Short Q-Resolution Proofs with Homomorphisms Friedrich Slivovsky 1 - - PowerPoint PPT Presentation

Short Q-Resolution Proofs with Homomorphisms

Friedrich Slivovsky1 Stefan Szeider1 Ankit Shukla2 TU Wien, Vienna, Austria1 JKU, Linz, Austria2 June 30, 2020

Focus of Analysis: Proof Complexity

Goal: Compare the strength of the proof systems and show an exponential separation.

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res A B

A is stronger than B. A has short proofs.

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Proof Complexity for QBF Symmetries

We introduce new QBF proof systems that use homomorphisms, generalizing work on symmetries [Kauers and Seidl, 2018].

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res A B

A is stronger than B. A has short proofs.

[ Ankit Shukla ] 3/34

Proof Complexity for QBF Symmetries and Homomorphisms: Our results

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res A B

A is stronger than B. A has short proofs.

[ Ankit Shukla ] 4/34

Outline of the talk

1 Present symmetries and homomorphisms.

◮ Awesome diagrams. ◮ in CNF (propositional level). ◮ in PCNF (QBF level).

2 Introduce the homomorphism rules for QBF. 3 Outline of two results:

1 ⋆ Lower bound for homomorphisms. 2 ⋆ Exponential separation between proof systems.

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Symmetry in Mathematics

Symmetry refers to an object that is invariant under some transformations. “I know it when I see it” (threshold test for obscenity)

• Justice Potter Stewart, Jacobellis v. Ohio

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Global Vs Local: Symmetries and Homomorphisms

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c)

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements)

[ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements) σ(F) =

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements) σ(F) = σ(a ∨ b) ∧ σ(a ∨ c) ∧ σ(b ∨ c)

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements) σ(F) = σ(a ∨ b) ∧ σ(a ∨ c) ∧ σ(b ∨ c) = (a ∨ c) ∧ (a ∨ b) ∧ (b ∨ c)

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements) σ(F) = σ(a ∨ b) ∧ σ(a ∨ c) ∧ σ(b ∨ c) = (a ∨ c) ∧ (a ∨ b) ∧ (b ∨ c) = (a ∧ b) ∧ (a ∨ c) ∧ (b ∨ c)

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Symmetries in CNF [Krishnamurthy, 1985]

Literal permutation that preserves the CNF. F σ(F) = F

σ

Example (Symmetries: Injective mapping)

F = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c) σ: a → a, b → c, c → b (a → a, ... preserves complements) σ(F) = σ(a ∨ b) ∧ σ(a ∨ c) ∧ σ(b ∨ c) = (a ∨ c) ∧ (a ∨ b) ∧ (b ∨ c) = (a ∧ b) ∧ (a ∨ c) ∧ (b ∨ c) = F (σ preserve the CNF).

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Symmetries in CNF [Krishnamurthy, 1985, Urquhart, 1999, Arai and Urquhart, 2000]

Literal permutation that preserves the CNF. F σ(F) = F

σ (a) Global Symmetry

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Symmetries in CNF [Krishnamurthy, 1985, Urquhart, 1999, Arai and Urquhart, 2000]

Literal permutation that preserves the CNF. F σ(F) = F

σ (a) Global Symmetry

F1 ⊆ F σ(F1) ⊆ F

σ (b) Local Symmetry

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Homomorphisms in CNF [Szeider, 2005]

⋆ Allow non-injective mapping. ⋆ Hence possible: |ϕ(C)| < |C| and |ϕ(F)| < |F|. F Start with F ϕ(F) ⊆ F

ϕ (a) Global Homomorphism

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Homomorphisms in CNF [Szeider, 2005]

⋆ Allow non-injective mapping. ⋆ Hence possible: |ϕ(C)| < |C| and |ϕ(F)| < |F|. F Start with F ϕ(F) ⊆ F

