1. SAT’07 Conference, Lisbon; 30.05.2007
Towards a Better Understanding of the Functionality of a Conflict-Driven SAT Solver
Nachum Dershowitz¹
Ziyad Hanna²
Alexander Nadel¹,²
1Tel-Aviv University
2Intel
SAT’07 Conference, Lisbon;

2. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time

3. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?

4. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
Decision Tree
Chaff is based on backtrack search DPLL, so the most natural way to explain its behaviour is to construct a decision tree, characterizing its run.

5. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
Implication Graph
Decision Tree
Also, constructing an implication graph is helpfull for explanatory purposes, since it visualizes the BCP process.

6. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
b’ + f’ + c
a’ + f’ + c’
f + g
g + e’ + c
g + e’ + c’
f + e + c’
I’ll explain on an example what difficulties one encounters when trying to construct a decision tree, characterizing a run of a modern SAT solver.
g’ + e + c …

7. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’
f + g
g + e’ + c a
g + e’ + c’
f + e + c’
g’ + e + c …

8. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b
f + g
g + e’ + c a
g + e’ + c’
f + e + c’
g’ + e + c …

9. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b
f + g
g + e’ + c a
g + e’ + c’
f + e + c’
g’ + e + c …

10. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b e
f + g
g + e’ + c a
g + e’ + c’ e’
f + e + c’
g’ + e + c …

11. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b e
f + g
g + e’ + c a f
g + e’ + c’ e’
f + e + c’
g’ + e + c …

12. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b c e
f + g
g + e’ + c a f
g + e’ + c’ e’ c’
f + e + c’
g’ + e + c …

13. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b c e
f + g
g + e’ + c a f
g + e’ + c’ e’ c’
f + e + c’
You have a conflict here, so plain DPLL would just flip the value of e and continue with backtracking.
g’ + e + c
dec. level 2 …

14. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: a’ + b’ + f’ a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b c e
f + g
g + e’ + c a f
g + e’ + c’ e’ c’
f + e + c’
Chaff needs doing a failure-driven assertion (goes down to GRASP). How would one show on the decision tree that the decision variable has changed?
a’ + b’ + f’
g’ + e + c
dec. level 2 …

15. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: a’ + b’ + f’ a
b’ + f’ + c
dec. level 1 1
a’ + f’ + c’ b c e
f + g f 1
g + e’ + c a f
g + e’ + c’ e’ c’
f + e + c’
… to continue, as if f had been a decision variable.
a’ + b’ + f’
g’ + e + c
dec. level 2 …

16. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f 1 a
g + e’ + c
g + e’ + c’
f + e + c’
a’ + b’ + f’
g’ + e + c …

17. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g 1 a
g + e’ + c
g + e’ + c’
f + e + c’
a’ + b’ + f’
g’ + e + c …

18. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g 1 a
g + e’ + c e
g + e’ + c’ 1
f + e + c’ e
a’ + b’ + f’
g’ + e + c …

19. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g 1 a
g + e’ + c c’ e
g + e’ + c’ 1
f + e + c’ e
a’ + b’ + f’ c
g’ + e + c …

20. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b
1UIP: g + e’ f’ e
f + g f g 1 a
g + e’ + c c’ e
g + e’ + c’ 1
f + e + c’ e
a’ + b’ + f’ c
g’ + e + c
g’ + e’ …

21. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g 1 a
g + e’ + c e’ e
g + e’ + c’ 1
f + e + c’
a’ + b’ + f’
g’ + e + c
g’ + e’ …

22. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g c 1 a
g + e’ + c e’ e
g + e’ + c’ c’ 1
f + e + c’
a’ + b’ + f’
g’ + e + c
g’ + e’ …

23. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: f a
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g c 1 a
g + e’ + c e’ e
g + e’ + c’ c’ 1
f + e + c’
We already have a node with f, should it be re-used? Or, should a be replaced with f, but what about the left subtree of a, should we delete it or to leave it as is?
a’ + b’ + f’
g’ + e + c
g’ + e’ f …

24. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: f a f
b’ + f’ + c 1
a’ + f’ + c’ b f’ e
f + g f g c 1 a
g + e’ + c e’ e
g + e’ + c’ c’ 1
f + e + c’
One solution would be replacing a by f as if f had been a decision variable of decision level 1, but then you have two f’s in the same path for two paths.
a’ + b’ + f’
g’ + e + c
g’ + e’ f …

25. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: f a f
b’ + f’ + c 1
a’ + f’ + c’ b f’
f + g g c a
g + e’ + c e’
g + e’ + c’ c’
f + e + c’
So maybe one should just drop the whole subtree, but then we loose information regarding the exploration itself.
g’ + e + c …

26. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q1: How can one explain to a student the behavior of a Chaff-like SAT solver?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP: f a
b’ + f’ + c 1
a’ + f’ + c’ b f f’ e
f + g f g c 1 a
g + e’ + c e’ e
g + e’ + c’ c’ 1
f + e + c’
A better solution in this case is as follows:
a’ + b’ + f’
g’ + e + c
g’ + e’ f …

27. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q2. Why 1UIP scheme is so much faster than AllUIP scheme?

