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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 2 / 27

Dependencies

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

Dependencies

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

DQBF

We consider (closed, prenex) dependency quantified Boolean formulas of the following form (a.k.a. S-form DQBF): Ψ =

universal variable

• ∀u1 · · · ∀um

existential variable

• ∃x1(Sx1) · · · ∃xn

support set

• (Sxn)
• prefix

· C1 ∧ · · · ∧ Cr

• matrix

u1 ∨

literal

• ¬x2 ∨¬u3 ∨ x4
• clause

A DQBF is true if there exist functions fxi : {0, 1}Sxi → {0, 1} whose substitution for xi yields a propositional tautology.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 3 / 27

QBF

DQBF extends QBF: Φ =

block

• ∀U1 ∃X1∀U2∃X2 · · · ∀Uk∃Xk · C1 ∧ · · · ∧ Cr

If xi ∈ Xi, then Sxi =

j<i Uj.

A DQBF is a QBF if and only if the support sets are linearly ordered under inclusion.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 4 / 27

Applications

Deciding whether a given QBF is true is PSPACE-complete.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 5 / 27

Spurious Dependencies

Consider the formula ∀u∃x({u}) · (x ∨ u) ∧ (x ∨ ¬u).

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27

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´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 6 / 27

Dependency Schemes

A dependency scheme as defined for QBF is a mapping: D : Φ → D(Φ) ⊆ Dtrv(Φ) = {(x, y) | x < y}

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27

Dependency Schemes

A dependency scheme as defined for QBF is a mapping: D : Φ → D(Φ) ⊆ Dtrv(Φ) = {(x, y) | x < y} Prominent dependency schemes are the standard Dstd and the reflexive resolution-path Drrs;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 7 / 27

Dependency Schemes

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

Dependency Schemes

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

Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones

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Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones A derivation in a proof system is a sequence of clauses each

• f which can be derived from previous clauses using the rules

ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ1 ⊥ ... ...

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27

Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones A derivation in a proof system is a sequence of clauses each

• f which can be derived from previous clauses using the rules

A refutation is a derivation of the empty clause

ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ1 ⊥ ... ...

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27

Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones A derivation in a proof system is a sequence of clauses each

• f which can be derived from previous clauses using the rules

A refutation is a derivation of the empty clause In particular, we are interested in ∀Exp+Res and Q-Res

ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ1 ⊥ ... ...

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27

Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones A derivation in a proof system is a sequence of clauses each

• f which can be derived from previous clauses using the rules

A refutation is a derivation of the empty clause 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;

ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ1 ⊥ ... ...

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27

Proof Systems

A proof system is a set of rules that prescribe how to derive new clauses from existing ones A derivation in a proof system is a sequence of clauses each

• f which can be derived from previous clauses using the rules

A refutation is a derivation of the empty clause 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; A ∀Exp+Res refutation is a resolution refutation of the universally expanded formula (a.k.a. Shannon expansion);

ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ1 ⊥ ... ...

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 8 / 27

Sound Use of Dependency Schemes

Dependency analysis using a dependency scheme D in reasoning based on a proof system P is captured by adding D to P, resulting in a proof system P(D);

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 9 / 27

Sound Use of Dependency Schemes

Dependency analysis using a dependency scheme D in reasoning based on a proof system P is captured by adding D to P, resulting in a proof system P(D); The goal is to show that P(D) is sound and stronger than just P;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 9 / 27

Sound Use of Dependency Schemes

Dependency analysis using a dependency scheme D in reasoning based on a proof system P is captured by adding D to P, resulting in a proof system P(D); The goal is to show that P(D) is sound and stronger than just P; Defining P(D) and proving its soundness can be highly non-trivial.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 9 / 27

Sound Use of Dependency Schemes

Dependency analysis using a dependency scheme D in reasoning based on a proof system P is captured by adding D to P, resulting in a proof system P(D); The goal is to show that P(D) is sound and stronger than just P; Defining P(D) and proving its soundness can be highly non-trivial.

