[PPT] - Multi-Linear Strategy Extraction for QBF Expansion Proofs via Local PowerPoint Presentation

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Multi-Linear Strategy Extraction for QBF Expansion Proofs via Local Soundness

Matthias Schlaipfer Friedrich Slivovsky Georg Weissenbacher Florian Zuleger July 6, 2020

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Outline

1. Motivation
2. QBF: Semantics and expansion proofs
3. Local soundness of ∀-Exp+Res
4. Circuit extraction in multi-linear time
5. Conclusion

Matthias Schlaipfer 1

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Motivation

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Motivation

❼ Strategy Extraction

❼ Strategy: witness of unsatisfiability, universal variable assignments ❼ Practically useful, e.g. program synthesis ❼ Better understanding of proof systems: strategy extraction for Q-resolution has led to new lower bound techniques ❼ Expansion-based solvers faster on certain classes of problems

❼ Local soundness

❼ Inductive correctness argument at the inference level ❼ Previous proofs “global”, or via a detour to FOL (not multi-linear time) ❼ Existing “local” proof is not polynomial time ❼ Allows for proof/strategy isomorphism a la Curry-Howard and Craig interpolation

Matthias Schlaipfer 2

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QBF

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QBF

We consider QBFs in PCNF

Q1vi . . . Qnvn

quantifier prefix

. ϕ

matrix

with Qi ∈ {∀, ∃}, dom(vi) = {0, 1}.

Example ∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2)

Matthias Schlaipfer 3

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QBF game semantics

❼ Players assign variables following the order of the quantifier prefix ❼ Existential player tries to satisfy the matrix ❼ Universal player tries to falsify the matrix Example

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) ∃x1∀u∃x2.(¬u ∨ x2) ∧ (¬u ∨ ¬x2) x1 = 1 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 1 ⊥ ⊥ x2 = 0 x2 = 1 ∃x1∀u∃x2.(u ∨ x2) ∧ (u ∨ ¬x2) x1 = 0 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 0 ⊥ ⊥ x2 = 0 x2 = 1

x1 u1 x2 ⊥ ⊥ u1 x2 ⊥ ⊥

1 1 1 1

Matthias Schlaipfer 4

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QBF game semantics

❼ Players assign variables following the order of the quantifier prefix ❼ Existential player tries to satisfy the matrix ❼ Universal player tries to falsify the matrix Example

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) ∃x1∀u∃x2.(¬u ∨ x2) ∧ (¬u ∨ ¬x2) x1 = 1 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 1 ⊥ ⊥ x2 = 0 x2 = 1 ∃x1∀u∃x2.(u ∨ x2) ∧ (u ∨ ¬x2) x1 = 0 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 0 ⊥ ⊥ x2 = 0 x2 = 1

x1 u1 x2 ⊥ ⊥ u1 x2 ⊥ ⊥

1 1 1 1

Matthias Schlaipfer 4

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QBF game semantics

❼ Players assign variables following the order of the quantifier prefix ❼ Existential player tries to satisfy the matrix ❼ Universal player tries to falsify the matrix Example

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) ∃x1∀u∃x2.(¬u ∨ x2) ∧ (¬u ∨ ¬x2) x1 = 1 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 1 ⊥ ⊥ x2 = 0 x2 = 1 ∃x1∀u∃x2.(u ∨ x2) ∧ (u ∨ ¬x2) x1 = 0 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 0 ⊥ ⊥ x2 = 0 x2 = 1

x1 u1 x2 ⊥ ⊥ u1 x2 ⊥ ⊥

1 1 1 1

Matthias Schlaipfer 4

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QBF game semantics

❼ Players assign variables following the order of the quantifier prefix ❼ Existential player tries to satisfy the matrix ❼ Universal player tries to falsify the matrix Example

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) ∃x1∀u∃x2.(¬u ∨ x2) ∧ (¬u ∨ ¬x2) x1 = 1 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 1 ⊥ ⊥ x2 = 0 x2 = 1 ∃x1∀u∃x2.(u ∨ x2) ∧ (u ∨ ¬x2) x1 = 0 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 0 ⊥ ⊥ x2 = 0 x2 = 1

x1 u1 x2 ⊥ ⊥ u1 x2 ⊥ ⊥

1 1 1 1

Matthias Schlaipfer 4

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QBF game semantics

❼ Players assign variables following the order of the quantifier prefix ❼ Existential player tries to satisfy the matrix ❼ Universal player tries to falsify the matrix Example

