1. Adnan Darwiche Computer Science Department UCLA
Searching while Keeping a Trace The Evolution from SAT to Knowledge Compilation
Adnan Darwiche
Computer Science Department
UCLA
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2. Searching while Keeping a Trace The evolution from SAT to Knowledge Compilation
Satisfiability (SAT)
Knowledge compilation
The connection: The trace of search
Implications
Open questions
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3. Satisfiability (SAT) Is a set of Boolean constraints satisfiable?
Input to SAT is typically a CNF
SAT is mostly solved by DPLL search
A & okX => B
A & okX => B
B & okY => C
B & okY => C
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4. Satisfiability (SAT)
SAT Solvers: Significant growth in last decade; many solvers publicly available (source code); millions of variables & constraints not uncommon.
Applications: Verification, planning, diagnosis, CAD, non-propositional reasoning (e.g., SMTs), …
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5. Knowledge Compilation
A & okX => B
A & okX => B
B & okY => C
B & okY => C
Compiled
Structure
Compiler
Evaluator (Polytime)
Queries
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6. Knowledge Compilation
A & okX => B
A & okX => B
B & okY => C
B & okY => C ?
Compiler
Evaluator
(Polytime)
Queries
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7. Knowledge Compilation
A & okX => B
A & okX => B
B & okY => C
B & okY => C
.....
Prime Implicates
OBDD …
Compiler
Evaluator
(Polytime)
Queries
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8. Knowledge Compilation Map
What’s the space of possible target compilation languages?
Can it be synthesized in a semantically systematic way?
How do the languages compare?
Succinctness (relative size)
Operations they support in polytime
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9. Applications Diagnosis Planning Probabilistic reasoning
Is this a normal behavior?
What are the possible faults?
Planning
Can this goal be achieved?
Generate a set of plans
Probabilistic reasoning
What is the probability of X given Y
Non-monotonic reasoning (penalty logics)
Does X follow preferentially from Y
Formal verification / CAD:
Is it possible that the design will exhibit behavior X?
Are two designs equivalent?
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10. Knowledge Compilation Map
For a given application: identify needed operations
Choose most succinct language that supports desired operations
Compile knowledge base into chosen language
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11. A Knowledge Compilation MAP
Succinctness
Polytime Operations
Consistency (CO)
Validity (VA)
Clausal entailment (CE)
Sentential entailment (SE)
Implicant testing (IP)
Equivalence testing (EQ)
Model Counting (CT)
Model enumeration (ME)
Projection (exist. quantification)
Conditioning
Conjoin, Disjoin, Negate
Negation Normal Form or
Decomposability
Determinism
Smoothness
Flatness
Decision
Ordering
and
and or or or or
and
and
and
and
and
and
and
and A B
 B A C
 D D
 C
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12. Negation Normal Form A B  B A C  D D  C and or
rooted DAG (Circuit)
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13. Negation Normal Form Decomposability Determinism Smoothness Flatness
and or
Decomposability
Determinism
Smoothness
Flatness
Decision
Ordering
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14. Decomposability or and and or or or or and and and and and and and and
C,D or or or or
and
and
and
and
and
and
and
and A B
 B A C
 D D
 C
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15. Determinism or and and or or or or and and and and and and and and AB
 B A C
 D D
 C
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16. Decision or a b X
and
and X a X b
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17. Binary Decision Diagrams (BDDs)or
and X1
 X1 X2
 X2 X3
 X3
true
false X1 X2 X3 1
Decision + decomposability = FBDD
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18. Binary Decision Diagrams (BDDs)or
and X1
 X1 X2
 X2 X3
 X3
true
false X1 X2 X3 1
Decision + decomposability + ordering = OBDD
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19. NNF Subsets (2002) NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD DNF CNF
CO, CE, ME
d-NNF
s-NNF
DNNF
f-NNF
VA,IP,CT
BDD
d-DNNF
EQ?
