Reasoning Engines for Rigorous System Engineering Block 3: Quantified Boolean Formulas and DepQBF 2. Basic Deduction Concepts for Quantified Boolean Formulas Uwe Egly Florian Lonsing Knowledge-Based Systems Group Institute of Information Systems Vienna University of Technology U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 1 / 40
Outline A resolution calculus for QBFs in PCNF 1 Long distance resolution 2 Gentzen/sequent systems for arbitrary QBFs 3 U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 2 / 40
Why do we need a resolution calculus for QBFs? We need a QSAT solver in our rapid implementation approach. Why not Q-resolution (Q-res)? Although you will usually not see it, but in nearly every QDPLL solver, there is Q-res inside. Some QDPLL solvers deliver Q-res clause proofs (“refutations”) as certificates for unsatisfiability. Some even deliver Q-res cube “proofs” as certificates for satisfiability. From such proofs, one can generate witness functions (as mentioned earlier). U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 3 / 40
A resolution calculus for QBFs: The definition of resolvents Definition (propositional resolvent) Given two clauses C 1 and C 2 and a pivot variable p with p ∈ C 1 and ¬ p ∈ C 2 , resolution produces the resolvent C r = ( C 1 \ { p } ) ∪ ( C 2 \ {¬ p } ) . Definition (Q-resolution with existential pivot variable) 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 := C 1 ⊗ C 2 . U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 4 / 40
A resolution calculus for QBFs: The quantification level Definition ( Quantification level) Let Q be a sequence of quantifiers. Associate to each alternation its level as follows. The left-most quantifier block gets level 1, and each alternation increments the level. Example ( QBF with 4 quantification levels and 3 quantifier alternations) ∀ x 1 ∀ x 2 ∃ y 1 ∃ y 2 ∃ y 3 ∀ x 3 ∃ y 4 ϕ � �� � ���� � �� � ���� level 1 level 2 level 3 level 4 An ordering between variables is defined according to their occurrence in the quantifier prefix and extended to literals. For instance, x 2 < y 4 as well as x 1 < ¬ x 3 . U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 5 / 40
A resolution calculus for QBFs: Universal reduction Definition (universal reduction (UR)) Given a clause C , UR on C produces the clause UR ( C ) := C \ { ℓ ∈ C | q ( ℓ ) = ∀ and ∀ ℓ ′ ∈ C with q ( ℓ ′ ) = ∃ : ℓ ′ < ℓ } , where < is the linear variable ordering given by the quantifier prefix. Universal reduction deletes “trailing” universal literals from clauses. Clauses are shortened by UR. Example Given Φ := ∀ y ∃ x 1 ∀ z ∃ x 2 . ( x 1 ∨ z ) ∧ ( ¬ y ∨ ¬ x 1 ) ∧ ( ¬ y ∨ x 2 ) , we have � �� � C UR ( C ) := x 1 . U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 6 / 40
A resolution calculus for QBFs Definition ( Q-resolution calculus) The Q-resolution (Q-res) calculus consists of the Q-resolution rule and the universal reduction rule. Remark 1 Resolution operations are only allowed over existential literals. 2 Tautological resolvents are never generated. We will relax these requirements later on. U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 7 / 40
Soundness and completeness or Q-resolution Theorem ( Kleine Büning, Karpinski, Flögel, Inf. Comput., 1995) A QBF in PCNF without tautological clauses is false iff there is a derivation of the empty clause � (= a refutation) in the Q-resolution calculus. Example Let Φ be ∃ a ∀ x ∃ b ∀ y ∃ c . C 1 ∧ · · · ∧ C 6 with C 1 : a ∨ b ∨ y ∨ c C 2 : a ∨ x ∨ b ∨ y ∨ ¬ c C 3 : x ∨ ¬ b C 4 : ¬ y ∨ c C 5 : ¬ a ∨ ¬ x ∨ b ∨ ¬ c C 6 : ¬ x ∨ ¬ b U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 8 / 40
A Q-resolution refutation of Φ ( C 5 ) ( C 4 ) C 1 C 2 ¬ y ∨ c ¬ a ∨ ¬ x ∨ b ∨ ¬ c a ∨ x ∨ b ∨ y R R ¬ a ∨ ¬ x ∨ b ∨ ¬ y ( C 3 ) ( C 6 ) UR UR a ∨ x ∨ b x ∨ ¬ b ¬ a ∨ ¬ x ∨ b ¬ x ∨ ¬ b R R a ∨ x ¬ a ∨ ¬ x UR UR a ¬ a R � Example (again) Let Φ be ∃ a ∀ x ∃ b ∀ y ∃ c . C 1 ∧ · · · ∧ C 6 with C 1 : a ∨ b ∨ y ∨ c C 2 : a ∨ x ∨ b ∨ y ∨ ¬ c C 3 : x ∨ ¬ b C 4 : ¬ y ∨ c C 5 : ¬ a ∨ ¬ x ∨ b ∨ ¬ c C 6 : ¬ x ∨ ¬ b U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 9 / 40
