On Propositional QBF Expansions and Q-Resolution s Janota 1 Joao - - PowerPoint PPT Presentation

On Propositional QBF Expansions and Q-Resolution

Mikol´ aˇ s Janota1 Joao Marques-Silva1,2

1 INESC-ID/IST, Lisbon, Portugal 2 CASL/CSI, University College Dublin, Ireland

SAT 2013, July 8-12

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 1 / 15

QBF Solving

Other DPLL-Based

QuBE depQBF GhostQ CirQit

Expansion- Based

quantor nenofex sKizzo

Careful Expansion

AReQS RAReQS

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 2 / 15

Quantified Boolean Formula (QBF)

• an extension of SAT with quantifiers

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF)

• an extension of SAT with quantifiers

Example

∀y1y2∃x1x2. (¯ y1 ∨ x1) ∧ (y2 ∨ ¯ x2)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF)

• an extension of SAT with quantifiers

Example

∀y1y2∃x1x2. (¯ y1 ∨ x1) ∧ (y2 ∨ ¯ x2)

• we consider prenex form with maximal blocks of variables

∀U1∃E2 . . . ∀U2N−1∃E2N. φ

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Quantified Boolean Formula (QBF)

• an extension of SAT with quantifiers

Example

∀y1y2∃x1x2. (¯ y1 ∨ x1) ∧ (y2 ∨ ¯ x2)

• we consider prenex form with maximal blocks of variables

∀U1∃E2 . . . ∀U2N−1∃E2N. φ

Solving

• DPLL — Q-Resolution (QuBE, depqbf, etc.)
• Expansion — ?? (Quantor, sKizzo, Nenofex)
• “Careful” expansion (AReQS,RAReQS)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 3 / 15

Q-resolution

Q-resolution = Q-resolution rule + ∀-reduction

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution

Q-resolution = Q-resolution rule + ∀-reduction

Q-resolution rule

C1, C2 with one and only one complementary literal l, where l is existential

• derive C1 ∪ C2 {l,¯

l}

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution

Q-resolution = Q-resolution rule + ∀-reduction

Q-resolution rule

C1, C2 with one and only one complementary literal l, where l is existential

• derive C1 ∪ C2 {l,¯

l}

∀-reduction

• if k ∈ C is universal with highest level in C, remove k from C

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Q-resolution

Q-resolution = Q-resolution rule + ∀-reduction

Q-resolution rule

C1, C2 with one and only one complementary literal l, where l is existential

• derive C1 ∪ C2 {l,¯

l}

∀-reduction

• if k ∈ C is universal with highest level in C, remove k from C

Tautologous resolvents are generally unsound!

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 4 / 15

Expansion

∀x. Φ = Φ[x/0] ∧ Φ[x/1] ∃x. Φ = Φ[x/0] ∨ Φ[x/1]

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

Expansion

∀x. Φ = Φ[x/0] ∧ Φ[x/1] ∃x. Φ = Φ[x/0] ∨ Φ[x/1]

Fresh variables in order to keep prenex form

∃e1∀u2∃e3. (¯ e1 ∨ e3) ∧ (¯ e3 ∨ e1) ∧ (u2 ∨ e3) ∧ (¯ u2 ∨ ¯ e3)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

Expansion

∀x. Φ = Φ[x/0] ∧ Φ[x/1] ∃x. Φ = Φ[x/0] ∨ Φ[x/1]

Fresh variables in order to keep prenex form

∃e1∀u2∃e3. (¯ e1 ∨ e3) ∧ (¯ e3 ∨ e1) ∧ (u2 ∨ e3) ∧ (¯ u2 ∨ ¯ e3) ∃e1eu2/0

3

eu2/1

3

. (¯ e1 ∨ eu2/0

3

) ∧ (¯ eu2/0

3

∨ e1) ∧ (¯ e1 ∨ eu2/1

3

) ∧ (¯ eu2/1

3

∨ e1) ∧ eu2/0

3

∧ ¯ eu2/1

3

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 5 / 15

How to prove by expansion?

• 1. Expand all universal variables
• 2. Refute by propositional resolution

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion?

• 1. Expand all universal variables
• 2. Refute by propositional resolution

Why only universals?

• 1. conjunction of CNF is still CNF
• 2. ∃-expansion “doing the work of resolution”

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion?

• 1. Expand all universal variables
• 2. Refute by propositional resolution

Why only universals?

