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1/30 SAT Solver Changki Hong @ PSWLAB
SAT Solver
Daniel Kroening, Ofer Strichman
2/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
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SAT made some progress…
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4/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
5/30 SAT Solver Changki Hong @ PSWLAB
Two main categories of SAT solver Davis-Putnam-Loveland-Logemann (DPLL) frame-
work The solver can be thought of as traversing and backtrack-
ing on a binary tree. Internal nodes represent partial assignments. Leave nodes represent full assignments.
Stochastic search The solver guesses a full assignment, and then, if the
formula is evaluated to FALSE under this assignment, starts to flip values of variables according to some heuristics.
DPLL solvers are considered better in most cases according to the annual competition.
6/30 SAT Solver Changki Hong @ PSWLAB
Status of a clause A clause can be
Satisfied: at least one literal is satisfied Unsatisfied: all literals are assigned but non are
satisfied Unit: all but one literals are assigned but none are
satisfied Unresolved: all other cases
Example: C = (x1 ∨ x2 ∨ x3)x1 x2 x3 C
1 0 Satisfied
0 0 0 Unsatisfied
0 0 Unit
0 Unresolved
7/30 SAT Solver Changki Hong @ PSWLAB
A Basic SAT algorithm
While (true){
if (Decide() == FALSE) return (SAT);
while (BCP() == “conflict”) {
backtrack-level = Analyze_Conflict();
if (backtrack-level < 0) return (UNSAT);
else BackTrack(backtrack-level);
}}
Choose the next variable and value.Return False if all
variables are assigned
Apply repeatedly the unit clause rule.
Return False if reached a conflict
Backtrack until no conflict.
Return False if impossible
8/30 SAT Solver Changki Hong @ PSWLAB
Given in CNF: (x ∨ y ∨ z) ∧ (¬x ∨ y) ∧ (¬y ∨ z) ∧ (¬x ∨ ¬y ∨ ¬z)
Decide()
BCP()
Analyze_Conflict()
A Basic SAT algorithm
(y)∧ (¬y ∨ z ) ∧ (¬y ∨ ¬z )(y ∨ z) ∧ (¬y ∨ z
)
(z) ∧ (¬z )
( ) ( )(y) ∧ (¬y)
( )
( )
( )
( )
x = 1
x = 0
y = 1
y = 0
z = 1 z = 0
z = 1 z = 0 y = 1
y = 0
X
X X X X
9/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
10/30 SAT Solver Changki Hong @ PSWLAB
Boolean Constraints Propagation (BCP) BCP is repeated application of the unit clause
rule until either a conflict is encountered or there are no more implications.
Each assignment is associated with the deci-sion level at which it occurred. notation : x=v@d
x ∈ {0,1} is assigned to v at decision level d
The process of BCP is best illustrated with an implication graph.
11/30 SAT Solver Changki Hong @ PSWLAB
Implication graph Def: An implication graph is a labeled directed
acyclic graph G(V, E), where: V represents the literals of the current partial assignment.
Each node is labeled with the literal that it represents and the decision level at which it entered the partial assignment.
E with E = { (vi, vj) | vi, vj ∈ V, ¬vi ∈ Antecedent(vj) } denotes the set of directed edges where each edge (vi, vj) is labeled with Antecedent(vj). Def: For a given unit clause C with an unassigned literal l,
we say that l is implied by C and that C is the antecedent clause of l, denoted by Antecedent(l).
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5
5 x6=1@6
Implication graphs and conflict clause
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)
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)
Current truth assignment: {x9=0@1 ,x10=0@3, x11=0@3, x12=1@2, x13=1@2}
Current decision assignment: {x1=1@6}
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6
con-flict
x9=0@1
x1=1@6
x10=0@3
x11=0@3
x5=1@6
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4
2
2
x3=1@6
1
x2=1@6
3
3
x4=1@6
We learn the conflict clause 10 = (¬x1 ∨ x9 ∨ x11 ∨ x10 )
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Implication graph, flipped assignment
x1=0@6
x11=0@3
x10=0@3
x9=0@1
x7=1@6
x12=1@2
7
7
x8=1@68
10
10
10 9
9
’
x13=1@2
9
Due to the conflict clause
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)
10 : (x1 x9 x11 x10)
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)
10 : (x1 x9 x11 x10) Another conflict clause:11 = (x13 ∨ x12 ∨ x11 ∨ x10 ∨ x9)
con-flict
14/30 SAT Solver Changki Hong @ PSWLAB
Non-chronological backtracking
Non-chrono-logical backtrack-ing
x1
4
5
6
’
Decision level
Which assignments caused the conflicts ? x9= 0@1
x10= 0@3
x11= 0@3
x12= 1@2
x13= 1@2
Backtrack to DL = 3
3
These assignmentsAre sufficient forCausing a conflict.
