SAT 2013 Nested Boolean Functions as Models for QBFs Uwe Bubeck and Hans Kleine Büning University of Paderborn July 11, 2013
Outline • Introduction: QBF and (Counter-)Models • Free Variables and Models • NBF Representation • Conclusion Uwe Bubeck NBFs as Models for QBFs 2
Section 1 Introduction: QBF and (Counter-)Models
Quantified Boolean Formulas QBF extends propositional logic by allowing universal and existential quantifiers over propositional variables. Semantics of closed QBF: ∃𝑧 Φ 𝑧 is true if and only if Φ 𝑧/0 is true or Φ 𝑧/1 is true. ∀𝑦 Φ 𝑦 is true if and only if Φ 𝑦/0 is true and Φ 𝑦/1 is true. Uwe Bubeck NBFs as Models for QBFs 4
Tree Models ∀𝑦 1 ∃𝑧 1 ∀𝑦 2 ∃𝑧 2 𝑦 1 ∨ ¬𝑧 1 ∧ ¬𝑦 1 ∨ 𝑧 2 ∧ 𝑧 1 ∨ 𝑦 2 ∨ ¬𝑧 2 ∧ ¬𝑦 2 ∨ 𝑧 2 𝑦 1 𝑦 1 = 0 𝑦 1 = 1 𝑧 1 𝑧 1 𝑧 1 = 0 𝑧 1 = 1 𝑦 2 𝑦 2 𝑦 2 = 0 𝑦 2 = 1 𝑦 2 = 0 𝑦 2 = 1 𝑧 2 𝑧 2 𝑧 2 𝑧 2 𝑧 2 = 0 𝑧 2 = 1 𝑧 2 = 1 𝑧 2 = 1 Uwe Bubeck NBFs as Models for QBFs 9
Function Models 1/2 QBF as a 2-player game: ∃ and ∀ player alternatingly choose assignments for variables in prefix order. 𝑦 1 𝑦 1 = 0 𝑦 1 = 1 𝑧 1 𝑧 1 𝑧 1 = 0 𝑧 1 = 1 𝑦 2 𝑦 2 𝑦 2 = 0 𝑦 2 = 1 𝑦 2 = 0 𝑦 2 = 1 𝑧 2 𝑧 2 𝑧 2 𝑧 2 𝑧 2 = 0 𝑧 2 = 1 𝑧 2 = 1 𝑧 2 = 1 Uwe Bubeck NBFs as Models for QBFs 10
Function Models 2/2 ∀𝑦 1 ∃𝑧 1 ∀𝑦 2 ∃𝑧 2 … ∀𝑦 𝑜 ∃𝑧 𝑜 𝜚 𝑦 1 , … , 𝑦 𝑜 , 𝑧 1 , … , 𝑧 𝑜 = true if and only if ∀𝑦 1 … ∀𝑦 𝑜 𝜚 𝑦 1 , … , 𝑦 𝑜 , 𝑔 1 𝑦 1 , … , 𝑔 𝑜 𝑦 1 , … , 𝑦 𝑜 = true for some 𝑔 1 , … , 𝑔 𝑜 (Skolem functions). ∀𝑦 1 ∃𝑧 1 ∀𝑦 2 ∃𝑧 2 … ∀𝑦 𝑜 ∃𝑧 𝑜 𝜚 𝑦 1 , … , 𝑦 𝑜 , 𝑧 1 , … , 𝑧 𝑜 = false if and only if ∃𝑧 1 … ∃𝑧 𝑜 𝜚( 1 (), 2 y 1 , … , 𝑜 (y 1 , … , 𝑧 𝑜−1 ), 𝑧 1 , … , 𝑧 𝑜 ) = false for some 1 , … , 𝑜 (Herbrand functions) . Uwe Bubeck NBFs as Models for QBFs 11
Motivation • Important applications: solver certificates, explanations, ... • Balabanov and Jiang (2012): Extract Skolem model from cube-resolution proof, Herbrand countermodel from clause-resolution proof. • Problem: compact representation 𝑄 ≠ Π 2 𝑄 ) (no polynomial-size propositional encoding if Σ 2 • Contributions: direct polynomial-size encoding by NBFs (counter)models parameterized by free variables Uwe Bubeck NBFs as Models for QBFs 12