Theorem ([SS16])

A QBF is false if, and only if, it has a Q(Drrs,Dstd)-Res refutation.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 9 / 27

Recap

We are

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 10 / 27

Recap

We are trying to solve a DQBF;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 10 / 27

Recap

We are trying to solve a DQBF; identify as many spurious dependencies as possible;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 10 / 27

Recap

We are trying to solve a DQBF; identify as many spurious dependencies as possible; while maintaining soundness of the proof system.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 10 / 27

Overview of Contributions

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 11 / 27

Overview of Contributions

1 A clean DQBF-centric definition of dependency schemes along with a characterisation of

when a dependency scheme can be used in any DQBF proof system;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 11 / 27

Overview of Contributions

1 A clean DQBF-centric definition of dependency schemes along with a characterisation of

when a dependency scheme can be used in any DQBF proof system;

2 A new, tautology-free dependency scheme Dtf that generalizes the to-date strongest

known resolution-path dependency scheme;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 11 / 27

Overview of Contributions

1 A clean DQBF-centric definition of dependency schemes along with a characterisation of

when a dependency scheme can be used in any DQBF proof system;

2 A new, tautology-free dependency scheme Dtf that generalizes the to-date strongest

known resolution-path dependency scheme;

3 DQBF-genuine exponential separations of ∀Exp+Res with and without Drrs and Dtf; Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 11 / 27

Overview of Contributions

1 A clean DQBF-centric definition of dependency schemes along with a characterisation of

when a dependency scheme can be used in any DQBF proof system;

2 A new, tautology-free dependency scheme Dtf that generalizes the to-date strongest

known resolution-path dependency scheme;

3 DQBF-genuine exponential separations of ∀Exp+Res with and without Drrs and Dtf; 4 QBF-genuine exponential separations of Q-Res with Drrs and with Dtf. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 11 / 27

DQBF Dependency Schemes

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 12 / 27

Dependency Schemes for DQBF

We define a DQBF dependency scheme D as a mapping between DQBFs that

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 13 / 27

Dependency Schemes for DQBF

We define a DQBF dependency scheme D as a mapping between DQBFs that

1 preserves the matrix and does not enlarge support sets:

D(Ψ) ≤ Ψ

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 13 / 27

Dependency Schemes for DQBF

We define a DQBF dependency scheme D as a mapping between DQBFs that

1 preserves the matrix and does not enlarge support sets:

D(Ψ) ≤ Ψ

2 is polynomial-time computable. Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 13 / 27

Dependency Schemes for DQBF

We define a DQBF dependency scheme D as a mapping between DQBFs that

1 preserves the matrix and does not enlarge support sets:

D(Ψ) ≤ Ψ

2 is polynomial-time computable.

We say that a dependency scheme D is fully exhibited if D(Ψ)

tr

≡ Ψ for every DQBF Ψ.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 13 / 27

Parameterising Proof Systems

Definition (P(D))

Let P be a DQBF proof system and let D be a dependency scheme. A P(D) refutation of a DQBF Ψ is a P refutation of D(Ψ).

Proposition

Given a DQBF proof system P and a dependency scheme D, P(D) is sound and complete if, and only if, D is fully exhibited.

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The Tautology-free Dependency Scheme

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Definition (Drrs [SS16] and Dtf)

The reflexive resolution path dependency scheme (Drrs) is defined as the mapping Ψ → Ψ′, where Ψ := ∀u1 · · · ∀um∃x1(Sx1) · · · ∃xn(Sxn) · ψ , Ψ′ := ∀u1 · · · ∀um∃x1(S′

x1) · · · ∃xn(S′ xn) · ψ ,

and S′

xi is the set of universal variables u ∈ Sxi for which there exists a sequence C1, . . . , Ck of

clauses in ψ and a sequence p1, . . . , pk−1 of existential literals satisfying the following conditions: (a) u ∈ C1 and u ∈ Ck; (b) for some j ∈ [k − 1], xi = var(pj); (c) for each j ∈ [k − 1], pj ∈ Cj, pj ∈ Cj+1, and u ∈ Svar(pj); (d) for each j ∈ [k − 2], var(pj) = var(pj+1). The tautology-free dependency scheme (Dtf) adds to Drrs the condition (e) for each j ∈ [k − 1], (Cj ∪ Cj+1)↾I∃(Ψ) is non-tautological.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 16 / 27

Dtf Intuition

Drrs identifies potential information flows between variables as resolution paths;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 17 / 27

