Localizing Quantifiers for DQBF Aile Ge-Ernst C. Scholl, R. Wimmer - - PowerPoint PPT Presentation

Localizing Quantifiers for DQBF

Aile Ge-Ernst

• C. Scholl, R. Wimmer

Albert-Ludwigs-University Freiburg Concept Engineering Freiburg

Formal Methods in Computer Aided Design San Jos´ e, CA, USA, Oct 25, 2019

Motivation

Prenex QBF vs. DQBF

Quantified Boolean Formulas (QBF) in prenex form: ψ := Q1v1Q2v2 ... Qnvn : ϕ ◮ Q1v1Q2v2 ... Qnvn: quantifier prefix ◮ Qi: quantifier ∀ or ∃ ◮ v1, v2, ... , vn: Boolean variables ◮ ϕ: matrix, which is a quantifier-free Boolean formula ◮ linearly ordered dependencies: Each existential variable depends on all universal variables to the left of it.

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Prenex QBF vs. DQBF

Quantified Boolean Formulas (QBF) in prenex form: ψ := Q1v1Q2v2 ... Qnvn : ϕ ◮ Q1v1Q2v2 ... Qnvn: quantifier prefix ◮ Qi: quantifier ∀ or ∃ ◮ v1, v2, ... , vn: Boolean variables ◮ ϕ: matrix, which is a quantifier-free Boolean formula ◮ linearly ordered dependencies: Each existential variable depends on all universal variables to the left of it.

Example: ∀x1∃y1∀x2∃y2 : ϕ

◮ y1 depends on x1 ◮ y2 depends on x1 and x2

∀x1 ∃y1 ∀x2 ∃y2

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QBF vs. DQBF

Dependency Quantified Boolean Formulas (DQBF) in prenex form: Ψ := ∀x1∀x2 ... ∀xn∃y1(Dy1)∃y2(Dy2) ... ∃ym(Dym) : ϕ ◮ Dyj: dependency sets of ∃-variables yj ◮ Dependencies are stated explicitly. ◮ Variable order in the prefix irrelevant.

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QBF vs. DQBF

Dependency Quantified Boolean Formulas (DQBF) in prenex form: Ψ := ∀x1∀x2 ... ∀xn∃y1(Dy1)∃y2(Dy2) ... ∃ym(Dym) : ϕ ◮ Dyj: dependency sets of ∃-variables yj ◮ Dependencies are stated explicitly. ◮ Variable order in the prefix irrelevant.

Example: ∀x1∀x2∃y1(x1)∃y2(x2) : ϕ

◮ y1 depends only on x1 ◮ y2 depends only on x2

x1 x2 y1 x1 y2 x2

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QBF vs. DQBF

◮ DQBFs encodings are more compact than their QBF counterpart. ◮ DQBFs can succinctly express problems involving decisions under partial information.

Example:

◮ y1 depends only on x1 and x3 ◮ y2 depends only on x2 and x3

⇒ ∀x1∀x2∀x3∃y1(x1, x3)∃y2(x2, x3) : ϕ

≡

BB1 BB2 X1 X2 Y1 Y2 Specification Implementation ≡ 1? Miter X3

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QBF vs. DQBF

◮ However, until now only prenex DQBFs have been investigated (except for a seminal theoretical work1). ◮ Our contribution:

◮ Definition of non-closed non-prenex DQBF. ◮ Use rules to simplify non-closed non-prenex DQBF. ◮ Experiments to verify effectiveness.

1Valeriy Balabanov, Hui-Ju Katherine Chiang, and Jie-Hong R. Jiang. “Henkin

quantifiers and Boolean formulae: A certification perspective of DQBF”. In: Theoretical Computer Science 523 (2014), pp. 86–100. doi: 10.1016/j.tcs.2013.12.020.