Section 2 Free Variables and Models
Semantics of Free Variables Closed QBF: either true or false Open QBF: valuation depends on the free variables: True QBF with Closed Truth Assignment Evaluation free vars QBF False Φ(𝑨 1 , … , 𝑨 𝑠 ) 𝑢: 𝑨 1 , … , 𝑨 r → 0,1 Φ 𝑢(𝑨 1 ), … , 𝑢(𝑨 r ) ≈ Propositional Formula Uwe Bubeck NBFs as Models for QBFs 14
Free Variables and Models True QBF with Closed Truth Assignment Evaluation free vars QBF False Multiple models and/or countermodels How are the models and countermodels for different assignments to the free variables related to each other? Uwe Bubeck NBFs as Models for QBFs 15
Complete Equivalence Models 1/2 Idea: Replace all quantified variables with functions over the free variables. ∀𝑤 𝑜 ∃𝑤 𝑜−1 … ∀𝑤 2 ∃𝑤 1 𝜚 𝑤 1 , … , 𝑤 𝑜 , 𝑨 1 , … , 𝑨 𝑠 ≈ 𝜚 ℎ 1 (𝑨 1 , … , 𝑨 𝑠 ), … , ℎ 𝑜 (𝑨 1 , … , 𝑨 𝑠 ), 𝑨 1 , … , 𝑨 𝑠 Uwe Bubeck NBFs as Models for QBFs 16
Complete Equivalence Models 2/2 Why bother about models parameterized by free variables? Non-prenex QBF: ∀𝑏∃𝑐 ∀𝑑∃𝑒 𝛽 𝑏, 𝑐, 𝑑, 𝑒 ∧ ∀𝑦∃𝑧 𝛾 𝑏, 𝑐, 𝑦, 𝑧 Open QBF with free vars 𝑏, 𝑐 . e.g. precompute partial certificate. Uwe Bubeck NBFs as Models for QBFs 17
Section 3 NBF Representation
Nested Boolean Functions A Nested Boolean Function (NBF) [Cook/Soltys 1999] is a sequence of functions 𝐺 = 𝑔 0 , … , 𝑔 𝑙 with • initial functions 𝑔 0 , … , 𝑔 𝑢 given as propositional formulas 𝑗 𝑦 𝑗 ≔ 𝑔 𝑗 , … , 𝑔 𝑗 • compound functions 𝑔 𝑘 0 𝑔 𝑘 1 𝑦 1 𝑘 𝑠 𝑦 𝑠 for previously defined functions 𝑔 𝑘0 , … , 𝑔 𝑘𝑠 . Example: parity of Boolean variables 𝑔 0 𝑞 1 , 𝑞 2 ≔ ¬𝑞 1 ∧ 𝑞 2 ∨ (𝑞 1 ∧ ¬𝑞 2 ) 𝑔 1 𝑞 1 , 𝑞 2 , 𝑞 3 , 𝑞 4 ≔ 𝑔 0 𝑔 0 𝑞 1 , 𝑞 2 , 𝑔 0 𝑞 3 , 𝑞 4 𝑔 2 𝑞 1 , … , 𝑞 16 ≔ 𝑔 1 𝑔 1 𝑞 1 , … , 𝑞 4 , … , 𝑔 1 𝑞 13 , … , 𝑞 16 Uwe Bubeck NBFs as Models for QBFs 19
Quantifier Encoding in NBF 1/2 Φ 𝒜 ≔ ∃𝑦 𝜚(𝑦, 𝒜) QBF: Φ 𝒜 ≈ 𝐺 1 (𝒜) 𝐺 0 𝑦, 𝒜 ≔ 𝜚 𝑦, 𝒜 NBF: 𝐺 0 (𝐺 0 1, 𝒜 , 𝒜) 𝐺 1 𝒜 ≔ = 1 if x = 1 is a satisfying choice = 𝐺 0 1, 𝒜 = 1 Uwe Bubeck NBFs as Models for QBFs 20
Quantifier Encoding in NBF 1/2 Φ 𝒜 ≔ ∃𝑦 𝜚(𝑦, 𝒜) QBF: 𝐺 0 𝑦, 𝒜 ≔ 𝜚 𝑦, 𝒜 NBF: 𝐺 0 (𝐺 0 1, 𝒜 , 𝒜) 𝐺 1 𝒜 ≔ = 0 if x = 1 is not satisfying = 𝐺 0 0, 𝒜 Uwe Bubeck NBFs as Models for QBFs 21
