On Propositional QBF Expansions and Q-Resolution s Janota 1 Joao - PowerPoint PPT Presentation

On Propositional QBF Expansions and Q-Resolution s Janota 1 Joao Marques-Silva 1 , 2 Mikol´ aˇ 1 INESC-ID/IST, Lisbon, Portugal 2 CASL/CSI, University College Dublin, Ireland SAT 2013, July 8-12 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 1 / 15

quantor nenofex Other sKizzo QBF Solving Expansion- CirQit Based GhostQ AReQS DPLL-Based Careful depQBF Expansion RAReQS QuBE Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 2 / 15

Quantified Boolean Formula (QBF) • an extension of SAT with quantifiers Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF) • an extension of SAT with quantifiers Example ∀ y 1 y 2 ∃ x 1 x 2 . (¯ y 1 ∨ x 1 ) ∧ ( y 2 ∨ ¯ x 2 ) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF) • an extension of SAT with quantifiers Example ∀ y 1 y 2 ∃ x 1 x 2 . (¯ y 1 ∨ x 1 ) ∧ ( y 2 ∨ ¯ x 2 ) • we consider prenex form with maximal blocks of variables ∀U 1 ∃E 2 . . . ∀U 2 N − 1 ∃E 2 N . φ Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF) • an extension of SAT with quantifiers Example ∀ y 1 y 2 ∃ x 1 x 2 . (¯ y 1 ∨ x 1 ) ∧ ( y 2 ∨ ¯ x 2 ) • we consider prenex form with maximal blocks of variables ∀U 1 ∃E 2 . . . ∀U 2 N − 1 ∃E 2 N . φ Solving • DPLL — Q-Resolution ( QuBE , depqbf , etc.) • Expansion — ?? ( Quantor , sKizzo , Nenofex ) • “Careful” expansion ( AReQS , RAReQS ) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Q-resolution Q-resolution = Q-resolution rule + ∀ -reduction Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution Q-resolution = Q-resolution rule + ∀ -reduction Q-resolution rule C 1 , C 2 with one and only one complementary literal l , where l is existential • derive C 1 ∪ C 2 � { l , ¯ l } Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution Q-resolution = Q-resolution rule + ∀ -reduction Q-resolution rule C 1 , C 2 with one and only one complementary literal l , where l is existential • derive C 1 ∪ C 2 � { l , ¯ l } ∀ -reduction • if k ∈ C is universal with highest level in C , remove k from C Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution Q-resolution = Q-resolution rule + ∀ -reduction Q-resolution rule C 1 , C 2 with one and only one complementary literal l , where l is existential • derive C 1 ∪ C 2 � { l , ¯ l } ∀ -reduction • if k ∈ C is universal with highest level in C , remove k from C Tautologous resolvents are generally unsound! Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Expansion ∀ x . Φ = Φ[ x / 0] ∧ Φ[ x / 1] ∃ x . Φ = Φ[ x / 0] ∨ Φ[ x / 1] Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

Expansion ∀ x . Φ = Φ[ x / 0] ∧ Φ[ x / 1] ∃ x . Φ = Φ[ x / 0] ∨ Φ[ x / 1] Fresh variables in order to keep prenex form ∃ e 1 ∀ u 2 ∃ e 3 . (¯ e 1 ∨ e 3 ) ∧ (¯ e 3 ∨ e 1 ) ∧ ( u 2 ∨ e 3 ) ∧ (¯ u 2 ∨ ¯ e 3 ) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

Expansion ∀ x . Φ = Φ[ x / 0] ∧ Φ[ x / 1] ∃ x . Φ = Φ[ x / 0] ∨ Φ[ x / 1] Fresh variables in order to keep prenex form ∃ e 1 ∀ u 2 ∃ e 3 . (¯ e 1 ∨ e 3 ) ∧ (¯ e 3 ∨ e 1 ) ∧ ( u 2 ∨ e 3 ) ∧ (¯ u 2 ∨ ¯ e 3 ) ∃ e 1 e u 2 / 0 e u 2 / 1 e 1 ∨ e u 2 / 0 e u 2 / 0 . (¯ ) ∧ (¯ ∨ e 1 ) ∧ 3 3 3 3 e 1 ∨ e u 2 / 1 e u 2 / 1 (¯ ) ∧ (¯ ∨ e 1 ) ∧ 3 3 e u 2 / 0 ∧ 3 e u 2 / 1 ¯ 3 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

