Improved static symmetry breaking for SAT

Jo Devriendt, Bart Bogaerts,
Maurice Bruynooghe, Marc  Denecker
 

University of Leuven / Aalto University

Or: the story of BreakID

Outline

• What's up with symmetry in SAT?
• Our improvements
  • Row interchangeability detection
  • Stabilizer chain symmetry breaking
  • Efficiency optimizations
• Future ideas for symmetry in SAT

What's up with symmetry in SAT?

• CNF theory T, literal l, variable x, assignment α
   
• Symmetry σ: permutation of literals
   
  • commutes with negation        σ(l) = ¬σ(¬l)
  • preserves satisfaction    σ(α) ⊨ T iff α ⊨ T
    • syntactically fixes T                   σ(T) = T
       
• Set of symmetries Σ generate a mathematical group <Σ> under ◦ (composition)

What's up with symmetry in SAT?

• E.g. pigeonhole problem
  
  
   
• Set of holes {h₂,h₂,...,h_n}
• Set of pigeons {p₁ ,p₂,...,p_n+1}
• Symmetries σ on pigeons and holes
  e.g., swapping h₁ and h₂
   
• Symmetry group <Σ> of all permutations on pigeons and/or holes

Symmetry detection in SAT

• Saucy [1]
  • Convert CNF to colored graph
  • Automorphisms are syntactical symmetries
  • Result: set Σ of generator symmetries for <Σ>
     
• Very efficient!

Symmetry breaking in SAT

• Dynamic symmetry breaking
• Static symmetry breaking
   
• lex-leader symmetry breaking formula sbf(σ) for σ
  • based on variable order
  • 
     
• Shatter preprocessor [2]
  • Given Σ, construct sbf(σ) for each σ∈Σ
  • Linear encoding of sbf(σ) into clauses 

avoid symmetrical parts of the search space

What's up with symmetry in SAT?

Let's try Shatter on the pigeonhole problem...

What's up with symmetry in SAT?

Let's try Shatter on the pigeonhole problem...

Only 2 more instances solved?

• Problem lies with generator symmetries Σ
• <Σ> is not completely broken by conjunction of sbf(σ), σ∈Σ
• For pigeonhole, there does exist some small Σ' for which sbf(σ), σ∈Σ' breaks <Σ> completely

BreakID tries to exploit symmetry group structure

• Detect row interchangeability symmetry subgroups of <Σ>
• Symmetry breaking based on stabilizer chain of <Σ>
• Small performance optimizations

BreakID: symmetry breaking preprocessor similar to Shatter

Detecting row interchangeability

• Row interchangeability: common form of symmetry
  • Stems from interchangeable objects
  • Variables can be ordered as rows in matrix
  • All permutations of rows are symmetries
• Can be broken completely by constructing sbf only for consecutive row swaps [3]
  • Assuming appropriate variable ordering

Detecting row interchangeability

Variable rows for 4 pigeons, 3 holes:

Detecting row interchangeability

• Input: CNF theory T, Σ detected by Saucy
• Output: variable matrix M such that rows are interchangeable and <M>⊆<Σ>

1. extract σ₁, σ₂ ∈Σ that form 2 subsequent row swaps
   • forms initial 3-rowed variable matrix M
2. apply every σ∈Σ to all detected rows r∈M so far
   • images σ(r) disjoint of M are candidates to extend M
   • test if swap r ↔ σ(r) is a symmetry by syntactical check on T
   • if success, extend M with σ(r)
3. use Saucy to extend Σ with new symmetry generators by fixing all variable nodes with variable in M, first row excepted

Let's try BreakID on the pigeonhole problem...

• Detects full pigeon subsymmetry
• Poly performance
• 100+ holes are no problem
   

Detecting row interchangeability

Stabilizer chain symmetry breaking

• Recall sbf(σ):
  • for i=0:
  • Binary symmetry breaking clause
     
• x is stabilized by <Σ> iff σ(x)=x for all σ∈<Σ>
• x' ∈ orbit(x) under <Σ> iff there exists σ∈<Σ> s.t. σ(x)=x'
   
• Given symmetry group <Σ> with smallest non-stabilized variable x, with x' ∈ orbit(x) under <Σ>,
                 is logical consequence of sbf(σ) for all σ∈<Σ>
   
• Moreover, all binary clauses in sbf(σ) for all σ∈<Σ> are derived this way [4]

Stabilizer chain symmetry breaking

• <Σ> has subgroups that have other smallest non-stabilized variables, depending on variable order
• Create stabilizer chain of <Σ> along variable ordering:
   
• next subgroup G_i is the stabilizer subgroup stabilizing the next non-stabilized variable in ordering
  • G_i have different smallest non-stabilized variables x
• for each i: Orbit(x) under G_i leads to binary symmetry breaking clauses
  • Derives all binary symmetry breaking clauses of <Σ> under variable ordering

Stabilizer chain symmetry breaking

Occupies(p₁ ,h₁) < Occupies(p₂ ,h₁) < Occupies(p₃ ,h₁) < Occupies(p₄ ,h₁) < ...

BreakID combines row interchangeability detection with stabilizer chain symmetry breaking (approximative)

Symmetry breaking in SAT competitions?

• BreakID overhead is low enough to be competitive, even on asymmetric instances
• Participated succesfully in SAT13, SAT15 competitions
• Experimental results on SAT14 instances
  (Glucose as base solver)

Summary

• BreakID follows in Shatter's footsteps (after ~10 years!)
• Derive group structure information
  • Row interchangeability detection
  • Stabilizer chain based symmetry breaking
    • Both are approximative algorithms
• Efficiency optimizations ensure competitiveness

Much work remains...

• Combination with computational algebra should improve group structure detection
  • Intertwine with graph automorphism detection?
• Good heuristic for variable ordering
   
• Dynamic symmetry breaking should outperform static symmetry breaking
  • Completeness result on row interchangeability for dynamic symmetry breaking?
• First-order level symmetry detection [6]

Thanks for your attention!

Questions?

[1] Symmetry and Satisfiability: An Update - 2010 - Katebi e.a.
[2] Efficient Symmetry-Breaking for Boolean Satisfiability - 2006 - Aloul e.a.
[3] On the importance of row symmetry - 2014 - Devriendt e.a.
[4] Automatic generation of constraints for partial symmetry breaking - 2011 - Jefferson & Petrie
[5] Symmetry and satisfiability - 2009 - Sakallah
[6] On Local Domain Symmetry for Model Expansion - 2016 - Devriendt e.a. (accepted for ICLP)