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Knowledge Compilation

Knowledge Compilation. Jinbo Huang NICTA and ANU. S l ides made by Adnan Darwiche and Jinbo Huang. B. C. A. Y. X. A  okX   B  A  okX  B B  okY   C  B  okY  C. KB =. Propositional Logic. Is A,  C a normal device behavior?. A  okX   B

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Knowledge Compilation

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  1. Knowledge Compilation Jinbo Huang NICTA and ANU A. Darwiche Slides made by Adnan Darwiche and Jinbo Huang

  2. B C A Y X A okX B A  okX  B B  okY C B  okY  C KB = Propositional Logic Is A, C a normal device behavior? A. Darwiche

  3. A okX B A  okX  B B  okY C B  okY  C A, C, okX, okY A okX B A  okX  B B  okY C B  okY  C Satisfiability Algorithm Propositional Reasoning Is A, C a normal device behavior? A. Darwiche

  4. Satisfiability (SAT) • SAT Solvers: Significant growth in last decade; many solvers publicly available (source code); millions of clauses not uncommon. • Applications: Verification, planning, diagnosis, CAD, non-propositional reasoning (e.g., SMT), … A. Darwiche

  5. Compiled Structure Compiler A okX B A  okX  B B  okY C B  okY  C Queries Evaluator(Polytime) Knowledge Compilation A. Darwiche

  6. ? Compiler A okX B A  okX  B B  okY C B  okY  C Queries Evaluator (Polytime) Knowledge Compilation A. Darwiche

  7. A okX B A  okX  B B  okY C B  okY  C Queries Evaluator (Polytime) Knowledge Compilation ..... Prime Implicates OBDD … Compiler A. Darwiche

  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 A. Darwiche

  9. Applications • Diagnosis • Is this a normal behavior? • What are the possible faults? • Planning • Can this goal be achieved? • Generate plan with highest reward • Generate plan with highest success probability • Probabilistic reasoning • What is the probability of X given Y • Formal verification / CAD: • Is it possible that the design will exhibit behavior X? • Are two designs equivalent? A. Darwiche

  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 A. Darwiche

  11. Agenda • Part I: Languages • Part II: Operations • Part III: Compilers • Part IV: Applications A. Darwiche

  12. Part I: Languages A. Darwiche

  13. 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 Succinctness A Knowledge Compilation MAP 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

  14. Propositional Logic • Literal • Clause • Term • CNF: Conjunctive Normal Form • DNF: Disjunctive Normal Form A. Darwiche

  15. Propositional Logic • Truth assignment (TA) • TA satisfies sentence (model) • Following TA is not a model A. Darwiche

  16. 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 Succinctness A Knowledge Compilation MAP 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

  17. Queries • Consistency (CO) • Validity (VA) • Sentential entailment (SE) • Clausal entailment (CE): KB implies clause • Implicant testing (IP): term implies KB • Equivalence testing (EQ) • Model Counting (CT) • Model enumeration (ME) A. Darwiche

  18. Transformations • Projection (existential quantification) • Conditioning • Conjoin • Disjoin • Negate A. Darwiche

  19. Representation vs Compilation Languags • Representation Language (intuitive):CNFDNF • Target Compilation Language (tractable): Binary Decision Diagrams (BDDs)DNNF A. Darwiche

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

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

  22. Decomposability or and and A,B C,D or or or or and and and and and and and and A B  B A C  D D  C A. Darwiche

  23. NNF Subsets NNF CO, CE, ME DNNF A. Darwiche

  24. 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

  25. Smoothness 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

  26. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF VA,IP,CT d-DNNF EQ? sd-DNNF A. Darwiche

  27. or and and and and X Y Z X Y Z Flatness Nested vs Flat languages A. Darwiche (XY Z)(ZXY) (YZX) (XY Z)

  28. Simple Conjunction or and and and and X Y Z X Y Z Simple conjunction implies decomposability A. Darwiche

  29. Simple Disjunction and or or or X Y Z X A. Darwiche

  30. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF f-NNF VA,IP,CT d-DNNF EQ? DNF CNF sd-DNNF A. Darwiche

  31. Resolution Prime Implicates (PI) • A CNF such that: • No clause subsumes another • If a clause is implied by the CNF, it must be implied by a clause in the CNF • CNF: • PI: A. Darwiche

  32. Consensus Prime Implicants (IP) • A DNF such that: • No term subsumes another • If a term implies the DNF, it must imply a term in the DNF • DNF: • IP: A. Darwiche

  33. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF f-NNF VA,IP,CT d-DNNF EQ? DNF CNF sd-DNNF CO,CE,MEVA,IP,SE,EQ VA,IP, SE,EQ IP PI A. Darwiche

  34. Decision or and and X a X b a, b: Are decision nodes A. Darwiche

  35. Decision or and and  X1 X1 or or and and and and  X2  X2 X2 X2 or or and and and and X3 X3  X3  X3 true false A. Darwiche

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

  37. or X1 and and  X1 X1 or or X2 X2 and and and and  X2  X2 X2 X2 X3 X3 or or and and and and X3 X3  X3  X3 1 0 true false Binary Decision Diagrams(BDDs) Decision implies determinism A. Darwiche

  38. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF f-NNF VA,IP,CT BDD d-DNNF EQ? DNF CNF sd-DNNF CO,CE,MEVA,IP,SE,EQ VA,IP, SE,EQ IP PI A. Darwiche

  39. or X1 and and  X1 X1 or or X2 X2 and and and and  X2  X2 X2 X2 X3 X3 or or and and and and X3 X3  X3  X3 1 0 true false Test once property Decision + decomposability = FBDD Binary Decision Diagrams(BDDs) A. Darwiche

  40. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF f-NNF VA,IP,CT BDD d-DNNF EQ? DNF CNF FBDD EQ? sd-DNNF CO,CE,MEVA,IP,SE,EQ SE,EQ VA,IP, SE,EQ MODS IP PI A. Darwiche

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

  42. NNF Subsets NNF CO, CE, ME d-NNF s-NNF DNNF f-NNF VA,IP,CT BDD d-DNNF EQ? DNF CNF FBDD EQ? sd-DNNF CO,CE,MEVA,IP,SE,EQ SE,EQ SE,EQ VA,IP, SE,EQ MODS IP OBDD PI A. Darwiche

  43. OBDD Example: Odd Parity X1 X2 X2 X3 X3 X4 X4 1 0 A. Darwiche Symmetric Functions

  44. L1 at least as succinct as L2 L1 <= L2 L1 is more succinct than L2 L1 < L2 Language Succinctness Size p(n) Size n A. Darwiche

  45. or and and A or or or or B B C C and and and and and and and and D D A B  B A C  D D  C 1 0 Odd Parity Function A. Darwiche

  46. PI <= DNNF DNF <= OBDD OBDD <= DNF DNNF <=? PI NNF DNNF CNF * = sd-DNNF d-DNNF DNF FBDD PI IP OBDD MODS

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

  48. Separating Functions • OBDD/FBDD: • Hidden weighted bit function hwb(x1,..,xn) • DNNF/DNF: • odd parity function parity(x1,..,xn) • DNNF/OBDD: • Distinct integers function distinct(x1,..,xn) A. Darwiche

  49. Agenda • Part I: Languages • Part II: Operations • Part III: Compilers • Part IV: Applications A. Darwiche

  50. Part II: Operations A. Darwiche

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