering Proof Width
Every UNSAT CNF with a short proof MUST have a narrow proof
[Ben-Sasson & Wigderson’01]
If a CNF has proof width k, we can find a proof in nO(k)
Just derive all resolvents of size <= k

22. Width-Based Restart Policy
Depends on the size of conflict clauses
Intuition:
Bias the solver to focus on narrow proofs
Long clauses = bad solver behavior
Restart when a lot of long clauses have been learned
A clause is violating if its size > W
Restart if there are > N violating clauses

23. Width-Based Restarts on CNFs with Small Widths
Grid-pebbling formula (UNSAT)
Running time
Conflicts

24. Width-Based Restarts on CNFs with Small Widths
Grid-pebbling formula (SAT)
Running time
Conflicts

25. Width-Based Restart in Practice
Widths not known up front
Cannot assume they will be small
Must eventually allow (many) long clauses to be learned
Set W to small initial value & gradually increase it using different policies

26. Incrementing Width Threshold
Arithmetic: W += C
Geometric: W *= C
Inner-outer Geo: keep two thresholds
Luby’s: W = x,x,2x,x,x,2x,4x,…

27. Performance on INDUS
175 problems from SAT’07 (INDUS), 1800-second timeout/problem
Policy
Solved
Total
SAT
UNSAT
Size-based, Luby’s
107 49 58
Width-based, arith.
100 46 54
Width-based, geo.
108 45 63
Width-based, inner-outer 90 36
Width-based, Luby’s
103 47 56

28. Combining Size & Width for Restarting
Bias the solver to construct short and narrow proofs
Size-based policy based on Luby’s series was combined with various width-based policies
Width & size thresholds are enforced independently

29. Performance of Hybrid Policies on INDUS
Policy
Solved
Total
SAT
UNSAT
Size-based, Luby’s
107 49 58
Width-based, arith.
100 46 54
Width-based, geo.
108 45 63
Width-based, inner-outer 90 36
Width-based, Luby’s
103 47 56
Size (Luby’s) + Width (arith.)
110 48 62
Size (Luby’s) + Width (geo.)
115 52
Size (Luby’s) + Width (in-out)
106 43
Size (Luby’s) + Width (Luby’s)
114 60
Rsat
Rsat-ws-3
Rsat-ws-1
Rsat-ws-2

30. More Results 427 264 290 279 275 Family Total Rsat Rsat-ws-1 (Geo.)
(Luby’s)
Rsat-ws-3
(Arith.)
SAT-07 INDUS
175
107
115
114
110
SAT-Race’06
100 86 90 88 84
Dlx-iq-unsat-1.0 32 11 18 14 17
Fvp-unsat-1.0,2.0,3.0 26
Liveness-sat-1.0 10 5 7 6
Liveness-unsat-2.0 9 3
Pipe-ooo-1.0,1.1 29 12 13
Pipe-unsat-1.0,1.1 27 16
Vliw-unsat-2.0,4.0 2
427
264
290
279
275

31. Performance Profiles on INDUS problems
SAT
UNSAT

32. Performance Profiles on CRAFTED problems
SAT
UNSAT

33. Conclusions Demonstrated benefits of width-based restart policies
Could perform as well as, if not better than, size-based ones
Initial study on combining width & size for better performance on various problem families

34. Thank you