[PPT] - Polluted Resolution and other Combined Proof Search Methods for PowerPoint Presentation

SLIDE 1

C. Nalon

WOLLI, 2015 – 1 / 22

Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics

Cláudia Nalon

nalon@unb.br University of Brasília WOLLI, 2015

SLIDE 2

C. Nalon

WOLLI, 2015 – 1 / 22

Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics

A Modal-Layered Resolution Calculus for K - Tableaux 2015

Cláudia Nalon

nalon@unb.br University of Brasília

Ullrich Hustadt Clare Dixon

U.Hustadt@liverpool.ac.uk C.Dixon@liverpool.ac.uk University of Liverpool WOLLI, 2015

SLIDE 3

Motivation

⊲ Motivation

Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 2 / 22

Kn, the smallest multi-modal normal logic, extends propositional logic

with a fixed, finite set of modal operators.

Formally, the set of well-formed formulae, WFFKn, is the least set such

that:

–

p ∈ P = {p, q, p′, q′, p1, q1, . . .} and true are in WFFKn;

–

if ϕ and ψ are in WFFKn, then so are ¬ϕ, (ϕ ∧ ψ), and

a ϕ for

each a ∈ An = {1, . . . , n}.

Formulae are interpreted, as usual, with respect to Kripke structures:

W, w0, R1, . . . , Rn, π where M, w | =

a ϕ if, and only if, for all w′, wRaw′ implies M, w′ |

= ϕ.

Abbreviations: false = ¬true, (ϕ ∨ ψ) = ¬(¬ϕ ∧ ¬ψ),

(ϕ → ψ) = (¬ϕ ∨ ψ), and ♦

a ϕ = ¬ a ¬ϕ.

SLIDE 4

Reasoning Tasks

Motivation

⊲ Reasoning Tasks

Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 3 / 22

W, w0, R1, . . . , Rn, π

For local satisfiability, formulae are interpreted with respect to the root
f M, that is, w0. A formula ϕ is locally satisfied in M, denoted by

M | =L ϕ, if M, w0 | = ϕ.

The formula ϕ is locally satisfiable if there is a model M such that

M, w0 | = ϕ.

A formula ϕ is globally satisfied in M, if for all w ∈ W, M, w |

= ϕ.

A formula ϕ is globally satisfiable if there is a model M such that M

globally satisfies ϕ, denoted by M | =G ϕ.

Given a set of formulae Γ and a formula ϕ, the local satisfiability of ϕ

under the global constraints Γ consists of showing that there is a model that globally satisfies the formulae in Γ and that there is a world in this model that satisfies ϕ.

SLIDE 5

Complexity

Motivation Reasoning Tasks

⊲ Complexity

Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 4 / 22

Local satisfiability: PSPACE-complete;
Global satisfiability: EXPTIME-complete;
Local satisfiability under global constraints: EXPTIME-complete.

SLIDE 6

Proof Methods

Motivation Reasoning Tasks Complexity

⊲ Proof Methods

Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 5 / 22

Translation into first-order logic;
Sequent calculus;
Tableaux;
Inverse method;
BDD;
SAT;
Resolution;
. . .

SLIDE 7

Implementation

C. Nalon

WOLLI, 2015 – 6 / 22

$./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds

SLIDE 8

Implementation

C. Nalon

WOLLI, 2015 – 6 / 22

$./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds

SLIDE 9

Implementation

C. Nalon

WOLLI, 2015 – 6 / 22

$./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds

SLIDE 10

Implementation

C. Nalon

WOLLI, 2015 – 6 / 22

$./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds $./prover -i benchmarks/lwb/k_branch_p.03.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.49 seconds

SLIDE 11

Implementation

C. Nalon

WOLLI, 2015 – 6 / 22

$./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds $./prover -i benchmarks/lwb/k_branch_p.03.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.49 seconds $./prover -i benchmarks/lwb/k_branch_p.04.ksp -fsub -ires -bnfsimp -bsub -unit -ple ^C 118.26 seconds

SLIDE 12

Example

Motivation Reasoning Tasks Complexity Proof Methods Implementation

⊲ Example

Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 7 / 22

♦♦p ∧ ¬p

1. start → t0 2. t0 → ♦t1 3. t1 → ♦p 4. t0 → ¬p

SLIDE 13

Previous work

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example

⊲ Previous work

The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 8 / 22

Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based

heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. 199-203. IOS Press (2000).

