Variable Dependencies & Q-resolution Friedrich Slivovsky & - - PowerPoint PPT Presentation

Variable Dependencies & Q-resolution

Friedrich Slivovsky & Stefan Szeider

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3 F PCNF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

plain QDPLL

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3 F

Dependency Scheme

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3 F

Dependency Scheme

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3 F

Dependency Scheme x1 x2 x3 y1 y2 z1 z2 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3 F

x1 x2 x3 y1 y2 z1 z2 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

x1 x2 x3 y1 y2 z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

x2 x3 y1 y2 z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

x3 y1 y2 z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

x3 y2 z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

y2 z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

z1 z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

DepQBF

∃x1∃x2∃x3∀y1∀y2∃z1∃z2∃z3

DepQBF

more freedom in decision making

DepQBF

more freedom in decision making shorter learned clauses

DepQBF

more freedom in decision making shorter learned clauses more unit clauses

DepQBF

prefix order

formula

DepQBF

prefix order

formula true Q-Term resolution proof

DepQBF

prefix order

formula true Q-Term resolution proof false Q-resolution refutation

DepQBF

prefix order

formula true Q(D)-Term resolution proof false Q(D)-resolution refutation

DepQBF

dependency scheme D

Cumulative Dependency Schemes

Cumulative Dependency Schemes allow sound reordering.

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering.

standard resolution-path triangle quadrangle strict standard

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering.

standard resolution-path triangle quadrangle strict standard

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering. Q(D)-res. unsound

standard resolution-path triangle quadrangle strict standard

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering. Q(D)-res. unsound

standard resolution-path triangle quadrangle strict standard

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering. Q(D)-res. unsound

reflexive resolution-path

Q(D)-res. sound

standard resolution-path triangle quadrangle strict standard

Cumulative Dependency Schemes But Q(D)-resolution can be unsound. allow sound reordering. Q(D)-res. unsound

reflexive resolution-path

Q(D)-res. sound

Our Result(s)

Our Result(s)

Q(Drrs)-resolution is sound.

Our Result(s)

Q(Drrs)-resolution is sound. (reflexive) resolution-path dependency scheme

Our Result(s)

Q(Drrs)-resolution is sound. Corollary Q(Dstd)-resolution is sound. (reflexive) resolution-path dependency scheme

Our Result(s)

Q(Drrs)-resolution is sound. Corollary Q(Dstd)-resolution is sound. (reflexive) resolution-path dependency scheme standard dependency scheme

Q-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution

Q-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C universal reduction

Q-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C universal reduction

∀u Q1x1 Qnxn

Q-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C universal reduction

∀u Q1x1 Qnxn

C

Q-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C universal reduction sound and complete for PCNF formulas

∀u Q1x1 Qnxn

C

Q-resolution

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e u ∨ y

¬e e

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e u ∨ y

¬e e

z ∨ x ∨ u

y ¬y

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e u ∨ y

¬e e

z ∨ x

u

z ∨ x ∨ u

y ¬y

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e ¬x

¬u

u ∨ y

¬e e

z ∨ x

u

z ∨ x ∨ u

y ¬y

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e ¬x

¬u

u ∨ y

¬e e

z ∨ x

u

z

x ¬x

z ∨ x ∨ u

y ¬y

¬x ∨ ¬u z ∨ x ∨ ¬y y ∨ e u ∨ ¬e ∀z ∃x ∀u ∃y ∃e ¬x

¬u

u ∨ y

¬e e

z ∨ x

u

z

x ¬x

z ∨ x ∨ u

y ¬y

⊥

z

Q(D)-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution

Q(D)-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C D-reduction

Q(D)-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C D-reduction every x in C is independent of u

Q(D)-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C D-reduction every x in C is independent of u u x D(F):

Q(D)-resolution

∃e C ∨ e ¬e ∨ C' C ∨ C' resolution ∀u C ∨ u C D-reduction every x in C is independent of u u x D(F):

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e y

u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e y

u y ind. of u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e y

u

¬e ∨ ¬y

x ¬x y ind. of u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e ¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x y ind. of u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e ¬y

¬u

¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x y ind. of u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e ¬y

¬u

¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x y ind. of u y ind. of u

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e ¬y

¬u

¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x

⊥

y ¬y y ind. of u y ind. of u

Q(Drrs)-resolution is sound.

