Symmetry exploitation for combinatorial problems
November 6, 2017, KTH, Stockholm
Bart Bogaerts, KU Leuven
(joint work with Jo Devriendt, Maurice Bruynooghe, Marc Denecker)
Setting
- Given:
- A logical representation of knowledge
- (possibly) some instance (a partial structure)
- An inference task to solve
- Search:
- A solution to your problem
- E.g.,
- SAT: determine satisfiability of a propositional theory in CNF
- FO-MX: find a model of a first-order theory that expands some input structure
- CP: find a(n) (optimal) solution of a set of constraints
- MIP, SMT, ASP, ...
Problem
- If your specification is highly symmetric, complete solvers tend to get lost
- How to exploit this symmetry?
Overview
- Improved Static Symmetry Breaking for SAT (and ASP and QBF) SAT 2016
- Symmetric explanation learning: Effective dynamic symmetry handling for SAT SAT 2017
- On Local Domain Symmetry for Model Expansion ICLP 2016
Part I
Improved static symmetry breaking for SAT
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 {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
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!
Static symmetry breaking in SAT
-
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; add to T
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 the 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 (locally) 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 ))
(Greedily) detecting row interchangeability
- Input: CNF theory T, Σ detected by Saucy
- Output: variable matrix M such that rows are interchangeable and <M>⊆<Σ>
- extract σ 1 , σ 2 ∈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>
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 [4]
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
- 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 in SAT13, SAT15, SAT16 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?
- BreakID is easy to fool with redundant clauses... Insight in properties of real-world problems?
- Extend these structure ideas to dynamic symmetry exploitation?
- Completeness result on row interchangeability for dynamic symmetry breaking?
[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)
Part II
Symmetric explanation learning: Effective dynamic symmetry handling for SAT
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
- 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>:
- infeasible [6] (simpy too many σ)
- 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
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
-
Compatible with other inference tasks, such as:
- "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?
[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.
Part III:
On Local Domain Symmetry for Model Expansion
An open research problem:
Do 7825 pigeons fit in 7824 holes?
Do 7825 pigeons fit in 7824 holes?
- impossible for SAT solvers
- trivial by first-order reasoning
Background: IDP
- A knowledge base system
-
Language: extensions first-order logic
- Types
- Inductive definitions
- Aggregates
- Partial functions
- Multiple types of inference
- Model expansion
- Model checking
- Propagation (various forms)
- Entailment
- Querying
- Definition evaluation
-
Language: extensions first-order logic
Model expansion in IDP
- Given: theory T, partial structure S (interprets some symbols) with finite domain
-
Ground and solve:
- Eliminate quantifiers in T (using S)
- Solve the resulting problem using MinisatID
(an extension of Minisat with inductive
definitions, integer variables, ...)
- Without special treatment: symmetries pose the same problem as for SAT.
Symmetry exploitation
- We know how to exploit symmetry in the underlying solver (let's assume we break them statically)
- (more could, and should, happen here, e.g., exploit them during grounding; that is not our focus)
- How to detect symmetry?
- Efficiently: preferably using syntactic properties of the first-order representation
- get the "right" symmetries
- yields row interchangeability of the ground theory
Symmetry exploitation
- We know how to exploit symmetry in the underlying solver (let's assume we break them statically)
- (more could happen here, e.g., exploit them during grounding; that is not our focus)
- How to detect symmetry?
- Efficiently: preferably using syntactic properties of the first-order representation
- get the "right" symmetries
- yields row interchangeability of the ground theory
Global domain symmetry
- A domain permutation is a bijection
- Each domain permutation induces a structure transformation (renaming) such that
- Each such structure transformation is a symmetry of any theory
- If furthermore, preserves S, then it is also a symmetry of MX(T,S)
Local domain symmetry
- Sometimes, we can exchange domain elements locally
- Given a set A of argument positions
- and A induce a structure transformation that performs only on those positions in A
- When is a symmetry of T?
- When is a symmetry of MX(T,S)?
Local domain symmetry
- If for each occurrence of an expression
in T,
then is a symmetry of T. - If furthermore preserves S, it is a symmetry of MX(T,S).
Local domain symmetry
- If for each occurrence of an expression
in T, where neither P nor f are interpreted by S
and preserves S, then is a symmetry of MX(T,S).
Efficiently breakable?
- Given a set A of argument positions such that A contains at most one argument position per predicate
- Assume D' is a subset of the domain such that for each permutation of D', is a symmetry of MX(T,S)
- Then, the group of all such symmetries can be broken by lex-leader constraints for swaps of two domain elements in D'
- Essentially, boils down to row interchangeability for the grounding
Experimental results
- Much more efficient than symmetry detection on the ground level.
- However, restricted to certain kinds of symmetries.
Conclusion
- Presented three recent developments in symmetry exploitation:
- SAT: detection & static symmetry breaking
- SAT: dynamic symmetry exploitation (symmetric learning)
- FO: symmetry detection
- Work well to speed up search
- Open questions:
- More robust static symmetry breaking
- Completeness guarantees for dynamic approaches
- More extensive FO detection (different types of symmetries)
Symmetry exploitation for combinatorial problems
By krr
Symmetry exploitation for combinatorial problems
KTH invited talk
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