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Outline

Semantics of QBF and DQBF Non-Closed Non-Prenex DQBF Quantifier Localization for DQBF Experiments

Semantics of QBF and DQBF

Semantics of Closed Prenex (D)QBF

QBF: ∀x1∃y1∀x2∃y2 : ϕ is satisfied iff there are functions sy1 and sy2 such that replacing y1 with sy1(x1) and y2 with sy2(x1, x2) yields a tautology. DQBF: ∀x1∀x2∃y1(x1)∃y2(x2) : ϕ is satisfied iff there are functions sy1 and sy2 such that replacing y1 with sy1(x1) and y2 with sy2(x2) yields a tautology. ⇒ sy1 and sy2 are called Skolem function.

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Semantics of Closed Prenex (D)QBF

QBF: ∀x1∃y1∀x2∃y2 : ϕ is satisfied iff there are functions sy1 and sy2 such that replacing y1 with sy1(x1) and y2 with sy2(x1, x2) yields a tautology. DQBF: ∀x1∀x2∃y1(x1)∃y2(x2) : ϕ is satisfied iff there are functions sy1 and sy2 such that replacing y1 with sy1(x1) and y2 with sy2(x2) yields a tautology. ⇒ sy1 and sy2 are called Skolem function.

Example: ∀x1∀x2∃y1(x1)∃y2(x2) : (x1 ≡ y1) ∧ (x2 ≡ y2)

◮ satisfiable with sy1(x1) = x1 and sy2(x2) = ¯ x2 ◮ ⇒ (x1 ≡ x1) ∧ (x2 ≡ ¯ x2) is a tautology

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

• ∀x1∃y1∀x2 :
• (x1 ∨ ¯

y1 ∨ x2) ∧

• ∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

• ∃y : (ψ1 ∧ ψ2) ≈ (ψ1 ∧ (∃y : ψ2)),

if y / ∈ Vψ1

∀x1∃y1∀x2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧

• ∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

• ∃y : (ψ1 ∧ ψ2) ≈ (ψ1 ∧ (∃y : ψ2)),

if y / ∈ Vψ1

∀x1∃y1∀x2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧

• ∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2) ∀x1∃y1 : ∀x2 : (x1 ∨ ¯ y1 ∨ x2)

• ∧
• ∀x2∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

• ∃y : (ψ1 ∧ ψ2) ≈ (ψ1 ∧ (∃y : ψ2)),

if y / ∈ Vψ1

∀x1∃y1∀x2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧

• ∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

∀x : (ψ1 ∧ ψ2) ≈ ((∀x : ψ1) ∧ (∀x : ψ2))

∀x1∃y1 : ∀x2 : (x1 ∨ ¯ y1 ∨ x2)

• ∧
• ∀x2∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: ∀x1∃y1∀x2∃y2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧ (y1 ∨ ¯ x2 ∨ ¯ y2)

• ∃y : (ψ1 ∧ ψ2) ≈ (ψ1 ∧ (∃y : ψ2)),

if y / ∈ Vψ1

∀x1∃y1∀x2 :

• (x1 ∨ ¯

y1 ∨ x2) ∧

• ∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2)

∀x : (ψ1 ∧ ψ2) ≈ ((∀x : ψ1) ∧ (∀x : ψ2))

∀x1∃y1 : ∀x2 : (x1 ∨ ¯ y1 ∨ x2)

• ∧
• ∀x2∃y2 : (y1 ∨ ¯

x2 ∨ ¯ y2) ◮ Obtain an equisatisfiable non-prenex QBF. ◮ Gain: Reduced cost for quantifier elimination. ◮ See e. g. optimized (pre-)image computation in Symbolic Model Checking with early quantification.

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Non-Closed Non-Prenex QBF

Quantifier Localization for QBF: Quantifier localization rules like ∃y : (ψ1 ∧ ψ2) ≈ (ψ1 ∧ (∃y : ψ2)), if y / ∈ Vψ1 can be easily proved by an alternative characterization of the semantics. Alternative characterization using symbolic quantifier elimination: ◮ (∃y : ψ) ≡ (ψ[0/y] ∨ ψ[1/y]) ◮ (∀x : ψ) ≡ (ψ[0/x] ∧ ψ[1/x]) ⇒ Possibly doubling the size of the formula for each quantification.

