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

SLIDE 1

Compression of Propositional Resolution Proofs by Lowering Subproofs

Joseph Boudou1 Bruno Woltzenlogel Paleo2

1Université Paul Sabatier, Toulouse 2Vienna University of Technology

SMT Workshop, 2013

SLIDE 2

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

SLIDE 3

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

SLIDE 4

Introduction Proofs’ representation

Proof as a tree

¯ b,c ¯ a,b

b

¯ a,c ¯ a,b ¯ a, ¯ b, ¯ c

¯ b

¯ a, ¯ c

¯ c

¯ a a

a

⊥

Proof as a directed acyclic graph (DAG)

⊥ ¯ a a ¯ a,c ¯ a,¯ c ¯ b,c ¯ a,b ¯ a,¯ b,¯ c

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 4 / 22

SLIDE 5

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

SLIDE 6

Introduction Proofs’ representation

Resolution Given two proofs ϕL and ϕR 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 ϕL and

ϕR on ℓ, noted ψ = ϕL ⊙ℓ ϕR, is such that:

◮ ψ’s nodes are the union of ϕL and ϕR nodes plus a new

root node;

◮ there is an edge from ρ(ψ) to ρ(ϕL) labeled with ¯

ℓ;

◮ there is an edge from ρ(ψ) to ρ(ϕR) labeled with ℓ; ◮ ψ’s conclusion is

Γ

L \ {¯

ℓ}

∪ (Γ

R \ {ℓ}).

¯ b b ¯ a a ⊙¯

c

bc a ¯ abc = c ¯ b bc b ¯ a a ¯ abc

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 6 / 22

SLIDE 7

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 \ (ϕ1,...,ϕn) denotes the deletions of subproofs ϕ1,...,ϕn from the proof ψ.

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 7 / 22

SLIDE 8

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

SLIDE 9

Redundancies and corresponding algorithms Vertical redundancy

Regular proof

Definition (Tseitin 1970)

A proof is regular iff 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

SLIDE 10

Redundancies and corresponding algorithms Vertical redundancy

RecyclePivotsWithIntersection (RPI)

Partial Regularization

◮ Delete an outgoing edge labeled with ℓ iff ¯

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

SLIDE 11

Redundancies and corresponding algorithms Vertical redundancy

⊥ ¯ c c ¯ b,c b a ¯ a,b a, ¯ c a,c ¯ a, ¯ b,c

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 11 / 22

SLIDE 12

Redundancies and corresponding algorithms Vertical redundancy

⊥ ¯ c c

c

¯ b,c b a ¯ a,b a, ¯ c a,c

c

¯ a, ¯ b,c

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 11 / 22

SLIDE 13

Redundancies and corresponding algorithms Vertical redundancy

⊥ ¯ c c

c

¯ b,c b a ¯ a,b a, ¯ c a,c

c

¯ a, ¯ b,c Original proof ⊥ ¯ c c ¯ b,c b a ¯ a,b a, ¯ c a,c ¯ a, ¯ b,c Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 11 / 22

SLIDE 14

Redundancies and corresponding algorithms Vertical redundancy

⊥ ¯ c c

c

¯ b,c b a ¯ a,b a, ¯ c a,c

c

¯ a, ¯ b,c Original proof

5 resolutions

⊥ ¯ c c ¯ b,c b,c a ¯ a,b a, ¯ c a,c ¯ a, ¯ b,c Compressed proof

4 resolutions

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 11 / 22

SLIDE 15

Redundancies and corresponding algorithms Horizontal redundancy

Definition

A node is an horizontal redundancy iff it has at least two incoming edges labeled with the same literal.

Reducing horizontal redundancy

◮ postponing resolution until resolvents are resolved.

Example

⊥ ¯ c c ¯ b,c b,c ¯ a, ¯ b,c a,c

a

¯ a,b

a

⊥ ¯ c c ¯ a,c a,c ¯ a, ¯ b,c ¯ a,b

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 12 / 22

SLIDE 16

Redundancies and corresponding algorithms Horizontal redundancy

LowerUnits (LU)

Definition (Unit)

A unit is a subproof with a conclusion clause having exactly

ne 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

SLIDE 17

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 18

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof ⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 19

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof ¯ b ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 20

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof ¯ b ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 21

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof a, ¯ b ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 22

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof ¯ a a, ¯ b a,b a, ¯ b, ¯ c ¯ b,c Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 23

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof a ¯ a a, ¯ b a,b a, ¯ b, ¯ c ¯ b,c Compressed proof

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 24

Redundancies and corresponding algorithms Horizontal redundancy

⊥ ¯ c c ¯ b, ¯ c b ¯ b,c a, ¯ b, ¯ c ¯ a a,b Original proof

5 resolutions

⊥ a ¯ a a, ¯ b a,b a, ¯ b, ¯ c ¯ b,c Compressed proof

3 resolutions

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 14 / 22

SLIDE 25

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

SLIDE 26

LowerUnivalents Principles

Goals

◮ Lower more subproofs. ◮ Allow fast combination after RPI.

Idea

◮ If a unit with conclusion clause {a} is already marked for

lowering, a subproof with conclusion clause {¯ a,b} may be lowered too.

J. Boudou, B. Woltzenlogel Paleo

Compression of Propositional Resolution Proofs SMT 2013 16 / 22

SLIDE 27

LowerUnivalents Principles

Definition (Valent literal)

In a proof ψ, a literal ℓ is valent for the subproof ϕ iff ¯ ℓ belongs to the conclusion of ψ \ (ϕ) but not to the conclusion of ψ.

Definition (Univalent subproof)

A subproof ϕ with conclusion Γ is univalent w.r.t. a set ∆ of literals iff ϕ has exactly one valent literal ℓ, ℓ ∆ and Γ ⊆ ∆ ∪ {ℓ}. ℓ is called the univalent literal of ϕ w.r.t. ∆.

Theorem