Q(Drrs)-resolution is sound. Proof strategy

Q(Drrs)-resolution refutation of F Q(Drrs)-resolution is sound. Proof strategy

Q(Drrs)-resolution refutation of F Q(Drrs)-resolution is sound. Proof strategy r e w r i t i n g

Q(Drrs)-resolution refutation of F Q(Drrs)-resolution is sound. Q-resolution refutation of F Proof strategy r e w r i t i n g

Q(Drrs)-resolution refutation of F Q(Drrs)-resolution is sound. Q-resolution refutation of F Proof strategy r e w r i t i n g D-reduction

Q(Drrs)-resolution refutation of F Q(Drrs)-resolution is sound. Q-resolution refutation of F Proof strategy r e w r i t i n g D-reduction universal reduction

delaying D-reductions ⊥ u ∨ … …

u

… …

proof sketch

⊥ u ∨ … … … delaying D-reductions

proof sketch

⊥ u ∨ … … … delaying D-reductions u

proof sketch

⊥ u ∨ … … … delaying D-reductions u

proof sketch

u ∨ … … … delaying D-reductions u

proof sketch

u ∨ … … … delaying D-reductions u u

proof sketch

u ∨ … … … delaying D-reductions u u ⊥

proof sketch

⊥ u ∨ … … … delaying D-reductions u ¬u

proof sketch

⊥ u ∨ … … … delaying D-reductions u ¬u tautological clause

proof sketch

⊥ u ∨ … …

u

… … ¬u

proof sketch

⊥ u ∨ … …

u

… … ¬u

proof sketch

y

⊥ u ∨ … …

u

… … ¬u

proof sketch

y

⊥ u ∨ … …

u

… … ¬u

proof sketch

cycle in implication graph

y

⊥ u ∨ … …

u

… … ¬u

proof sketch

cycle in implication graph

x y

⊥ u ∨ … …

u

… … ¬u

proof sketch

cycle in implication graph

∀u Q1x1 Qnxn ∃x x y

⊥ u ∨ … …

u

… … ¬u

proof sketch

cycle in implication graph

∀u Q1x1 Qnxn ∃x x y

• r

in Drrs(F) u y

⊥ u ∨ … …

u

… … ¬u

proof sketch

cycle in implication graph

∀u Q1x1 Qnxn ∃x x y

contradiction

• r

in Drrs(F) u y

⊥ u ∨ … …

u

… … ¬u

proof sketch

∀u Q1x1 Qnxn ∃x

…

x ¬x

x ∨ … ¬x ∨ … …

⊥ u ∨ … …

u

… … ¬u

proof sketch

…

x ¬x

x ∨ … ¬x ∨ … … lower resolution

⊥ … ¬u

proof sketch

x ¬x

x ∨ … ¬x ∨ … lower resolution u ∨ … …

u

⊥ … ¬u

proof sketch

x ¬x

x ∨ … ¬x ∨ … recurse on subderivations u ∨ … …

u

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e ¬y

¬u

¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x

⊥

y ¬y

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬u

¬u ∨ ¬y

¬e e

y

u

¬e ∨ ¬y

x ¬x

⊥

y ¬y

¬u

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬u

¬u ∨ ¬y

¬e e

y

u x ¬x

⊥

y ¬y

¬u ¬u ∨ x

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬u ¬e e

y

u x ¬x

⊥

y ¬y

¬u ¬u ∨ x ¬x

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬u ¬e e

y

u x ¬x

⊥

y ¬y

¬u ¬u ∨ x ¬x

proof by example

¬x ∨ ¬y u ∨ y ¬e ∨ x ¬u ∨ e ∃x ∀u ∃y ∃e

¬u ¬e e u x ¬x

⊥

y ¬y

¬u ¬u ∨ x ¬x ∨ u ¬x

• pen problems

dependency schemes for universal expansion

• pen problems

fast certificate extraction from Q(D)-resolution proofs dependency schemes for universal expansion

• pen problems

fast certificate extraction from Q(D)-resolution proofs relative complexity of Q(D)-resolution proof systems dependency schemes for universal expansion

• pen problems