1. State-of-the-art in SAT solvers
Vibhav Gogate

2. SAT formulas
A set of propositional variables and clauses involving variables
(x1+x2’+x3) and (x2+x1’+x4)
x1,x2, x3 and x4 are variables (True or false)
Literals: Variable and its negation
x1 and x1’
A clause is satisfied if one of the literals is true
x1=true satisfies clause 1
x1=false satisfies clause 2
Solution: An assignment that satisfies all clauses

3. SAT solvers Given 10 minutes of time Started with DPLL (1962)
Able to solve variable problems
Satz (Chu Min Li, 1995)
Able to solve some 1000 variable problems
Chaff (Malik et al., 2001)
Intelligently hacked DPLL , Won the 2004 competition
Able to solve some variable problems
Current state-of-the-art
Minisat and SATELITEGTI (Chalmer’s university, )
Jerusat and Haifasat (Intel Haifa, 2002)
Ace (UCLA, )

4. DPLL Example {p,r},{p,q,r},{p,r} {T,r},{T,q,r},{T,r}
p=T
p=F
{T,r},{T,q,r},{T,r}
{F,r},{F,q,r},{F,r}
SIMPLIFY
SIMPLIFY
{q,r}
{r},{r}
SIMPLIFY {}

5. DPLL Algorithm as seen by SAT solver
While (1) {
if (decide_next_branch()) { //1. Branching
while (deduce()==conflict) { //2. Deducing
blevel=analyze_conflicts() // 3. Learning
if (blevel < 0)
return UNSAT
else backtrack(blevel) // 4. Backtracking }
else RETURN UNSAT;

8. Chaff implementation While (1) {
if (decide_next_branch()) { //1. Branching
while (deduce()==conflict) { //2. Deducing
blevel=analyze_conflicts() // 3. Learning
if (blevel < 0)
return UNSAT
else backtrack(blevel) // 4. Backtracking }
else RETURN UNSAT;
Use conflict-directed backjumping + Learning

9. Learning
Adding information about the instance into the solution process without changing the satisfiability of the problem.
In CNF representation it is accomplished by adding clauses into the clause database
Knowledge of failure may help search in other spaces
Learning is very effective in pruning the search space for structured problems
It is of limited use for random instances
Why? Still an open question

10. Chaff implementation While (1) {
if (decide_next_branch()) { //1. Branching
while (deduce()==conflict) { //2. Deducing
blevel=analyze_conflicts() // 3. Learning
if (blevel < 0)
return UNSAT
else backtrack(blevel) // 4. Backtracking }
else RETURN UNSAT;
Boolean constraint propagation: Main factor

11. Naive Implementation of Deduce or Unit propagation
Check every clause after an assignment is made and reduce it if possible
Repeat if a unit clause is generated (implication)
After backtrack, revert all clauses to their original form as they were before.
Very slow.
A solver would spend 85-90% of the time doing unit propagation
Why not speed it up?

28. Chaff implementation While (1) {
if (decide_next_branch()) { //1. Branching
while (deduce()==conflict) { //2. Deducing
blevel=analyze_conflicts() // 3. Learning
if (blevel < 0)
return UNSAT
else backtrack(blevel) // 4. Backtracking }
else RETURN UNSAT;
Variable ordering heuristics

32. Other issues Clause Deletion Winner of SAT competition 2004.
Learned clauses slows down the bcp and eat up memory
Delete clauses periodically
Various heuristics for this purpose
Winner of SAT competition 2004.

33. But Chaff no longer State of the Art
More hacking
Minisat (2006, winner of SAT RACE 2006)
Based on chaff but a better faster implementation
Some new things like conflict analysis and minimization but basically same as chaff

34. Benchmarks
Random
Crafted
Industrial

40. SAT race 2006

41. SAT race 2006