GRASP-an efficient SAT solver

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GRASP-an efficient SAT solver

• Pankaj Chauhan

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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 x10, x21, x30, x41
SATisfying assignment!
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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 PNP!

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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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Outline

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

Please interrupt me if anything is not clear!
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Terminology

• CNF formula ?
• x1,, xn n variables
• ?1,, ?m m clauses
• Assignment A
• Set of (x,v(x)) pairs
• A lt 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 x10, x21, x30, x41
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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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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)
• xv_at_d ? x is assigned to v at decision level d

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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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Backtracking Search in Action
?1 (x2 ? x3) ?2 (?x1 ? ?x4) ?3 (?x2 ? x4)
x1 1_at_1
? x4 0_at_1
? x2 0_at_1
? x3 1_at_1
? x3 1_at_2
(x1,1), (x2,0), (x3,1) , (x4,0)
(x1,0), (x2,0), (x3,1)
No backtrack in this example!
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Backtracking Search in Action
Add a clause
?1 (x2 ? x3) ?2 (?x1 ? ?x4) ?3 (?x2 ?
x4) ?4 (?x1 ? x2 ? ?x3)
? x4 0_at_1
? x2 0_at_1
? x3 1_at_1
? x3 1_at_2
conflict
(x1,0), (x2,0), (x3,1)
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Davis-Putnam revisited
Deduction
Decision
Backtrack
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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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GRASP search template
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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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GRASP Deduction

• Boolean Constraint Propagation using implication
  graphs
• E.g. for the clause ? (x ? ?y), if y1, then
  we must have x1
• 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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Implication Graphs

• Nodes are variable assignments xv(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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Example Implication Graph
Current truth assignment x90_at_1 ,x100_at_3,
x110_at_3, x121_at_2, x131_at_2 Current decision
assignment x11_at_6
x100_at_3
?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)
x11_at_6
x90_at_1
x110_at_3
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GRASP Deduction Process
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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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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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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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Learning (some math)

• For any node of an IG x, partition A(x) into
• ?(x) (y,v(y)) ?A(x)?(y)lt?(x)
• ?(x) (y,v(y)) ?A(x)?(y)?(x)
• Conflicting assignment AC(?) causesof(?), where

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Learning (some math)

• Deriving conflicting clause ?C(?) from the
  conflicting assignment AC(?) is straight forward
• For our IG,

AC(?) x11_at_6, x9 0_at_1, x10 0_at_3, x11 0_at_3
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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 gt parameter, dont
  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 x10

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Continued IG
Decision level

3
x81_at_6
x90_at_1
?8
?C(?)

?9
x100_at_3
5
?
x131_at_2
x10_at_6
?C(?)
?9
?9
?7
?C(?)
x110_at_3
x71_at_6
x1
?7
6
x121_at_2
Due to ?C(?)


?
?
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Backtracking

• Conflict Directed Backtracking
• Now deriving a new IG after setting x10 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-1 chronological backtrack
• ? lt d-1 non-chronological backtrack

AC(?) x9 0_at_1, x10 0_at_3, x11 0_at_3, x121_at_2,
x131_at_2 ?C(?) (x9 ? x10 ? x11 ? ? x12 ? ?
x13)
? max?(x)(x,v(x))?AC(?)
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Procedure Diagnose()
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Is that all?

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