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) ∃x1∀u∃x2.(¬u ∨ x2) ∧ (¬u ∨ ¬x2) x1 = 1 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 1 ⊥ ⊥ x2 = 0 x2 = 1 ∃x1∀u∃x2.(u ∨ x2) ∧ (u ∨ ¬x2) x1 = 0 ∃x1∀u∃x2.(x2) ∧ (¬x2) u = 0 ⊥ ⊥ x2 = 0 x2 = 1

x1 u1 x2 ⊥ ⊥ u1 x2 ⊥ ⊥

1 1 1 1

Matthias Schlaipfer 4

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Strategies

x1 u x2 ⊥ ⊥ 1 u x2 ⊥ ⊥ 1 1 1 fu(x1) = x1 A strategy . . . ❼ . . . represents a set of assignments determined by its paths ❼ . . . represents functions: one for each universal variable ❼ . . . is winning for the universal player when all paths falsify the matrix ϕ We refer to strategies by P and Q

Matthias Schlaipfer 5

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Expansion proofs (∀-Exp+Res)

1. Instantiation of universal variables:

(∀-exp) {l[τ] | l ∈ C, l is existential}

2. Resolution on existential variables

C1 ∨ xσ C2 ∨ ¬xσ (res) C1 ∨ C2 Example ∃x1∀u∃x2.x1 ∨ u ∨ x2

∀-exp

− − − → x1 ∨ x0/u

2

, we write x1 ∨ x¬u

2

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) (∀-exp) x1 ∨ x¬u

2

(∀-exp) x1 ∨ ¬x¬u

2

(res) x1 (∀-exp) ¬x1 ∨ xu

2

(∀-exp) ¬x1 ∨ ¬xu

2

(res) ¬x1 (res)

Matthias Schlaipfer

6

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Expansion proofs (∀-Exp+Res)

1. Instantiation of universal variables:

(∀-exp) {l[τ] | l ∈ C, l is existential}

2. Resolution on existential variables

C1 ∨ xσ C2 ∨ ¬xσ (res) C1 ∨ C2 Example ∃x1∀u∃x2.x1 ∨ u ∨ x2

∀-exp

− − − → x1 ∨ x0/u

2

, we write x1 ∨ x¬u

2

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) (∀-exp) x1 ∨ x¬u

2

(∀-exp) x1 ∨ ¬x¬u

2

(res) x1 (∀-exp) ¬x1 ∨ xu

2

(∀-exp) ¬x1 ∨ ¬xu

2

(res) ¬x1 (res)

Matthias Schlaipfer

6

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Expansion proofs (∀-Exp+Res)

1. Instantiation of universal variables:

(∀-exp) {l[τ] | l ∈ C, l is existential}

2. Resolution on existential variables

C1 ∨ xσ C2 ∨ ¬xσ (res) C1 ∨ C2 Example ∃x1∀u∃x2.x1 ∨ u ∨ x2

∀-exp

− − − → x1 ∨ x0/u

2

, we write x1 ∨ x¬u

2

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) (∀-exp) x1 ∨ x¬u

2

(∀-exp) x1 ∨ ¬x¬u

2

(res) x1 (∀-exp) ¬x1 ∨ xu

2

(∀-exp) ¬x1 ∨ ¬xu

2

(res) ¬x1 (res)

Matthias Schlaipfer

6

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Expansion proofs (∀-Exp+Res)

1. Instantiation of universal variables:

(∀-exp) {l[τ] | l ∈ C, l is existential}

2. Resolution on existential variables

C1 ∨ xσ C2 ∨ ¬xσ (res) C1 ∨ C2 Example ∃x1∀u∃x2.x1 ∨ u ∨ x2

∀-exp

− − − → x1 ∨ x0/u

2

, we write x1 ∨ x¬u

2

∃x1∀u∃x2.(x1 ∨ u ∨ x2) ∧ (x1 ∨ u ∨ ¬x2) ∧ (¬x1 ∨ ¬u ∨ x2) ∧ (¬x1 ∨ ¬u ∨ ¬x2) (∀-exp) x1 ∨ x¬u

2

(∀-exp) x1 ∨ ¬x¬u

2

(res) x1 (∀-exp) ¬x1 ∨ xu

2

(∀-exp) ¬x1 ∨ ¬xu

2

(res) ¬x1 (res)

