SAT Solving

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SAT Solving
Presented by Avi Yadgar
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The SAT Problem

• Given a Boolean formula , look for
  assignment A for such that
  .
• A is a solution for .
• A partial assignment assigns a subset of .
• CNF representation of
• is a conjunction of clauses
• A clause is a disjunction of literals

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The SAT Problem

• ? (v1? v2? ? v5) ? (?v1? v7? ? v9) (v5? v7)
• A satisfies ? A satisfies all its
  clauses.
• Each clause needs one true literal
• NP-Complete
• Many applications
• Very efficient solvers
• Search / Proof engine

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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.

v1 0 v2 1 v3 ?
v1 ?
cl ( v3)
v3 1
?v2 ?
v1 0
?v2 0
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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.
• cl implies and is its antecedent.
• v1 and v2 are the antecedent variables of
• BCP() Calculates all the possible implications.

v1 0 v2 1 v3 ?
v3 1
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DPLL Davis Putnam Logemann Loveland Backtrack
Search

• Choose a decision variable and value.

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DPLL Davis Putnam Logemann Loveland Backtrack
Search

• Choose a decision variable and value.
• Run bcp()

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DPLL Davis Putnam Logemann Loveland Backtrack
Search

• Choose a decision variable and value.
• Run bcp()

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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.

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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.

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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().

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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().

13
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().

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DPLL Davis Putnam Logemann Loveland Backtrack
Search

• Termination
• No unassigned variables SAT
• No decision variable to flip un-SAT

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Optimized BCP

• Solver spends 90 in BCP

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The SAT Problem - Resolution
v2? ? v5

• c1 (v1? v2? ? v5) , c2(?v1? v7? ? v9)

v7? ? v9
cres(c1 , c2) (v2? ? v5 ? v7? ? v9)
? ? cres ? ? ? ? cres
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v9
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v5
v9
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v5
v9
v8
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v5
v9
v8
v12
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v9
v8
v12
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v4
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v4
v1
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v19
v4
v1
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v19
v4
v1
v6
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v19
v4
v1
v15
v6
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
v19
v3
v4
v1
v15
v6
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
Conflict
v19
v3
v4
v1
v15
v6
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SAT Solving - Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12

v4
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Learning Cuts

• CUT A bi-partition of the graph.
• The conflict is on the conflict side.
• The decisions are on the reason side
• No edge from the conflict side to the reason
  side.
• The edges which cut the cut represent the reason
  to the conflict.

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Learning Cuts
v9
Conflict
v4
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Learning Conflict Clauses
Reason Side
Conflict Side
v7
v5
v3
v9
(v5,v7,v3) (v7,v15,v3)
Conflict
(v7,v5,v15)
v4
v15

• v7,v5,v15 are the reason for the conflict.
• Adding the clause will
  prevent it in the future.

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Learning Conflict Clauses
v1
0
1
v2
v2
The clause (v2,v3) is created after a conflict
0
0
1
1
v3
v3
v3
v3
0
0
The search tree is pruned accordingly
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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.

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Learning Implication Graph
(v9,v5) (v9,v8) (v9,v12) (v5,v7) (v5,v7,v3) (v
4,v1) (v1,v12,v19) (v1,v6) (v6,v19,v8,v15) (v
7,v15,v3)
v7
v5
v3
v9
v8
v12
Conflict
v19
v3
v4
v1
v15
v6
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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.

Conflict Clause (v10,v7,v2,v9)
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Non Chronological Backtracking
v8
v8
1
1
v2
v2
0
0
v4
v9
0
1
v19
1
v6
1
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Branching - VSIDS(Variable State Independent
Decaying Sum)

• Counter for each literal
• Updated when clauses are added
• Branch on the literal with the highest rank
• Divide counters periodically
• Favors recent conflict clauses.
• Recent gained knowledge

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Restarts

• After a fixed interval
• Lengthen interval after each restart
• Nullify all branching and implications
• Keep conflict clauses
• Leads to a new branching order
• New score for the literals
• Hopefully a better order

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Branching for BMC

• Add static ordering
• Forward
• backward
• Branch on latches \ inputs

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Branching for BMC

• Create independent sub-problems

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un-SAT Core
Conflict
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un-SAT Core
v2
Conflict
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un-SAT Core
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un-SAT Core

• Easily obtained from a DPLL SAT solver
• Requires saving the resolution graph

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Branching BerkMin \HaifaSAT

• SAT as abstraction / refinement
• Stack of added conflict clauses
• Order clauses chronologically (BerkMin)
• Order on clauses topologically (HaifaSAT)
• Branch on a literal from the top clause

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The Rise and Fall of Parallel SAT Solving

• VERY appealing!
• Obstacles
• Dependencies
• Communication
• Parallelized BCP()
• Centralized
• Appealing since bcp() is 90 of the runtime
• Useless since bcp() is only 90 of the runtime

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The Rise and Fall of Parallel SAT Solving

• Parallelize for BMC
• Create independent formulae
• Take advantage of symmetry
• Share learned clauses
• Inference is local