♦♦p ∧ ¬p =

⇒ ♦♦p2 ∧ ¬p1

SLIDE 14

Previous work

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example

⊲ Previous work

The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 8 / 22

Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based

heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. 199-203. IOS Press (2000).

♦♦p ∧ ¬p =

⇒ ♦♦p2 ∧ ¬p1 p ∧ ¬p = ⇒ p0 ∧ ¬p1

SLIDE 15

Previous work

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example

⊲ Previous work

The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 8 / 22

Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based

heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. 199-203. IOS Press (2000).

♦♦p ∧ ¬p =

⇒ ♦♦p2 ∧ ¬p1 p ∧ ¬p = ⇒ p0 ∧ ¬p1

Areces, C., de Nivelle, H., de Rijke, M.: Prefixed Resolution: A

Resolution Method for Modal and Description Logics. In: Ganzinger, H. (ed.) Proc. CADE-16. LNAI, vol. 1632, pp. 187-201. Springer, Berlin (Jul 7-10 1999).

–

Formulae labelled by either constants or pair of constants.

–

The inference rule for ♦ generates new labels.

–

The inference rule for corresponds to propagation.

SLIDE 16

The main idea

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work

⊲ The main idea

The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 9 / 22

The calculus should allow for both local and modal reasoning.
A formula to be tested for (un)satisfiability is translated into a normal

form, where labels refer to the modal level they occur.

Inference rules are then applied by modal level.

SLIDE 17

The Normal Form

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea

⊲ The Normal Form

Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 10 / 22

After translation we have formulae of the form: ml : ϕ where ml ∈ N, denoting that ϕ holds at the modal level ml; or ∗ : ϕ which denotes that ϕ holds everywhere in the model. That is, satisfiability

f labelled formulae is given by:
M∗ |

=L ml : ϕ if, and only if, for all worlds w ∈ W such that depth(w) = ml, we have M∗, w | =L ϕ;

M∗ |

=L ∗ : ϕ if, and only if, M∗ | =L

∗ ϕ.

SLIDE 18

Clauses

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form

⊲ Clauses

Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 11 / 22

Literal clause

ml : r

b=1 lb

Positive a-clause

ml : l′ →

a l

Negative a-clause

ml : l′ → ♦

a l

where ml ∈ N ∪ {∗} and l, l′, lb ∈ L. Positive and negative a-clauses are together known as modal a-clauses; the index a may be omitted if it is clear from the context.

SLIDE 19

Transformation Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses

⊲

Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 12 / 22

ρ(ml : t → ϕ ∧ ϕ′) = ρ(ml : t → ϕ) ∧ ρ(ml : t → ϕ′) ρ(ml : t →

a ϕ)

= (ml : t →

a ϕ), if ϕ is a literal

= (ml : t →

a t′) ∧ ρ(ml + 1 : t′ → ϕ), otherwise

ρ(ml : t → ♦

a ϕ)

= (ml : t → ♦

a ϕ), if ϕ is a literal

= (ml : t → ♦

a t′) ∧ ρ(ml + 1 : t′ → ϕ), otherwise

ρ(ml : t → ϕ ∨ ϕ′) = (ml : ¬t ∨ ϕ ∨ ϕ′), if ϕ′ is a disjunction of literals = ρ(ml : t → ϕ ∨ t′) ∧ ρ(ml : t′ → ϕ′), otherwise

