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Experiences with QBF Solvers

Experiences with QBF Solvers. Sharad Malik Princeton University BMC Workshop Edinburgh July 11, 2005. Acknowledgements. Daijue Tang Yinlei Yu Zhaohui Fu Yogesh Mahajan Darsh Ranjan Lintao Zhang (now at Microsoft Research). Outline. QBF QBF algorithms

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Experiences with QBF Solvers

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  1. Experiences with QBF Solvers Sharad Malik Princeton University BMC Workshop Edinburgh July 11, 2005

  2. Acknowledgements • Daijue Tang • Yinlei Yu • Zhaohui Fu • Yogesh Mahajan • Darsh Ranjan • Lintao Zhang (now at Microsoft Research)

  3. Outline • QBF • QBF algorithms • Satisfiability driven learning • Long distance resolution • Validating QBF solvers • 2QBF • Sequential circuit state space diameter problem • 2QBF algorithms • Analysis

  4. Outline • QBF • QBF algorithms • Satisfiability driven learning • Long distance resolution • Validating QBF solvers • 2QBF • Sequential circuit state space diameter problem • 2QBF algorithms • Analysis

  5. Quantification Level 1 Quantification Level n Problem Formulation • Quantified Boolean Formula F: Q1X1······ QnXn where Qi (i=1,···, n) is either  or ,  is a propositional formula • Example: ue(u+e’)(u’+e) e4e5u1u2u3e1e2e3f(e1,e2,e3,e4,e5,u1,u2,u3) • QBF Problem: Is F satisfiable? • P-Space Complete, theoretically harder than NP-Complete problems such as SAT.

  6. Motivations • QBF has practical applications: • AI Planning • Sequential Circuit Verification • QBF has some similarities with SAT • Modern SAT solvers are very efficient and widely used in many fields • May be able to leverage SAT techniques

  7. Outline • QBF • QBF algorithms • Satisfiability driven learning • Long distance resolution • Validating QBF solvers • 2QBF • Sequential circuit state space diameter problem • 2QBF algorithms • Analysis

  8. Basic QBF Algorithms • Resolution based • Plaisted’s Algorithm • Search based

  9. a + b + c’ + f g + h’ + c + f a + b + g + h’ a + b + g + h’ + f Resolution • Resolution of a pair of clauses with exactly ONE incompatible variable

  10. Resolution Based QBF Algorithm false [BKF95] Hans Kleine Buning, Marek Karpinski, and Andreas Flogel. Resolution for Quantified Boolean Formulas. Information and Computation 117(1): 12-18 (1995).[Biere05]A. Biere. Resolve and Expand. In Proc. 7th Intl. Conf. on Theory and Applications of Satisfiability Testing (SAT'04), Lecture Notes in Computer Science (LNCS), Springer 2005.

  11. Plaisted’s Algorithm enumerate conflict assignments of u1 e1 through DPLL search cut: e2 e3 [PBZ03] David A. Plaisted, Armin Biere, Yunshan Zhu, A satisfiability procedure for quantified Boolean formulae, Discrete Applied Mathematics 130 (2003) 291-328.

  12. Search Based QBF Algorithms • Work by gradually assigning variables • A partial assignment  [KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)

  13. Search Based QBF Algorithms • Work by gradually assigning variables • A partial assignment  • Undetermined • Continue search [KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)

  14. Search Based QBF Algorithms • Work by gradually assigning variables • A partial assignment  • Undetermined • Conflict • Backtrack • Record the reason [KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)

  15. Search Based QBF Algorithms • Work by gradually assigning variables • A partial assignment  • Undetermined • Conflict • Satisfied • Backtrack • Determine the covered satisfying space [KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)

  16. Search Based QBF Algorithms • Work by gradually assigning variables • A partial assignment  • Undetermined • Conflict • Satisfied • The majority of QBF solvers are search based, the DPLL algorithm is an example of this

  17. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) Unknown True (1) False(0)

  18. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 Unknown True (1) False(0)

  19. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 Unknown True (1) u = 1 False(0)

  20. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 Conflicting Node Unknown True (1) u = 1 False(0)

  21. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 Backtrack Unknown True (1) u = 1 False(0)

  22. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 e = 0 Unknown True (1) u = 1 False(0)

  23. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 e = 0 Satisfying Node Unknown True (1) u = 1 u = 1 False(0)

  24. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 e = 0 Backtrack Unknown True (1) u = 1 u = 1 False(0)

  25. Basic DPLL Flow for QBF ey (e + y)(e’ + y’) e = 1 e = 0 Unknown True (1) u = 1 u = 1 u = 0 False(0)

  26. Basic DPLL Flow for QBF eu (e + u)(e’ + u’) e = 1 e = 0 Unknown True (1) u = 1 u = 1 u = 0 False(0)

  27. Basic DPLL Flow for QBF False eu (e + u)(e’ + u’) e = 1 e = 0 Unknown True (1) u = 1 u = 1 u = 0 False(0)

  28. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) Unknown True (1) False(0)

