1. The document demonstrates how a CDCL SAT solver works on a small example problem with 9 variables and 7 clauses. 2. It shows the step-by-step deductions made by the solver as it decides variable assignments, deduces implications, and detects a conflict. 3. When a conflict is found, the solver performs conflict analysis to determine the reason for the inconsistency and learns a new clause, which is added to its clause database.

1. How a CDCL SAT
Solver works
Masahiro Sakai
Twitter: @masahiro_sakai

2. Abstract
• Demonstrating how a modern SAT
solver works on a small example.
• Target Algorithm
• Conflict-Driven Clause Learning
(CDCL)
• 1-UIP learning + far-backtracking
• details are omitted efficient data
(decision heuristics,
structure, etc.,)

3. An Example Problem
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

4. Decide Implication Graph
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

5. Decide Implication Graph
x1
x1@1
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

6. Decide Implication Graph
x1
x2 x2@2
x1@1
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

7. Decide Implication Graph
x1 x3@3
x2 x2@2
x1@1
x3
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

8. Decide Implication Graph
x1 x3@3
x2 x2@2
x1@1
x3
x4@4
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

9. Deduce Implication Graph
x1 x3@3
x2 x2@2
x1@1
x3
x4@4
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

10. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3
ω1
x4@4
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

11. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3
ω1
x4@4
x4
ω2
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

12. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3 ω3
ω1
x4@4 x7@4
x4
ω2 ω3
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

13. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3 ω3
ω1
x4@4 x7@4
x4
ω2 ω3 ω4
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

14. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2 ω5
x1@1
x5@4 x9@4
x3 ω3 ω5
ω1
x4@4 x7@4
x4
ω2 ω3 ω4
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

15. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2 ω5
x1@1
x5@4 x9@4
x3 ω3 ω5 ω6
ω1
x4@4 x7@4
x4
ω2 ω3 ω4 ω6
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

16. Deduce Implication Graph
x1 x3@3
x2 ω1 x2@2 ω5
x1@1
x5@4 x9@4
x3 ω3 ω5 ω6
ω1
x4@4 x7@4
x4
ω2 ω3 ω4 ω6
x6@4 x8@4
Clause DB Conflict
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

17. Diagnose
Implication Graph
x3@3
ω1 x2@2 ω5
x1@1
x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

18. Diagnose
Implication Graph
Reason
ω6: x3@3 Side
¬x8 ¬x9
ω1 x2@2 ω5
x1@1
x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

19. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
x1@1
x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

20. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
x1@1
x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

21. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 x1@1
x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

22. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 ω5: x1@1
¬x2 ¬x7 x9 x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

23. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 ω5: x1@1
¬x2 ¬x7 x9 x5@4 x9@4
ω3 ω5 ω6
ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

24. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 ω5: x1@1
¬x2 ¬x7 x9 x5@4 x9@4
ω3 ω5 ω6
¬x2 ¬x7 ω1
x4@4 x7@4
ω2 ω3 ω4 ω6
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

25. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 ω5: x1@1
¬x2 ¬x7 x9 x5@4 x9@4
ω3 ω5 ω6
¬x2 ¬x7 ω1
x4@4 x7@4
It contains one literal
in the current ω3 ω6
ω2 ω4
decision level (i.e. 4).
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

26. Diagnose
Implication Graph
Reason
ω6: ω4: x3@3 Side
¬x8 ¬x9 ¬x7 x8
ω1 x2@2 ω5
¬x7 ¬x9 ω5: x1@1
¬x2 ¬x7 x9 x5@4 x9@4
ω3 ω5 ω6
¬x2 ¬x7 ω1
x4@4 x7@4
It contains one literal
in the current ω3 ω6
ω2 ω4
decision level (i.e. 4).
x6@4 x8@4 Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9 New clause is learnt !

27. Backtrack Implication Graph
Learnt clause:
¬x2 ¬x7 x3@3
ω1 x2@2 ω5
x2 was set at level 2 x1@1
x7 was set at level 4 x5@4 x9@4
ω3 ω5 ω6
Assertion level of this
clause (which is the ω1
second highest level)
x4@4 x7@4
is 2. ω6
ω2 ω3 ω4
Backtrack to level 2! x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

28. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2 ω5
x1@1
x5@4 x9@4
x3 ω3 ω5 ω6
ω1
x4@4 x7@4
x4
ω2 ω3 ω4 ω6
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

29. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2 ω5
x1@1
x5@4 x9@4
x3 ω3 ω5
ω1
x4@4 x7@4
x4
ω2 ω3 ω4
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

30. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3 ω3
ω1
x4@4 x7@4
x4
ω2 ω3 ω4
x6@4 x8@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

31. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3 ω3
ω1
x4@4 x7@4
x4
ω2 ω3
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

32. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3
ω1
x4@4
x4
ω2
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

33. Backtrack Implication Graph
x1 x3@3
x2 ω1 x2@2
x1@1
x5@4
x3
ω1
x4@4
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