SLIDE 20

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules

⊲ Inference Rules

Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 13 / 22

[LRES] ml : D ∨ l ml′ : D′ ∨ ¬l σ({ml, ml′}) : D ∨ D′ [MRES] ml : l1 →

a l

ml′ : l2 →

♦

a ¬l

σ({ml, ml′}) : ¬l1 ∨ ¬l2

SLIDE 21

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules

⊲ Inference Rules

Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 14 / 22

[GEN1] ml1 : l′

1

→

a ¬l1

. . . mlm : l′

m

→

a ¬lm

mlm+1 : l′ →

♦

a ¬l

mlm+2 : l1 ∨ . . . ∨ lm ∨ l ml : ¬l′

1 ∨ . . . ∨ ¬l′ m ∨ ¬l′

where ml = σ({ml1, . . . , mlm+1, mlm+2 − 1}) l′

1, . . . , l′ m, l′

SLIDE 22

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules

⊲ Inference Rules

Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 14 / 22

[GEN1] ml1 : l′

1

→

a ¬l1

. . . mlm : l′

m

→

a ¬lm

mlm+1 : l′ →

♦

a ¬l

mlm+2 : l1 ∨ . . . ∨ lm ∨ l ml : ¬l′

1 ∨ . . . ∨ ¬l′ m ∨ ¬l′

where ml = σ({ml1, . . . , mlm+1, mlm+2 − 1}) l′

1, . . . , l′ m, l′

¬l a

SLIDE 23

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules

⊲ Inference Rules

Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 14 / 22

[GEN1] ml1 : l′

1

→

a ¬l1

. . . mlm : l′

m

→

a ¬lm

mlm+1 : l′ →

♦

a ¬l

mlm+2 : l1 ∨ . . . ∨ lm ∨ l ml : ¬l′

1 ∨ . . . ∨ ¬l′ m ∨ ¬l′

where ml = σ({ml1, . . . , mlm+1, mlm+2 − 1}) l′

1, . . . , l′ m, l′

¬l, ¬l1, . . . , ¬lm a

SLIDE 24

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules

⊲ Inference Rules

Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 14 / 22

[GEN1] ml1 : l′

1

→

a ¬l1

. . . mlm : l′

m

→

a ¬lm

mlm+1 : l′ →

♦

a ¬l

mlm+2 : l1 ∨ . . . ∨ lm ∨ l ml : ¬l′

1 ∨ . . . ∨ ¬l′ m ∨ ¬l′

where ml = σ({ml1, . . . , mlm+1, mlm+2 − 1}) l′

1, . . . , l′ m, l′

¬l, ¬l1, . . . , ¬lm l1 ∨ . . . ∨ lm ∨ l a

SLIDE 25

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules

⊲ Inference Rules

Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 15 / 22

[GEN2] ml1 : l′

1

→

a l1

ml2 : l′

2

→

a ¬l1

ml3 : l′

3

→

♦

a l2

σ({ml1, ml2, ml3}) : ¬l′

1 ∨ ¬l′ 2 ∨ ¬l′ 3

SLIDE 26

Inference Rules

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules

⊲ Inference Rules

Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 16 / 22

[GEN3] ml1 : l′

1

→

a ¬l1

. . . mlm : l′

m

→

a ¬lm

mlm+1 : l′ →

♦

a l

mlm+2 : l1 ∨ . . . ∨ lm ml : ¬l′

1 ∨ . . . ∨ ¬l′ m ∨ ¬l′

where ml = σ({ml1, . . . , mlm+1, mlm+2 − 1})

SLIDE 27

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

SLIDE 28

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond]

SLIDE 29

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond] 12. 1 : ¬tall ∨ ¬t3 [LRES, 11, 3, t1]

SLIDE 30

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond] 12. 1 : ¬tall ∨ ¬t3 [LRES, 11, 3, t1] 13. 1 : ¬t3 ∨ ¬t2 ∨ ¬female [LRES, 7, 12, tall]

SLIDE 31

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond] 12. 1 : ¬tall ∨ ¬t3 [LRES, 11, 3, t1] 13. 1 : ¬t3 ∨ ¬t2 ∨ ¬female [LRES, 7, 12, tall] 14. 1 : male ∨ ¬t2 ∨ ¬t3 [LRES, 13, 1, tall]

