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 an algebraic group <Σ> under ◦ (composition)

What's up with symmetry in SAT?

• E.g. pigeonhole problem
  
  
   
• Set of holes {h₁,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>
     
• BreakID [7]
  • Find "the right" generators

Symmetry exploitation in SAT

Two classes of techniques

• Dynamic symmetry exploitation
• Static symmetry exploitation/breaking

avoid symmetrical parts of the search space

Symmetry exploitation in SAT

• lex-leader symmetry breaking formula sbf(σ) for σ
  • based on variable order
  • 
     
• Shatter and BreakID preprocessors [2,7]
  • Given Σ, construct sbf(σ) for each σ∈Σ
  • Linear encoding of sbf(σ) into clauses
  • Very good results in recent SAT competitions

static symmetry breaking

Symmetry exploitation in SAT

• Takes symmetries into account during search
• Requires dedicated solvers
  • SymChaff [3]
  • Symmetry propagation for SAT (SP) [4]
  • Symmetric Learning Scheme (SLS) [5]
  • ...

dynamic symmetry exploitation

Symmetric learning

• Modern SAT solvers learn new clauses during search
• Typically by resolution, but the idea works for any proof system
• Learnt clauses are logical consequences of the theory
• Whenever c is a consequence of T, so is σ(c) 
• Why not also learn σ(c)? 

Symmetric learning

• Learning σ(c) for all  σ∈<Σ>:
  1. infeasible [6] (simpy too many σ)
  2. How to ensure that no duplicate/subsumed clauses are learnt?
• Various algorithms exist that make some choice on which symmetries to apply (SP, SLS)
• None of them perform as well as state-of-the art static symmetry breaking
• We present a novel algorithm in this family: symmetric explanation learning (SEL)

Symmetric explanation learning

• Idea: we prefer to learn clauses with a high chance of propagation
  • Even better: we prefer to only learn clauses that propagate at least once
• Idea: symmetries typically permute only a fraction of the literals
• Hence, if c is unit, σ(c) has a good chance of being unit as well
  • This notion of "interesting" candidate learned clause is dynamic

Symmetric explanation learning

• Proposal: whenever c propagates, store σ(c) in a separate clause store θ for each σ∈Σ
• Propagation on θ happens with low priority: only if standard unit propagation is at a fixpoint
• Whenever a clause in θ propagates, we learn it (we upgrade it to a "normal" learned clause)
• Whenever we backtrack over the level where c propagated, we remove all σ(c) from θ

SEL: Properties

• Duplicate/subsumed clauses are never learnt
• Combinatorial explosion of learnt clauses is contained by:
  • Only considering candidates σ(c) for σ∈Σ (not σ∈<Σ>) and c a learnt clause
  • Removing candidates upon backtracking
• Still allows for transitive effects (learning σ'(σ(c)) after learning σ(c))

SEL: Optimizations

• Standard two-watched literal scheme applied to θ
• Do not store σ(c) if it is satisfied at the time of c's propagation
• Store a simplification of σ(c) in θ:
  • Remove all false literals at the time of c's propagation
  • Restore original σ(c) when upgrading

SEL vs SLS

• SLS: learn σ(c) for each learned clause c and each σ∈Σ
  • possibly learns useless clauses
  • no (intrinsic) mechanism to avoid duplicates
  • no transitive effects (possibly misses useful clauses)
• SEL:
  • possibly misses useful clauses

SEL vs SP

• SP: tracks symmetrical literals that can be propagated
  • uses notion of weak activity of symmetry
  • learns σ(c) as reason of symmetrical propagation
• SEL:
  • whenever SP learns σ(c), so does SEL (not vice versa)

Notation

• State:
•   : assignment (list of literals)
•   : decision literals
•   : learned clause store
•   : explanation (maps propagated literals to the clause that propagated them)
•   : symmetrical learned clause store

SEL: Experiments

• Limits:
  • 5000s 
  • 16GB
• Benchmarks
  • app14, hard14
  • app16, hard16
  • highly symmetric instances
• Solvers:
  • Glucose 4.0
  • Glucose 4.0 + BreakID
  • Glucose 4.0 + SEL
  • Glucose 4.0 + SP
  • Glucose 4.0 + SLS

If SEL does not outperform static symmetry breaking... Why bother?

Symmetry exploitation in SAT

Static symmetry breaking:

• Compatible with any solver (simple preprocessing!) 
• Discards models a priori

Dynamic symmetry exploitation (SEL):

• Simple idea
• Requires dedicated solvers
• Does not discard models a priori
  • Compatible with other inference tasks, such as:
    • Model counting
    • (partial) MAXSAT
    • Optimization
• "Lazy"

static vs dynamic

Summary

• We presented the first general-purpose dynamic symmetry exploitation technique +/- on par with static symmetry breaking
• Based on symmetric learning; propagation-directed selection of learnt clauses
• No completeness results yet. Under which conditions on Σ is SEL guaranteed to never visit two symmetrical assignments? 

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.
[7] Improved Static Symmetry Breaking for SAT - 2016 - Devriendt e.a.

An open research problem:

Do 7825 pigeons fit in 7824 holes? 

• impossible for SAT solvers
• trivial on the first-order level

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.
[7] Improved Static Symmetry Breaking for SAT - 2016 - Devriendt e.a.

(really, this time)