Efficient Clause Learning for Quantified Boolean Formulas via QBF Pseudo Unit Propagation Florian Lonsing 1 Uwe Egly 1 Allen Van Gelder 2 1 Vienna University of Technology http://www.kr.tuwien.ac.at/staff/{egly,lonsing} 2 University of California at Santa Cruz, USA http://www.cse.ucsc.edu/~avg This work is supported by the Austrian Science Fund (FWF) under grant S11409-N23. 1 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Overview (1/2) Conflict-Driven Clause Learning (CDCL): [SS96] Crucial for the performance of modern SAT solvers. Resolution proofs, trimming the search space. Extensions of CDCL for SAT to QBF: QCDCL. Traditional QCDCL for QBF: [ZM02, GNT02, GNT06, Let02] Like CDCL is based on resolution, QCDCL is based on Q-resolution. Q-resolution derivation of the clause to be learned. Tautological resolvents must be avoided explicitly. Problem: Common approach to avoiding tautologies in traditional QCDCL has an exponential worst case [VG12]. The derivation of a single learned clause might have an exponential number of intermediate resolvents. 2 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Overview (1/2) Conflict-Driven Clause Learning (CDCL): [SS96] Crucial for the performance of modern SAT solvers. Resolution proofs, trimming the search space. Extensions of CDCL for SAT to QBF: QCDCL. Traditional QCDCL for QBF: [ZM02, GNT02, GNT06, Let02] Like CDCL is based on resolution, QCDCL is based on Q-resolution. Q-resolution derivation of the clause to be learned. Tautological resolvents must be avoided explicitly. Problem: Common approach to avoiding tautologies in traditional QCDCL has an exponential worst case [VG12]. The derivation of a single learned clause might have an exponential number of intermediate resolvents. 2 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Overview (2/2) Our Work: efficient polynomial time procedure for QCDCL. QCDCL based on QBF Pseudo Unit Propagation (QPUP) [VG12]: carefully select the order of resolution steps in QCDCL to avoid tautologies. Learn a single non-tautological clause in polynomial time. QPUP-based QCDCL is compatible with other approaches (e.g. Alexandra’s talk). Implementation in the search-based QBF solver DepQBF. 3 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Quantified Boolean Formulae (QBF) Syntax Prenex CNF: quantifier-free CNF over quantified Boolean variables. PCNF ψ := Q 1 x 1 . . . Q n x n . φ , where Q i ∈ {∃ , ∀} , no free variables. Q i x i ≤ Q i + 1 x i + 1 : variables are linearly ordered. Example A CNF: ( x ∨ ¬ y ) ∧ ( ¬ x ∨ y ) , and a PCNF: ∀ x ∃ y . ( x ∨ ¬ y ) ∧ ( ¬ x ∨ y ) . Search-based QBF Solving with Clause Learning: Implicitly enumerate paths in a semantic tree by recursive variable instantiation. Terminology “QCDCL”: conflict-driven clause learning (CDCL) for QBF. Learn clauses at unsatisfiable (i.e. conflicting) branches in the search tree. Like CDCL in SAT: QCDCL is based on resolution for QBF. 4 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Resolution for QBF Q-Resolution: Combination of universal reduction and propositional resolution. Sound and refutational-complete proof system for QBF: Q-resolution proofs. Definition ([BKF95]) Given a clause C , universal reduction (UR) on C produces the clause UR ( C ) := C \ { l ∈ L ∀ ( C ) | ∀ l ′ ∈ L ∃ ( C ) : var ( l ′ ) < var ( l ) } , where < is the linear variable ordering given by the quantifier prefix. Universal reduction deletes trailing universal literals from clauses. Definition ([BKF95]) Let C 1 , C 2 be non-tautological clauses where v ∈ C 1 , ¬ v ∈ C 2 for an ∃ -variable v . Tentative Q-resolvent of C 1 and C 2 : C 1 ⊗ C 2 := ( UR ( C 1 ) ∪ UR ( C 2 )) \ { v , ¬ v } . If { x , ¬ x } ⊆ C 1 ⊗ C 2 for some variable x , then no Q-resolvent exists. Otherwise, the non-tautological Q-resolvent is C := UR ( C 1 ⊗ C 2 ) . 