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**Adnan Darwiche Computer Science Department UCLA**

Searching while Keeping a Trace The Evolution from SAT to Knowledge Compilation Adnan Darwiche Computer Science Department UCLA A. Darwiche

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**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 A. Darwiche

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**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 A. Darwiche

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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), … A. Darwiche

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**Knowledge Compilation**

A & okX => B A & okX => B B & okY => C B & okY => C Compiled Structure Compiler Evaluator (Polytime) Queries A. Darwiche

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**Knowledge Compilation**

A & okX => B A & okX => B B & okY => C B & okY => C ? Compiler Evaluator (Polytime) Queries A. Darwiche

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**Knowledge Compilation**

A & okX => B A & okX => B B & okY => C B & okY => C ..... Prime Implicates OBDD … Compiler Evaluator (Polytime) Queries A. Darwiche

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**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 A. Darwiche

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**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? A. Darwiche

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**Knowledge Compilation Map**

For a given application: identify needed operations Choose most succinct language that supports desired operations Compile knowledge base into chosen language A. Darwiche

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**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 A. Darwiche

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**Negation Normal Form A B B A C D D C and or**

rooted DAG (Circuit) A. Darwiche

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**Negation Normal Form Decomposability Determinism Smoothness Flatness**

and or Decomposability Determinism Smoothness Flatness Decision Ordering A. Darwiche

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**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 A. Darwiche

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**Determinism or and and or or or or and and and and and and and and A**

B B A C D D C A. Darwiche

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Decision or a b X and and X a X b A. Darwiche

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**Binary Decision Diagrams (BDDs)**

or and X1 X1 X2 X2 X3 X3 true false X1 X2 X3 1 Decision + decomposability = FBDD A. Darwiche

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**Binary Decision Diagrams (BDDs)**

or and X1 X1 X2 X2 X3 X3 true false X1 X2 X3 1 Decision + decomposability + ordering = OBDD A. Darwiche

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**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 A. Darwiche

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**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) A. Darwiche

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**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: A. Darwiche

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**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) A. Darwiche

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**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 A. Darwiche

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**The Language of Search Exhaustive DPLL Knowledge Compiler Record Trace**

Variations Languages A. Darwiche

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**Trace of DPLL X Y X Y Z X Y Z X Y 1 Z 1 unsat unsat sat**

Y 1 Z unsat 1 unsat sat A. Darwiche

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**Exhaustive DPLL Run to Exhaustion X Y X Y Z X Y Z X 1 Y**

1 Y Y 1 1 Z Z unsat sat 1 1 unsat sat unsat sat A. Darwiche

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**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 A. Darwiche

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**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 A. Darwiche

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**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 A. Darwiche

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**Dealing with Redundancy**

X Y Y Z Z unsat unsat unsat sat A. Darwiche

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**Dealing with Redundancy**

Simply point to existing node Do not create X Y Y Z Z unsat sat A. Darwiche

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This is an OBDD! X Y Y Z 1 A. Darwiche

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**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 A. Darwiche

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**A Non-traditional OBDD Compiler**

X Exhaustive DPLL, Fixed variable order, Unique nodes X Y X Y Z X Y Z Compile Y Y Z 1 New complexity guarantees A. Darwiche

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**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 A. Darwiche

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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 A. Darwiche

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**Dealing with Redundancy**

Level One: Unique nodes (done) Level Two: Avoid redundant compilation (searches) A. Darwiche

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**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 A. Darwiche

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**Formula Caching Majercik and Litmman, 1998 Darwiche, 2002**

Bacchus et al, 2003, 2004 Huang, 2004 Sang, Kautz, Beam, 2004, 2005 Thurley, 2006 A. Darwiche

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Beyond BDDs… Plain DPLL FBDD Fixed Variable Ordering OBDD A. Darwiche

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**Decomposition (Component Analysis)**

Solve disjoint subproblems independently d-DNNF Combine as AND node A. Darwiche

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**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 A. Darwiche

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**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 A. Darwiche

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FBDD vs d-DNNF d-DNNF more succinct than FBDD (effectiveness of decomposition) Deterministic equivalence test open Probabilistic equivalence test apply Other queries same… A. Darwiche

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**The Language of Search Fixed Variable Ordering OBDD Plain DPLL FBDD**

Allowing Decomposition d-DNNF Other languages: deterministic DNF A. Darwiche

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**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) A. Darwiche

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**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 A. Darwiche

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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 A. Darwiche

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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) A. Darwiche

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**Limitation of DPLL: General determinism**

or Decision nodes (d-DNNF’) and X X or Deterministic nodes (d-DNNF) A B C and or A B A B A. Darwiche

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**Beyond DPLL: Decomposability (D) without determinism (d)**

or DNNF: CO, CE, ME, exist quant and X3 or and X1 X2 A. Darwiche

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**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 A. Darwiche

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A. Darwiche

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**Knowledge Compilation Map, with Pierre Marquis DPLL with a Trace, **

Language of Search, with Jinbo Huang A. Darwiche

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**Multi-Linear Functions Arithmetic Circuits**

* + Factoring A B A. Darwiche

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**Propositional theory: Multi-linear function:**

MLFsACs CNFsd-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 A. Darwiche

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