Combining Component Caching and Clause Learning for Effective Model Counting

1
Combining Component Caching and Clause Learning
for Effective Model Counting

• Tian Sang
• University of Washington
• Fahiem Bacchus (U Toronto), Paul Beame (UW),
• Henry Kautz (UW), Toniann Pitassi (U Toronto)

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Why SAT?

• Prototypical P complete problem
• Natural encoding for counting problems
• Test-set size
• CMOS power consumption
• Can encode probabilistic inference

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Generality
NP complete

• SAT
• SAT
• Bayesian Networks
• Bounded-alternation Quantified Boolean formulas
• Quantified Boolean formulas
• Stochastic SAT

P complete
PSPACE complete
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Our Approach

• Good old Davis-Putnam-Logemann-Loveland
• Clause learning (no good-caching)
• Bounded component analysis
• Formula caching

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DPLL with Clause Learning
DPLL(F) while exists unit clause (y) ? F F
? Fy if F is empty, report satisfiable and
halt if F contains the empty clause Add a
conflict clause C to F return false choose a
literal x return DPLL(Fx) DPLL(F?x)
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Conflict Graph
Known Clauses (p ? q ? a) (? a ? ? b ? ? t) (t ?
?x1) (t ? ?x2) (t ? ?x3) (x1 ? x2 ? x3 ? y) (x2 ?
?y)
Current decisions p ? false q ? false b ? true
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Component Analysis

• Can use DPLL to count models
• Just dont stop when first assignment is found
• If formula breaks into separate components (no
  shared variables), can count each separately and
  multiply results
• SAT(C1 ? C2) SAT(C1) SAT(C2)
• (Bayardo Shrag 1996)

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Formula Caching

• New idea cache number of models of residual
  formulas at each node
• Bacchus, Dalmao Pitassi 2003
• Beame, Impagliazzo, Pitassi, Segerlind 2003
• Matches time/space tradeoffs of best known exact
  probabilistic inference algorithms

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SAT with Component Caching

• SAT(F) // Returns fraction of all truth
• // assignments that satisfy F
• a 1
• for each G ? to_components(F)
• if (G ?) m 1
• else if (? ? G) m 0
• else if (in_cache(G)) m cache_value(G)
• else select v ? G
• m ½ SAT(Gv)
• ½ SAT(G?v)
• insert_cache(G,m)
• if (m 0) return 0
• a a m
• return a

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Putting it All Together

• Goal combine
• Clause learning
• Component analysis
• Formula caching
• to create a practical SAT algorithm
• Not quite as straightforward as it looks!

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Issue 1 How Much to Cache?

• Everything
• Infeasible often gt 10,000,000 nodes
• Only sub-formulas on current branch
• Linear space
• Similar to recursive conditioning
• Darwiche 2002
• Can we do better?

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Age versus Cumulative Hits
age time elapsed since the entry was cached
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Efficient Cache Management

• Age-bounded caching
• Separate-chaining hash table
• Lazy deletion of entries older than K when
  searching chains
• Constant amortized time

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Issue 2 Interaction of Component Analysis
Clause Learning

• As clause learning progresses, formula becomes
  huge
• 1,000 clauses ? 1,000,000 learned clauses
• Finding connected components becomes too costly
• Components using learned clauses unlikely to
  reoccur!

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Bounded Component Analysis

• Use only clauses derived from original formula
  for
• Component analysis
• Keys for cached entries
• Use all the learned clauses for unit propagation
• Can this possibly be sound?

Almost!
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Safety Theorem
G?
F?
A2
A1
A3
Then ? can be extended to satisfy G? It is
safe to use learned clauses for unit propagation
for SAT sub-formulas

• Given
• original formula F
• learned clauses G
• partial assignment ?
• F? is satisfiable
• Ai is a component of F?
• ? satisfies Ai

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UNSAT Sub-formulas

• But if F? is unsatisfiable, all bets are off...
• Without component caching, there is still no
  problem because the final value is 0 in any
  case
• With component caching, could cause incorrect
  values to be cached
• Solution
• Flush siblings ( their descendents) of UNSAT
  components from cache

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Safe Caching Clause Learning Implementation

• ...
• else if (? ? G) m 0
• add a conflict clause
• ...
• if (m0)
• flush_cache( siblings(G) )
• if (G is not last child of F)
  flush_cache(G)
• return 0
• a a m
• ...

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Evaluation

• Implementation based on zChaff
• (Moskewicz, Madigan, Zhao, Zhang, Malik 2001)
• Benchmarks
• Random formulas
• Pebbling graph formulas
• Circuit synthesis
• Logistics planning

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Random 3-SAT, 75 Variables
sat/unsat threshhold
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Random 3-SAT Results
75V, R1.0
75V, R1.4
75V, R1.6
75V, R2.0
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Results Pebbling Formulas
X means time-out after 12 hours
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Summary

• A practical exact model-counting algorithm can be
  built by the careful combination of
• Bounded component analysis
• Component caching
• Clause learning
• Outperforms the best previous algorithm by orders
  of magnitude

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Whats Next?

• Better heuristics
• component ordering
• variable branching
• Incremental component analysis
• Currently consumes 10-50 of run time!
• Applications to Bayesian networks
• Compiler for discrete BN to weighted SAT
• Direct BN implementation
• Applications to other P problems
• Testing, model-based diagnosis,

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Questions?
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Results Planning Formulas
X means time-out after 12 hours
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Results Circuit Synthesis
X means time-out after 12 hours
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Bayesian Nets to Weighted Counting

• Introduce new vars so all internal vars are
  deterministic

A
B
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Bayesian Nets to Weighted Counting

• Introduce new vars so all internal vars are
  deterministic

A
B
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Bayesian Nets to Weighted Counting

• Weight of a model is product of variable weights
• Weight of a formula is sum of weights of its
  models

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Bayesian Nets to Weighted Counting

• Let F be the formula defining all internal
  variables
• Pr(query) weight(F query)

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Bayesian Nets to Counting

• Unweighted counting is case where all non-defined
  variables have weight 0.5
• Introduce sets of variables to define other
  probabilities to desired accuracy
• In practice just modify SAT algorithm to
  weighted SAT