How to prove by expansion? 1. Expand all universal variables 2. Refute by propositional resolution Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion? 1. Expand all universal variables 2. Refute by propositional resolution Why only universals? 1. conjunction of CNF is still CNF 2. ∃ -expansion “doing the work of resolution” Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion? 1. Expand all universal variables 2. Refute by propositional resolution Why only universals? 1. conjunction of CNF is still CNF 2. ∃ -expansion “doing the work of resolution” Partial Expansions Only certain values may be needed: ∀ u ∃ e . ( u ∨ e ) ∧ ( u ∨ ¯ e ) ∧ (¯ u ∨ e ) (false) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion? 1. Expand all universal variables 2. Refute by propositional resolution Why only universals? 1. conjunction of CNF is still CNF 2. ∃ -expansion “doing the work of resolution” Partial Expansions Only certain values may be needed: ∀ u ∃ e . ( u ∨ e ) ∧ ( u ∨ ¯ e ) ∧ (¯ u ∨ e ) (false) ∃ e u / 0 e u / 1 . e u / 0 ∧ ¯ e u / 0 ∧ e u / 1 (false full) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion? 1. Expand all universal variables 2. Refute by propositional resolution Why only universals? 1. conjunction of CNF is still CNF 2. ∃ -expansion “doing the work of resolution” Partial Expansions Only certain values may be needed: ∀ u ∃ e . ( u ∨ e ) ∧ ( u ∨ ¯ e ) ∧ (¯ u ∨ e ) (false) ∃ e u / 0 e u / 1 . e u / 0 ∧ ¯ e u / 0 ∧ e u / 1 (false full) ∃ e u / 0 . e u / 0 ∧ ¯ e u / 0 (false partial) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

Recursive Partial Expansion ∀U 1 ∃E 2 ∀U 3 ∃E 4 φ Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

Recursive Partial Expansion � . . . ∃E 2 ∃E 2 ∀U 3 ∀U 3 ∃E 4 ∃E 4 φ φ Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

Recursive Partial Expansion � . . . ∃E 2 ∃E 2 � � . . . . . . ∃E 4 ∃E 4 ∃E 4 ∃E 4 φ φ φ φ Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

∀ Exp+Res Proof: ( T , π ) Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

∀ Exp+Res Proof: ( T , π ) (1) Expansion tree T : for each block of variables it tells us how to expand it. T u 1 / 0 u 1 / 1 u 2 / 1 u 2 / 0 u 2 / 1 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

∀ Exp+Res Proof: ( T , π ) (1) Expansion tree T : for each block of variables it tells us how to expand it. T u 1 / 0 u 1 / 1 u 2 / 1 u 2 / 0 u 2 / 1 (2) Propositional Resolution Refutation π of expansion resulting from the expansion tree T . Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

Performing Expansion • For a clause C = e i ∨ u ∨ e k , for τ = τ 1 , . . . , τ n τ 1 ,...,τ i / 2 τ 1 ,...,τ k / 2 E ( τ 1 , . . . , τ n , C ) = e ∨ e if u [ τ ] = 0 i k E ( τ 1 , . . . , τ n , C ) = 1 if u [ τ ] = 1 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 9 / 15

Performing Expansion • For a clause C = e i ∨ u ∨ e k , for τ = τ 1 , . . . , τ n τ 1 ,...,τ i / 2 τ 1 ,...,τ k / 2 E ( τ 1 , . . . , τ n , C ) = e ∨ e if u [ τ ] = 0 i k E ( τ 1 , . . . , τ n , C ) = 1 if u [ τ ] = 1 • For an expansion tree T and a matrix φ consider the union of clauses E ( τ, C ) for all branches τ ∈ T and C ∈ φ . Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 9 / 15

From Tree Q-resolution to ∀ Exp+Res ¯ x , y ∈ D 2 x ∈ D 1 x ∨ C 1 x ∨ C 2 ¯ C 1 ∨ C 2 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀ Exp+Res x τ 1 ,...,τ j ∈ E ( D 1 , τ ) ¯ x µ 1 ,...,µ j , y µ 1 ,...,µ k ∈ E ( D 1 , µ ) x τ 1 ,...,τ j ∨ C ′ x τ 1 ,...,τ j ∨ C ′ ¯ 1 2 C ′ 1 ∨ C ′ 2 µ 1 , . . . , µ j = τ 1 , . . . , τ j Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀ Exp+Res x τ 1 ,...,τ j ∈ E ( D 1 , τ ) ¯ x µ 1 ,...,µ j , y µ 1 ,...,µ k ∈ E ( D 1 , µ ) x τ 1 ,...,τ j ∨ C ′ x τ 1 ,...,τ j ∨ C ′ ¯ 1 2 C ′ 1 ∨ C ′ 2 µ 1 , . . . , µ j = τ 1 , . . . , τ j y ∨ C 3 ¯ y ∨ C 4 C 3 ∨ C 4 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀ Exp+Res x τ 1 ,...,τ j ∈ E ( D 1 , τ ) ¯ x µ 1 ,...,µ j , y µ 1 ,...,µ k ∈ E ( D 1 , µ ) x τ 1 ,...,τ j ∨ C ′ x τ 1 ,...,τ j ∨ C ′ ¯ 1 2 y ∈ D 3 ¯ C ′ 1 ∨ C ′ 2 µ 1 , . . . , µ j = τ 1 , . . . , τ j y ∨ C 3 ¯ y ∨ C 4 C 3 ∨ C 4 Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15