Dtf Intuition

Drrs identifies potential information flows between variables as resolution paths; A resolution path is a sequence of clauses which can trigger unit propagation under a suitable assignment;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 17 / 27

Dtf Intuition

Drrs identifies potential information flows between variables as resolution paths; A resolution path is a sequence of clauses which can trigger unit propagation under a suitable assignment; If a resolution path connects u and x, then assigning u may affect the choices for x;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 17 / 27

Dtf Intuition

Drrs identifies potential information flows between variables as resolution paths; A resolution path is a sequence of clauses which can trigger unit propagation under a suitable assignment; If a resolution path connects u and x, then assigning u may affect the choices for x; However, certain resolution paths are blocked: they contain tautologies on variables that are “already assigned at the time” u is assigned, such as the independent existential variables I∃(Ψ).

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 17 / 27

Example (Drrs vs. Dtf)

∀u∃x(∅)∃z({u}) · (x ∨ u ∨ z) ∧ (¬x ∨ ¬u ∨ ¬z)

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 18 / 27

Example (Drrs vs. Dtf)

∀u∃x(∅)∃z({u}) · (x ∨ u ∨ z) ∧ (¬x ∨ ¬u ∨ ¬z) The two clauses (x ∨ u ∨ z) and (¬x ∨ ¬u ∨ ¬z) constitute a resolution path that connects u and z. Indeed, if x is set to false, the first clause simplifies to the implication ¬u = ⇒ z, and if x is set to true, the second clause simplifies to u = ⇒ ¬z. The value

• f u may potentially force either value of z.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 18 / 27

Example (Drrs vs. Dtf)

∀u∃x(∅)∃z({u}) · (x ∨ u ∨ z) ∧ (¬x ∨ ¬u ∨ ¬z) The two clauses (x ∨ u ∨ z) and (¬x ∨ ¬u ∨ ¬z) constitute a resolution path that connects u and z. Indeed, if x is set to false, the first clause simplifies to the implication ¬u = ⇒ z, and if x is set to true, the second clause simplifies to u = ⇒ ¬z. The value

• f u may potentially force either value of z.

Accordingly, Drrs identifies z as truly dependent on u.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 18 / 27

Example (Drrs vs. Dtf)

∀u∃x(∅)∃z({u}) · (x ∨ u ∨ z) ∧ (¬x ∨ ¬u ∨ ¬z) The two clauses (x ∨ u ∨ z) and (¬x ∨ ¬u ∨ ¬z) constitute a resolution path that connects u and z. Indeed, if x is set to false, the first clause simplifies to the implication ¬u = ⇒ z, and if x is set to true, the second clause simplifies to u = ⇒ ¬z. The value

• f u may potentially force either value of z.

Accordingly, Drrs identifies z as truly dependent on u. But x has to be set “before” z, because it does not depend on anything. Hence one of the implications is always killed. In other words, the union of the clauses, restricted to independent existential variables, is a tautology.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 18 / 27

Example (Drrs vs. Dtf)

∀u∃x(∅)∃z({u}) · (x ∨ u ∨ z) ∧ (¬x ∨ ¬u ∨ ¬z) The two clauses (x ∨ u ∨ z) and (¬x ∨ ¬u ∨ ¬z) constitute a resolution path that connects u and z. Indeed, if x is set to false, the first clause simplifies to the implication ¬u = ⇒ z, and if x is set to true, the second clause simplifies to u = ⇒ ¬z. The value

• f u may potentially force either value of z.

Accordingly, Drrs identifies z as truly dependent on u. But x has to be set “before” z, because it does not depend on anything. Hence one of the implications is always killed. In other words, the union of the clauses, restricted to independent existential variables, is a tautology. Dtf detects the tautology and concludes that z is independent of u. Indeed x → 0 and z → 1 is a model that exhibits this.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 18 / 27

Properties of Dtf

Proposition

Dtf is a monotone dependency scheme, i.e. Ψ ≤ Ψ′ = ⇒ Dtf(Ψ) ≤ Dtf(Ψ′).

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 19 / 27

Properties of Dtf

Proposition

Dtf is a monotone dependency scheme, i.e. Ψ ≤ Ψ′ = ⇒ Dtf(Ψ) ≤ Dtf(Ψ′).