SLIDE 32

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond] 12. 1 : ¬tall ∨ ¬t3 [LRES, 11, 3, t1] 13. 1 : ¬t3 ∨ ¬t2 ∨ ¬female [LRES, 7, 12, tall] 14. 1 : male ∨ ¬t2 ∨ ¬t3 [LRES, 13, 1, tall] 15. 0 : ¬t0 [GEN1, 10, 6, 8, 14, male, t2, t3]

SLIDE 33

Example

C. Nalon

WOLLI, 2015 – 17 / 22

1. ∗ : female ∨ male 2. ∗ : ¬female ∨ ¬male 3. ∗ : ¬tall ∨ t1 4. ∗ : t1 →

c blond

5. 0 : t0 6. 0 : t0 →

c t2

7. 1 : ¬t2 ∨ ¬female ∨ tall 8. 0 : t0 → ♦

c t3

9. 1 : t3 → ♦

c ¬blond

10. 0 : t0 →

c ¬male

11. 1 : ¬t1 ∨ ¬t3 [MRES, 9, 4, blond] 12. 1 : ¬tall ∨ ¬t3 [LRES, 11, 3, t1] 13. 1 : ¬t3 ∨ ¬t2 ∨ ¬female [LRES, 7, 12, tall] 14. 1 : male ∨ ¬t2 ∨ ¬t3 [LRES, 13, 1, tall] 15. 0 : ¬t0 [GEN1, 10, 6, 8, 14, male, t2, t3] 16. 0 : false [LRES, 15, 5, t0]

SLIDE 34

Negative Resolution

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example

⊲

Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 18 / 22

A literal is negative if is of the form ¬p, where p ∈ P.
A clause C is negative if all literals in C are negative.
Negative resolution restricts the application of the inference rules by

requiring that one of the clauses being resolved is negative.

For completeness, we need to change the normal form:

ρ(ml : t →

a ¬p)

= (ml : t →

a t′) ∧ ρ(ml + 1 : t′ → ¬p)

ρ(ml : t → ♦

a ¬p)

= (ml : t → ♦

a t′) ∧ ρ(ml + 1 : t′ → ¬p)

SLIDE 35

Ordered Resolution

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution

⊲

Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 19 / 22

Let Φ be a set of clauses and PΦ be the set of propositional symbols
ccurring in Φ.
Let ≻ be a well-founded and total ordering on PΦ.
This ordering can be extended to literals LΦ occurring in Φ by setting

¬p ≻ p and p ≻ ¬q whenever p ≻ q, for all p, q ∈ PΦ.

A literal l is said to be maximal with respect to a clause C ∨ l if, and
nly if, there is no l′ occurring in C such that l′ ≻ l.
Two clauses C ∨ l and C′ ∨ ¬l can be resolved if, and only if, l is

maximal with respect to C and ¬l is maximal with respect to C′.

For completeness, we have to make sure that every literal occurring in

the scope of a modal operator is minimal with respect to the other literals occurring at the same modal level.

For the running example (k_branch_p.04), negative resolution reports

unsatisfiability in 4.14 seconds whilst ordered resolution takes 0.05 seconds.

SLIDE 36

LWB – K_T4P

C. Nalon

WOLLI, 2015 – 20 / 22

Figure 1: Unsatisfiable Formulae Figure 2: Satisfiable Formulae

SLIDE 37

QBF

C. Nalon

WOLLI, 2015 – 21 / 22

SLIDE 38

Conclusion and Future Work

Motivation Reasoning Tasks Complexity Proof Methods Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF

⊲

Conclusion and Future Work

C. Nalon

WOLLI, 2015 – 22 / 22

We have presented a terminating, sound, and complete (non-natural,

polluted) calculus for Kn.

Negative and ordered resolution, together with layering, are also

complete.

Implementation is still work in progress, but results seem to be

promising.

We are considering other refinements as negative ordered resolution, for