ϕ (a) Global Homomorphism

F1 ⊆ F ϕ(F1) ⊆ F

ϕ (b) Local Homomorphism

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z)

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z) F1 ⊆ F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y)

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z) F1 ⊆ F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ϕ: x → z, y → z, a → b, v → v (x → z, . . . )

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z) F1 ⊆ F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ϕ: x → z, y → z, a → b, v → v (x → z, . . . ) ϕ(F1) = (b ∨ z) ∧ (v ∨ z ∨ z) ∧ (b ∨ z)

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z) F1 ⊆ F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ϕ: x → z, y → z, a → b, v → v (x → z, . . . ) ϕ(F1) = (b ∨ z) ∧ (v ∨ z ∨ z) ∧ (b ∨ z) = (b ∨ z) ∧ (v ∨ z)

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Local Homomorphism Example

Allow non-injective mapping. F1 ⊆ F ϕ(F1) ⊆ F

ϕ

Example (Homomorphism; Non-injective mapping)

F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ∧ (b ∨ z) ∧ (v ∨ z) F1 ⊆ F = (a ∨ x) ∧ (v ∨ x ∨ y) ∧ (a ∨ y) ϕ: x → z, y → z, a → b, v → v (x → z, . . . ) ϕ(F1) = (b ∨ z) ∧ (v ∨ z ∨ z) ∧ (b ∨ z) = (b ∨ z) ∧ (v ∨ z) ⊆ F

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Going to a Higher Level: SAT to QBF

PCNF: Quantified Boolean Formulas (QBFs)

Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀x1x2∃y1 . ((x1 ∨ x2 ∨ y1) ∧ (x1 ∨ y1))

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PCNF: Quantified Boolean Formulas (QBFs)

Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀x1x2∃y1 . ((x1 ∨ x2 ∨ y1) ∧ (x1 ∨ y1)) ⋆ Matrix (CNF)

[ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs)

Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀x1x2∃y1 . ((x1 ∨ x2 ∨ y1) ∧ (x1 ∨ y1)) ⋆ Matrix (CNF)

[ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs)

Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀x1x2∃y1 . ((x1 ∨ x2 ∨ y1) ∧ (x1 ∨ y1)) ⋆ Matrix (CNF) ◮ For all values of x1 and x2, is there a value for y1 such that the formula evaluates to true?

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PCNF: Quantified Boolean Formulas (QBFs)

Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀x1x2∃y1 . ((x1 ∨ x2 ∨ y1) ∧ (x1 ∨ y1)) ⋆ Matrix (CNF) ◮ For all values of x1 and x2, is there a value for y1 such that the formula evaluates to true? ◮ Linearly ordered dependencies: assign variables based on the quantified prefix order. Can prefix order be relaxed to increase freedom?

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PCNF and Dependency Scheme [Samer and Szeider, 2009]

Definition: Dependency Scheme [Lonsing, 2012] A mapping D that associates each PCNF formula Φ with a relation DΦ ⊆ { (x, y) : x <Φ y } called the dependency relation

• f Φ with respect to D.

Definition: Dtrv

Φ

The Dtrv associates each PCNF formula Φ with the relation Dtrv

Φ = { (x, y) : x <Φ y }. [ Ankit Shukla ] 14/34

A Stronger Dependency Scheme (Drrs)

Theorem: Drrs [Beyersdorff and Blinkhorn, 2016] The Reflexive Resolution-Path Dependency Scheme Drrs is a stronger dependency scheme than Trivial Dependency Scheme.

Drrs

Φ

Dstd

Φ

Dtrv

Φ

Dependency defined via resolution path Dependency induced by quantifier scoping Dependency induced by quantifier prefix

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Q-resolution [Kleine B¨ uning et al., 1995]

1 Input clause:

(Axiom) C C a clause in φ

2 Resolution:

C1 ∨ e e ∨ C2 (Res) C1 ∨ C2 e is existential and C1 ∨ C2 non-tautologous

3 Universal reduction:

C ∨ u (∀-Red) C u is universal and u quantified after C.