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Non-Close Non-Prenex DQBF?

Quantifier Localization for DQBF? But what about DQBF? ◮ An alternative characterization of the semantics based on symbolic quantifier elimination does not work. ◮ How to define the semantics of non-closed non-prenex DQBFs like

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• which are not just a Boolean combination of closed prenex DQBFs?

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Non-Closed Non-Prenex DQBF

Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF Basic Idea: Define semantics based on Skolem functions like for prenex DQBFs. ◮ The Skolem function of an ∃-variable ∃y(Dy) may depend on universal variable x

◮ if x is in y’s dependency set Dy and ◮ if ② is in the scope of ∀①.

◮ The only admissable Skolem functions for free variables are constants 0 or 1. ◮ A DQBF is satisfiable, if

◮ replacing ∃- and free variables by their Skolem functions and ◮ omitting all quantifiers

yields a tautological formula.

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Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF:

Example:

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
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Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF:

Example:

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound

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Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF:

Example:

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound free free

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Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF:

Example:

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound free free

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound free free sy1(x1)

×

sy2(x2)

×

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Non-Closed Non-Prenex DQBF

Semantics of Non-Closed Non-Prenex DQBF:

Example:

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound free free

• ∀x1∃y1(x1, x2) :
• (x1 ⊕ y1) ∧ ¬x2
• ∧
• ∀x2∃y2(x1, x2) :
• (x2 ⊕ y2) ∧ ¬x1
• bound

bound free free sy1(x1)

×

sy2(x2)

×

⇒ The formula satisfiable with sy1(x1) = ¯ x1, sx2 = 0, sy2(x2) = ¯ x2, sx1 = 0.

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Quantifier Localization for DQBF

Taking Advantage of Quantifier Localization

◮ The proofs of quantifier localization rules for DQBF are more involved, for details see paper. ◮ Here we only give an illustration.

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Taking Advantage of Quantifier Localization

◮ The proofs of quantifier localization rules for DQBF are more involved, for details see paper. ◮ Here we only give an illustration. Overview: ◮ Use quantifier localization rules to push quantifiers as deep into the formula as possible. ◮ If successful, this enables symbolic quantifier elimination. ◮ Then push remaining quantifiers back and solve the resulting simplified prenex DQBF.

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Taking Advantage of Quantifier Localization

• 1. Step: Combine subcircuits into macrogates

◮ Increase the degree of freedom to apply localization rules. ◮ Summarize all consecutive disjunctions/conjunctions into one multi-input disjunction/conjunction.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 x1 x2 y1 x1 x2 y2 x1 y2

∀x1∀x2∃y1(x1)∃y2(x2)

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Taking Advantage of Quantifier Localization

• 1. Step: Combine subcircuits into macrogates

◮ Increase the degree of freedom to apply localization rules. ◮ Summarize all consecutive disjunctions/conjunctions into one multi-input disjunction/conjunction.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 x1 x2 y1 x1 x2 y2 x1 y2

∀x1∀x2∃y1(x1)∃y2(x2)

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

◮ Localization rules sometimes similar to QBF localization rules. ◮ More complicated to prove, have to pay attention to dependencies!

