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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

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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

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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)

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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

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae QBF

QBF: formal definition

Definition (QBF) A QBF Π is an expression of the form Q1X1 . . . QnXnΦ, (n ≥ 0)

◮ X1 . . . Xn sets of propositional variables ◮ Φ a propositional formula on those variables ◮ Qi(0 ≤ i ≤ n) an existential ∃ or universal ∀ quantifier

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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 ∃ y1, y2 [(y1 ∨ y2) ∧ (¬y2 ∨ x)∧ (¬y1 ∨ ¬y2) ∧ (y2 ∨ ¬x)]

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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 ∃ y1, y2 [(y1 ∨ y2)∧(¬y2 ∨ x) / / / / / / / / / / / / / / /∧ (¬y1 ∨ ¬y2) ∧ (y2∨¬x / / / / / /)]

⊤ y2 ¬y1 x

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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 ∃ y1, y2 [(y1 ∨ y2) ∧ (¬y2∨x / / / /)∧ (¬y1 ∨ ¬y2)∧(y2 ∨ ¬x) / / / / / / / / / / / / / / /]

⊤ ⊤ y2 ¬y1 ¬y2 y1 x

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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 ∃ y1, y2 [(y1 ∨ y2) ∧ (¬y2 ∨ x)∧ (¬y1 ∨ ¬y2) ∧ (y2 ∨ ¬x)]

⊤ ⊤ y2 ¬y1 ¬y2 y1 x

≡ ∃ y1 ∀ x ∃ y2 [(y1 ∨ y2) ∧ (¬y2 ∨ x)∧ (¬y1 ∨ ¬y2) ∧ (y2 ∨ ¬x)]

y1 x ¬y2 ⊥ ∗ ⊤

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Negation normal form

Definition (NNF [Darwiche 1999]) A formula in NNFPS 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

b a c d ¬a ¬b ¬d ¬c ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Properties [Darwiche 1999]

◮ Decomposability ◮ Determinism ◮ Smoothness ◮ Decision ◮ Ordering

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Fragments of NNFPS: examples

Example

b a c d ¬a ¬b ¬d ¬c ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Fragments of NNFPS: examples

Decomposability: if C1, . . . , Cn are the children of and-node C, then Var(Ci) ∩ Var(Cj) = ∅ for i = j Example Decomposability

b a c d ¬a ¬b ¬d ¬c ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Fragments of NNFPS: examples

Determinism: if C1, . . . , Cn are the children of or-node C, then Ci ∧ Cj | = false for i = j Example Determinism

b a c d ¬a ¬b ¬d ¬c ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Fragments of NNFPS: examples

Smoothness: if C1, . . . , Cn are the children of or-node C, then Var(Ci) = Var(Cj) Example Smoothness

b a c d ¬a ¬b ¬d ¬c ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Target fragments Other propositional fragments

Fragments of NNFPS: definitions

Definition (Propositional fragments [Darwiche & Marquis 2001])

◮ DNNF: NNFPS + decomposability. ◮ d-DNNF: NNFPS + decomposability and determinism. ◮ FBDD: NNFPS + decomposability and decision. ◮ OBDD<: NNFPS + decomposability, decision and ordering. ◮ MODS: DNF ∩ d-DNNF + smoothness.

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Complexity results Complexity landscape

Complexity results for qbf

Fragment Complexity PROPPS (general case) PSPACE-c CNF PSPACE-c DNF PSPACE-c d-DNNF PSPACE-c DNNF PSPACE-c FBDD PSPACE-c OBDD< PSPACE-c OBDD< (compatible prefix) ∈ P PI PSPACE-c IP PSPACE-c MODS ∈ P

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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 DNNF f-NNF BDD FBDD OBDD OBDD< MODS DNF CNF d-DNNF IP PI sd-DNNF s-NNF

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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<

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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

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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

φ = x y ⊥ z ⊥ ⊤

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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 y ⊥ ∃z φ = ⊤

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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 y ⊤ ¬∃z φ = ⊥

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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 ⊤ ⊥ ∃y¬∃z φ =

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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 ⊥ ⊤ ¬∃y¬ ∃z φ = ∀y

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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∀y∃z φ = ⊤ ⇒ Σ is valid

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Conclusion and perspectives

Conclusion

◮ Presentation of propositional fragments ◮ Numerous intractable fragments ◮ Some tractability results (under very restrictive conditions)

Perspectives

◮ Study of other (polynomial) complete or incomplete fragments ◮ Integrate specialized algorithms into our solver ◮ A sharper analysis of the prefix in order to improve our solver

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Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae That’s all folks!

Any questions ?

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