SAT Solvers

1. SAT Solvers

2. The SAT Problem • Given a Boolean formula , look for assignmentAfor such that . • A is a solution for . • A partial assignment assigns a subset of . • CNF representation of : • is a conjunction of clauses: • A clause is a disjunction of literals: • A satisfies ↔ A satisfies all its clauses.

3. Boolean Constraint Propagation • Unit Clause: A clause with exactly one unassigned literal, while all the rest are false. • Asserts the value of the unassigned variable. • cl implies and is its antecedent. • a and c are the antecedent variables of • BCP(): Calculates all the possible implications. Returns conflict / no-conflict. a = 0 b = ? c = 1 b = 0

4. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value.

5. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp()

6. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp()

7. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp() • If a conflict occurs • Flip the highest decision variable not yet flipped.

8. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp() • If a conflict occurs • Flip the highest decision variable not yet flipped. • Mark as flipped.

9. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp() • If a conflict occurs • flip the highest decision variable not yet flipped. • Mark as flipped • Run bcp().

10. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp() • If a conflict occurs • flip the highest decision variable not yet flipped. • Mark as flipped • Run bcp().

11. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Choose a decision variable and value. • Run bcp() • If a conflict occurs • flip the highest decision variable not yet flipped. • Mark as flipped • Run bcp().

12. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Termination • No unassigned variables – SAT

13. DPLL: Davis Putnam Logemann Loveland Backtrack Search • Termination • No unassigned variables – SAT • No decision variable to flip – un-SAT

14. The SAT Problem - Resolution • Given a Boolean formula in CNF, for clauses and . • For

15. x3 ¬x5 ¬x1 x12 x7 ¬x8 x19 ¬x6 ¬x3 ¬x9 x4 Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8, ¬x7,¬x3)

16. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

17. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

18. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

19. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

20. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

21. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

22. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

23. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

24. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

25. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

26. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 x19 ¬x3 x4 ¬x1 ¬x6

27. Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 Conflict x19 ¬x3 x4 ¬x1 ¬x6

28. x12 x7 ¬x6 x3 ¬x3 ¬x5 ¬x8 x19 ¬x1 Learning: Cuts ¬x9 Conflict x4

29. ¬x3 ¬x6 x12 x19 ¬x8 ¬x1 Learning: Conflict Clauses Reason Side Conflict Side • x7,¬x5,x15 are the reason for the conflict. • Adding the clause will prevent it in the future. (x5,¬x7,x3) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 Conflict x4

30. Learning: Conflict Clauses X1 0 1 X2 X2 The clause (x2,x3) is created after a conflict 0 1 0 1 X3 X3 X3 X3 0 0 The search tree is pruned accordingly

31. 1 UIP 2 UIP Learning: Implication Graph (x9,¬x5) (x9,¬x8) (x9,x12) (x5,x7) (x5,¬x7,x3) (¬x4,¬x1) (x1,¬x12,x19) (x1,¬x6) (x6,¬x19,x8,¬x7,¬x3) x7 ¬x5 x3 ¬x9 ¬x8 x12 Conflict x19 ¬x3 x4 ¬x1 ¬x6

32. Non –Chronological Backtracking • Backtrack multiple levels instead of one. • Use conflict clause to determine the level • Backtrack to the minimum level where the clause is still asserting. • Emphasis on recent learning. Conflict Clause (x10,¬x7,x2,x9)

33. Learning: Conflict Clauses • Prevent the reason to the conflict. • Consists of the negation to the reason literals. • Prunes the search tree. • Different cuts yield different conflict clauses. • We choose cuts such that: • Conflict clause includes one variable from the top level.  It is a unit clause after backtracking one level. • The new problem is equivalent to the original.