15/30 SAT Solver Changki Hong @ PSWLAB
Non-chronological backtracking So the rule is: backtrack to the largest deci-
sion level in the conflict clause. x1 = 0
x2 = 0
x3 = 1
x4 = 0
x5 = 0
x7 = 1
x9 = 0
x6 = 0
...x5 = 1
x9 = 1
x3 = 0
16/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
17/30 SAT Solver Changki Hong @ PSWLAB
Conflict clauses Def: A clause is asserting if the clause con-
tains all value 0 literals; and among them only one is assigned at current decision level.
After backtracking, this clause will become a unit clause and force the literal to assume an-other value, thus bringing the search to a new space.
Modern solvers only consider Asserting Clauses.
18/30 SAT Solver Changki Hong @ PSWLAB
Unique Implication Points (UIP’s) Definition: 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.
The First-UIP is the closest UIP to the conflict.
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con-flict
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UIPUIP
Conflict clauses and Resolution
The Binary-resolution :
Example:
20/30 SAT Solver Changki Hong @ PSWLAB
Conflict clauses and resolution
This function is to return TRUE if and only if cl con-tains the negation of the first UIP as its single literal at the current decision level
21/30 SAT Solver Changki Hong @ PSWLAB
Conflict clauses and resolution
Resolution order : x4, x5, x6, x7Since the c5 contains the negation of the first UIP as its single literal at the current decision level, the stop criterion is met.c5 is asserting clause.
First UIP
22/30 SAT Solver Changki Hong @ PSWLAB
Resolution graph
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10 9
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con-flict
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con-flict
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Resolution Graph
23/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
24/30 SAT Solver Changki Hong @ PSWLAB
Compute for every clause w and every variable l (in each phase):
J(l) :=
Choose a variable l that maximizes J(l). This gives an exponentially higher weight
to literals in shorter clauses.
,
||2l
Decision heuristics - JW
Jeroslow-Wang method
25/30 SAT Solver Changki Hong @ PSWLAB
Choose the assignment that satisfies the largest number of currently unsatisfied clauses Cxp – # unresolved clauses in which x appears Let x be the literal with Cxp
Let y be the literal with Cyp
If Cxp > Cyp choose x, Otherwise choose y
Requires l (#literals) queries for each decision.
DLIS (Dynamic Largest Individual Sum)
Decision heuristics - DLIS
26/30 SAT Solver Changki Hong @ PSWLAB
Decision heuristics – VSIDS
(Implemented in Chaff)
VSIDS (Variable State Independent Decay-ing Sum)
1. Each variable in each polarity has a counter initialized to 0.
2. When a clause is added, the counters are updated.
3. The unassigned variable with the highest counter is chosen.
4. Periodically, all the counters are divided by a constant.
27/30 SAT Solver Changki Hong @ PSWLAB
Decision heuristics – VSIDS (cont’d)
Chaff holds a list of unassigned variables sorted by the counter value.
Updates are needed only when adding con-flict clauses.
Thus decision is made in constant time.
28/30 SAT Solver Changki Hong @ PSWLAB
Decision Heuristics - Berkmin Keep conflict clauses in a stack Choose the first unresolved clause in the stack
If there is no such clause, use VSIDS Choose from this clause a variable + value
according to some scoring (e.g. VSIDS)
This gives absolute priority to conflicts.
29/30 SAT Solver Changki Hong @ PSWLAB
Contents Introduction The DPLL framework BCP and Implication Graph Conflict Clauses and Resolution Decision Heuristics References
30/30 SAT Solver Changki Hong @ PSWLAB
References Decision Procedures – Daniel Kroening and Ofer Strichman The Quest for Efficient Boolean Satisfiability Solvers – Lintao
Zhang and Sharad Malik Efficient Conflict Driven Learning in a Boolean Satisfiability
Solver – Lintao Zhang, Conor F. Madigan, Matthew H. Moskewicz and Sharad Malik