34. Backtrack Implication Graph
x1 x3@3
x2 x2@2
x1@1
x3
x4@4
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

35. Backtrack Implication Graph
x1 x3@3
x2 x2@2
x1@1
x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

36. Backtrack Implication Graph
x1
x2 x2@2
x1@1
x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

37. Deduce Implication Graph
x1
x2 x2@2
x1@1
x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

38. Deduce Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1
x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

39. Decide Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1
x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

40. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1
x3 x3
x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

41. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1
x3 x3
x4@4
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

42. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1
x3 x3
x4@4
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

43. Deduce Implication Graph
x1 x3@3
ω8
x2 ω1 x2@2 ¬x7@2
x1@1
x5@4
x3 x3
ω1
x4@4
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

44. Deduce Implication Graph
x1 x3@3
ω8
x2 ω1 x2@2 ¬x7@2
x1@1
x5@4
x3 x3
ω1
x4@4
x4 x4
ω2
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

45. Deduce Implication Graph
x1 x3@3
ω8
x2 ω1 x2@2 ¬x7@2
x1@1
x5@4
x3 x3 ω3 ω3
ω1
x4@4
x4 x4
ω2 ω3
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

46. Deduce Implication Graph
x1 x3@3
ω8
x2 ω1 x2@2 ¬x7@2
x1@1
x5@4
x3 x3 ω3 ω3
ω1
x4@4
x4 x4
ω2 ω3
x6@4
Clause DB Conflict
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

47. Diagnose
Implication Graph
x3@3
ω8
ω1 x2@2 ¬x7@2
x1@1
x5@4 ω3 ω3
ω1
x4@4
ω2 ω3
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

48. Diagnose
Implication Graph
Reason
ω3: x3@3 Side
¬x5 ¬x6 x7 ω8
ω1 x2@2 ¬x7@2
x1@1
x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

49. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
x1@1
x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

50. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
x1@1
x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

51. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 x1@1
x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

52. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 ω1: x1@1
¬x1 ¬x4 x5 x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

53. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 ω1: x1@1
¬x1 ¬x4 x5 x5@4 ω3 ω3
ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

54. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 ω1: x1@1
¬x1 ¬x4 x5 x5@4 ω3 ω3
¬x1 ¬x4 x7 ω1
x4@4
ω2 ω3 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

55. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 ω1: x1@1
¬x1 ¬x4 x5 x5@4 ω3 ω3
¬x1 ¬x4 x7 ω1
x4@4
It contains one literal
in the current ω3
decision level (i.e. 4). ω2 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8
• ω5: ¬x2 ¬x7 x9

56. Diagnose
Implication Graph
Reason
ω3: ω2: x3@3 Side
¬x5 ¬x6 x7 ¬x4 x6 ω8
ω1 x2@2 ¬x7@2
¬x4 ¬x5 x7 ω1: x1@1
¬x1 ¬x4 x5 x5@4 ω3 ω3
¬x1 ¬x4 x7 ω1
x4@4
It contains one literal
in the current ω3
decision level (i.e. 4). ω2 Conflict
x6@4 Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 New clause is learnt !

57. Backtrack Implication Graph
Learnt clause:
¬x1 ¬x4 x7 x3@3
ω8
ω1 x2@2 ¬x7@2
x1 was set at level 1
x1@1
x4 was set at level 4 x5@4
x7 was set at level 2 ω3 ω3
ω1
Assertion level of this x4@4
clause is 2.
ω2 ω3
Backtrack to level 2! x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

58. Backtrack Implication Graph
x1 x3@3
ω8
x2 ω1 x2@2 ¬x7@2
x1@1
x5@4
x3 x3 ω3 ω3
ω1
x4@4
x4 x4
ω2 ω3
x6@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

59. Backtrack Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1
x3 x3
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

60. Deduce Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1
x3 x3
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

61. Deduce Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 ¬x4@2
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

62. Decide Implication Graph
x1
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 ¬x4@2
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

63. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 ¬x4@2
x4 x4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

64. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 ¬x4@2
x4 x4 x5
x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

65. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 ¬x4@2
x4 x4 x5
x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

66. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 ω3
x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

67. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 ω3
x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

68. Decide Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x8@5 ω3
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

69. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x8@5 ω3
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

70. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
ω6 ¬x6@4
x4 x4 x5 x8@5 ¬x9@5 ω3
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

71. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
ω6 ¬x6@4
x4 x4 x5 x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

72. Deduce Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
ω6 ¬x6@4
x4 x4 x5 x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
x8
Clause DB Conflict
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

73. Diagnose
Implication Graph
x3@3
ω8
x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
¬x4@2
ω6 ¬x6@4
x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

74. Diagnose
Implication Graph
ω7: x3@3
¬x8 x9 ω8
x2@2 ¬x7@2
x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
ω6 ¬x6@4
x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

