Symmetric explanation learning: Effective dynamic symmetry handling for SAT

Jo Devriendt, Bart Bogaerts,
Maurice Bruynooghe, Marc  Denecker
 

University of Leuven

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 {h1,h2,...,hn}
  • Set of pigeons {p1 ,p2,...,pn+1}
  • Symmetries σ on pigeons and holes
    e.g., swapping h1 and h2
     
  • Symmetry group <Σ> of all permutations on pigeons and/or holes
\forall p \colon \exists h \colon Occupies(p,h)
p:h:Occupies(p,h)\forall p \colon \exists h \colon Occupies(p,h)
\forall p~p'~h\colon Occupies(p,h) \land Occupies(p',h) \Rightarrow p=p'
p p h:Occupies(p,h)Occupies(p,h)p=p\forall p~p'~h\colon Occupies(p,h) \land Occupies(p',h) \Rightarrow p=p'
\sigma \colon Occupies(p_i,h_1) \leftrightarrow Occupies(p_i,h_2)
σ:Occupies(pi,h1)Occupies(pi,h2)\sigma \colon Occupies(p_i,h_1) \leftrightarrow Occupies(p_i,h_2)

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

\forall i\colon (\forall j < i \colon x_j \Leftrightarrow \sigma(x_j)) \Rightarrow \neg x_i \vee \sigma(x_i)
i:(j<i:xjσ(xj))¬xiσ(xi)\forall i\colon (\forall j < i \colon x_j \Leftrightarrow \sigma(x_j)) \Rightarrow \neg x_i \vee \sigma(x_i)
x_0 < x_1 < \ldots < x_n
x0<x1<<xnx_0 < x_1 < \ldots < x_n

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, SEL)
  • None of them performs as well as state-of-the art static symmetry breaking
  • We present a novel algorithm in this family: symmetric explanation learning

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 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)) if σ'(c) propagates at least once)

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) if needed

SEL vs SLS

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

SEL vs SP

  • SP: similar idea to SEL: heuristic to guarantee that symmetric learnt clause propagates at least once
  • SP: based on weak activity
  • SP vs 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


\langle\alpha,\gamma,\Delta,\mathcal{E},\Theta\rangle
α,γ,Δ,E,Θ\langle\alpha,\gamma,\Delta,\mathcal{E},\Theta\rangle
\alpha
α\alpha
\gamma
γ\gamma
\Delta
Δ\Delta
\mathcal{E}
E\mathcal{E}
\Theta
Θ\Theta

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? 

Do 7825 pigeons fit in 7824 holes? 

  • impossible for SAT solvers
  • trivial on the first-order level
\forall p \colon \exists h \colon Occupies(p,h)
p:h:Occupies(p,h)\forall p \colon \exists h \colon Occupies(p,h)
\forall p~p'~h\colon Occupies(p,h) \land Occupies(p',h) \Rightarrow p=p'
p p h:Occupies(p,h)Occupies(p,h)p=p\forall p~p'~h\colon Occupies(p,h) \land Occupies(p',h) \Rightarrow p=p'

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)

SEL

By krr

SEL

SAT17

  • 40
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