Quantifier Encoding in NBF 2/2 Φ 𝒜 ≔ ∀𝑧∃𝑦 𝜚 𝑦, 𝑧, 𝒜 QBF: 𝐺 0 𝑦, 𝑧, 𝒜 ≔ 𝜚 𝑦, 𝑧, 𝒜 𝐺 1 𝑧, 𝒜 ≔ 𝐺 0 𝐺 0 1, 𝑧, 𝒜 , 𝑧, 𝒜 NBF: 𝐺 2 𝒜 ≔ 𝐺 1 𝐺 1 0, 𝒜 , 𝒜 = 0 if y = 0 is not satisfying = 𝐺 1 0, 𝒜 = 0 Uwe Bubeck NBFs as Models for QBFs 22
Quantifier Encoding in NBF 2/2 Φ 𝒜 ≔ ∀𝑧∃𝑦 𝜚 𝑦, 𝑧, 𝒜 QBF: 𝐺 0 𝑦, 𝑧, 𝒜 ≔ 𝜚 𝑦, 𝑧, 𝒜 𝐺 1 𝑧, 𝒜 ≔ 𝐺 0 𝐺 0 1, 𝑧, 𝒜 , 𝑧, 𝒜 NBF: 𝐺 2 𝒜 ≔ 𝐺 1 𝐺 1 0, 𝒜 , 𝒜 = 1 if y = 0 is satisfying = 𝐺 1 1, 𝒜 Uwe Bubeck NBFs as Models for QBFs 23
Quantifier Encoding in NBF 2/2 Φ 𝒜 ≔ ∀𝑧∃𝑦 𝜚 𝑦, 𝑧, 𝒜 QBF: 𝐺 0 𝑦, 𝑧, 𝒜 ≔ 𝜚 𝑦, 𝑧, 𝒜 𝐺 1 𝑧, 𝒜 ≔ 𝐺 0 𝐺 0 1, 𝑧, 𝒜 , 𝑧, 𝒜 NBF: 𝐺 2 𝒜 ≔ 𝐺 1 𝐺 1 0, 𝒜 , 𝒜 → Concise representation of QDPLL branching: innermost call of 𝐺 𝑗 is the first branch, outermost call of 𝐺 𝑗 is the second branch, or a repetition of the first one if it is already conclusive. Uwe Bubeck NBFs as Models for QBFs 24
Complete Equiv. Model in NBF 1/2 First branch determines which branch is conclusive. → this is our witness, i.e. (counter)model. Uwe Bubeck NBFs as Models for QBFs 25
Complete Equiv. Model in NBF 1/2 First branch determines which branch is conclusive. → this is our witness, i.e. (counter)model. Φ 𝒜 ≔ ∀𝑧∃𝑦 𝜚 𝑦, 𝑧, 𝒜 QBF: ℎ 𝑦 (𝒜) 𝐺 0 𝑦, 𝑧, 𝒜 ≔ 𝜚 𝑦, 𝑧, 𝒜 𝐺 1 𝑧, 𝒜 ≔ 𝐺 0 𝐺 0 1, 𝑧, 𝒜 , 𝑧, 𝒜 NBF: 𝐺 2 𝒜 ≔ 𝐺 1 𝐺 1 0, 𝒜 , 𝒜 ℎ 𝑧 (𝒜) Uwe Bubeck NBFs as Models for QBFs 27
Complete Equiv. Model in NBF 1/2 Φ 𝒜 ≔ ∀𝑧∃𝑦 𝜚 𝑦, 𝑧, 𝒜 QBF: ℎ 𝑦 (𝒜) 𝐺 0 𝑦, 𝑧, 𝒜 ≔ 𝜚 𝑦, 𝑧, 𝒜 𝐺 1 𝑧, 𝒜 ≔ 𝐺 0 𝐺 0 1, ℎ 𝑧 (𝒜), 𝒜 , 𝑧, 𝒜 NBF: 𝐺 2 𝒜 ≔ 𝐺 1 𝐺 1 0, 𝒜 , 𝒜 ℎ 𝑧 (𝒜) Uwe Bubeck NBFs as Models for QBFs 28
Complete Equiv. Model in NBF 2/2 In general: Φ 𝒜 ≔ 𝑅 𝑜 𝑤 𝑜 … 𝑅 1 𝑤 1 𝜚 𝑤 1 , … , 𝑤 𝑜 , 𝒜 Complete equivalence model: ℎ 𝑗 𝒜 ≔ 𝐺 𝑗−1 0, ℎ 𝑗+1 𝒜 , … , ℎ 1 𝒜 , 𝒜 , if 𝑅 𝑗 = ∀ 𝐺 𝑗−1 1, ℎ 𝑗+1 𝒜 , … , ℎ 1 𝒜 , 𝒜 , if 𝑅 𝑗 = ∃ Clearly polynomial size, which is not possible with a 𝑄 ≠ Π 2 𝑄 . propositional encoding if Σ 2 Admittedly more difficult to evaluate. But: Equiv. model checking PSPACE-hard even if ℎ 𝑗 𝒜 ∈ {0,1} . Uwe Bubeck NBFs as Models for QBFs 29
Section 4 Conclusion
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