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

Short Q-Resolution Proofs with Homomorphisms Friedrich Slivovsky 1 Stefan Szeider 1 Ankit Shukla 2 TU Wien, Vienna, Austria 1 JKU, Linz, Austria 2 June 30, 2020

Focus of Analysis: Proof Complexity Goal : Compare the strength of the proof systems and show an exponential separation . LH( D ) LH LS( D ) GH( D ) A LS GH GS( D ) B A is stronger than B. Q-Res ( D ) GS A has short proofs. Q-Res [ Ankit Shukla ] 2/34

Proof Complexity for QBF Symmetries We introduce new QBF proof systems that use homomorphisms, generalizing work on symmetries [Kauers and Seidl, 2018]. LH( D ) LH LS( D ) GH( D ) A LS GH GS( D ) B A is stronger than B. Q-Res ( D ) GS A has short proofs. Q-Res [ Ankit Shukla ] 3/34

Proof Complexity for QBF Symmetries and Homomorphisms: Our results LH( D ) LH LS( D ) GH( D ) A LS GH GS( D ) B A is stronger than B. Q-Res ( D ) GS A has short proofs. Q-Res [ 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. [ Ankit Shukla ] 5/34

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 [ Ankit Shukla ] 6/34

Global Vs Local: Symmetries and Homomorphisms [ Ankit Shukla ] 7/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F Example (Symmetries: Injective mapping) F = ( a ∨ b ) ∧ ( a ∨ c ) ∧ ( b ∨ c ) [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F 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. σ σ ( F ) = F F Example (Symmetries: Injective mapping) F = ( a ∨ b ) ∧ ( a ∨ c ) ∧ ( b ∨ c ) σ : a � → a , b � → c , c � → b ( a � → a , ... preserves complements) σ ( F ) = [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F 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 ) [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F 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 ) [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F 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 ) [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985] Literal permutation that preserves the CNF. σ σ ( F ) = F F 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 ) ( σ preserve the CNF). = F [ Ankit Shukla ] 8/34

Symmetries in CNF [Krishnamurthy, 1985, Urquhart, 1999, Arai and Urquhart, 2000] Literal permutation that preserves the CNF. σ σ ( F ) = F F (a) Global Symmetry [ Ankit Shukla ] 9/34

Symmetries in CNF [Krishnamurthy, 1985, Urquhart, 1999, Arai and Urquhart, 2000] Literal permutation that preserves the CNF. σ σ σ ( F ) = F σ ( F 1 ) ⊆ F F 1 ⊆ F F (a) Global Symmetry (b) Local Symmetry [ Ankit Shukla ] 9/34

Homomorphisms in CNF [Szeider, 2005] ⋆ Allow non-injective mapping. ⋆ Hence possible: | ϕ ( C ) | < | C | and | ϕ ( F ) | < | F | . Start with F ϕ ϕ ( F ) ⊆ F F (a) Global Homomorphism [ Ankit Shukla ] 10/34

Homomorphisms in CNF [Szeider, 2005] ⋆ Allow non-injective mapping. ⋆ Hence possible: | ϕ ( C ) | < | C | and | ϕ ( F ) | < | F | . Start with F ϕ ϕ ( F ) ⊆ F F (a) Global Homomorphism ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F (b) Local Homomorphism [ Ankit Shukla ] 10/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) [ Ankit Shukla ] 11/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) F 1 ⊆ F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) [ Ankit Shukla ] 11/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) F 1 ⊆ F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ϕ : x � → z , y � → z , a � → b , v � → v ( x � → z , . . . ) [ Ankit Shukla ] 11/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) F 1 ⊆ F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ϕ : x � → z , y � → z , a � → b , v � → v ( x � → z , . . . ) ϕ ( F 1 ) = ( b ∨ z ) ∧ ( v ∨ z ∨ z ) ∧ ( b ∨ z ) [ Ankit Shukla ] 11/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) F 1 ⊆ F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ϕ : x � → z , y � → z , a � → b , v � → v ( x � → z , . . . ) ϕ ( F 1 ) = ( b ∨ z ) ∧ ( v ∨ z ∨ z ) ∧ ( b ∨ z ) = ( b ∨ z ) ∧ ( v ∨ z ) [ Ankit Shukla ] 11/34

Local Homomorphism Example Allow non-injective mapping. ϕ F 1 ⊆ F ϕ ( F 1 ) ⊆ F Example (Homomorphism; Non-injective mapping) F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ∧ ( b ∨ z ) ∧ ( v ∨ z ) F 1 ⊆ F = ( a ∨ x ) ∧ ( v ∨ x ∨ y ) ∧ ( a ∨ y ) ϕ : x � → z , y � → z , a � → b , v � → v ( x � → z , . . . ) ϕ ( F 1 ) = ( b ∨ z ) ∧ ( v ∨ z ∨ z ) ∧ ( b ∨ z ) = ( b ∨ z ) ∧ ( v ∨ z ) ⊆ F [ Ankit Shukla ] 11/34

Going to a Higher Level: SAT to QBF

PCNF: Quantified Boolean Formulas (QBFs) Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀ x 1 x 2 ∃ y 1 . (( x 1 ∨ x 2 ∨ y 1 ) ∧ ( x 1 ∨ y 1 )) [ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs) Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀ x 1 x 2 ∃ y 1 . (( x 1 ∨ x 2 ∨ y 1 ) ∧ ( x 1 ∨ y 1 )) ⋆ Matrix (CNF) [ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs) Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀ x 1 x 2 ∃ y 1 . (( x 1 ∨ x 2 ∨ y 1 ) ∧ ( x 1 ∨ y 1 )) ⋆ Matrix (CNF) [ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs) Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀ x 1 x 2 ∃ y 1 . (( x 1 ∨ x 2 ∨ y 1 ) ∧ ( x 1 ∨ y 1 )) ⋆ Matrix (CNF) ◮ For all values of x 1 and x 2 , is there a value for y 1 such that the formula evaluates to true? [ Ankit Shukla ] 13/34

PCNF: Quantified Boolean Formulas (QBFs) Propositional logic formulas + quantifiers ∀∃ ⋆ Prefix ∀ x 1 x 2 ∃ y 1 . (( x 1 ∨ x 2 ∨ y 1 ) ∧ ( x 1 ∨ y 1 )) ⋆ Matrix (CNF) ◮ For all values of x 1 and x 2 , is there a value for y 1 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? [ Ankit Shukla ] 13/34

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 D Φ ⊆ { ( x , y ) : x < Φ y } called the dependency relation of Φ with respect to D . Definition: D trv Φ The D trv associates each PCNF formula Φ with the relation D trv Φ = { ( x , y ) : x < Φ y } . [ Ankit Shukla ] 14/34