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

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A. Darwiche 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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A. Darwiche Is a set of Boolean constraints satisfiable? Input to SAT is typically a CNF SAT is mostly solved by DPLL search Satisfiability (SAT) A & okX => B B & okY => C

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A. Darwiche 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), … Satisfiability (SAT)

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

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

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

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A. Darwiche Knowledge Compilation Map Whats 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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A. Darwiche Diagnosis 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? Applications

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A. Darwiche 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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A. Darwiche 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 Decomposability Determinism Smoothness Flatness Decision Ordering Negation Normal Form A B B AC D D C and or and or A Knowledge Compilation MAP

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A. Darwiche A B B AC D D C and or and or rooted DAG (Circuit) Negation Normal Form

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A. Darwiche A B B AC D D C and or and or Decomposability Determinism Smoothness Flatness Decision Ordering Negation Normal Form

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A. Darwiche A B B AC D D C and or and or A,B C,D Decomposability

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A. Darwiche A B B AC D D C and or and or Determinism

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A. Darwiche X X and or and X Decision

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A. Darwiche X1 X2 X3 1 0 or and X1 or and X2 and X3 or truefalse Binary Decision Diagrams (BDDs) Decision + decomposability = FBDD

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A. Darwiche X1 X2 X3 1 0 or and X1 or and X2 and X3 or truefalse Binary Decision Diagrams (BDDs) Decision + decomposability + ordering = OBDD

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A. Darwiche NNF d-NNFs-NNFf-NNF sd-DNNF DNNF CO, CE, ME d-DNNF VA,IP,CT EQ? CNF DNF IP PI CO,CE,ME VA,IP,SE,EQ VA,IP, SE,EQ BDD FBDD EQ? OBDD SE,EQ MODS SE,EQ NNF Subsets (2002)

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A. Darwiche OBDD FBDD d-DNNF DNNF Space Efficiency (succinctness) Tractable Operations NNF decomposability determinism decision ordering Diagnosis, Non-mon Probabilistic reasoning Tractability & Succinctness

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A. Darwiche 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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A. Darwiche x y ¬x ¬z ¬w z ¬v v w z Terminating condition for recursion: empty set (satisfied), or empty clause (contradiction) SAT? x y ¬x ¬z w z v = false SAT? x y ¬x ¬z ¬w z v = true SAT? SAT by DPLL Search Unit resolution Conflict-directed backtracking Clause learning Branching heuristics Restarts

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A. Darwiche 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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A. Darwiche Knowledge Compiler Exhaustive DPLL Record Trace VariationsLanguages The Language of Search

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A. Darwiche Trace of DPLL X Y Z unsat sat X Y X Y Z

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A. Darwiche X YY ZZ unsat sat unsat sat X Y X Y Z Run to Exhaustion Exhaustive DPLL

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A. Darwiche X or X and Y Y0 or Y Y and or 1 Z Z X YY ZZ unsat sat unsat sat Trace of DPLL:a Formula

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A. Darwiche X or X and Y Y0 or Y Y and or 1 Z Z Equivalent to original CNF Tractable (e.g., count models) Trace of DPLL: a Formula

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A. Darwiche X YY ZZ unsat sat Level One: Do not record redundant portions of trace Level Two: Try not to solve equivalent subproblems Dealing with Redundancy

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A. Darwiche X YY ZZ unsat sat Dealing with Redundancy

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A. Darwiche X YY ZZ unsatsat Do not createSimply point to existing node Dealing with Redundancy

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A. Darwiche X YY Z 01 This is an OBDD!

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A. Darwiche X YY Z 01 X or X and 0 or Y Y and or 1 Z This is an OBDD! NNF + decision, decomposability, ordering

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A. Darwiche X Y X Y Z Compile 0 X YY Z 1 A Non-traditional OBDD Compiler Exhaustive DPLL, Fixed variable order, Unique nodes New complexity guarantees

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A. Darwiche 0 X YZ Z 1 Y FBDD Compile Exhaustive DPLL, Dynamic variable order, Unique nodes X Y X Y Z NNF + decision, decomposability

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A. Darwiche 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 FBDD vs OBDD

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

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A. Darwiche Redundant Compilation x 5 x 6 x 4 x 5 x 6 x 1 x 3 x 4 x 5 x 2 x 3 x 1 x 2 x 3 X1X1 X2X2 X2X2 X3X3 X3X x 5 x 6 x 4 x 5 x 6 x 5 x 6 x 4 x 5 x 6 Formula Caching: complexity guarantees

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

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A. Darwiche Combine as AND node d-DNNF Decomposition (Component Analysis) Solve disjoint subproblems independently

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A. Darwiche A B C A D E B C D E B C and 0 A B 1 DB C D E Deterministic Decomposable Negation Norm Form (d-DNNF)

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A. Darwiche or and A A or and B C or and D E or BD and Deterministic Decomposable Negation Norm Form (d-DNNF)

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

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A. Darwiche Plain DPLLFBDD Fixed Variable Ordering OBDD Allowing Decomposition d-DNNF The Language of Search Other languages: deterministic DNF

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A. Darwiche 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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A. Darwiche 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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A. Darwiche 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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A. Darwiche 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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A. Darwiche or and X X Decision nodes (d-DNNF) Deterministic nodes (d-DNNF) or A B C and or A B A B Limitation of DPLL: General determinism

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A. Darwiche Beyond DPLL: Decomposability (D) without determinism (d) or and X1X1 X2X2 X 3 DNNF: CO, CE, ME, exist quant

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A. Darwiche 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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A. Darwiche

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A. Darwiche Knowledge Compilation Map, with Pierre Marquis DPLL with a Trace, Language of Search, with Jinbo Huang

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A. Darwiche Multi-Linear Functions Arithmetic Circuits AB * * ** + ++ * ** * Factoring

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A. Darwiche a c + a b c + c Multi-linear function:Propositional theory: c ^ (a b) Encode c b 1 a 1 Arithmetic Circuit Decode c b b a a Smooth d-DNNF Compile MLFs ACs CNFs d-DNNF

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