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 1: disjunction + ∃

◮ ∃y(Dy) : (ψ1 ∨ ψ2) ≈

• ∃y(Dy) : ψ1
• ∨
• ∃y(Dy) : ψ2
• ◮ ∃y(Dy) : (ψ1 ∨ ψ2) ≡
• ψ1 ∨ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 1: disjunction + ∃

◮ ∃y(Dy) : (ψ1 ∨ ψ2) ≈

• ∃y(Dy) : ψ1
• ∨
• ∃y(Dy) : ψ2
• ◮ ∃y(Dy) : (ψ1 ∨ ψ2) ≡
• ψ1 ∨ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

Example: ∃y(Dy) : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 x1 x2 y1 x1 x2 y2 x1 y2

∀x1∀x2∃y1(x1)∃y2(x2)

→

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 x1 x2 y1 x1 x2 y2 x1 y2

∀x1∀x2 ∃y1(x1) ∃y1(x1) ∃y2(x2)

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 2: disjunction + ∀

◮ ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)
• , if x /

∈ Vψ1, x / ∈ Dy for all ∃-variables y ∈ Vψ1

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 2: disjunction + ∀

◮ ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)
• , if x /

∈ Vψ1, x / ∈ Dy for all ∃-variables y ∈ Vψ1 ◮ Proof that condition “x / ∈ Dy for all ∃-variables y ∈ Vψ1” is needed: ∀x1∀x2 : ((∃y(x2) : (x1 ≡ y)) ∨ (x1 ≡ x2)) is satisfiable, but: ∀x1 : ((∃y(x2) : (x1 ≡ y)) ∨ (∀x2 : (x1 ≡ x2))) is unsatisfiable.

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 2: disjunction + ∀

◮ ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)
• , if x /

∈ Vψ1, x / ∈ Dy for all ∃-variables y ∈ Vψ1 ◮ Proof that condition “x / ∈ Dy for all ∃-variables y ∈ Vψ1” is needed: ∀x1∀x2 : ((∃y(x2) : (x1 ≡ y)) ∨ (x1 ≡ x2)) is satisfiable, but: ∀x1 : ((∃y(x2) : (x1 ≡ y)) ∨ (∀x2 : (x1 ≡ x2))) is unsatisfiable.

Example: ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)

, if ...

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 x1 x2 y1 x1 x2 y2 x1 y2

∀x1∀x2 ∃y1(x1) ∃y1(x1) ∃y2(x2) → ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y1(x1) ∃y1(x1) ∃y2(x2) white

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 2: disjunction + ∀

◮ ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)
• , if x /

∈ Vψ1, x / ∈ Dy for all ∃-variables y ∈ Vψ1 ◮ Proof that condition “x / ∈ Dy for all ∃-variables y ∈ Vψ1” is needed: ∀x1∀x2 : ((∃y(x2) : (x1 ≡ y)) ∨ (x1 ≡ x2)) is satisfiable, but: ∀x1 : ((∃y(x2) : (x1 ≡ y)) ∨ (∀x2 : (x1 ≡ x2))) is unsatisfiable.

Example: ∀x : (ψ1 ∨ ψ2) ≡

• ψ1 ∨ (∀x : ψ2)

, if ...

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y1(x1) ∃y1(x1) ∃y2(x2) white

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 3: conjunction + ∃

◮ ∃y(Dy) : (ψ1 ∧ ψ2) ≡

• ψ1 ∧ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 3: conjunction + ∃

◮ ∃y(Dy) : (ψ1 ∧ ψ2) ≡

• ψ1 ∧ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

Example: ∃y(Dy) : (ψ1 ∧ ψ2) ≡

• ψ1 ∧ (∃y(Dy) : ψ2)
• , if y /

∈ Vψ1

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y1(x1) ∃y1(x1) ∃y2(x2) white

→

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 4: conjunction + ∀

◮ ∀x : (ψ1 ∧ ψ2) ≈

• ∀x : ψ1
• ∧
• ∀x : ψ2
• ◮ ∀x : (ψ1 ∧ ψ2) ≈
• ψ−x

1

∧ (∀x : ψ2)

• if x /

∈ Vψ1 (ψ−x

1

means to delete x from all dependency sets from ψ1)

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Taking Advantage of Quantifier Localization

• 2. Step: Quantifier Localization

Case 4: conjunction + ∀

◮ ∀x : (ψ1 ∧ ψ2) ≈

• ∀x : ψ1
• ∧
• ∀x : ψ2
• ◮ ∀x : (ψ1 ∧ ψ2) ≈
• ψ−x

1

∧ (∀x : ψ2)

• if x /

∈ Vψ1 (ψ−x

1

means to delete x from all dependency sets from ψ1)

Example: (rule here not applicable ...)

