1. GRASP-an efficient SAT solver
Pankaj Chauhan

2. SATisfying assignment!
What is SAT?
Given a propositional formula in CNF, find an assignment to boolean variables that makes the formula true!
E.g.
1 = (x2  x3)
2 = (x1  x4)
3 = (x2  x4)
A = {x1=0, x2=1, x3=0, x4=1}
SATisfying assignment!
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3. What is SAT? Solution 1: Search through all assignments!
n variables  2n possible assignments, explosion!
SAT is a classic NP-Complete problem, solve SAT and P=NP!
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4. Why SAT? Fundamental problem from theoretical point of view
Numerous applications
CAD, VLSI
Optimization
Model Checking and other type of formal verification
AI, planning, automated deduction
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5. Outline Terminology Basic Backtracking Search GRASP
Pointers to future work
Please interrupt me if anything is not clear!
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6. Terminology CNF formula  Assignment A x1,…, xn: n variables
1,…, m: m clauses
Assignment A
Set of (x,v(x)) pairs
|A| < n  partial assignment {(x1,0), (x2,1), (x4,1)}
|A| = n  complete assignment {(x1,0), (x2,1), (x3,0), (x4,1)}
|A= 0  unsatisfying assignment {(x1,1), (x4,1)}
|A= 1  satisfying assignment {(x1,0), (x2,1), (x4,1)}
1 = (x2  x3)
2 = (x1  x4)
3 = (x2  x4)
A = {x1=0, x2=1, x3=0, x4=1}
Manually write down the examples of all types of assignments
For unsat, same as A but with x1=1
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7. Terminology Assignment A (contd.)
|A= X  unresolved {(x1,0), (x2,0), (x4,1)}
An assignment partitions the clause database into three classes
Satisfied, unsatisfied, unresolved
Free literals: unassigned literals of a clause
Unit clause: #free literals = 1
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8. Basic Backtracking Search
Organize the search in the form of a decision tree
Each node is an assignment, called decision assignment
Depth of the node in the decision tree  decision level (x)
 x is assigned to v at decision level d
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9. Basic Backtracking Search
Iterate through:
Make new decision assignments to explore new regions of search space
Infer implied assignments by a deduction process. May lead to unsatisfied clauses, conflict! The assignment is called conflicting assignment.
Conflicting assignments leads to backtrack to discard the search subspace
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10. Backtracking Search in Action
1 = (x2  x3)
2 = (x1  x4)
3 = (x2  x4) x1
x1 = x2
x2 =
x1 =
 x4 =
 x2 =
Say which step is which.
 x3 =
 x3 =
{(x1,1), (x2,0), (x3,1) , (x4,0)}
{(x1,0), (x2,0), (x3,1)}
No backtrack in this example!
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11. Backtracking Search in Action
Add a clause
1 = (x2  x3)
2 = (x1  x4)
3 = (x2  x4)
4 = (x1  x2  x3) x1
x1 =
x1 =
 x4 = x2
x2 =
 x2 =
 x3 =
 x3 =
conflict
{(x1,0), (x2,0), (x3,1)}
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12. Davis-Putnam revisited
Deduction
Decision
Backtrack
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13. GRASP
GRASP is Generalized seaRch Algorithm for the Satisfiability Problem (Silva, Sakallah, ’96)
Features:
Implication graphs for BCP and conflict analysis
Learning of new clauses
Non-chronological backtracking!
