Shatter: Efficient SymmetryBreaking for Boolean Satisfiability

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Shatter Efficient Symmetry-Breaking for Boolean
Satisfiability

• Fadi A. AloulIgor L. Markov, Karem A. Sakallah
• The University of Michigan

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Motivation

• Many powerful SAT solvers are currently available
• Yet, many EDA instances remain hard to solve
• Recent work pointed out that breaking symmetries
  can speed up search
• E.g.

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Previous Work

• In 1996, Crawford et al.
• Laid theoretical foundation for detecting and
  breaking symmetries in CNF formulas
• In 2002, Aloul et al.
• Extended the framework to handle phase shift
  symmetries and their composition with
  permutational symmetries
• Presented efficient construction of
  symmetry-breaking predicates

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Symmetries in SAT

• Permutations of variables that preserve clauses
• e.g., symmetries of
• ? (a b c)(d e
  f)include

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Why Break Symmetries?
SAT Solver
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Why Break Symmetries?
SAT Solver
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Why Break Symmetries?
SAT Solver
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Symmetry Detection and Breaking Flow
CNF instance
Returns Generators which implicitly represents
the group of symmetries
? (ab)(bc)(ca)
DetectSymmetries
?1 (ab)(ab)?2 (bc)(bc)
BreakSymmetries
? (ab)(bc)(ca) (ab)(bc)
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Symmetry Detection and Breaking Flow
CNF instance
Returns Generators which implicitly represents
the group of symmetries
? (ab)(bc)(ca)
DetectSymmetries
?1 (ab)(ab)?2 (bc)(bc)
BreakSymmetries
? (ab)(bc)(ca) (ab)(bc)
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Outline

• Symmetry-Breaking Predicate (SBP) construction by
  Crawford et al.
• Efficient SBP constructions
• Experimental results
• Conclusions

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Full Symmetry Breaking

• Lex-leader formula Crawford et al. 96
• Given a group of symmetries
  defined over totally-ordered variables
  
• For each symmetry , construct a permutation
  predicate

Image of variable xi under ?
PP(?) size5n clauses0.5n2 13.5n literals
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Linear-Sized PPs
PP(?) size5n clauses0.5n2 13.5n literals
PP(?) size4n clauses14n literals
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Linear-Sized Tautology-Free PPs

• Variables that map to themselves (i.e.
  )lead to
• Assume maps to itself

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Linear-Sized Tautology-Free PPs

• Variables that map to themselves (i.e.
  )lead to
• Assume maps to itself

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Linear-Sized Tautology-Free PPs

• Variables that map to themselves (i.e.
  )lead to
• Assume maps to itself

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Linear-Sized Tautology-Free PPs

• Variables that map to themselves (i.e.
  )lead to
• Assume maps to itself

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Partial Symmetry-Breaking (1)

• Full symmetry breaking may not speed up search
  because
• Exponential number of symmetries
• Their SBPs may be redundant
• Partial symmetry breaking provides a better
  trade-off
• Consider first k-variables from each permutation
• e.g. if k1

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Partial Symmetry-Breaking (1)

• Full symmetry breaking may not speed up search
  because
• Exponential number of symmetries
• Their SBPs may be redundant
• Partial symmetry breaking provides a better
  trade-off
• Consider first k-variables from each permutation
• e.g. if k1

1
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Partial Symmetry-Breaking (2)

• Instead of breaking all symmetries, break only
• Generators
• Generators and their powers
• Generators and their pair-wise compositions

Group of symmetries
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Experimental Results
ofbits thatmap tothemselves
Generators consisted of cycles of size 2 only
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Experimental Results
Break generators only
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Experimental Results
Total size of generator-only SBPs using various
SBP constructions
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Experimental Results
Total search runtimes for all instances when only
k bits are considered from each generator(using
linear tautology-free construction)
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Experimental Results
SBP statistics for various symmetry-breaking
candidatesusing linear tautology-free
construction
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Conclusions