75. Diagnose
Implication Graph
ω7: ω6: x3@3
¬x8 x9 ¬x8 ¬x9 ω8
x2@2 ¬x7@2
x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
ω6 ¬x6@4
x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

76. Diagnose
Implication Graph
ω7: ω6: x3@3
¬x8 x9 ¬x8 ¬x9 ω8
x2@2 ¬x7@2
x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
ω6 ¬x6@4
x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

77. Diagnose
Implication Graph
ω7: ω6: x3@3
¬x8 x9 ¬x8 ¬x9 ω8
x2@2 ¬x7@2
¬x8 x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
ω6 ¬x6@4
x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

78. Diagnose
Implication Graph
ω7: ω6: x3@3
¬x8 x9 ¬x8 ¬x9 ω8
x2@2 ¬x7@2
¬x8 x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
It contains one literal ω6 ¬x6@4
in the current x8@5 ¬x9@5 ω3
decision level (i.e. 5).
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9

79. Diagnose
Implication Graph
ω7: ω6: x3@3
¬x8 x9 ¬x8 ¬x9 ω8
x2@2 ¬x7@2
¬x8 x1@1 ω9
Reason ω9 ω3
Side ¬x4@2
It contains one literal ω6 ¬x6@4
in the current x8@5 ¬x9@5 ω3
decision level (i.e. 5).
ω7 ω7 x5@4
Conflict
Side
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 •New ¬x2 ¬x7 learnt !
ω8: clause is
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

80. Backtrack Implication Graph
Learnt clause:
¬x8 x3@3
ω8
x2@2 ¬x7@2
x8 was set at level 5 x1@1 ω9
ω9 ω3
Assertion level of this ¬x4@2
clause (which is the
second highest level) ω6 ¬x6@4
is 0 by convention. x8@5 ¬x9@5 ω3
Backtrack to level 0 ω7 ω7 x5@4
(root level)
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

81. Backtrack Implication Graph
x1 x3@3
ω8
x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 ¬x4@2
ω6 ¬x6@4
x4 x4 x5 x8@5 ¬x9@5 ω3
ω7 ω7 x5@4
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

82. Backtrack Implication Graph
x1
x2
x3 x3 x3
x4 x4 x5
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

83. Deduce Implication Graph
x1
x2
x3 x3 x3
x4 x4 x5
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

84. Deduce Implication Graph
x1
x2
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

85. Decide Implication Graph
x1
x2
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

86. Decide Implication Graph
x1 x1
x2 x1@1
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

87. Decide Implication Graph
x1 x1
x2 x2 x2@2
x1@1
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

88. Deduce Implication Graph
x1 x1
x2 x2 x2@2
x1@1
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

89. Deduce Implication Graph
x1 x1
ω8
x2 x2 x2@2 ¬x7@2
x1@1
x3 x3 x3
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

90. Deduce Implication Graph
x1 x1
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 ¬x4@2
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

91. Decide Implication Graph
x1 x1
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 ¬x4@2
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

92. Decide Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 x3 ¬x4@2
x4 x4 x5 ¬x8@0
x8
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

93. Decide Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 x3 ¬x4@2
x4 x4 x5 x5 ¬x8@0
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

94. Deduce Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9
x3 x3 x3 x3 ¬x4@2
x4 x4 x5 x5 ¬x8@0
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

95. Deduce Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

96. Decide Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

97. Decide Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

98. Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

99. Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3
All variables are
¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
assigned without
Clause DB
• ω1: •¬x1 ¬x4 x5 ω6: ¬x8 ¬x9
•
•
conflict
ω2:
ω3:
•
•
¬x4
¬x5
x6
¬x6 x7
ω7: ¬x8
ω8: ¬x2
x9
¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

100. Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

101. Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
Clause DB Satisfiable
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

102. Implication Graph
x1 x1 x3@3
ω8
x2 x2 x2@2 ¬x7@2
x1@1 ω9
ω9 ω3
x3 x3 x3 x3 ¬x4@2
¬x6@4
x4 x4 x5 x5 ¬x8@0 ω3
x8 x9 x9@5 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

103. Implication Graph
Found a model
x3@3
ω8
M= x2@2 ¬x7@2
x1@1 ω9
{ x1, x2, x3, ω9 ω3
¬x4@2
¬x4, x5, ¬x6 ¬x6@4
¬x8@0 ω3
¬x7, ¬x8, x9 }
x9@5 x5@4
Clause DB
• ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9
• ω2: ¬x4 x6 • ω7: ¬x8 x9
• ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7
• ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7
• ω5: ¬x2 ¬x7 x9 • ω10: ¬x8

104. fin.

105. References
• J. P. Marques-Silva and K. A. Sakallah,
"GRASP: a search algorithm for
propositional satisfiability," Computers,
IEEE Transactions on, vol. 48, no. 5, pp.
506-521, May 1999.
• Handbook of Satisfiability, A. Biere,
M. Heule, H. Van Maaren, and T. Walsh,
Eds. IOS Press, Feb. 2009.