Compression of Propositional Resolution Proofs by Lowering Subproofs - PowerPoint PPT Presentation

Compression of Propositional Resolution Proofs by Lowering Subproofs Joseph Boudou 1 Bruno Woltzenlogel Paleo 2 1 Université Paul Sabatier, Toulouse 2 Vienna University of Technology SMT Workshop, 2013

Overview Introduction Motivations Proofs’ representation Redundancies and corresponding algorithms Vertical redundancy Horizontal redundancy LowerUnivalents Principles Algorithm and implementation Experiments Conclusion J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 2 / 22

Introduction Motivations Why compressing propositional part of SMT proofs? SMT solvers are embeded in other tools ◮ Sledgehammer’s extension to SMT solver ◮ SMTCoq Some tools need the proof to be ◮ checked; ◮ tranlated; ◮ analysed. J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 3 / 22

Introduction Proofs’ representation Proof as a tree ¯ a , ¯ ¯ ¯ ¯ b , ¯ b , c a , b a , b c b ¯ b ¯ a , c a , ¯ ¯ c ¯ c ¯ a a a ⊥ Proof as a directed acyclic graph (DAG) ¯ a , ¯ a ,b ¯ b ,c ¯ b , ¯ c ¯ a ,c a , ¯ ¯ c a a ¯ ⊥ J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 4 / 22

Introduction Proofs’ representation Proof Definition (Proof) A proof ψ is a directed acyclic graph ◮ having a root noted ρ ( ψ ) ; ◮ with nodes labeled with clauses; ◮ with edges oriented from the resolvent to the premise; ◮ with edges labeled with the premise’s literal removed in the resolvent; ◮ which is either an axiom or a resolution proof. Definition (Axiom) An axiom is a proof with only one node. J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 5 / 22

Introduction Proofs’ representation Resolution Given two proofs ϕ L and ϕ R with conclusion Γ L and Γ R and a literal ℓ s.t. ¯ ℓ ∈ Γ L and ℓ ∈ Γ R , the resolution proof ψ of ϕ L and ϕ R on ℓ , noted ψ = ϕ L ⊙ ℓ ϕ R , is such that: ◮ ψ ’s nodes are the union of ϕ L and ϕ R nodes plus a new root node; ◮ there is an edge from ρ ( ψ ) to ρ ( ϕ L ) labeled with ¯ ℓ ; ◮ there is an edge from ρ ( ψ ) to ρ ( ϕ R ) labeled with ℓ ; � L \ { ¯ � ◮ ψ ’s conclusion is ℓ } ∪ ( Γ R \ { ℓ } ) . Γ b ¯ a ¯ a abc a a abc ¯ b ¯ a ¯ ⊙ ¯ = bc b c ¯ bc b c J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 6 / 22

Introduction Proofs’ representation Deletion Deletion of an edge ◮ The resolvent is replaced by the other premise. ◮ Some subsequent resolutions may have to be deleted too. Deletion of a subproof ϕ ◮ Deletion of every edge coming to ρ ( ϕ ) . ◮ The operation is commutative and associative. Notation ψ \ ( ϕ 1 ,...,ϕ n ) denotes the deletions of subproofs ϕ 1 ,...,ϕ n from the proof ψ . J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 7 / 22

Redundancies and corresponding algorithms Introduction Redundancies and corresponding algorithms Vertical redundancy Horizontal redundancy LowerUnivalents Conclusion J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 8 / 22

Redundancies and corresponding algorithms Vertical redundancy Regular proof Definition (Tseitin 1970) A proof is regular i ff on every path from its root to any of its axiom, any literal appears at most once as edge label. Theorem (Goerdt 1990) Given a set of axioms and a clause Γ , the smallest regular proof of Γ might be exponentially bigger than the smallest irregular proof of Γ . J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 9 / 22

Redundancies and corresponding algorithms Vertical redundancy RecyclePivotsWithIntersection (RPI) Partial Regularization ◮ Delete an outgoing edge labeled with ℓ i ff ¯ ℓ appears on every path from the root to the node. Definition (Safe literal) A literal is safe for a node η if it can be added to η ’s clause without changing proof’s conclusion. Two traversals ↑ Collect safe literals and mark edges to be deleted. ↓ Delete edges. J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 10 / 22