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A Q-resolution Derivation Example

Example (Q-resolution derivation)

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x)

(a ∨ c ∨ w) (a ∨ c ∨ w ∨ z) C1 : (a ∨ y ∨ z) C2 : (c ∨ y ∨ w) (c ∨ z) C3 : (c ∨ z ∨ u)

Q-resolution derivation of (a ∨ c ∨ w) from formula Q.φ.

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Our Contribution: Homomorphisms for QBFs

Definition: Homomorphisms: HomΦ(φ, φ). Start Φ = Q.φ in PCNF. For any L ⊆ lit(Φ), a map ϕ : L → lit(Φ)

1 Mapping is between variables of same quantifier type.

For every ℓ ∈ L, QtypeΦ(ℓ) = QtypeΦ(ϕ(ℓ)).

2 ∀-literal mapping is injective.

If ϕ(ℓ) = ϕ(ℓ′) and QtypeΦ(ℓ) = ∀, then ℓ = ℓ′.

3 Relative position of the variables in the quantifier prefix

is preserved.

4 ϕ(φ) ⊆ φ.

(ϕ preserves the complements).

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Homomorphisms for QBFs

Definition: D-Homomorphisms- HomD

Φ(φ, φ)

Start Φ = Q.φ in PCNF. For any L ⊆ lit(Φ), a map ϕ : L → lit(Φ) D: dependency scheme for which Q(D)-resolution is sound.

1 For every ℓ ∈ L, QtypeΦ(ℓ) = QtypeΦ(ϕ(ℓ)). 2 If ϕ(ℓ) = ϕ(ℓ′) and QtypeΦ(ℓ) = ∀, then ℓ = ℓ′. 3 If ℓ, ℓ′ ∈ L and (var(ϕ(ℓ)), var(ϕ(ℓ′))) ∈ DΦ then

(var(ℓ), var(ℓ′)) ∈ DΦ.

4 ϕ(φ) ⊆ φ.

(ϕ preserves the complements).

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Local and Global Homomorphisms

φ Start with Φ = Q.φ ϕ(φ) ⊆ φ

ϕ (a) Global Homomorphism

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Local and Global Homomorphisms

φ Start with Φ = Q.φ ϕ(φ) ⊆ φ

ϕ (a) Global Homomorphism

φ1 ⊆ φ ϕ(φ1) ⊆ φ

ϕ (b) Local Homomorphism

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x)

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x) φ1 = {C1, C2, C3} ⊆ φ.

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x) φ1 = {C1, C2, C3} ⊆ φ. ϕ : a → a, c → b, y → x, w → x, z → x, u → u

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x) φ1 = {C1, C2, C3} ⊆ φ. ϕ : a → a, c → b, y → x, w → x, z → x, u → u ϕ(φ1) =

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x) φ1 = {C1, C2, C3} ⊆ φ. ϕ : a → a, c → b, y → x, w → x, z → x, u → u ϕ(φ1) = ...

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Local Homomorphism Example

Example (Local Homomorphism: Homφ(φ1, φ))

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} C1 = (a ∨ y ∨ z), C2 = (c ∨ y ∨ w), C3 = (c ∨ z ∨ u), C4 = (b ∨ x), C5 = (a ∨ x) φ1 = {C1, C2, C3} ⊆ φ. ϕ : a → a, c → b, y → x, w → x, z → x, u → u ϕ(φ1) = {C4, C5} ⊆ φ. ϕ ∈ Homφ(φ1, φ).

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Our Contribution: The Homomorphism Rule (HR)

Proof System: Q-resolution + HR

4 Homomorphism Rule:

C ϕ (HR) ϕ(C) From an already derived clause C and a homomorphism ϕ of Φ = Q.φ, the clause ϕ(C) can be derived. ⋆ Local HR (LH): If ϕ ∈ Hom(φ1 ⊆ φ, φ) then derive ϕ(C). ⋆ Global HR (GH): If ϕ ∈ Hom(φ, φ) then derive ϕ(C).