5 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Boolean Constraint Propagation for QBF (QBCP) Generate assignments by assumptions, unit clause rule, universal reduction (UR). Like BCP for SAT: antecedent clauses and implication graphs. Like CDCL for SAT: QCDCL is based on the implication graph given by QBCP. Example (assignments, implication graphs) Assignment A := {} . p cnf 5 4 e 1 3 4 0 Assumption: A := A ∪ { 1 } . a 5 0 Clause (-1 2) is unit under A e 2 0 A := A ∪ { 2 } = { 1 , 2 } -1 2 0 3 5 -2 0 ante ( 2 ) := (-1 2) 4 -5 -2 0 Clause (3 5 -2) is unit under A and UR. -3 -4 0 A := A ∪ { 3 } = { 1 , 2 , 3 } ante ( 3 ) := (3 5 -2) Implication graph: Clause (4 5 -2) is unit under A and UR. A := A ∪ { 4 } = { 1 , 2 , 3 , 4 } ante ( 4 ) := (4 5 -2) Clause (-3 -4) is conflicting under A . ante ( ∅ ) := (-3 -4) 6 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Boolean Constraint Propagation for QBF (QBCP) Generate assignments by assumptions, unit clause rule, universal reduction (UR). Like BCP for SAT: antecedent clauses and implication graphs. Like CDCL for SAT: QCDCL is based on the implication graph given by QBCP. Example (assignments, implication graphs) Assignment A := {} . p cnf 5 4 e 1 3 4 0 Assumption: A := A ∪ { 1 } . a 5 0 Clause (-1 2) is unit under A e 2 0 A := A ∪ { 2 } = { 1 , 2 } -1 2 0 3 5 -2 0 ante ( 2 ) := (-1 2) 4 -5 -2 0 Clause (3 5 -2) is unit under A and UR. -3 -4 0 A := A ∪ { 3 } = { 1 , 2 , 3 } ante ( 3 ) := (3 5 -2) Implication graph: Clause (4 5 -2) is unit under A and UR. A := A ∪ { 4 } = { 1 , 2 , 3 , 4 } 1 ante ( 4 ) := (4 5 -2) Clause (-3 -4) is conflicting under A . ante ( ∅ ) := (-3 -4) 6 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Boolean Constraint Propagation for QBF (QBCP) Generate assignments by assumptions, unit clause rule, universal reduction (UR). Like BCP for SAT: antecedent clauses and implication graphs. Like CDCL for SAT: QCDCL is based on the implication graph given by QBCP. Example (assignments, implication graphs) Assignment A := {} . p cnf 5 4 e 1 3 4 0 Assumption: A := A ∪ { 1 } . a 5 0 Clause (-1 2) is unit under A e 2 0 A := A ∪ { 2 } = { 1 , 2 } -1 2 0 3 5 -2 0 ante ( 2 ) := (-1 2) 4 -5 -2 0 Clause (3 5 -2) is unit under A and UR. -3 -4 0 A := A ∪ { 3 } = { 1 , 2 , 3 } ante ( 3 ) := (3 5 -2) Implication graph: Clause (4 5 -2) is unit under A and UR. A := A ∪ { 4 } = { 1 , 2 , 3 , 4 } 1 2 ante ( 4 ) := (4 5 -2) Clause (-3 -4) is conflicting under A . ante ( ∅ ) := (-3 -4) 6 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Boolean Constraint Propagation for QBF (QBCP) Generate assignments by assumptions, unit clause rule, universal reduction (UR). Like BCP for SAT: antecedent clauses and implication graphs. Like CDCL for SAT: QCDCL is based on the implication graph given by QBCP. Example (assignments, implication graphs) Assignment A := {} . p cnf 5 4 e 1 3 4 0 Assumption: A := A ∪ { 1 } . a 5 0 Clause (-1 2) is unit under A e 2 0 A := A ∪ { 2 } = { 1 , 2 } -1 2 0 3 5 -2 0 ante ( 2 ) := (-1 2) 4 -5 -2 0 Clause (3 5 -2) is unit under A and UR. -3 -4 0 A := A ∪ { 3 } = { 1 , 2 , 3 } ante ( 3 ) := (3 5 -2) Implication graph: Clause (4 5 -2) is unit under A and UR. A := A ∪ { 4 } = { 1 , 2 , 3 , 4 } 1 2 3 ante ( 4 ) := (4 5 -2) Clause (-3 -4) is conflicting under A . ante ( ∅ ) := (-3 -4) 6 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
Boolean Constraint Propagation for QBF (QBCP) Generate assignments by assumptions, unit clause rule, universal reduction (UR). Like BCP for SAT: antecedent clauses and implication graphs. Like CDCL for SAT: QCDCL is based on the implication graph given by QBCP. Example (assignments, implication graphs) Assignment A := {} . p cnf 5 4 e 1 3 4 0 Assumption: A := A ∪ { 1 } . a 5 0 Clause (-1 2) is unit under A e 2 0 A := A ∪ { 2 } = { 1 , 2 } -1 2 0 3 5 -2 0 ante ( 2 ) := (-1 2) 4 -5 -2 0 Clause (3 5 -2) is unit under A and UR. -3 -4 0 A := A ∪ { 3 } = { 1 , 2 , 3 } ante ( 3 ) := (3 5 -2) Implication graph: Clause (4 5 -2) is unit under A and UR. A := A ∪ { 4 } = { 1 , 2 , 3 , 4 } 1 2 3 ante ( 4 ) := (4 5 -2) 4 Clause (-3 -4) is conflicting under A . ante ( ∅ ) := (-3 -4) 6 Florian Lonsing, Uwe Egly, Allen Van Gelder Efficient Clause Learning for Quantified Boolean Formulas via QPUP
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