‫מבוא לאימות תוכנה‬
‫תרגול ‪11‬‬
‫‪SAT Solvers‬‬
The SAT Problem
Boolean Satisfiability Problem
• Given a Boolean formula (v̅), look for
assignment A for v̅ such that (A(v̅))=true .
– A is a solution for (v̅).
• A partial assignment assigns a subset of v̅.
• CNF representation of (v̅):
– (v̅) is a conjunction of clauses: (v̅)=cl1cl2… 
cln.
– A clause is a disjunction of literals: cli=(lit1…litl).
– A satisfies (v̅) ↔ A satisfies all its clauses.
SAT Usage
•
•
•
•
Verification: symbolic algorithms, CBMC
Automatic theorem proving
Solve any NP problem
Artificial Intelligence: planning, automated
deduction
and many more…
SAT Made Some Progress
100000
10000
Vars
1000
100
10
1
1960
1970
1980
1990
Year
2000
2010
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.
a=0
b=?
c=1
cl = (a  b 
c)
b=0
– cl implies b and is its antecedent.
– a and c are the antecedent variables of b.
• BCP(): Calculates all the possible implications.
Returns conflict / no-conflict.
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
Running on the formula:
¬𝑥6 ∨ ¬𝑥14 ∧ ¬𝑥6 ∨ 𝑥3 ∧
x1 ∨ ¬𝑥3 ∨ 𝑥18 ∧ ¬𝑥18 ∨ 𝑥4 ∧
¬𝑥4 ∨ 𝑥14 ∨ ¬𝑥2 ∧ ¬𝑥8 ∨ ¬𝑥10 ∧
¬𝑥8 ∨ 𝑥10 ∨ 𝑥7 ∧ 𝑥9 ∨ 𝑥5 ∨ 𝑥10 ∧
¬𝑥5 ∨ ¬𝑥3 ∨ ¬𝑥9 ∧ ¬𝑥9 ∨ 𝑥15
∧ ¬𝑥15 ∨ 𝑥14 ∨ ¬𝑥9 ∧
(𝑥8 ∨ 𝑥11 )
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
x6
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
• Run bcp()
x6
¬x14 x3
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
• Run bcp()
x6
¬x14 x3
¬x1
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
• Run bcp()
x6
¬x14 x3
¬x1 x18 x4 ¬x2
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
• Run bcp()
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Choose a decision
variable and value.
• Run bcp()
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8 ¬x10 x7
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.
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8 ¬x10 x7
¬x9
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.
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8 ¬x10 x7
¬x9 x5 ¬x3
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.
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8 ¬x10 x7
x9**
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().
x6
¬x14 x3
¬x1 x18 x4 ¬x2
x8 ¬x10 x7
x9** x15 x14
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().
x6
¬x14 x3
¬x1
x18 x4 ¬x2
¬x8**
x9** x15 x14
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().
x6
¬x14 x3
¬x1
x18 x4 ¬x2
¬x8** x11
DPLL: Davis Putnam Logemann Loveland
Backtrack Search
• Termination
– No unassigned variables – SAT
– No decision variable to flip – un-SAT
The SAT Problem - Resolution
• Given a Boolean formula (v̅) in CNF, for
clauses cl1=(A  v) and cl2=(B  v) .
– resolution(cl1,cl2)=(A  B)
• For c=resolution(c1,c2)
– (v̅)  c
– (v̅)  (v̅)  c
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
Conflict
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: Cuts
x7
¬x5
x3
¬x9
¬x8
x12
Conflict
x19
x4
¬x1
¬x6
¬x3
Learning: Conflict Clauses
Reason Side
¬x5
x3
¬x9
¬x8
x12
Conflict
x19
x4
Conflict Side
x7
¬x1
¬x3
¬x6
• x7, x5, x8,x19, x6 are the reason for the conflict.
• Adding the clause (x5  x7  x6  x19  x8) will
prevent it in the future.
Learning: Conflict Clauses
X1
0
X2
The clause (x2,x3) is
0
created after a conflict
X3
0
1
X2
1
0
X3
X3
1
X3
0
The search tree is
pruned accordingly
Which cut should we choose\ which conflict
clause should we choose?
Learning: Asserting Clause
Asserting Clause: a Conflict Clause with a single
literal at the largest decision level.
– Backtracking (to the right level) makes it a Unit
clause.
Modern solvers only consider Asserting Clauses.
Learning: Implication Graph
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
Conflict
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
¬x3
(x1,¬x12,x19)
(x1,¬x6)
¬x6
(x6,¬x19,x8,¬x7,¬x3)
Learning: UIP
A Unique Implication Point (UIP) is an internal
node in the Implication Graph that all paths
from the decision to the conflict node go
through it.
Learning: UIP
(x9,¬x5)
(x9,¬x8)
x7
¬x5
x3
¬x9
(x9,x12)
(x5,x7)
¬x8
x12
Conflict
(x5,¬x7,x3)
(¬x4,¬x1)
x19
x4
¬x1
1 UIP
(x1,¬x12,x19)
(x1,¬x6)
¬x6
2 UIP
¬x3
(x6,¬x19,x8,¬x7,¬x3)
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.
x8
¬x10 x3 x7 x14
¬x2
x5
¬x4
x21 ¬x12 ¬x15
x19
x18 x32
¬x6
x16 ¬x9 ¬x14
x1
Conflict Clause
(x10,¬x7,x2,x9)
x8
¬x10 x3
¬x2
x5
x7
x1 x9
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.