Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Sylvie Coste-Marquis Daniel Le Berre Florian Letombe Pierre Marquis CRIL, CNRS FRE 2499 Lens, Universit´ e d’Artois, France Monday, July 11, 2005 1/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Introduction The qbf problem ◮ Canonical PSPACE-complete problem ◮ Can be used in many AI areas: planning, nonmonotonic reasoning, paraconsistent inference, abduction, etc ◮ High complexity, both in theory and in practice ◮ A possible solution: tractable classes ◮ Instances of those tractable classes hard for current qbf solvers (e.g. (renamable) Horn benchmarks) 2/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Introduction Outline QBF Target fragments Negation normal form Other propositional fragments Complexity results Complexity landscape A glimpse at some proofs A polynomial case Conclusion and perspectives 3/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF QBF: formal definition Definition (QBF) A QBF Π is an expression of the form ( n ≥ 0) Q 1 X 1 . . . Q n X n Φ , ◮ X 1 . . . X n sets of propositional variables ◮ Φ a propositional formula on those variables ◮ Q i (0 ≤ i ≤ n ) an existential ∃ or universal ∀ quantifier 4/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ∀ ) Example ∀ x ∃ y 1 , y 2 [( y 1 ∨ y 2 ) ∧ ( ¬ y 2 ∨ x ) ∧ ( ¬ y 1 ∨ ¬ y 2 ) ∧ ( y 2 ∨ ¬ x )] 5/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ∀ ) Example ∀ x ∃ y 1 , y 2 [( y 1 ∨ y 2 ) ∧ ( ¬ y 2 ∨ x ) / ∧ / / / / / / / / / / / / / / ( ¬ y 1 ∨ ¬ y 2 ) ∧ ( y 2 ∨¬ x / / / / / /)] x y 2 ¬ y 1 ⊤ 5/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ∀ ) Example ∀ x ∃ y 1 , y 2 [( y 1 ∨ y 2 ) ∧ ( ¬ y 2 ∨ x / / / /) ∧ ( ¬ y 1 ∨ ¬ y 2 ) ∧ ( y 2 ∨ ¬ x ) /] / / / / / / / / / / / / / / x ¬ y 2 y 2 ¬ y 1 y 1 ⊤ ⊤ 5/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ∀ ) Example ∀ x ∃ y 1 , y 2 ∃ y 1 ∀ x ∃ y 2 [( y 1 ∨ y 2 ) ∧ ( ¬ y 2 ∨ x ) ∧ [( y 1 ∨ y 2 ) ∧ ( ¬ y 2 ∨ x ) ∧ �≡ ( ¬ y 1 ∨ ¬ y 2 ) ∧ ( y 2 ∨ ¬ x )] ( ¬ y 1 ∨ ¬ y 2 ) ∧ ( y 2 ∨ ¬ x )] y 1 x ¬ y 2 y 2 x ¬ y 1 ∗ ¬ y 2 y 1 ⊤ ⊤ ⊥ ⊤ 5/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Negation normal form Definition ( NNF [Darwiche 1999]) A formula in NNF PS is a rooted DAG where: ◮ each leaf node is labeled with true , false , x or ¬ x , x ∈ PS ◮ each internal node is labeled with ∧ or ∨ and can have arbitrarily many children Example ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ¬ a ¬ b ¬ d ¬ c b a c d 6/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Properties [Darwiche 1999] ◮ Decomposability ◮ Determinism ◮ Smoothness ◮ Decision ◮ Ordering 7/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Fragments of NNF PS : examples Example ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ¬ a ¬ b ¬ d ¬ c b a c d 8/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Fragments of NNF PS : examples Decomposability : if C 1 , . . . , C n are the children of and-node C , then Var ( C i ) ∩ Var ( C j ) = ∅ for i � = j Example Decomposability ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ¬ a ¬ b ¬ d ¬ c b a c d 8/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Fragments of NNF PS : examples Determinism : if C 1 , . . . , C n are the children of or-node C , then C i ∧ C j | = false for i � = j Example Determinism ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ¬ a ¬ b ¬ d ¬ c b a c d 8/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Fragments of NNF PS : examples Smoothness : if C 1 , . . . , C n are the children of or-node C , then Var ( C i ) = Var ( C j ) Example Smoothness ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ¬ a ¬ b ¬ d ¬ c b a c d 8/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments Fragments of NNF PS : definitions Definition (Propositional fragments [Darwiche & Marquis 2001]) ◮ DNNF : NNF PS + decomposability. ◮ d-DNNF : NNF PS + decomposability and determinism. ◮ FBDD : NNF PS + decomposability and decision. ◮ OBDD < : NNF PS + decomposability, decision and ordering. ◮ MODS : DNF ∩ d-DNNF + smoothness. 9/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results Complexity landscape Complexity results for qbf Fragment Complexity PROP PS (general case) PSPACE-c PSPACE-c CNF PSPACE-c DNF PSPACE-c d-DNNF PSPACE-c DNNF PSPACE-c FBDD PSPACE-c OBDD < ∈ P OBDD < (compatible prefix) PSPACE-c PI PSPACE-c IP ∈ P MODS 10/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-NNF s-NNF DNNF f-NNF BDD d-DNNF FBDD sd-DNNF DNF CNF OBDD OBDD < MODS IP PI 11/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case ◮ Prefix compatible: < extension of the variable ordering induced by the prefix of the QBF ◮ Eliminating quantifiers from the innermost to the outermost ◮ Eliminating existential quantifiers ◮ Eliminating universal quantifiers ( ∀ x ≡ ¬∃ x ¬ ) ◮ Reduce the OBDD < at each elimination step ◮ Remark: Negation in constant time in OBDD < 12/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = ∃ x ∀ y ∃ z φ φ ≡ ( x ∨ y ) ∧ z 13/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = ∃ x ∀ y ∃ z φ φ ≡ ( x ∨ y ) ∧ z x φ = y ⊥ z ⊥ ⊤ 13/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = ∃ x ∀ y ∃ z φ φ ≡ ( x ∨ y ) ∧ z x ∃ z φ = y ⊥ ⊤ 13/15
Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = ∃ x ∀ y ∃ z φ φ ≡ ( x ∨ y ) ∧ z x ¬∃ z φ = y ⊤ ⊥ 13/15
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