  29. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) u = 1 Unknown True (1) False(0)

  30. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) u = 1 Unknown True (1) e = 1 False(0)

  31. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) u = 1 Unknown True (1) e = 1 e = 0 False(0)

  32. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) u = 1 Unknown True (1) e = 1 e = 0 False(0)

  33. Basic DPLL Flow for QBF ue (u + e)(u’ + e’) u = 1 u = 0 Unknown True (1) e = 1 e = 1 e = 0 False(0)

  34. Basic DPLL Flow for QBF True ue (u + e)(u’ + e’) u = 1 u = 0 Unknown True (1) e = 1 e = 1 e = 0 False(0)

  35. Naïve DPLL Based Approach • Works on a CNF database • Backtracking is chronological • No learning is possible • In contrast, learning is critical for efficient SAT

  36. Quaffle’s Approach • A new data structure for the database • New deduction rules for this data structure • Conflict driven learning and satisfaction driven learning • Non-chronological backtracking by assertion

  37. Outline • QBF • QBF algorithms • Satisfiability driven learning [ZM02] • Long distance resolution • Validating QBF solvers • 2QBF • Sequential circuit state space diameter problem • 2QBF algorithms • Analysis [ZM02] L. Zhang and S. Malik. Towards Symmetric Treatment of Conflicts And Satisfaction in Quantified Boolean Satisfiability Solver. In Proc. of 8th International Conference on Principles and Practice of Constraint Programming (CP2002).

  38. CNF/DNF/ACNF/ADNF Let  = C1 C2…Cm = S1 + S2 +…+ Sn Then:  = (C1 C2…Cm + S1 + S2 +…+ Sn ) = C1 C2…Cm (S1 + S2 +…+ Sn) = (C1 C2…Cm + AnySubset{ S1, S2,…,Sn}) = (AnySubset{ C1,C2,…,Cm})(S1+ S2 +…+ Sn)

  39. ACNF • Definition: A Propositional formula  is said to be in Augmented CNF (ACNF)if •  = C1 C2…Cm+ S1 + S2 +……+ Sk • Where Ci’s are clauses, and Sj’s are cubes. • Each Sjis contained in the clause termC1 C2…Cm. i.e.i{1,2…k}, Si C1 C2…Cm • In ACNF, cubes are redundant • Example: f = a’b’c’+a’bc+ab’c+abc’ [DNF] = (a’+b’+c’)(a’+b+c)(a+b’+c)(a+b+c’) [CNF] = (a’+b’+c’)(a’+b+c)(a+b’+c)(a+b+c’) + a’b’c’+ ab’c [ACNF]

  40. free literal Clause Deduction Example F = (…)(…)(e1+e2’+u1+u2’)(…) • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula true literal false literal

  41. free literal Clause Deduction Example F = (…)(…)(e1+e2’+u1+u2’)(…) • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula Free literals are all univeral,conflicting clause! true literal false literal

  42. free literal Clause Deduction Example F = (…)(…)(e1+e2’+u1+u2’)(…) • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula If u1 and u2 have higher quantification level than e2,unit clause! true literal false literal

  43. free literal Clause Deduction Example • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula If u1 and u2 have higher quantification level than e2,unit clause! F = (…)(…)(e1+e2’+u1+u2’)(…) true literal Implication! false literal

  44. Unknown True (1) False(0) Deduction and Search Space Pruning

  45. Unknown True (1) False(0) Deduction and Search Space Pruning

  46. Deduction and Search Space Pruning Unknown True (1) Unit Clause False(0) Conflict

  47. free literal Cube Deduction Example F = (…)(…)(e1+e2’+u1+u2’)(…) + e3e4’u3 +… • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula true literal false literal

  48. free literal Cube Deduction Example • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula Free literals are all existential, satisfying cube! F = (…)(…)(e1+e2’+u1+u2’)(…) + e3e4’u3 +… true literal false literal

  49. free literal Cube Deduction Example • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula If e3 has higher quantification level than u3, unit cube! F = (…)(…)(e1+e2’+u1+u2’)(…) + e3e4’u3 +… true literal false literal

  50. free literal Cube Deduction Example • e1, e2, e3…… existential literals • Try to satisfy the formula • u1, u2, u3…… universal literals • Try to falsify the formula If e3 has higher quantification level than u3, unit cube! F = (…)(…)(e1+e2’+u1+u2’)(…) + e3e4’u3 +… true literal false literal Implication!

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