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On Deciding MUS Membership with QBF s Janota 1 Joao Marques-Silva 1 , 2 Mikol a 1 INESC-ID/IST, Lisbon, Portugal 2 CASL/CSI, University College Dublin, Ireland (INESC-ID & UCD) cmMUS 1 / 18 CNF and Unsatisfiability y , { x y ,

  1. On Deciding MUS Membership with QBF s Janota 1 Joao Marques-Silva 1 , 2 Mikol´ aˇ 1 INESC-ID/IST, Lisbon, Portugal 2 CASL/CSI, University College Dublin, Ireland (INESC-ID & UCD) cmMUS 1 / 18

  2. CNF and Unsatisfiability ¬ y , { x ∨ y , ¬ x , z } (INESC-ID & UCD) cmMUS 2 / 18

  3. CNF and Unsatisfiability ¬ y , { x ∨ y , ¬ x , z } (INESC-ID & UCD) cmMUS 2 / 18

  4. CNF and Unsatisfiability ¬ y , { x ∨ y , ¬ x , z } MUS An UNSAT set of clauses that becomes SAT by removing any clause is called m inimally u nsatisfiable s et (MUS) (INESC-ID & UCD) cmMUS 2 / 18

  5. CNF and Unsatisfiability ¬ y , { x ∨ y , ¬ x , z } MUS An UNSAT set of clauses that becomes SAT by removing any clause is called m inimally u nsatisfiable s et (MUS) MUS-Membership IN: a clause ω and a CNF φ Q: Is there an MUS ψ ⊆ φ such that ω ∈ ψ ? (INESC-ID & UCD) cmMUS 2 / 18

  6. Motivation Restoring Consistency Removing a clause that is not part of any MUS, will certainly not restore consistency. (INESC-ID & UCD) cmMUS 3 / 18

  7. Motivation Restoring Consistency Removing a clause that is not part of any MUS, will certainly not restore consistency. Product Configuration When configuring a product, some sets of its features result in an inconsistent configuration. Clearly, it is useful for the user(s) to know if a feature is relevant for the inconsistency. (INESC-ID & UCD) cmMUS 3 / 18

  8. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } (INESC-ID & UCD) cmMUS 4 / 18

  9. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } (INESC-ID & UCD) cmMUS 4 / 18

  10. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } (INESC-ID & UCD) cmMUS 4 / 18

  11. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } (INESC-ID & UCD) cmMUS 4 / 18

  12. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } (INESC-ID & UCD) cmMUS 4 / 18

  13. How Hard Is It? x 1 , x 1 → z , { x 2 , x 2 → z , y 1 , y 1 → ¬ z , y 2 , y 2 → ¬ z , ω } MUS-Membership is Σ P 2 -complete [Kul07] (INESC-ID & UCD) cmMUS 4 / 18

  14. Approaches to the Problem QBF 2 , ∃ , O ( n 2 ) QBF 3 , ∃ , O ( n ) MUS-membership MSS-membership QBF 2 , ∃ , O ( n ) Circ-Infer, O ( n ) [JMS11] [JGMS10] (INESC-ID & UCD) cmMUS 5 / 18

  15. Quantifying over Subsets Relaxation φ ∗ = { c ∨ r c | c ∈ φ } (INESC-ID & UCD) cmMUS 6 / 18

  16. Quantifying over Subsets Relaxation φ ∗ = { c ∨ r c | c ∈ φ } Relaxing Clauses Example φ = { x ∨ y , ¬ x , ¬ y } φ ∗ = { r 1 ∨ x ∨ y , r 2 ∨ ¬ x , r 3 ∨ ¬ y } (INESC-ID & UCD) cmMUS 6 / 18

  17. Quantifying over Subsets Relaxation φ ∗ = { c ∨ r c | c ∈ φ } Relaxing Clauses Example φ = { x ∨ y , ¬ x , ¬ y } φ ∗ = { r 1 ∨ x ∨ y , r 2 ∨ ¬ x , r 3 ∨ ¬ y } r 1 = 0 r 1 ∨ x ∨ y r 2 = 0 r 2 ∨ ¬ x r 3 = 1 r 3 ∨ ¬ y (INESC-ID & UCD) cmMUS 6 / 18

  18. Quantifying over Subsets Relaxation φ ∗ = { c ∨ r c | c ∈ φ } Relaxing Clauses Example φ = { x ∨ y , ¬ x , ¬ y } φ ∗ = { r 1 ∨ x ∨ y , r 2 ∨ ¬ x , r 3 ∨ ¬ y } r 1 = 0 r 1 ∨ x ∨ y r 2 = 0 r 2 ∨ ¬ x r 3 = 1 r 3 ∨ ¬ y (INESC-ID & UCD) cmMUS 6 / 18