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

Additional: Swap the order of variables ◮ ∃y1(Dy1) ∃y2(Dy2) : ψ ≡ ∃y2(Dy2) ∃y1(Dy1) : ψ ◮ ∀x1 ∀x2 : ψ ≡ ∀x2 ∀x1 : ψ ◮ ∀x ∃y(Dy) : ψ ≡ ∃y(Dy) ∀x : ψ, if x / ∈ Dy

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

y1 x1 y1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨

1 x1 1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2 ∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 1: existential quantifier ◮ ψ := ψ1 ⋄ (∃y(Dy) : ψ2) ≈ ψ′ := ψ1 ⋄ (ψ2[0/

y] ∨ ψ2[1/ y]),

◮ if ψ2 does not contain quantifiers and ◮ ψ2 only includes free variables, ∀-variables from Dy or ∃-variables y ′ with Dy′ ⊆ Dy

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 2: universal quantifier ◮ ∀x : ψ ≡ ψ[0/

x] ∧ ψ[1/ x], if ψ does not contain quantifiers.

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 2: universal quantifier ◮ ∀x : ψ ≡ ψ[0/

x] ∧ ψ[1/ x], if ψ does not contain quantifiers.

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1 ∀x2∃y2(x2) ∃y1(x1) white

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Taking Advantage of Quantifier Localization

• 3. Step: Quantifier Elimination

Case 2: universal quantifier ◮ ∀x : ψ ≡ ψ[0/

x] ∧ ψ[1/ x], if ψ does not contain quantifiers.

◮ Drag variable back if elimination is not possible.

Example:

∨ ∨ ∨ ∧ ∧ ∧ ∨ ∨

1 x1 x1 x2 x2 y2 x1 y2

∀x1∀x2∃y2(x2) ∃y1(x1) white

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Experiments

Experiments

◮ 4811 Benchmarks ◮ Intensive preprocessing using HQSpre ... ◮ Focused on those 974 instances that reach our algorithm (neither solved nor failed in HQSpre) ◮ local elimination of variables on 840 instances ◮ 66918 local eliminations in total ◮ HQS solved 531, HQSnp solved 689 ⇒ increase of 29.8%

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Experiments

◮ 4811 Benchmarks ◮ Intensive preprocessing using HQSpre ... ◮ Focused on those 974 instances that reach our algorithm (neither solved nor failed in HQSpre) ◮ local elimination of variables on 840 instances ◮ 66918 local eliminations in total ◮ HQS solved 531, HQSnp solved 689 ⇒ increase of 29.8% ◮ QBFEVAL’18: 334 benchmarks ◮ 68 reach our algorithm ◮ HQS solved 22, HQSnp solved 30 ⇒ increase of 36.4%

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Experiments

HQS vs. HQSnp - Solved Instances (out of 974)

531 689

100 200 300 400 500 600 700 800 500 1,000 1,500 Number of solved instances Runtime in seconds

HQS HQSnp

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Experiments

HQS vs. HQSnp - Computation Time

10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 HQS time in seconds HQSnp time in seconds

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Conclusion

Conclusion

◮ Defined syntax and semantics for non-closed non-prenex DQBFs. ◮ Stated rules for transformation from prenex to non-prenex DQBFs. ◮ Stated rules to perform local elimination on subcircuits. ⇒ Significantly increased the number of solved instances.

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Future Work

◮ Introduce estimates on costs and benefits.

◮ E. g. estimation on the size of the subcircuits resulting from symbolic quantifier elimination. ◮ Optimize the choice of order of pushing.

◮ Find limits for the growth of circuit sizes.

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