28. What is the Motivation behind this Work?
Answering 2 questions that puzzled me for some time
Q2. Why 1UIP scheme is so much faster than AllUIP scheme?
a’ + b
Implication Graph
Decision Tree
e + f
1UIP of dl 1
1UIP: a’ + b’ + f’ a
b’ + f’ + c 1
a’ + f’ + c’ b c e
f + g
g + e’ + c a f
g + e’ + c’ e’ c’
f + e + c’
AllUIP clauses are always shorter than 1UIP clauses, but
It was shown in [ZMMM’01] and that 1UIP outperforms AllUIP
author = {Lintao Zhang and Conor F. Madigan and Matthew H. Moskewicz and Sharad Malik},
title = {Efficient conflict driven learning in a boolean satisfiability solver},
booktitle = {ICCAD'01},
a’ + b’ + f’
g’ + e + c …
AllUIP: a’ + f’

29. Results: 1 / 3
Characterization of decision tree construction and pruning process by a conflict-driven SAT solver

30. Results: 1 / 3
Characterization of decision tree construction and pruning process by a conflict-driven SAT solver
Each flipped variable is associated with a left decision sub-tree, explored while refuting corresponding left subspace
A way to create this decision tree for a newly flipped variable is shown in the paper
The solver maintains a forest of left decision subtrees, joined into one tree after the last conflict
It is building and pruning the decision forest on-the-fly
Two types of pruning
Forward pruning: conflict clause recording
Backward pruning: conflict-directed backjumping

31. Results: 2 / 3
Explaining why 1UIP is preferable over other schemes

32. Results: 2 / 3 Explaining why 1UIP is preferable over other schemes b
1UIP is preferable over AllUIP
Better forward pruning
Less decisions & flips will be required in future
Fr(b): fraction of all clauses, recorded before flipping b, that contain b (b is a flipped variable)
All such clauses will be satisfied after flipping b; thus they will not contribute to BCP b
Fr(b) = 2/3 1 c d 1
b’ + c’
c + d
b’ + c + d

33. Results: 2 / 3 Explaining why 1UIP is preferable over other schemes
1UIP is preferable over AllUIP
Better forward pruning
Less decisions & flips will be required in future
Fr(b): fraction of all clauses, recorded before flipping b, that contain b (b is a flipped variable)
All such clauses will be satisfied after flipping b
Average Fr(b) for AllUIP is twice as higher as for 1UIP

34. 1UIP vs. AllUIP (Timeout is 4 hours)
Instance
1UIP
AllUIP
Time
10-3 Conf Fr
NSk
10-3 Conf.
4pipe 51
101
0.41
0.19
11930
29985
0.83
0.24
5pipe 50 85
0.40
0.18
t/0
28185
0.84
0.21
8pipe_k
2426
1478
0.37
t/o
13192
0.81
9pipe_k
1493
640
0.20
6678
0.86
0.26
longmult10
485
237
0.13
590
379
longmult11
559
273
0.14
690
471
0.25
rotmul
578
615
0.52
0.16
992
1576
0.27
term1mul
2173
1585
0.55
0.15
2875
3059

35. Results: 2 / 3 Explaining why 1UIP is preferable over other schemes
1UIP is preferable over AllUIP
Better forward pruning
Fr(b): fraction of all clauses, recorded before flipping b, that contain b (b is a flipped variable)
All such clauses will be satisfied after flipping b
Fr(b) for AllUIP is twice as higher than for 1UIP
1UIP is preferable over UIP-2 (2nd UIP of the last decision level)
Better backward pruning: more flipped variables and respective decision tree nodes are skipped
1UIP scheme corrects mistakes on-the-fly

36. 1UIP vs. UIP-2 (Timeout is 4 hours)
Instance
1UIP
UIP-2
Time
10-3 Conf Fr
NSk
10-3 Conf.
4pipe 51
101
0.41
0.19
148
308
0.38
0.14
5pipe 50 85
0.40
0.18
347
562
0.33
8pipe_k
2426
1478
0.37
0.21
t/o
10129
0.26
0.13
9pipe_k
1493
640
0.20
6040
0.27
0.16
longmult10
485
237
513
261
0.34
0.11
longmult11
559
273
756
346
0.35
rotmul
578
615
0.52
1186
1371
0.48
term1mul
2173
1585
0.55
0.15
5213
3750
0.54

37. Results: 3 / 3 Local conflict clause recording a b f’ e g c a e’ e c’
If there exists a flipped variable, dominated by the first UIP
Suppose last such variable f defines a new decision level (like GRASP does)
Record a 1UIP clause wrt new decision level
Normal 1UIP clause is recorded and used as usual
a’ + b
e + f
Implication Graph
Decision Tree
b’ + f’ + c
1UIP: f
LCC: e a f
a’ + f’ + c’ 1 b
f + g f’ e f g c
g + e’ + c 1 a e’
g + e’ + c’ e
f + e + c’ c’ 1
a’ + b’ + f’
g’ + e + c …
g + e’ f

38. LCC: What Problem are We Aiming to Solve?
Asserting clause contains e’. It is consistent with the rightmost path a 1
Asserting clause for flipping b contained b’ It is inconsistent with the left-most path. It is consistent with the right-most path b 1 c e 1 d f
LTree(B) 1
Let P be a path in LTree(a), having N nodes
Leftmost path is inconsistent with N asserting conflict clauses, recorded while exploring LTree(a)
Rightmost path is inconsistent with none of the asserting conflict clauses
If leftmost path of LTree(a) is encountered after backtracking beyond a, it will be immediately pruned away. Rightmost path will not be pruned.
We want to help forward pruning rightmost paths and all the paths that have few left edges.

39. Local Conflict Clause Recording Impact
Family
Default
Default + LCC
Time in sec.
t/o
sat04-maris03-gripper-sat
2238
986
sat04-goldb03_hard_eq_chk
30336 2
15353
sat04-maris03-gripper-unsat
30135 4
17842
velev_fvp_unsat.3.0
18199
10928
velev_fvp_sat.3.0
9041
7155
velev_vliw_sat.2.0
5970
4715
barrel
260
226
velev_pipe_unsat_1.0
15880
13094
velev_pipe_unsat_4.0
17260
14810
longmult
5413
5076
velev_vliw_sat_4.0
5116
6882

40. Thanks for your attention! (See you in poster session)