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### GRASP-an efficient SAT solver

Pankaj Chauhan

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!

15-398: GRASP and Chaff

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!

15-398: GRASP and Chaff

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

15-398: GRASP and Chaff

Outline

- Terminology
- Basic Backtracking Search
- GRASP
- Pointers to future work

Please interrupt me if anything is not clear!

15-398: GRASP and Chaff

Terminology

- CNF formula
- 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 unsatisfyingassignment {(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}

15-398: GRASP and Chaff

Terminology

- Assignment A (contd.)
- |A= X unresolved {(x1,0), (x2,0), (x4,1)}
- An assignment partitions the clause databaseinto three classes
- Satisfied, unsatisfied, unresolved
- Free literals: unassigned literals of a clause
- Unitclause: #free literals = 1

15-398: GRASP and Chaff

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)
- [email protected] xis assigned tovat decision leveld

15-398: GRASP and Chaff

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

15-398: GRASP and Chaff

x1

x1 = [email protected]

x2

x2 = [email protected]

Backtracking Search in Action1 = (x2 x3)

2 = (x1 x4)

3 = (x2 x4)

x1 = [email protected]

x4 = [email protected]

x2 = [email protected]

x3 = [email protected]

x3 = [email protected]

{(x1,1), (x2,0), (x3,1) , (x4,0)}

{(x1,0), (x2,0), (x3,1)}

No backtrack in this example!

15-398: GRASP and Chaff

x1

x1 = [email protected]

x1 = [email protected]

x2

x2 = [email protected]

Backtracking Search in ActionAdd a clause

1 = (x2 x3)

2 = (x1 x4)

3 = (x2 x4)

4 = (x1 x2 x3)

x4 = [email protected]

x2 = [email protected]

x3 = [email protected]

x3 = [email protected]

conflict

{(x1,0), (x2,0), (x3,1)}

15-398: GRASP and Chaff

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!

15-398: GRASP and Chaff

GRASP search template

15-398: GRASP and Chaff

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

15-398: GRASP and Chaff

GRASP Deduction

- Boolean Constraint Propagation using implication graphs

E.g. for the clause = (x y), ify=1, then we must havex=1

- For a variable xoccuring in a clause , assignment 0 to all other literals is calledantecedent assignmentA(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 ofx
- Antecedent assignment of a decision variable is empty

15-398: GRASP and Chaff

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)}

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4

1

4

3

6

conflict

6

3

2

5

2

5

Example Implication GraphCurrent truth assignment: {[email protected] ,[email protected], [email protected], [email protected], [email protected]}

Current decision assignment: {[email protected]}

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)

15-398: GRASP and Chaff

GRASP Deduction Process

15-398: GRASP and Chaff

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

15-398: GRASP and Chaff

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 occurrenceof again

15-398: GRASP and Chaff

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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(x,v(x)) if A(x) =

causesof(x) =

(x) [ causesof(y)]o/w

(y,v(y)) (x)

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

15-398: GRASP and Chaff

C() = xv(x)

(x,v(x)) AC()

Learning (some math)- Deriving conflicting clause C()from the conflicting assignment AC() is straight forward
- For our IG,

AC() = {[email protected], x9 [email protected], x10 = [email protected], x11 = [email protected]}

15-398: GRASP and Chaff

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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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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Continued IG

Decision level

3

C()

8

9

5

’

C()

9

9

C()

7

x1

7

6

Due to C()

’

15-398: GRASP and Chaff

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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Backtracking

Conflict Directed Backtracking contd.

- Backtrack level is given by
- = d-1chronological backtrack
- < d-1non-chronological backtrack

AC(’) = {x9 [email protected], x10 = [email protected], x11 = [email protected], [email protected], [email protected]}

C() = (x9 x10 x11 x12 x13)

= max{(x)|(x,v(x))AC(’)}

15-398: GRASP and Chaff

Procedure Diagnose()

15-398: GRASP and Chaff

Is that all?

- Huge overhead for constraint propagation
- Better decision heuristics
- Better learning, problem specific
- Better engineering!

Chaff

15-398: GRASP and Chaff

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