A resolution calculus for QBFs (cont’d) Is the following rule allowed/sound? Definition (QU-resolution with universal pivot variable) Let C 1 , C 2 be non-tautological clauses where v ∈ C 1 , ¬ v ∈ C 2 for an ∀ -variable v . Tentative QU-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 QU-resolvent exists. Otherwise, the non-tautological QU-resolvent is C := C 1 ⊗ C 2 . U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 10 / 40
A resolution calculus for QBFs (cont’d) Is the following rule allowed/sound? Definition (QU-resolution with universal pivot variable) Let C 1 , C 2 be non-tautological clauses where v ∈ C 1 , ¬ v ∈ C 2 for an ∀ -variable v . Tentative QU-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 QU-resolvent exists. Otherwise, the non-tautological QU-resolvent is C := C 1 ⊗ C 2 . YES. Q-resolution can be extended by this rule yielding QU-resolution! U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 10 / 40
A stronger resolution calculus for QBFs Definition ( QU-resolution calculus) The Q-resolution (Q-res) calculus consists of the Q-resolution rule, the QU-resolution rule and the universal reduction rule. The QU-resolution calculus is a slight extension of the Q-resolution calculus, but . . . it has the potential to enable shorter proofs. ➥ We will demonstrate this in the following. U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 11 / 40
A hard class of formulas for Q-resolution Definition ( Class (Ψ k ) k ≥ 1 of unsatisfiable QBFs) Ψ ( k ≥ 1 ) := ∃ d 1 ∃ e 1 ∀ x 1 ∃ d 2 ∃ e 2 ∀ x 2 · · · ∃ d k ∃ e k ∀ x k ∃ f 1 · · · ∃ f k . ( d 1 ∨ e 1 ) ∧ (1) ( d k ∨ x k ∨ f 1 ∨ · · · ∨ f k ) ∧ (2) ( e k ∨ x k ∨ f 1 ∨ · · · ∨ f k ) ∧ (3) � k − 1 j = 1 ( d j ∨ x j ∨ d j + 1 ∨ e j + 1 ) ∧ (4) � k − 1 j = 1 ( e j ∨ x j ∨ d j + 1 ∨ e j + 1 ) ∧ (5) � k j = 1 ( x j ∨ f j ) ∧ (6) � k j = 1 ( x j ∨ f j ) (7) U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 12 / 40
A hard class of formulas for Q-resolution Theorem ( Kleine Büning, Karpinski, Flögel, Inf. Comput., 1995) Any Q-resolution proof of Ψ k has at least 2 k resolution steps. Result is a bit surprising, because the existential part (in black) is Horn and propositional Horn clause sets have short (unit) resolution proofs. Short proofs are possible for Horn clause sets containing ∀ variables. ➥ Universal non-Horn part forces exponential proof length! U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 13 / 40
QU-resolution and the class (Ψ k ) k ≥ 1 In general: QU-res allows to derive clauses which Q-res cannot derive. In particular for formula Ψ k : QU-res allows to derive unit clauses. Key observation: unit clauses f i (1 ≤ i ≤ k ) obtained by QU-resolution allow for short proofs of Ψ k . Proposition ( Van Gelder 2012) Every formula Ψ k has a QU-resolution proof with O ( k ) resolution steps. U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 14 / 40
Short QU-res proofs for Ψ k ( k ≥ 1) Example ( Ψ 2 in QDIMACS format) c k=2 p cnf 8 9 Derive new unit clauses from all the binary clauses by e 1 2 0 QU-resolution over universal variables. The result are a 3 0 two clauses f 1 and f 2 ( 7 0 ) and ( 8 0 ). e 4 5 0 Observe: the unit clauses resulting from the previous a 6 0 step cannot be derived by Q-res. e 7 8 0 -1 -2 0 We derive ( 4 0 ) and ( 5 0 ) by Q-resolutions and UR. 1 -3 -4 -5 0 Use the new unit clauses to successively shorten all the 2 3 -4 -5 0 clauses of size four by unit resolution and universal 4 -6 -7 -8 0 reduction. Further unit clauses can be obtained this way. 5 6 -7 -8 0 3 7 0 Finally the empty clause is derived using ( -1 -2 0 ). -3 7 0 This resolution strategy can be applied to Ψ k for all k . 6 8 0 -6 8 0 U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 15 / 40
Outline A resolution calculus for QBFs in PCNF 1 Long distance resolution 2 Gentzen/sequent systems for arbitrary QBFs 3 U. Egly and F. Lonsing (TU Wien) QBFs and DepQBF: Deduction Concepts 16 / 40
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