SymChaff: A Structure-Aware Satisfiability Solver

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SymChaffA Structure-Aware Satisfiability Solver

• Ashish Sabharwal
• University of Washington, Seattle
• AAAI, July 2005

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SAT The Satisfiability Problem

• Input Boolean formula F in CNF
• F (x1 OR ?x2) AND (x1 OR ?x3 OR x4) AND
• Output satisfying assignment or unsat
• NP-complete, hence difficult but useful!

Problem instance
sat
CNF Encoder
F
SAT Solver
unsat
e.g. planning, verification, design,
domain dependent
general purpose
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SAT The Satisfiability Problem

• Numerous real-world applications
• hardware verification, testing, planning,
  scheduling, design automation,
• Dozens of academic/industrial SAT solvers
• e.g. Grasp MarquesSilva-Sakallah-96, Relsat
  Bayardo-Schrag-97, SATO Zhang-97, zChaff
  Moskewicz-etal-01, Berkmin Goldberg-Novikov-02
  , March-eq Huele-vanMaaren

Most dont exploit problem structure well!
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Planning Example Logistics
CityA
CityC
CityD
CityB
Each city has several boxes Each box has a
destination city Several trucks at a base station
Domain knowledge All trucks are equal!
Task use trucks to move boxes to their
resp. destinations
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SAT and Symmetry

• Natural symmetry in many domains, e.g.
• planning all trucks at a base station, all nails
  in assembly
• circuit design all wires connecting two switch
  boxes
• multi-processor scheduling all processors
• cache coherency protocols all caches
• If k trucks need to be sent from CityA to CityB,
  w.l.o.g. send the first k.
• Proposed techniques have several drawbacks

Easy for humans, hard for SAT solvers!
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Our Solution SymChaff
Problem instance
sat
F
CNF Encoder
SAT Solver
unsat

• Captures a wide class of natural symmetries by
  incorporating variable semantics
• Exploits high level problem structurerather than
  flat CNF formulation

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Previous Approaches 1/4

• Symmetry Breaking Predicates Crawford-etal-96
• e.g. Shatter Aloul-Markov-Sakallah-03

F
sat
SAT Solver
F
unsat
SBP
SBP generator

• SBPs can be prohibitively many, too large,
  difficult to compute (graph isomorphism)

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Previous Approaches 2/4

• pseudoBoolean tools counting constraints
• e.g. PBS Aloul-Ramani-Markov-Sakallah-02,
  pbChaff Dixon-Ginsberg-Parkes-04, Galena
  Chai-Kuehlmann-03

Problem instance
pbCNF
sat
pbCNF Encoder
pbSAT Solver
unsat

• some domains provably hard (e.g. clique color)
• implicit counting representation not always
  suitable

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Previous Approaches 3/4

• Handle symmetries dynamically in search
• e.g. sEqSatz Li-Jurkowiak-Purdom-02, another
  solver Dixon-Ginsberg-Luks-Parkes-04

F
sat
SYM-SAT Solver
unsat

• expensive group-theoretic computations

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Previous Approaches 4/4

• Domain specific solutions
• Testing MarquesSilva-Sakallah-97, model
  checking Shtrichman-04, planning Rintanen-03,
  Fox-Long-99

• Domain specific
• Cannot exploit advances in SAT solvers
• Fox-Long-99 Very similar to our technique
  yields non-optimal parallel plans

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Our Contribution

• New framework for symmetry in SAT
• Low overhead, top-down approach
• Seamless integration with current SAT solvers
• Foundation in many-sorted First Order logic
• SAT solver SymChaff that extends zChaff
• Outperforms previous approaches on several
  domains from theory, planning, and design
• Philosophical a tiny bit of non-CNF input can
  dramatically speed up SAT solvers

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Key Ideas

• Symmetry Representation
• Multi-way Branching
• Symmetric Learning

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Planning Example Logistics
CityA
CityC
CityD
CityB
Task use n trucks to move boxes to
their resp. destinations Problem variables
mv_tr3_cD_time1 load_boxA1_tr1_time1

Possible solution begin by sending some trucks
to cityA
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Idea 1 Multi-way Branching
Typical SAT solver - effectively explore 2n
branches for mv_tr?_cA
mv_tr1_cA
0
1
mv_tr2_cA
1
0
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Idea 1 Multi-way Branching

• Idea is extended to multi-class symmetry
• e.g. let boxA1, boxA2 be symmetric as well
• variable load_boxA1_tr1 symmetric tovariable
  load_boxA?_tr?
• Issues
• How is symmetry information provided?
• How is it maintained as we branch?

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Idea 2 Symmetry Representation

• Input to SymChaff includes
• Symmetry classes for objectse.g. tr1, , trn
  belong to class TR
• Semantics for variablese.g. mv_tr1_cA is indexed
  by class TR
• Symmetry sets of objects dynamic
• initially tr1, , trn is a symmetry set
• branch and send tr1, tr2 to cityAsplit/refine
  set into tr1, tr2, tr3, , trn
• backtrack re-unite these two sets

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Idea 3 Symmetric Learning
send j trucks to cityA

Fail
Fail too few trucks left for other cities!
SymChaff learns that these branches need not be
explored

• A sym-conflict clause is learnt
• NOTE this replaces the triedGroups approach of
  Fox-Long-99 doesnt suffer from
  non-optimality of parallel plans

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Experimental Results

• Compared SymChaff with
• zChaff(winner of SAT04 competition industrial
  category)
• March-eq-100(winner of SAT04 competition
  hand-made category)
• zChaff Shatter (using SBP)
• Galena and pbChaff (pseudoBoolean solvers) (on
  some domains see paper)
• Problem domains
• Theory pigeonhole, clique coloring
• Planning gripper, logistics (2)
• Circuit design channel routing

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Representative Runtime Samples
Best SAT solvers dont do very well!
Computing SBPsmay be expensive!
Computing SBPsmay not help!
Problem may bereally hard!
denotes gt 6 hours
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And all this from

• Original STRIPS specification
• (objects
• tr1 tr2 tr3
• boxA1 boxA2
• boxB1 boxB2
• ...)
• (actions
• ...)
• (init
• ...)
• ...

• New STRIPS specification
• (objects
• tr1 tr2 tr3 symTrucks
• boxA1 boxA2 symBoxA
• boxB1 boxB1 symBoxB
• ...)
• (actions
• ...)
• (init
• ...)
• ...

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Conclusion

• New framework that efficiently incorporates
  symmetry into SAT solvers
• Domain independent
• Low overhead
• Exploits high level problem description
• Can this framework help local search?
• Can we extend it to handle new symmetries that
  arise during the search? E.g. AirLock domain
  Fox-Long-02

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Ongoing and Future Work

• Use symmetry framework elsewhere
• local search techniques like Walksat
• improved variable selection heuristic
• Handle new symmetries as they arise
• e.g. AirLock domain of Fox-Long-02
• Easy extensions
• implement in pseudoBoolean solver
• create PDDL-to-SYM converter for planning
• Improvements to SymChaff
• other symmetric learning schemes
• dynamic selection in multi-way branches