Theorem

Dtf is fully exhibited.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 19 / 27

Properties of Dtf

Proposition

Dtf is a monotone dependency scheme, i.e. Ψ ≤ Ψ′ = ⇒ Dtf(Ψ) ≤ Dtf(Ψ′).

Theorem

Dtf is fully exhibited.

Proof.

By reduction to full exhibition of Drrs established for DQBF by Wimmer et al. [WSWB16]. If Ψ is true, pick a satisfying assignment α to I∃(Ψ), and restrict with it. Because Ψ[α] has no independent existential variables, Dtf reduces to Drrs and the theorem follows by full exhibition

• f Drrs.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 19 / 27

Properties of Dtf

Proposition

Dtf is a monotone dependency scheme, i.e. Ψ ≤ Ψ′ = ⇒ Dtf(Ψ) ≤ Dtf(Ψ′).

Theorem

Dtf is fully exhibited.

Corollary

Dtf can be plugged in into any proof system, in particular ∀Exp+Res.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 19 / 27

Separations

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 20 / 27

Genuine DQBF Separations

Extending the notion of genuine QBF hardness [Che17, BHP20], we define genuine DQBF separations as such separations, where the hardness is not witnessed by any embedded QBF family.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 21 / 27

Genuine DQBF Separations

Extending the notion of genuine QBF hardness [Che17, BHP20], we define genuine DQBF separations as such separations, where the hardness is not witnessed by any embedded QBF family.

Definition

Let P and Q be DQBF proof systems. We write Q ∗

p P when there exists a DQBF family

{Ψn}n∈N such that: (a) {Ψn}n∈N has polynomial-size Q refutations; (b) {Ψn}n∈N requires superpolynomial-size P refutations; (c) every QBF family {Φn}n∈N with Φn ≤ Ψn has polynomial-size P refutations. We write P <∗

p Q when both P ≤p Q and Q ∗ p P hold.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 21 / 27

Main Theorem

Theorem

∀Exp+Res <∗

p ∀Exp+Res(Drrs) <∗ p ∀Exp+Res(Dtf).

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Main Theorem

Theorem

∀Exp+Res <∗

p ∀Exp+Res(Drrs) <∗ p ∀Exp+Res(Dtf).

Definition (EQ0

n (adapted from [BBH19]))

EQ0

n := ΠEQ n

· ψEQ

n , where

ΠEQ

n

:= ∀u1 · · · ∀un∃x1(∅) · · · ∃xn(∅) ∃z1(u1) · · · ∃zn(un) , ψEQ

n

:= (z1 ∨ · · · ∨ zn) ∧

n

• i=1
• (xi ∨ ui ∨ zi) ∧ (xi ∨ ui ∨ zi)
• .

Human readably: there are xi and zi depending on ui such that for all values of the ui if ui = xi, then zi, but not all zi.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 22 / 27

First Separation

Theorem

{EQ0

n}n∈N requires exponential-size ∀Exp+Res refutations.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 23 / 27

First Separation

Theorem

{EQ0

n}n∈N requires exponential-size ∀Exp+Res refutations.

Proposition ( [BB19])

For all n, the dependency sets of Drrs(EQ0

n) are empty.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 23 / 27

First Separation

Theorem

{EQ0

n}n∈N requires exponential-size ∀Exp+Res refutations.

Proposition ( [BB19])

For all n, the dependency sets of Drrs(EQ0

n) are empty.

Theorem ( [BB19])

{EQ0

n}n∈N has linear-size ∀Exp+Res(Drrs) refutations.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 23 / 27

Second Separation

Definition (EQ1

n (adapted from [BB17]))

For each natural number n, EQ1

n := ΠEQ n

∃r(∅) ∃s({u1, . . . , un}) ·

• ψEQ

n

⊗ (r ∨ s)

• ∧
• ψEQ

n

⊗ (r ∨ s)

• ∧ (r ∨ s) ∧ (r ∨ s) .

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 24 / 27

Second Separation

Definition (EQ1

n (adapted from [BB17]))

For each natural number n, EQ1

n := ΠEQ n

∃r(∅) ∃s({u1, . . . , un}) ·

• ψEQ

n

⊗ (r ∨ s)

• ∧
• ψEQ

n

⊗ (r ∨ s)

• ∧ (r ∨ s) ∧ (r ∨ s) .