Redundancies and corresponding algorithms Vertical redundancy a , ¯ a , c c a , ¯ a ¯ ¯ a , b b , c ¯ b , c b c ¯ c ⊥ J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 11 / 22

Redundancies and corresponding algorithms Vertical redundancy a , ¯ a , c c c a , ¯ a ¯ ¯ a , b b , c ¯ b , c b c ¯ c c ⊥ J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 11 / 22

Redundancies and corresponding algorithms Vertical redundancy a , ¯ a , c a , ¯ a , c c c c a , ¯ a , ¯ a ¯ a ¯ ¯ a , b ¯ a , b b , c b , c ¯ ¯ b , c b b , c b c ¯ c c ¯ c c ⊥ ⊥ Original proof Compressed proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 11 / 22

Redundancies and corresponding algorithms Vertical redundancy a , ¯ a , c a , ¯ a , c c c c a , ¯ a , ¯ a ¯ a ¯ ¯ a , b ¯ a , b b , c b , c ¯ ¯ b , c b , c b b , c c ¯ c c ¯ c c ⊥ ⊥ Original proof Compressed proof 5 resolutions 4 resolutions J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 11 / 22

Redundancies and corresponding algorithms Horizontal redundancy Definition A node is an horizontal redundancy i ff it has at least two incoming edges labeled with the same literal. Reducing horizontal redundancy ◮ postponing resolution until resolvents are resolved. Example a , ¯ a , ¯ a , c ¯ ¯ ¯ b , c a , b ¯ b , c a , b a a ¯ ¯ a , c b , c a , c b , c c c c ¯ c ¯ ⊥ ⊥ J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 12 / 22

Redundancies and corresponding algorithms Horizontal redundancy LowerUnits (LU) Definition (Unit) A unit is a subproof with a conclusion clause having exactly one literal. Theorem A unit can always be lowered. Two traversals � Collect units with more than one resolvent. ↓ Delete units and reintroduce them at the bottom of the proof. J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 13 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ b , ¯ c a ¯ a , b ¯ ¯ b , ¯ c b b , c c c ¯ ⊥ J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ a , ¯ b , ¯ c a ¯ a , b b , ¯ c a ¯ a , b ¯ ¯ ¯ ¯ b , ¯ c b b , c b , ¯ c b b , c c c c ¯ c ¯ ⊥ ⊥ Original proof Compressed proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ a , ¯ ¯ a , b b , ¯ a ¯ a , b b , ¯ c a c ¯ ¯ ¯ ¯ b , ¯ b b , ¯ c b b , c c b , c ¯ c ¯ c c c ¯ ⊥ b Original proof Compressed proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ a , ¯ ¯ a , b b , ¯ a ¯ a , b b , ¯ c a c ¯ ¯ ¯ ¯ b , ¯ b b , ¯ c b b , c c b , c ¯ c ¯ c c c ¯ ⊥ b Original proof Compressed proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ a , ¯ b , ¯ a ¯ a , b b , ¯ a ¯ a , b c c ¯ ¯ ¯ ¯ b , ¯ c b b , c b , ¯ c b b , c ¯ c ¯ c c c a , ¯ ⊥ b Original proof Compressed proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ b , ¯ c a ¯ a , b a , ¯ ¯ b , ¯ c b , c ¯ ¯ a , ¯ b , ¯ c b b , c a , b b c c ¯ a ¯ ⊥ Compressed proof Original proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ b , ¯ c a ¯ a , b a , ¯ ¯ b , ¯ c b , c ¯ ¯ a , ¯ b , ¯ c b b , c a , b b c c ¯ a a ¯ ⊥ Compressed proof Original proof J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

Redundancies and corresponding algorithms Horizontal redundancy a , ¯ a , ¯ ¯ b , ¯ c a ¯ a , b b , ¯ c b , c ¯ ¯ a , ¯ a , b b , ¯ c b b , c b c a c ¯ a ¯ ⊥ ⊥ Original proof Compressed proof 5 resolutions 3 resolutions J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 14 / 22

LowerUnivalents Introduction Redundancies and corresponding algorithms LowerUnivalents Principles Algorithm and implementation Experiments Conclusion J. Boudou, B. Woltzenlogel Paleo Compression of Propositional Resolution Proofs SMT 2013 15 / 22