• 1. conjunction of CNF is still CNF
• 2. ∃-expansion “doing the work of resolution”

Partial Expansions

Only certain values may be needed: ∀u∃e. (u ∨ e) ∧ (u ∨ ¯ e) ∧ (¯ u ∨ e) (false)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion?

• 1. Expand all universal variables
• 2. Refute by propositional resolution

Why only universals?

• 1. conjunction of CNF is still CNF
• 2. ∃-expansion “doing the work of resolution”

Partial Expansions

Only certain values may be needed: ∀u∃e. (u ∨ e) ∧ (u ∨ ¯ e) ∧ (¯ u ∨ e) (false) ∃eu/0eu/1. eu/0 ∧ ¯ eu/0 ∧ eu/1 (false full)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

How to prove by expansion?

• 1. Expand all universal variables
• 2. Refute by propositional resolution

Why only universals?

• 1. conjunction of CNF is still CNF
• 2. ∃-expansion “doing the work of resolution”

Partial Expansions

Only certain values may be needed: ∀u∃e. (u ∨ e) ∧ (u ∨ ¯ e) ∧ (¯ u ∨ e) (false) ∃eu/0eu/1. eu/0 ∧ ¯ eu/0 ∧ eu/1 (false full) ∃eu/0. eu/0 ∧ ¯ eu/0 (false partial)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 6 / 15

Recursive Partial Expansion

∀U1 ∃E2 ∀U3 ∃E4 φ

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

Recursive Partial Expansion

• . . .

∃E2 ∃E2 ∀U3 ∃E4 φ ∀U3 ∃E4 φ

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

Recursive Partial Expansion

• . . .

∃E2 ∃E2

• . . .

∃E4 φ ∃E4 φ

• . . .

∃E4 φ ∃E4 φ

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 7 / 15

∀Exp+Res

Proof: (T , π)

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

∀Exp+Res

Proof: (T , π) (1) Expansion tree T : for each block of variables it tells us how to expand it. T u2/1 u1/0 u2/0 u2/1 u1/1

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

∀Exp+Res

Proof: (T , π) (1) Expansion tree T : for each block of variables it tells us how to expand it. T u2/1 u1/0 u2/0 u2/1 u1/1 (2) Propositional Resolution Refutation π of expansion resulting from the expansion tree T .

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 8 / 15

Performing Expansion

• For a clause C = ei ∨ u ∨ ek, for τ = τ1, . . . , τn

E (τ1, . . . , τn, C) = e

τ1,...,τi/2 i

∨ e

τ1,...,τk/2 k

if u[τ] = 0 E (τ1, . . . , τn, C) = 1 if u[τ] = 1

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 9 / 15

Performing Expansion

• For a clause C = ei ∨ u ∨ ek, for τ = τ1, . . . , τn

E (τ1, . . . , τn, C) = e

τ1,...,τi/2 i

∨ e

τ1,...,τk/2 k

if u[τ] = 0 E (τ1, . . . , τn, C) = 1 if u[τ] = 1

• For an expansion tree T and a matrix φ consider the union of

clauses E (τ, C) for all branches τ ∈ T and C ∈ φ.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 9 / 15

From Tree Q-resolution to ∀Exp+Res

C1 ∨ C2 x ∨ C1 ¯ x ∨ C2 x ∈ D1 ¯ x, y ∈ D2

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀Exp+Res

C ′

1 ∨ C ′ 2

xτ1,...,τj ∨ C ′

1

¯ xτ1,...,τj ∨ C ′

2

xτ1,...,τj ∈ E (D1, τ) ¯ xµ1,...,µj, yµ1,...,µk ∈ E (D1, µ) µ1, . . . , µj = τ1, . . . , τj

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀Exp+Res

C ′

1 ∨ C ′ 2

xτ1,...,τj ∨ C ′

1

¯ xτ1,...,τj ∨ C ′

2

xτ1,...,τj ∈ E (D1, τ) ¯ xµ1,...,µj, yµ1,...,µk ∈ E (D1, µ) µ1, . . . , µj = τ1, . . . , τj C3 ∨ C4 ¯ y ∨ C3 y ∨ C4

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀Exp+Res

C ′

1 ∨ C ′ 2

xτ1,...,τj ∨ C ′

1

¯ xτ1,...,τj ∨ C ′

2

xτ1,...,τj ∈ E (D1, τ) ¯ xµ1,...,µj, yµ1,...,µk ∈ E (D1, µ) µ1, . . . , µj = τ1, . . . , τj C3 ∨ C4 ¯ y ∨ C3 y ∨ C4 ¯ y ∈ D3