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Local Homomorphism Rule (LH)

Example (LH)

Q.φ = ∀a b ∃x ∀c ∃y z w ∀u. {C1, C2, C3, C4, C5} ϕ : a → a, c → b, y → x, w → x, z → x, u → u. ϕ ∈ Homφ(φ1, φ).

C : (a ∨ c ∨ w) (a ∨ c ∨ w ∨ z) C1 : (a ∨ y ∨ z) C2 : (c ∨ y ∨ w) (c ∨ z) C3 : (c ∨ z ∨ u)

By using the local homomorphism rule, we can obtain the clause ϕ(a ∨ c ∨ w) = (a ∨ b ∨ x) and add it to the matrix φ.

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Central Idea

⋆ Given Φ = Q.{C1, ..., Cn}. ⋆ ϕ ∈ Homφ(φ1, φ): ϕ({C1, .., Ck}) = {Ck+1, ..., Cn}.

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Central Idea

⋆ Given Φ = Q.{C1, ..., Cn}. ⋆ ϕ ∈ Homφ(φ1, φ): ϕ({C1, .., Ck}) = {Ck+1, ..., Cn}.

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Known Separation Results [Kauers and Seidl, 2018] [Blinkhorn and Beyersdorff, 2017]

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res

D = Drrs Dependency Scheme LS = Local Symmetry Rule GS = Global Symmetry Rule LH = Local Homomorphism Rule GH = Global Homomorphism Rule

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Separation Results for QBFs

We can lift the separation results of these systems from the propositional case [Szeider, 2005].

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res

D = Drrs Dependency Scheme LS = Local Symmetry Rule GS = Global Symmetry Rule LH = Local Homomorphism Rule GH = Global Homomorphism Rule

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Lower Bounds for LH and LH(Drrs)

Φ = Q.φ Hard for Q-Res Φ◦ = Q◦.φ◦ No non-trivial Hom construct a rigid formula Φ hard for Q-resolution = ⇒ Φ◦ hard example for LH Careful: ⋆⋆ The size of Φ◦ must be bounded by a polynomial.

• Do not introduce new dependencies.
• Existential quantifier mapping (refer to the paper).

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Construction of Φ◦ = Q◦.φ◦

◮ Φ = Q.φ = (e1 ∨ e2 ∨ u1) ∧ (e1 ∨ e4) ∧ . . . ) in 3CNF matrix.

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Construction of Φ◦ = Q◦.φ◦

◮ Φ = Q.φ = (e1 ∨ e2 ∨ u1) ∧ (e1 ∨ e4) ∧ . . . ) in 3CNF matrix. ◮ Introduce new xi: ((x1 ∨ x2 ∨ u1) ∧ (x3 ∨ x4) ∧ . . . }. plus “links” (x1 ∨ e1) ∧ (x2 ∨ e2) ∧ (x3 ∨ e1) . . .

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Construction of Φ◦ = Q◦.φ◦

◮ Φ = Q.φ = (e1 ∨ e2 ∨ u1) ∧ (e1 ∨ e4) ∧ . . . ) in 3CNF matrix. ◮ Introduce new xi: ((x1 ∨ x2 ∨ u1) ∧ (x3 ∨ x4) ∧ . . . }. plus “links” (x1 ∨ e1) ∧ (x2 ∨ e2) ∧ (x3 ∨ e1) . . . ◮ Make links longer (new yj,i): Lj = ((xi ∨ yj,1), (yj,1 ∨ yj,2), . . . , (yj,n, ei))