  19. Modeling Elements Membership ∃ R . ¬ r ω (INESC-ID & UCD) cmMUS 7 / 18

  20. Modeling Elements Membership ∃ R . ¬ r ω Unsat ∃ R . ∀ X . ¬ φ ∗ ( R , X ) (INESC-ID & UCD) cmMUS 7 / 18

  21. Modeling Elements Membership ∃ R . ¬ r ω Unsat ∃ R . ∀ X . ¬ φ ∗ ( R , X ) Subset R = { r 1 , . . . , r n } , R ′ = { r ′ 1 , . . . , r ′ n } R < R ′ ≡ � � r i ⇒ r ′ ¬ r i ∧ r ′ i ∧ i r i ∈ R r i ∈ R (INESC-ID & UCD) cmMUS 7 / 18

  22. Na¨ ıve Approaches Schema exists ψ ⊆ φ s.t. ω ∈ ψ and ψ is unsatisfiable and forall ψ ′ � ψ is satisfiable (INESC-ID & UCD) cmMUS 8 / 18

  23. Na¨ ıve Approaches Schema exists ψ ⊆ φ s.t. ω ∈ ψ and ψ is unsatisfiable and forall ψ ′ � ψ is satisfiable 3-level quantification ∃ R . ¬ r ω ∧ ( ∀ X . ¬ φ ∗ ( R , X )) ∧ ( ∀ R ′ . ( R < R ′ ) ⇒ ∃ X ′ .φ ∗ ( R ′ , X ′ )) (INESC-ID & UCD) cmMUS 8 / 18

  24. Na¨ ıve Approaches Schema exists ψ ⊆ φ s.t. ω ∈ ψ and ψ is unsatisfiable and forall ψ ′ � ψ is satisfiable 3-level quantification ∃ R . ¬ r ω ∧ ( ∀ X . ¬ φ ∗ ( R , X )) ∧ ( ∀ R ′ . ( R < R ′ ) ⇒ ∃ X ′ .φ ∗ ( R ′ , X ′ )) 2-level quantification, O ( n 2 ) ∃ R . ¬ r ω ∧ ( ∀ X . ¬ φ ∗ ( R , X )) ∧ r ω i ∈ R ( ¬ r ω i ⇒ ∃ X ω i .φ ∗ [ r ω i / 1]( R , X ω i )) � (INESC-ID & UCD) cmMUS 8 / 18

  25. Na¨ ıve Approaches Schema exists ψ ⊆ φ s.t. ω ∈ ψ and ψ is unsatisfiable and forall ψ ′ � ψ is satisfiable 3-level quantification ∃ R . ¬ r ω ∧ ( ∀ X . ¬ φ ∗ ( R , X )) ∧ ( ∀ R ′ . ( R < R ′ ) ⇒ ∃ X ′ .φ ∗ ( R ′ , X ′ )) 2-level quantification, O ( n 2 ) ∃ R . ¬ r ω ∧ ( ∀ X . ¬ φ ∗ ( R , X )) ∧ r ω i ∈ R ( ¬ r ω i ⇒ ∃ X ω i .φ ∗ [ r ω i / 1]( R , X ω i )) � 2-level quantification, O ( n 2 ), prefix form ∃ RX ω 1 . . . ∃ X ω n ∀ X . ¬ r ω ∧ ¬ φ ∗ ( R , X ) ∧ r ω i ∈ R ( ¬ r ω i ⇒ φ ∗ [ r ω i / 1]( R , X ω i )) � (INESC-ID & UCD) cmMUS 8 / 18

  26. Approaches to the Problem QBF 2 , ∃ , O ( n 2 ) QBF 3 , ∃ , O ( n ) MUS-membership MSS-membership QBF 2 , ∃ , O ( n ) Circ-Infer, O ( n ) [JMS11] [JGMS10] (INESC-ID & UCD) cmMUS 9 / 18

  27. From MUS-Membership to MSS-Membership MSS A set of clauses ψ ⊆ φ is a Maximally Satisfiable Subset (MSS) iff ψ is satisfiable and any set ψ ′ ⊆ φ such that ψ � ψ ′ is unsatisfiable. (INESC-ID & UCD) cmMUS 10 / 18

  28. From MUS-Membership to MSS-Membership MSS A set of clauses ψ ⊆ φ is a Maximally Satisfiable Subset (MSS) iff ψ is satisfiable and any set ψ ′ ⊆ φ such that ψ � ψ ′ is unsatisfiable. MSS-membership IN: A CNF formula φ and a clause ω ∈ φ . Q: Is there an MSS ψ of φ such that ω / ∈ ψ ? (INESC-ID & UCD) cmMUS 10 / 18

