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
- 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
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...
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>
- extract σ1, σ2 ∈Σ that form 2 subsequent row swaps
- forms initial 3-rowed variable matrix M
- 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)
- 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 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
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
- 3 clauses of size 3 instead of
- 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
- 2,044