FBDD
DNF
CNF
EQ?
sd-DNNF
CO,CE,ME VA,IP,SE,EQ
SE,EQ
SE,EQ
VA,IP, SE,EQ
OBDD
MODS IP PI
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20. Space Efficiency (succinctness)
Tractability & Succinctness
NNF
Tractable Operations
OBDD
FBDD
d-DNNF
DNNF
decomposability
Diagnosis,
Non-mon
Probabilistic
reasoning
determinism
decision
ordering
Space Efficiency (succinctness)
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21. Inference by Compiling to d-DNNF
Deterministic Conformant Planning
Blai Bonet and Hector Geffner (KR 2006)
Probabilistic Conformant Planning
Jinbo Huang (AIPS 2006)
Model-based diagnosis
Paul Elliott and Brian Williams (AAAI 2006)
Anthony Barrett (IJCAI 2005)
Databases (query re-write)
Yolife Arvelo, Blai Bonet and Maria Esther Vidal (AAAI 2006)
Inference in Bayesian Networks (2006 competition)
Mark Chavira, Adnan Darwiche (IJCAI 2005)
Inference in Probabilistic Relational Models
Mark Chavira, Adnan Darwiche and Manfred Jaeger (IJAR 2006)
c2d compiler:
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22. SAT by DPLL Search Unit resolution Conflict-directed backtracking
x  y
¬x  ¬z
¬w  z  ¬v
v  w  z
SAT?
x  y
¬x  ¬z
¬w  z
v = true
SAT?
x  y
¬x  ¬z
w  z
v = false
SAT?
Unit resolution
Conflict-directed backtracking
Clause learning
Branching heuristics
Restarts
Terminating condition for recursion:
empty set (satisfied), or empty clause (contradiction)
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23. Recent Trend: Exhaustive DPLL
Count number of models:
Model counters, e.g., relsat, cachet
Generate all/subset of models:
Image computation in model checking
SMTs (non-propositional reasoning)
Variations on DPLL Search
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24. The Language of Search Exhaustive DPLL Knowledge Compiler Record Trace
Variations
Languages
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25. Trace of DPLL X  Y X  Y  Z X  Y  Z X Y 1 Z 1 unsat unsat satY 1 Z
unsat 1
unsat
sat
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26. Exhaustive DPLL Run to Exhaustion X  Y X  Y  Z X  Y  Z X 1 Y1 Y Y 1 1 Z Z
unsat
sat 1 1
unsat
sat
unsat
sat
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27. Trace of DPLL: a Formula or X and and or X X or Y Y and and and and Z
unsat
sat Y Y Z Z Y Y 1
unsat
sat
unsat
sat
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28. Trace of DPLL: a Formula
Equivalent to original CNF
Tractable (e.g., count models)
and
and or X X or
and
and
and
and Y Y Z Z Y Y 1
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29. Dealing with Redundancy
Level One: Do not record redundant portions of trace
Level Two: Try not to solve equivalent subproblems X Y Y Z Z
unsat
sat
unsat
sat
unsat
sat
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30. Dealing with RedundancyX Y Y Z Z
unsat
unsat
unsat
sat
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31. Dealing with Redundancy
Simply point to existing node
Do not create X Y Y Z Z
unsat
sat
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32. This is an OBDD! X Y Y Z 1
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33. NNF + decision, decomposability, ordering
This is an OBDD! or X
and
and Y Y or X X or
and
and
and
and Z Z 1 Y Y 1
NNF + decision, decomposability, ordering
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34. A Non-traditional OBDD CompilerX
Exhaustive DPLL,
Fixed variable order,
Unique nodes
X  Y
X  Y  Z
X  Y  Z
Compile Y Y Z 1
New complexity guarantees
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35. FBDD X X  Y X  Y  Z X  Y  Z Y Z Compile Z Y 1
Exhaustive DPLL,
Dynamic variable order,
Unique nodes
X  Y
X  Y  Z
X  Y  Z Y Z
Compile Z Y 1
NNF + decision, decomposability
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36. FBDD vs OBDD
FBDD more succinct than OBDD (dynamic var ordering in sat)
Top-down vs bottom-up algorithms
OBDD: equivalence test (canonical)
FBDD: probabilistic equivalence test
Both allow model counting
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37. Dealing with Redundancy
Level One: Unique nodes (done)
Level Two: Avoid redundant compilation (searches)