Matthias Schlaipfer

6

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Strategy extraction

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Strategy extraction via local soundness

❼ Associate strategies with clauses in the proof: Clause [Strategy] ❼ Combine strategies locally along the proof derivation For a (∀-exp) inference: (∀-exp) C τ [ConstStrat(Π, τ)] ConstStrat(Π, τ) . . . follows Π, sets ∀-edges according to τ For a (res) rule with pivot xτ: [P] C1 ∨ ¬xτ C2 ∨ xτ [Q] (res) C1 ∨ C2 [P ⊔

xτQ]

Inspired by [Suda and Gleiss, 2018]

Matthias Schlaipfer 7

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Strategy extraction via local soundness

❼ Associate strategies with clauses in the proof: Clause [Strategy] ❼ Combine strategies locally along the proof derivation For a (∀-exp) inference: (∀-exp) C τ [ConstStrat(Π, τ)] ConstStrat(Π, τ) . . . follows Π, sets ∀-edges according to τ For a (res) rule with pivot xτ: [P] C1 ∨ ¬xτ C2 ∨ xτ [Q] (res) C1 ∨ C2 [P ⊔

xτQ]

Inspired by [Suda and Gleiss, 2018]

Matthias Schlaipfer 7

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Strategy extraction via local soundness

❼ Associate strategies with clauses in the proof: Clause [Strategy] ❼ Combine strategies locally along the proof derivation For a (∀-exp) inference: (∀-exp) C τ [ConstStrat(Π, τ)] ConstStrat(Π, τ) . . . follows Π, sets ∀-edges according to τ For a (res) rule with pivot xτ: [P] C1 ∨ ¬xτ C2 ∨ xτ [Q] (res) C1 ∨ C2 [P ⊔

xτQ]

Inspired by [Suda and Gleiss, 2018]

x1 u x2 ⊥ ⊥ u x2 ⊥ ⊥

1 1 1

ConstStrat(∃x1∀u∃x2, ¬u)

Matthias Schlaipfer 7

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Strategy extraction via local soundness

❼ Associate strategies with clauses in the proof: Clause [Strategy] ❼ Combine strategies locally along the proof derivation For a (∀-exp) inference: (∀-exp) C τ [ConstStrat(Π, τ)] ConstStrat(Π, τ) . . . follows Π, sets ∀-edges according to τ For a (res) rule with pivot xτ: [P] C1 ∨ ¬xτ C2 ∨ xτ [Q] (res) C1 ∨ C2 [P ⊔

xτQ]

Inspired by [Suda and Gleiss, 2018]

Matthias Schlaipfer 7

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Correctness of Combine—strategies

C1 [P1] · · · Cn [Pn] [Pwin]

We differentiate between ❼ Intermediate strategies Associated with initial and internal proof nodes ❼ Winning strategy (universal): Associated with the empty clause Each path falsifies the matrix ϕ, i.e., Pwin | = ¬ϕ We need to answer what it means to be an intermediate strategy

Matthias Schlaipfer 8

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Correctness of Combine—strategies

C1 [P1] · · · Cn [Pn] [Pwin]

We differentiate between ❼ Intermediate strategies Associated with initial and internal proof nodes ❼ Winning strategy (universal): Associated with the empty clause Each path falsifies the matrix ϕ, i.e., Pwin | = ¬ϕ We need to answer what it means to be an intermediate strategy

Matthias Schlaipfer 8

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Correctness of Combine—strategies

C1 [P1] · · · Cn [Pn] [Pwin]

We differentiate between ❼ Intermediate strategies Associated with initial and internal proof nodes ❼ Winning strategy (universal): Associated with the empty clause Each path falsifies the matrix ϕ, i.e., Pwin | = ¬ϕ We need to answer what it means to be an intermediate strategy

Matthias Schlaipfer 8

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Correctness of Combine

❼ Winning strategy: a strategy P such that P | = ¬ϕ ❼ Find invariant I(P, C) ending in P | = ¬ϕ, maintained by the Combine operator ❼ P and C are over different alphabets, C uses annotated literals Invariant

P | = enc(C) ⇒ ¬ϕ

enc(xτi

i ∨ C ′)

def

=

u∈τi

u

⇒ ¬xi
∧ enc(C ′).