Proposition

For each n, Drrs(EQ1

n) = EQ1 n and the dependency sets of Dtf(EQ1 n) are all empty.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 24 / 27

Second Separation

Definition (EQ1

n (adapted from [BB17]))

For each natural number n, EQ1

n := ΠEQ n

∃r(∅) ∃s({u1, . . . , un}) ·

• ψEQ

n

⊗ (r ∨ s)

• ∧
• ψEQ

n

⊗ (r ∨ s)

• ∧ (r ∨ s) ∧ (r ∨ s) .

Proposition

For each n, Drrs(EQ1

n) = EQ1 n and the dependency sets of Dtf(EQ1 n) are all empty.

Theorem

Hence, {EQ1

n}n∈N requires exponential-size ∀Exp+Res(Drrs) refutations, but has linear-size

∀Exp+Res(Dtf) refutations.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 24 / 27

Summary

We proposed a clean framework for DQBF dependency schemes and their proof complexity centered around the notion of full exhibition;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 25 / 27

Summary

We proposed a clean framework for DQBF dependency schemes and their proof complexity centered around the notion of full exhibition; We defined a novel dependency scheme Dtf based on the intuition about how resolution paths do and do not transfer information between variables;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 25 / 27

Summary

We proposed a clean framework for DQBF dependency schemes and their proof complexity centered around the notion of full exhibition; We defined a novel dependency scheme Dtf based on the intuition about how resolution paths do and do not transfer information between variables; We showed that Dtf is fully exhibited;

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 25 / 27

Summary

We proposed a clean framework for DQBF dependency schemes and their proof complexity centered around the notion of full exhibition; We defined a novel dependency scheme Dtf based on the intuition about how resolution paths do and do not transfer information between variables; We showed that Dtf is fully exhibited; We showed that the use of Dtf in both ∀Exp+Res and Q-Res results in exponentially shorter proofs compared to Drrs.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 25 / 27

Summary

We proposed a clean framework for DQBF dependency schemes and their proof complexity centered around the notion of full exhibition; We defined a novel dependency scheme Dtf based on the intuition about how resolution paths do and do not transfer information between variables; We showed that Dtf is fully exhibited; We showed that the use of Dtf in both ∀Exp+Res and Q-Res results in exponentially shorter proofs compared to Drrs. A short remark on the Equality formulas: the QBF template is hard for both proof system, but our DQBF version is only hard for ∀Exp+Res and becomes easy in Q-Res. Why?

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 25 / 27

References I

Joshua Blinkhorn and Olaf Beyersdorff, Shortening QBF proofs with dependency schemes, International Conference on Theory and Practice of Satisfiability Testing (SAT) (Serge Gaspers and Toby Walsh, eds.), Lecture Notes in Computer Science, vol. 10491, Springer, 2017, pp. 263–280. Olaf Beyersdorff and Joshua Blinkhorn, Dynamic QBF dependencies in reduction and expansion, ACM Transactions on Computational Logic 21 (2019), no. 2, 1–27. Olaf Beyersdorff, Joshua Blinkhorn, and Luke Hinde, Size, cost, and capacity: A semantic technique for hard random QBFs, Logical Methods in Computer Science 15 (2019), no. 1. Olaf Beyersdorff, Luke Hinde, and J´ an Pich, Reasons for hardness in QBF proof systems, ACM Transactions on Computation Theory 12 (2020), no. 2, 10:1–10:27. Hubie Chen, Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness, ACM Transactions on Computation Theory 9 (2017),

• no. 3, 15:1–15:20.

Olaf Beyersdorff, Joshua Blinkhorn, Tom´ aˇ s Peitl Strong (D)QBF Dependency Schemes via Tautology-free Resolution Paths 26 / 27

References II

Friedrich Slivovsky and Stefan Szeider, Soundness of Q-resolution with dependency schemes, Theoretical Computer Science 612 (2016), 83–101. Ralf Wimmer, Christoph Scholl, Karina Wimmer, and Bernd Becker, Dependency schemes for DQBF, International Conference on Theory and Practice of Satisfiability Testing (SAT) (Nadia Creignou and Daniel Le Berre, eds.), Lecture Notes in Computer Science, vol. 9710, Springer, 2016, pp. 473–489.

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