Knowledge Compilation Properties of TreeofBDDs

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Knowledge Compilation Properties of Tree-of-BDDs
Lucas Bordeaux, Youssef Hamadi

• Sathiamoorthy Subbarayan

Microsoft Research, Cambridge UK
IT University of Copenhagen Denmark
AAAI-2007, Vancouver
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Knowledge Compilation

• Compile a system model in to a form
• Forms BDD, DNNF, d-DNNF, Tree-of-BDDs
• Quickly answer interesting queries
• Applications configuration, verification,
  fault-trees, Bayesian networks, model-based
  diagnosis, etc.,

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Compilation Forms
Tree-of-BDDs even 1000 times smaller than d-DNNF
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Binary Decision Diagrams
A compressed representation of solutions
Answers many queries in polytime
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CNF to BDD
AND
BDD
CNF clauses
clause BDDs
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Tree-of-BDDs (ToB)
Tree Decomposition
CNF clauses
ToB
clause BDDs
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Example CNF to ToB
CNF clauses
Tree Decomposition
Group clause-BDDs
ToB
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Example Comparison
Exploits tree-width!
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Some Interesting Queries

• Consistency (CO)
• Validity (VA)
• Clausal Entailment (CE)
• Implicant Check (IM)
• Equality (EQ)
• Sentential Entailment (SE)
• Model Counting (CT)
• Model Enumeration (ME)

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Conditioning ToB

• Condition (ToB, term)
• //Restrict ToB by term
• Restrict each BDD with term
• Propagate

Worst case exponential in tree-width!
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Clausal Entailment, Implicant Check

• IsCE (ToB, clause)
• //Does ToB entails the clause
• Condition (ToB, clause)
• if (ToBfalse)
• return true
• else
• return false

IsIM (ToB, term) //Does the term entails ToB
Condition (ToB, term) if ( term entails each BDD
) return true else return false
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Complexity of Queries
Worst case exponential in w
Polytime
Polytime if PNP
Worst case exponential in (w1 w2)
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Query Complexity Vs. Size
Realistic instances often have low treewidth !
ToB exploits it well !!
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Experiments

• 91 realistic CNF instances
• 35 Mercedes car configuration problems
• 7 ISCAS85 circuits
• 17 ISCAS89 circuits
• 14 ISCAS93 Addendum circuits
• 13 ISCAS99 circuits
• 5 Bounded model checking circuits
• Compare with c2d, a d-DNNF compiler
• Simulate CE and IM queries on ToBs

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Tools, Instances, Presentation
http//www.itu.dk/sathi/tob/
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Compilation Overview
for successful compilations
We could not find any realistic instance where
d-DNNF beats ToB in compilation !
Realistic Industrial instance in SAT competition
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Details Cases hard for d-DNNF
Very low treewidth!
ToB even 1000 times smaller than d-DNNF!
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CE and IM Simulation
Quick responses!
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Compilation Time ToB vs. d-DNNF
ToB is much faster than d-DNNF
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Compilation Space ToB vs. d-DNNF
ToB is much smaller than d-DNNF
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Conclusion

• ToBs often very small in time and space
• Even 1000 times smaller than d-DNNF
• Still quick response to many queries
• Future work
• EQ Check, usefulness in verification ?
• Succinctness of ToB vs d-DNNF
• Transformations supported by ToB
• Adopting ToBs for multi-core CPUs

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Thanks!
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Query Complexity Vs. Size
Realistic instances often have low treewidth !
ToB exploits it well !!
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Propagation in ToB

• Use a rooted BFS tree
• Propagate up
• Makes each parent consistent with children
• Propagate down
• Makes each child consistent with parent

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Projection in ToB
Project (ToB, K) //Project ToB over K variables
Conjoin BDDs from bottom to top Early
quantification of non-K variables
Worst case exponential in (wK)
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Equivalence Checking
IsEQ (ToB1, ToB2) Project ToB1 over vars in
each node of ToB2 -Check whether the two BDDs
agree Project ToB2 over vars in each node of
ToB1 -Check whether the two BDDs agree
Worst case exponential in (w1w2)