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Construction of Φ◦ = Q◦.φ◦

◮ Φ = Q.φ = (e1 ∨ e2 ∨ u1) ∧ (e1 ∨ e4) ∧ . . . ) in 3CNF matrix. ◮ Introduce new xi: ((x1 ∨ x2 ∨ u1) ∧ (x3 ∨ x4) ∧ . . . }. plus “links” (x1 ∨ e1) ∧ (x2 ∨ e2) ∧ (x3 ∨ e1) . . . ◮ Make links longer (new yj,i): Lj = ((xi ∨ yj,1), (yj,1 ∨ yj,2), . . . , (yj,n, ei)) ◮ Give links a direction (no non-trivial Hom): (x1 ∨ yj,1 ∨ z1), (yj,1 ∨ yj,2 ∨ z1), . . . , (yj,n ∨ e1 ∨ z1), (z1)) 3 3 3 2 3 . . . 3

j+1

2 3 3 1

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Construction of Φ◦ = Q◦.φ◦

◮ Φ = Q.φ = (e1 ∨ e2 ∨ u1) ∧ (e1 ∨ e4) ∧ . . . ) in 3CNF matrix. ◮ Introduce new xi: ((x1 ∨ x2 ∨ u1) ∧ (x3 ∨ x4) ∧ . . . }. plus “links” (x1 ∨ e1) ∧ (x2 ∨ e2) ∧ (x3 ∨ e1) . . . ◮ Make links longer (new yj,i): Lj = ((xi ∨ yj,1), (yj,1 ∨ yj,2), . . . , (yj,n, ei)) ◮ Give links a direction (no non-trivial Hom): (x1 ∨ yj,1 ∨ z1), (yj,1 ∨ yj,2 ∨ z1), . . . , (yj,n ∨ e1 ∨ z1), (z1)) 3 3 3 2 3 . . . 3

j+1

2 3 3 1 ◮ Add xi,yj,i and zi (var∃,) at the same quantifier depth as ei.

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Result-2: Separation Result

We show an exponential separation between LH(D) and LH for the reflexive resolution-path dependency scheme.

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res

D = Drrs Dependency Scheme LS = Local Symmetry Rule GS = Global Symmetry Rule LH = Local Homomorphism Rule GH = Global Homomorphism Rule

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Separation Results

The result also provides separations between LS(D) and LS, GH(D) and GH, and GS(D) and GS

LH(D) LS(D) GH(D) GS(D) LH LS GH GS Q-Res(D) Q-Res

D = Drrs Dependency Scheme LS = Local Symmetry Rule GS = Global Symmetry Rule LH = Local Homomorphism Rule GH = Global Homomorphism Rule

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Separating LH(Drrs) from LH

Claim: Our assumption of the formula Φ with matrix in 3CNF does not affect certain semantic lower bound techniques. Proposition: [Beyersdorff et al., 2019] For each n ∈ N, EQ(n) has a Q-resolution refutation

• f size 2Ω(n).

Theorem: [Beyersdorff and Blinkhorn, 2020] For each n ∈ N, EQ(n) has a Q(Drrs)-resolution refutation

• f size O(n).

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Separating LH(Drrs) from LH

For (Φn)n∈N = EQ(n) construct Φ◦

EQ(n) with ⋆⋆.

Theorem: Separation There is an infinite sequence (Φn)n∈N of false formulas such that the shortest LH(Drrs)-refutation of Φn is polynomial in n but any LH-refutation of Φn has length 2Ω(n).

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Conclusion

Our systems strengthen [Kauers and Seidl, 2018] system :

1 Global symmetries to local symmetries, 2 Symmetries to homomorphisms, and 3 Quantifier-block preserving mappings to dependencies

preserving mappings with respect to a dependency system. Future Work: ◮ Dynamic homomorphism rule: consider homomorphisms involving derived clauses. ◮ Symmetry recomputation [Blinkhorn and Beyersdorff, 2019]: combined with our homomorphism systems.

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Do YOGA and Stay Healthy!! Thanks.