  29. MUS-Membership ↔ MSS-membership A clause ω belongs to some MUS iff there is some MSS that does not contain ω . (INESC-ID & UCD) cmMUS 11 / 18

  30. MUS-Membership ↔ MSS-membership A clause ω belongs to some MUS iff there is some MSS that does not contain ω . MSS-membership to QBF ∃ R ∃ X ∀ R ′ ∀ X ′ . r ω ∧ φ ∗ ( R , X ) ∧ ( R ′ < R ⇒ ¬ φ ∗ ( R ′ , X ′ )) (INESC-ID & UCD) cmMUS 11 / 18

  31. Minimal Models A model of a formula is V-minimal iff flipping any subset of 1-values of variables from V to 0, yields a non-model. (INESC-ID & UCD) cmMUS 12 / 18

  32. Minimal Models A model of a formula is V-minimal iff flipping any subset of 1-values of variables from V to 0, yields a non-model. x ∨ ( y ∧ z ) [1,1,1] [1,1,0] [1,0,1] [0,1,1] [1,0,0] [0,1,0] [0,0,1] [0,0,0] (INESC-ID & UCD) cmMUS 12 / 18

  33. Minimal Models A model of a formula is V-minimal iff flipping any subset of 1-values of variables from V to 0, yields a non-model. x ∨ ( y ∧ z ) [1,1,1] [1,1,0] [1,0,1] [0,1,1] [1,0,0] [0,1,0] [0,0,1] [0,0,0] (INESC-ID & UCD) cmMUS 12 / 18

  34. Minimal Models A model of a formula is V-minimal iff flipping any subset of 1-values of variables from V to 0, yields a non-model. x ∨ ( y ∧ z ) [1,1,1] [0,1,1] [1,1,0] [1,0,1] [0,1,1] [1,0,0] [1,0,0] [0,1,0] [0,0,1] [0,0,0] (INESC-ID & UCD) cmMUS 12 / 18

  35. Entailment in Circumscription CircInfer IN: τ and ψ be propositional formulas Q: Does ψ hold in all minimal models of τ . τ | = min ψ (INESC-ID & UCD) cmMUS 13 / 18

  36. Entailment in Circumscription CircInfer IN: τ and ψ be propositional formulas Q: Does ψ hold in all minimal models of τ . τ | = min ψ CircInfer , complexity = min ψ is in Π P Deciding τ | 2 -complete [EG93] (INESC-ID & UCD) cmMUS 13 / 18

  37. MSSes and Minimal Models . . . r 1 x φ = { x , ¬ x , z } r 2 . . . ¬ x . . . r 3 z (INESC-ID & UCD) cmMUS 14 / 18

  38. MSSes and Minimal Models . . . r 1 x φ = { x , ¬ x , z } r 2 . . . ¬ x . . . r 3 z {} , [1,1,1] { x } , [0,1,1] {¬ x } , [1,0,1] { z } , [1,1,0] { x , ¬ x } , [0,0,1] { x , z } , [0,1,0] {¬ x , z } , [1,0,0] { x , ¬ x , z } , [0,0,0] (INESC-ID & UCD) cmMUS 14 / 18

  39. MSSes and Minimal Models . . . r 1 x φ = { x , ¬ x , z } r 2 . . . ¬ x . . . r 3 z {} , [1,1,1] { x } , [0,1,1] {¬ x } , [1,0,1] { z } , [1,1,0] { x , ¬ x } , [0,0,1] { x , z } , [0,1,0] {¬ x , z } , [1,0,0] { x , ¬ x , z } , [0,0,0] (INESC-ID & UCD) cmMUS 14 / 18

  40. MSSes and Minimal Models . . . r 1 x φ = { x , ¬ x , z } r 2 . . . ¬ x . . . r 3 z {} , [1,1,1] { x } , [0,1,1] {¬ x } , [1,0,1] { z } , [1,1,0] { x , ¬ x } , [0,0,1] { x , z } , [0,1,0] { x , z } , [0,1,0] {¬ x , z } , [1,0,0] {¬ x , z } , [1,0,0] { x , ¬ x , z } , [0,0,0] (INESC-ID & UCD) cmMUS 14 / 18

  41. From MSS-Membership to CircInfer MSSes ↔ Min. Models MSSes correspond to R -minimal models of φ ∗ ( R , X ). (INESC-ID & UCD) cmMUS 15 / 18

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