Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths - PowerPoint PPT Presentation

Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths Olaf Beyersdorff Joshua Blinkhorn Tom´ aˇ s Peitl Friedrich-Schiller-Universit¨ at Jena, Germany June 25, 2020

Dependencies Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 2 / 27 aˇ s Peitl

Dependencies Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 2 / 27 aˇ s Peitl

Dependencies Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 2 / 27 aˇ s Peitl

DQBF We consider (closed, prenex) dependency quantified Boolean formulas of the following form (a.k.a. S-form DQBF ): support universal existential variable variable set ���� ���� ���� Ψ = ∀ u 1 · · · ∀ u m ∃ x 1 ( S x 1 ) · · · ∃ x n ( S x n ) · C 1 ∧ · · · ∧ C r � �� � � �� � matrix prefix literal ���� u 1 ∨ ¬ x 2 ∨¬ u 3 ∨ x 4 � �� � clause A DQBF is true if there exist functions f x i : { 0 , 1 } S xi → { 0 , 1 } whose substitution for x i yields a propositional tautology. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 3 / 27 aˇ s Peitl

QBF DQBF extends QBF: block ���� Φ = ∀ U 1 ∃ X 1 ∀ U 2 ∃ X 2 · · · ∀ U k ∃ X k · C 1 ∧ · · · ∧ C r If x i ∈ X i , then S x i = � j < i U j . A DQBF is a QBF if and only if the support sets are linearly ordered under inclusion. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 4 / 27 aˇ s Peitl

Applications Deciding whether a given QBF is true is PSPACE-complete. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27 aˇ s Peitl

Applications Deciding whether a given QBF is true is PSPACE-complete. Deciding whether a given DQBF is true is NEXP-complete. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27 aˇ s Peitl

Applications Deciding whether a given QBF is true is PSPACE-complete. Deciding whether a given DQBF is true is NEXP-complete. DQBFs can be used to model various real-world problems arising in areas such as formal verification, synthesis, automated design of circuits, or games such as chess. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27 aˇ s Peitl

Applications Deciding whether a given QBF is true is PSPACE-complete. Deciding whether a given DQBF is true is NEXP-complete. DQBFs can be used to model various real-world problems arising in areas such as formal verification, synthesis, automated design of circuits, or games such as chess. We are interested in solving DQBFs as efficiently as possible. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27 aˇ s Peitl

Spurious Dependencies Consider the formula ∀ u ∃ x ( { u } ) · ( x ∨ u ) ∧ ( x ∨ ¬ u ). Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27 aˇ s Peitl

Spurious Dependencies Consider the formula ∀ u ∃ x ( { u } ) · ( x ∨ u ) ∧ ( x ∨ ¬ u ). It is obviously true by setting x := 1. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27 aˇ s Peitl

Spurious Dependencies Consider the formula ∀ u ∃ x ( { u } ) · ( x ∨ u ) ∧ ( x ∨ ¬ u ). It is obviously true by setting x := 1. But that does not need the dependency on u . Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27 aˇ s Peitl

Spurious Dependencies Consider the formula ∀ u ∃ x ( { u } ) · ( x ∨ u ) ∧ ( x ∨ ¬ u ). It is obviously true by setting x := 1. But that does not need the dependency on u . Hence, the dependency of x on u is spurious. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27 aˇ s Peitl

Dependency Schemes A dependency scheme as defined for QBF is a mapping: D : Φ �→ D (Φ) ⊆ D trv (Φ) = { ( x , y ) | x < y } Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27 aˇ s Peitl

Dependency Schemes A dependency scheme as defined for QBF is a mapping: D : Φ �→ D (Φ) ⊆ D trv (Φ) = { ( x , y ) | x < y } Prominent dependency schemes are the standard D std and the reflexive resolution-path D rrs ; Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27 aˇ s Peitl

Dependency Schemes A dependency scheme as defined for QBF is a mapping: D : Φ �→ D (Φ) ⊆ D trv (Φ) = { ( x , y ) | x < y } Prominent dependency schemes are the standard D std and the reflexive resolution-path D rrs ; First proposed by Samer and Szeider for backdoor sets, the definition has since evolved to accommodate different use cases; each of the following tools supports a dependency scheme in some form: DepQBF, Qute, HQSpre, CaQE, Qesto; Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27 aˇ s Peitl

Dependency Schemes A dependency scheme as defined for QBF is a mapping: D : Φ �→ D (Φ) ⊆ D trv (Φ) = { ( x , y ) | x < y } Prominent dependency schemes are the standard D std and the reflexive resolution-path D rrs ; First proposed by Samer and Szeider for backdoor sets, the definition has since evolved to accommodate different use cases; each of the following tools supports a dependency scheme in some form: DepQBF, Qute, HQSpre, CaQE, Qesto; Because dependency schemes were created for QBF, dependencies are defined both ways. This turned out unnecessary in the analysis of refutational proof systems, and becomes meaningless in DQBF. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27 aˇ s Peitl

Proof Systems A proof system is a set of rules that prescribe how to derive new clauses from existing ones Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl

Proof Systems ϕ 1 A proof system is a set of rules that prescribe how to derive new clauses from existing ones ϕ 2 A derivation in a proof system is a sequence of clauses each ... of which can be derived from previous clauses using the rules ϕ 3 ϕ 4 ... ϕ 5 ϕ 6 ⊥ Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl

Proof Systems ϕ 1 A proof system is a set of rules that prescribe how to derive new clauses from existing ones ϕ 2 A derivation in a proof system is a sequence of clauses each ... of which can be derived from previous clauses using the rules ϕ 3 A refutation is a derivation of the empty clause ϕ 4 ... ϕ 5 ϕ 6 ⊥ Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl

Proof Systems ϕ 1 A proof system is a set of rules that prescribe how to derive new clauses from existing ones ϕ 2 A derivation in a proof system is a sequence of clauses each ... of which can be derived from previous clauses using the rules ϕ 3 A refutation is a derivation of the empty clause ϕ 4 In particular, we are interested in ∀ Exp+Res and Q-Res ... ϕ 5 ϕ 6 ⊥ Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl

Proof Systems ϕ 1 A proof system is a set of rules that prescribe how to derive new clauses from existing ones ϕ 2 A derivation in a proof system is a sequence of clauses each ... of which can be derived from previous clauses using the rules ϕ 3 A refutation is a derivation of the empty clause ϕ 4 In particular, we are interested in ∀ Exp+Res and Q-Res A Q-Res refutation is a sequence of clauses that are either ... existential resolvents or universal reducts; ϕ 5 ϕ 6 ⊥ Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl

Proof Systems ϕ 1 A proof system is a set of rules that prescribe how to derive new clauses from existing ones ϕ 2 A derivation in a proof system is a sequence of clauses each ... of which can be derived from previous clauses using the rules ϕ 3 A refutation is a derivation of the empty clause ϕ 4 In particular, we are interested in ∀ Exp+Res and Q-Res A Q-Res refutation is a sequence of clauses that are either ... existential resolvents or universal reducts; ϕ 5 A ∀ Exp+Res refutation is a resolution refutation of the ϕ 6 universally expanded formula (a.k.a. Shannon expansion); ⊥ Olaf Beyersdorff, Joshua Blinkhorn, Tom´ Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27 aˇ s Peitl