• Introduced more efficient CNF constructions of
  symmetry-breaking predicates
• Constructions lead to
• Empirical speedups
• Smaller memory requirements
• Described options for partial symmetry-breaking

http//vlsicad.eecs.umich.edu/BK/Slots/shatter/
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Thank You!
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Reduction to Graph Automorphism

• CNF formula ? colored graph
• Linear time and space
• Find graphs colored symmetries
• Worst-case exponential time
• Interpret graph symmetries found as symmetries of
  the CNF formula
• Permutational symmetries
• Phase-shift symmetries

Clauses A (-a ? b ? c) B (a ? -b ? -c) C (-b
? c)
Symmetry(a -a)(b -c)(-b c)
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Identifying Equivalence Classes on Set of Truth
Assignments
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Identifying Equivalence Classes on Set of Truth
Assignments
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Identifying Equivalence Classes on Set of Truth
Assignments
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Identifying Equivalence Classes on Set of Truth
Assignments
ab
c
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Symmetry-Breaking Predicates

• To restrict search
• Add clauses to the original CNF formula
• (symmetry-breaking clauses)
• They will pick at least one representative
  (including lex-leaders) from each equivalence
  class and prune search space

ab
c
Lex-leader of equivalence class
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Example
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Full Symmetry Breaking

• Lex-leader formula Crawford et al. 96.Given a
  group of symmetries defined
  over totally-ordered variables
  

Impractical
Permutation Predicate quadratic in the number of
variables n
Number of permutations in symmetry group is
exponentially large
. . .
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Efficient SBP Constructions (1)

• Elimination of tautologies
• Bits that map to themselves
• Last bit in each cycle
• Bits occurring after phase-shift variables

k eliminated bits Permutation Predicate
quadratic in n-k
. . .
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Efficient SBP Constructions (2)

• Linear construction of permutation predicates
  through chaining

k eliminated bits Permutation Predicate
linear in n-k
. . .
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Partial Symmetry Breaking (1)

• To avoid exponential number of clauses
• Create permutation predicates for subset of
  permutations

Generators or Powers of Generators or Composition
of Generators
. . .
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Partial Symmetry Breaking (2)

• To avoid exponential number of clauses
• Consider first k, instead of all, bits in a
  permutation

e.g. consider only first 5 bits in each
permutation
. . .
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Shatter Flow Diagram
CNF instance
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Symmetries in Search Space
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Symmetry Detection and Breaking Flow
CNF instance
? (ab)(bc)(ca)
Identify Symmetries
Break Symmetries
?1 (ac)(bc)(ca)?2 (ca)(abc)
SAT Solver
Our focus
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Symmetry Detection and Breaking Flow
CNF instance
DetectSymmetries
? (ab)(bc)(ca) (ac)(!bc)
?1 (ac)(bc)(ca)?2 (ca)(abc)
? (ab)(bc)(ca)
BreakSymmetries
Our focus
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Symmetry Detection and Breaking Flow
CNF instance
? (ab)(bc)(ca)
?1 (ac)(bc)(ca)?2 (ca)(abc)
? (ab)(bc)(ca) (ac)(!bc)
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Linear-Sized Tautology-Free PPs

• Variables that map to themselves (i.e.
  )lead to
• Assume maps to itself

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Motivation

• Many powerful SAT solvers are currently available
• Yet, many EDA instances remain hard to solve
• Recent work pointed out that breaking symmetries
  can speed up search
• In 1996, Crawford et al.
• Laid theoretical foundation for detecting and
  breaking symmetries in CNF formulas
• In 2002, Aloul et al.
• Extended the framework to handle phase shift
  symmetries and their composition with
  permutational symmetries

46
Outline

• Basic definitions
• Symmetry-Breaking Predicate (SBP) construction by
  Crawford et al.
• Efficient SBP constructions
• Experimental results
• Conclusions

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Permutations and Generators

• Number of symmetries can be exponentially large
• Represent the group of symmetries implicitly
• Elementary group theory proves
• If redundant generators are avoided
• A group with N elements can be represented by at
  most log2(N) generators
• Generators provide exponential compression of
  solution space