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14. GRASP search template 8/2/2019 15-398: GRASP and Chaff
Decide: simple one, choose the one with most # of sat. clauses
Deduce: do BCP
Diagnose: do learning
Erase: erase assignment at current decision level
Search tree is maintained implicitly in procedure call stack
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15. GRASP Decision Heuristics
Procedure decide()
Choose the variable that satisfies the most #clauses == max occurences as unit clauses at current decision level
Other possibilities exist
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16. GRASP Deduction
Boolean Constraint Propagation using implication graphs
E.g. for the clause  = (x  y), if y=1, then we must have x=1
For a variable x occuring in a clause , assignment 0 to all other literals is called antecedent assignment A(x)
E.g. for  = (x  y  z),
A(x) = {(y,0), (z,1)}, A(y) = {(x,0),(z,1)}, A(z) = {(x,0), (y,0)}
Variables directly responsible for forcing the value of x
Antecedent assignment of a decision variable is empty
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17. (x) = max{(y)|(y,v(y))A(x)}
Implication Graphs
Nodes are variable assignments x=v(x) (decision or implied)
Predecessors of x are antecedent assignments A(x)
No predecessors for decision assignments!
Special conflict vertices have A() = assignments to vars in the unsatisfied clause
Decision level for an implied assignment is
(x) = max{(y)|(y,v(y))A(x)}
Tell them it is assigned to each node in the search tree
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18. Example Implication Graph
Current truth assignment:
Current decision assignment:
1 = (x1  x2)
2 = (x1  x3  x9)
3 = (x2  x3  x4)
4 = (x4  x5  x10)
5 = (x4  x6  x11)
6 = (x5  x6)
7 = (x1  x7  x12)
8 = (x1 x8)
9 = (x7  x8   x13) 4 1 3 6 
conflict 2 5
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19. GRASP Deduction Process
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20. GRASP Conflict Analysis
After a conflict arises, analyze the implication graph at current decision level
Add new clauses that would prevent the occurrence of the same conflict in the future  Learning
Determine decision level to backtrack to, might not be the immediate one  Non-chronological backtrack
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21. Learning
Determine the assignment that caused the conflict, negation of this assignment is called conflict induced clause C()
The conjunct of this assignment is necessary condition for 
So adding C() will prevent the occurrence of  again
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22. Learning
Find C() by a backward traversal of the IG, find the roots of the IG in the transitive fanin of 
For our example IG,
C() = (x1  x9  x10  x11)
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23. Learning (some math) For any node of an IG x, partition A(x) into
(x) = {(y,v(y)) A(x)|(y)<(x)}
(x) = {(y,v(y)) A(x)|(y)=(x)}
Conflicting assignment AC() = causesof(), where
(x,v(x)) if A(x) = 
(x)  [  causesof(y) ]o/w
(y,v(y))  (x)
causesof(x) =
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24. Learning (some math)
Deriving conflicting clause C() from the conflicting assignment AC() is straight forward
For our IG,
C() =  xv(x)
(x,v(x))  AC()
AC() = x9 x10 = x11 =
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25. Learning
Unique implication points (UIPs) of an IG also provide conflict clauses
Learning of new clauses increases clause database size
Heuristically delete clauses based on a user parameter
If size of learned clause > parameter, don’t include it
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26. Backtracking Failure driven assertions (FDA):
If C() involves current decision variable, then after addition, it becomes unit clause, so different assignment for the current variable is immediately tried.
In our IG, after erasing the assignment at level 6, C() becomes a unit clause x1
This immediately implies x1=0
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27. Continued IG  ’ x1 ’ Decision level 3 x9=0@1 x8=1@6 C() 89 ’ 5
C() 9 9
C() 7 x1 7 6
Due to C()  ’
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28. Backtracking Conflict Directed Backtracking
Now deriving a new IG after setting x1=0 by FDA, we get another conflict ’
Non-chronological backtrack to decision level 3, because backtracking to any level 5, 4 would generate the same conflict ’
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29. Backtracking Conflict Directed Backtracking contd.
Backtrack level is given by
 = d-1 chronological backtrack
 < d-1 non-chronological backtrack
AC(’) = {x9 x10 = x11 =
C() = (x9  x10  x11   x12   x13)
 = max{(x)|(x,v(x))AC(’)}
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30. Procedure Diagnose()
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31. Is that all? Huge overhead for constraint propagation
Better decision heuristics
Better learning, problem specific
Better engineering!
Chaff
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