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38. Redundant Compilation
x5  x6
x4  x5  x6
x1  x3  x4  x5
x2  x3
x1  x2  x3 X1 1 X2 X2 1 X3 X3 1 1
x5  x6
x4  x5  x6
x5  x6
x4  x5  x6
Formula Caching: complexity guarantees
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39. Formula Caching Majercik and Litmman, 1998 Darwiche, 2002
Bacchus et al, 2003, 2004
Huang, 2004
Sang, Kautz, Beam, 2004, 2005
Thurley, 2006
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40. Beyond BDDs…
Plain DPLL
FBDD
Fixed Variable Ordering
OBDD
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41. Decomposition (Component Analysis)
Solve disjoint subproblems independently
d-DNNF
Combine as AND node
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42. and and A B D B D C E 1 Deterministic Decomposable Negation Norm Form
(d-DNNF)
A  B  C A  B  C
A  D  E A  D  E A
B  C
D  E
and
and
B  C
D  E B D B D C E 1
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43. or and and and A A and or or or or and and and and B D B D C E
Deterministic Decomposable Negation Norm Form
(d-DNNF) or
and
and
and A A
and or or or or
and
and
and
and B D B D C E
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44. FBDD vs d-DNNF
d-DNNF more succinct than FBDD (effectiveness of decomposition)
Deterministic equivalence test open
Probabilistic equivalence test apply
Other queries same…
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45. The Language of Search Fixed Variable Ordering OBDD Plain DPLL FBDD
Allowing Decomposition
d-DNNF
Other languages: deterministic DNF
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46. Relation to AND/OR Search (CP)
AND/OR graphs are deterministic and decomposable
AND/OR search algorithms are doing enough work to compile networks into (multi-valued equivalent of) d-DNNF
Capable of more than answering a single query (model counting, belief revision, etc)
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47. Implications SAT techniques harnessed for knowledge compilation
c2d compiler based on Rsat Solver (SAT-race 06)
Language properties (succinctness/tractability) help characterize power and limitations of search
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48. Understanding DPLL
Take any program X that runs exhaustive DPLL-style search
Examine traces, if traces  L, then
X can answer all queries tractable for L
X is hopeless on any input having no polynomial-size representation in L
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49. Power of DPLL
Traces of several model counters (Relsat, Cachet, e.g.) are in d-DNNF
Are doing enough work to
compile formulas into d-DNNF
solve tasks beyond model counting (e.g., minimum cardinality, probabilistic equivalent testing)
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50. Limitation of DPLL: General determinismor
Decision nodes (d-DNNF’)
and X X or
Deterministic nodes (d-DNNF)
A B C
and or
A B A B
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51. Beyond DPLL: Decomposability (D) without determinism (d)or
DNNF: CO, CE, ME, exist quant
and
X3 or
and X1 X2
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52. Summary Overview of recent results in knowledge compilation
Overview of recent trends in exhaustive DPLL search
A connection between SAT and knowledge compilation (search trace):
SAT techniques harnessed for compilation into various languages
Language properties (succinctness, tractability) characterize power and limitations of search algorithms
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53.
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54. Knowledge Compilation Map, with Pierre Marquis DPLL with a Trace,
Language of Search, with Jinbo Huang
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56. Multi-Linear Functions  Arithmetic Circuits* +
Factoring A B
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57. Propositional theory: Multi-linear function:
MLFsACs CNFsd-DNNF
Propositional theory:
c ^ (a   b)
a c + a b c + c
Multi-linear function:
Encode
Compile  *  + c c
Decode   * *  + b b b 1 a a a 1
Smooth d-DNNF
Arithmetic Circuit
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