Intuition: existential player responds to a universal play by setting its literal to false At the empty clause we have P | = enc()

1

⇒ ¬ϕ ≡ P | = ¬ϕ

Matthias Schlaipfer 9

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Correctness of Combine

❼ Winning strategy: a strategy P such that P | = ¬ϕ ❼ Find invariant I(P, C) ending in P | = ¬ϕ, maintained by the Combine operator ❼ P and C are over different alphabets, C uses annotated literals Invariant

P | = enc(C) ⇒ ¬ϕ

enc(xτi

i ∨ C ′)

def

=

u∈τi

u

⇒ ¬xi
∧ enc(C ′).

Intuition: existential player responds to a universal play by setting its literal to false At the empty clause we have P | = enc()

1

⇒ ¬ϕ ≡ P | = ¬ϕ

Matthias Schlaipfer 9

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The Combine operator, example fu1(x1), fu2(x1, x2)

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1

⊔

xu1

2

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

= x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1

| = enc(C1 ∨ ¬xu1

2 )

⇒ ¬ϕ | = enc(C2 ∨ xu1

2 )

⇒ ¬ϕ | = enc(C1 ∨ C2) ⇒ ¬ϕ ∧ ⇒ P Q P ⊔

xu1

2

Q C1 ∨ ¬xu1

2

C2 ∨ xu1

2

res C1 ∨ C2 =

Matthias Schlaipfer 10

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The Combine operator, example fu1(x1), fu2(x1, x2)

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1

⊔

xu1

2

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

= x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1

| = (u1 ⇒ x2) ∧ enc(C1) ⇒ ¬ϕ | = (u1 ⇒ ¬x2) ∧ enc(C2) ⇒ ¬ϕ | = enc(C1 ∨ C2) ⇒ ¬ϕ ∧ ⇒ P Q P ⊔

xu1

2

Q C1 ∨ ¬xu1

2

C2 ∨ xu1

2

res C1 ∨ C2 =

Matthias Schlaipfer 10

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The Combine operator, example fu1(x1), fu2(x1, x2)

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1

⊔

xu1

2

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

= x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1

| = ( u1 ⇒ x2 ) ∧ enc(C1) ⇒ ¬ϕ | = ( u1 ⇒ ¬x2) ∧ enc(C2) ⇒ ¬ϕ | = enc(C1 ∨ C2) ⇒ ¬ϕ ∧ ⇒ P Q P ⊔

xu1

2

Q C1 ∨ ¬xu1

2

C2 ∨ xu1

2

res C1 ∨ C2 =

Matthias Schlaipfer 10

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The Combine operator, example fu1(x1), fu2(x1, x2)

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1

⊔

xu1

2

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

= x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1

| = ( u1 ⇒ x2) ∧ enc(C1) ⇒ ¬ϕ | = ( u1 ⇒ ¬x2) ∧ enc(C2) ⇒ ¬ϕ | = enc(C1 ∨ C2) ⇒ ¬ϕ ∧ ⇒ P Q P ⊔

xu1

2

Q C1 ∨ ¬xu1

2

C2 ∨ xu1

2

res C1 ∨ C2 =

Matthias Schlaipfer 10

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The Combine operator, example fu1(x1), fu2(x1, x2)

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1

⊔

xu1

2

x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

= x1 u1 x2 u2 ⊥ u2 ⊥ u1 x2 u2 ⊥ u2 ⊥

1 1 1 1 1 1 1

| = ( u1 ⇒ x2 ) ∧ enc(C1) ⇒ ¬ϕ | = ( u1 ⇒ ¬x2 ) ∧ enc(C2) ⇒ ¬ϕ | = enc(C1 ∨ C2) ⇒ ¬ϕ ∧ ⇒ P Q P ⊔

xu1

2

Q C1 ∨ ¬xu1

2

C2 ∨ xu1

2

res C1 ∨ C2 =

Matthias Schlaipfer 10

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Multi-linear strategy extraction

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Combine using circuits

Strategy trees are not suitable for implementation—they branch at every existential node Theorem Given a ∀-Exp+Res derivation of length p from a PCNF formula Φ with n universal variables, a family of circuits implementing a universal winning strategy for Φ can be computed in time O(p · n).

Matthias Schlaipfer 11

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Combine using circuits

Strategy trees are not suitable for implementation—they branch at every existential node Theorem Given a ∀-Exp+Res derivation of length p from a PCNF formula Φ with n universal variables, a family of circuits implementing a universal winning strategy for Φ can be computed in time O(p · n).

Matthias Schlaipfer 11

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