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Tree Q-resolution to ∀Exp+Res

C ′

1 ∨ C ′ 2

xτ1,...,τj ∨ C ′

1

¯ xτ1,...,τj ∨ C ′

2

xτ1,...,τj ∈ E (D1, τ) ¯ xµ1,...,µj, yµ1,...,µk ∈ E (D1, µ) µ1, . . . , µj = τ1, . . . , τj C3 ∨ C4 ¯ yρ1,...,ρk ∨ C ′

3

yµ1,...,µk ∨ C ′

4

¯ yρ1,...,ρk ∈ E (D3, ρ) ρ1, . . . , ρk = µ1, . . . , µk

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 10 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥ u ∨ x u ∨ ¯ x u ⊥

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥ u ∨ x u ∨ ¯ x u ⊥ xτ ∨ yµ ¯ xτ ∨ ¯ yρ ¯ yρ ∨ yµ

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥ u ∨ x u ∨ ¯ x u ⊥ xτ ∨ yµ ¯ xτ ∨ ¯ yρ ¯ yρ ∨ yµ x ∨ ¯ u ∨ y ¯ x ∨ u ∨ ¯ y u ∨ ¯ u ∨ y ∨ ¯ y

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥ u ∨ x u ∨ ¯ x u ⊥ xτ ∨ yµ ¯ xτ ∨ ¯ yρ ¯ yρ ∨ yµ x ∨ ¯ u ∨ y ¯ x ∨ u ∨ ¯ y u ∨ ¯ u ∨ y ∨ ¯ y

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

From Expansion Refutation to Q-resolution

• Why don’t we just revert substitutions?

xτ ¯ xτ ⊥ u ∨ x u ∨ ¯ x u ⊥ xτ ∨ yµ ¯ xτ ∨ ¯ yρ ¯ yρ ∨ yµ x ∨ ¯ u ∨ y ¯ x ∨ u ∨ ¯ y u ∨ ¯ u ∨ y ∨ ¯ y

• Such a construction is possible if propositional resolution

follows the order of the prefix, starting with the innermost levels.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 11 / 15

What is hard for ∀Exp+Res

⊥ u1 ∨ e2 u1 ∨ e2 ∨ u3 ∨ u4 u1 ∨ e2 ∨ u3 ∨ u4 ∨ e5 u3 ∨ ¯ e5 u1 ∨ ¯ e2 u1 ∨ ¯ e2 ∨ u3 ∨ ¯ u4 u1 ∨ ¯ e2 ∨ u3 ∨ ¯ u4 ∨ e5

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 12 / 15

What Seems to Be Hard for Q-resolution

xi ∨ z ∨ C1

i

¯ xi ∨ ¯ z ∨ C2

i

x1 ∨ z ∨ ¯ y1 ¯ x1 ∨ ¯ z ∨ ¯ y1 x2 ∨ z ∨ y1 ¯ x2 ∨ ¯ z ∨ ¯ y1 x3 ∨ z ∨ ¯ y1 ¯ x3 ∨ ¯ z ∨ y1 x4 ∨ z ∨ y1 ¯ x4 ∨ ¯ z ∨ y1 z/0 z/1 x1 ∨ ¯ y z/0

1

¯ x1 ∨ ¯ y z/1

1

x2 ∨ y z/0

1

¯ x2 ∨ ¯ y z/1

1

x3 ∨ ¯ y z/0

1

¯ x3 ∨ y z/1

1

x4 ∨ y z/0

1

¯ x4 ∨ y z/1

1

Figure : Example formula for n = 1

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 13 / 15

Summary and Future Work

• We have defined a proof system based on “careful”

expansions and propositional resolution.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 14 / 15

Summary and Future Work

• We have defined a proof system based on “careful”

expansions and propositional resolution.

• Such system simulates tree Q-resolution.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 14 / 15

Summary and Future Work

• We have defined a proof system based on “careful”

expansions and propositional resolution.

• Such system simulates tree Q-resolution.
• Q-resolution can simulate a fragment of this system, when

variables are resolved “inside out”.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 14 / 15

Summary and Future Work

• We have defined a proof system based on “careful”

expansions and propositional resolution.

• Such system simulates tree Q-resolution.
• Q-resolution can simulate a fragment of this system, when

variables are resolved “inside out”.

• We conjecture that the systems are incomparable. Showing

such is the subject of future work.

Janota and Marques-Silva On Propositional QBF Expansions and Q-Resolution 14 / 15

Thank you for your attention! Questions?

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