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,...,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>
       
  • 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

\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

What's up with symmetry in SAT?

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

Only 2 more instances solved?

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

In general, Σ lacks information on structure of group <Σ>

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

Occupies(p1,h1) Occupies(p1,h2) Occupies(p1,h3)
Occupies(p2,h1) Occupies(p2,h2) Occupies(p2,h3)
Occupies(p3,h1) Occupies(p3,h2) Occupies(p3,h3)
Occupies(p4,h1) Occupies(p4,h2) Occupies(p4,h3)

Variable rows for 4 pigeons, 3 holes:

Symmetry due to interchangeable pigeons completely broken by

sbf(swap(p1,p2)) sbf(swap(p2,p3)) sbf(swap(p3,p4))

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

\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)
  • 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]
\neg x_0 \vee \sigma(x_0)
¬x0σ(x0)\neg x_0 \vee \sigma(x_0)
\neg x \vee x'
¬xx\neg x \vee x'

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 Gi is the stabilizer subgroup stabilizing the next non-stabilized variable in ordering
    • Gi have different smallest non-stabilized variables x
  • for each i: Orbit(x) under Gi leads to binary symmetry breaking clauses
    • Derives all binary symmetry breaking clauses of <Σ> under variable ordering
<\Sigma> = G_0 \supset G_1 \supset \ldots \supset 1
<Σ>=G0G11<\Sigma> = G_0 \supset G_1 \supset \ldots \supset 1

Stabilizer chain symmetry breaking

Occupies(p1,h1) Occupies(p1,h2) Occupies(p1,h3)
Occupies(p2,h1) Occupies(p2,h2) Occupies(p2,h3)
Occupies(p3,h1) Occupies(p3,h2) Occupies(p3,h3)
Occupies(p4,h1) Occupies(p4,h2) Occupies(p4,h3)

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

Stabilizer chain for pigeon symmetry consists of 4 subgroups

G0 permutes rows 1-2-3-4,
G1 permutes rows 2-3-4,
G2 swaps rows 3-4,
G3=1

Binary symmetry breaking clauses:

¬Occupies(p1 ,h1) v Occupies(p2 ,h1)
...
¬Occupies(p3 ,h1) v Occupies(p4 ,h1)

Further improvements on Shatter

  • More compact conversion of sbf to CNF [5]
    • 3 clauses of size 3 instead of
      4 clauses of sizes 3-4
  • Limit the size of sbf to 50
  • Limit symmetry detection time of Saucy

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)

BreakID

By krr

BreakID

SAT16

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