Models for QBFs Uwe Bubeck and Hans Kleine Bning University of - - PowerPoint PPT Presentation

β–Ά
models for qbfs
SMART_READER_LITE
LIVE PREVIEW

Models for QBFs Uwe Bubeck and Hans Kleine Bning University of - - PowerPoint PPT Presentation

Jan 27, 2023 413 likes β€’694 views

SAT 2013 Nested Boolean Functions as Models for QBFs Uwe Bubeck and Hans Kleine Bning University of Paderborn July 11, 2013 Outline Introduction: QBF and (Counter-)Models Free Variables and Models NBF Representation

slide-1
SLIDE 1

Nested Boolean Functions as Models for QBFs

SAT 2013 Uwe Bubeck and Hans Kleine BΓΌning University of Paderborn

July 11, 2013

slide-2
SLIDE 2

Outline

  • Introduction: QBF and (Counter-)Models
  • Free Variables and Models
  • NBF Representation
  • Conclusion

Uwe Bubeck NBFs as Models for QBFs 2

slide-3
SLIDE 3

Introduction: QBF and (Counter-)Models

Section 1

slide-4
SLIDE 4

Quantified Boolean Formulas

Uwe Bubeck NBFs as Models for QBFs 4

QBF extends propositional logic by allowing universal and existential quantifiers over propositional variables. Semantics of closed QBF: βˆƒπ‘§ Ξ¦ 𝑧 is true if and only if Ξ¦ 𝑧/0 is true or Ξ¦ 𝑧/1 is true. βˆ€π‘¦ Ξ¦ 𝑦 is true if and only if Ξ¦ 𝑦/0 is true and Ξ¦ 𝑦/1 is true.

slide-5
SLIDE 5

Tree Models

βˆ€π‘¦1βˆƒπ‘§1βˆ€π‘¦2βˆƒπ‘§2 𝑦1 ∨ ¬𝑧1 ∧ ¬𝑦1 ∨ 𝑧2 ∧ 𝑧1 ∨ 𝑦2 ∨ ¬𝑧2 ∧ ¬𝑦2 ∨ 𝑧2

Uwe Bubeck NBFs as Models for QBFs 9

𝑦1 𝑧1 𝑦2 𝑧1 𝑧2 𝑧2 𝑧2 𝑧2 𝑦2 𝑦1 = 0 𝑦1 = 1 𝑧1 = 0 𝑧1 = 1 𝑦2 = 0 𝑦2 = 1 𝑦2 = 0 𝑦2 = 1 𝑧2 = 0 𝑧2 = 1 𝑧2 = 1 𝑧2 = 1

slide-6
SLIDE 6

Function Models 1/2

Uwe Bubeck NBFs as Models for QBFs 10

QBF as a 2-player game: βˆƒ and βˆ€ player alternatingly choose assignments for variables in prefix order.

𝑦1 𝑧1 𝑦2 𝑧1 𝑧2 𝑧2 𝑧2 𝑧2 𝑦2 𝑦1 = 0 𝑦1 = 1 𝑧1 = 0 𝑧1 = 1 𝑦2 = 0 𝑦2 = 1 𝑦2 = 0 𝑦2 = 1 𝑧2 = 0 𝑧2 = 1 𝑧2 = 1 𝑧2 = 1

slide-7
SLIDE 7

Function Models 2/2

Uwe Bubeck NBFs as Models for QBFs 11

βˆ€π‘¦1βˆƒπ‘§1βˆ€π‘¦2βˆƒπ‘§2 … βˆ€π‘¦π‘œβˆƒπ‘§π‘œ 𝜚 𝑦1, … , π‘¦π‘œ, 𝑧1, … , π‘§π‘œ = true if and only if βˆ€π‘¦1 … βˆ€π‘¦π‘œ 𝜚 𝑦1, … , π‘¦π‘œ, 𝑔

1 𝑦1 , … , 𝑔 π‘œ 𝑦1, … , π‘¦π‘œ

= true for some 𝑔

1, … , 𝑔 π‘œ (Skolem functions).

βˆ€π‘¦1βˆƒπ‘§1βˆ€π‘¦2βˆƒπ‘§2 … βˆ€π‘¦π‘œβˆƒπ‘§π‘œ 𝜚 𝑦1, … , π‘¦π‘œ, 𝑧1, … , π‘§π‘œ = false

if and only if

βˆƒπ‘§1 … βˆƒπ‘§π‘œ 𝜚(𝑕1(), 𝑕2 y1 , … , π‘•π‘œ(y1, … , π‘§π‘œβˆ’1), 𝑧1, … , π‘§π‘œ) = false

for some 𝑕1, … , π‘•π‘œ (Herbrand functions).

slide-8
SLIDE 8

Motivation

Uwe Bubeck NBFs as Models for QBFs 12

  • Important applications: solver certificates, explanations, ...
  • Balabanov and Jiang (2012):

Extract Skolem model from cube-resolution proof, Herbrand countermodel from clause-resolution proof.

  • Problem: compact representation

(no polynomial-size propositional encoding if Ξ£2

𝑄 β‰  Ξ 2 𝑄)

  • Contributions:

ο€­ direct polynomial-size encoding by NBFs ο€­ (counter)models parameterized by free variables

slide-9
SLIDE 9

Free Variables and Models

Section 2

slide-10
SLIDE 10

Semantics of Free Variables

Uwe Bubeck NBFs as Models for QBFs 14

Closed QBF: either true or false Open QBF: valuation depends on the free variables:

QBF with free vars Truth Assignment Closed QBF Evaluation True False Propositional Formula β‰ˆ Ξ¦(𝑨1, … , 𝑨𝑠) 𝑒: 𝑨1, … , 𝑨r β†’ 0,1 Ξ¦ 𝑒(𝑨1), … , 𝑒(𝑨r)

slide-11
SLIDE 11

Free Variables and Models

Uwe Bubeck NBFs as Models for QBFs 15

How are the models and countermodels for different assignments to the free variables related to each other?

QBF with free vars Truth Assignment Closed QBF Evaluation True False

Multiple models and/or countermodels

slide-12
SLIDE 12

Complete Equivalence Models 1/2

Uwe Bubeck NBFs as Models for QBFs 16

Idea: Replace all quantified variables with functions over the free variables. βˆ€π‘€π‘œβˆƒπ‘€π‘œβˆ’1 … βˆ€π‘€2βˆƒπ‘€1 𝜚 𝑀1, … , π‘€π‘œ, 𝑨1, … , 𝑨𝑠 β‰ˆ 𝜚 β„Ž1(𝑨1, … , 𝑨𝑠), … , β„Žπ‘œ(𝑨1, … , 𝑨𝑠), 𝑨1, … , 𝑨𝑠

slide-13
SLIDE 13

Complete Equivalence Models 2/2

Uwe Bubeck NBFs as Models for QBFs 17

Why bother about models parameterized by free variables? Non-prenex QBF: βˆ€π‘βˆƒπ‘ βˆ€π‘‘βˆƒπ‘’ 𝛽 𝑏, 𝑐, 𝑑, 𝑒 ∧ βˆ€π‘¦βˆƒπ‘§ 𝛾 𝑏, 𝑐, 𝑦, 𝑧 e.g. precompute partial certificate. Open QBF with free vars 𝑏, 𝑐.

slide-14
SLIDE 14

NBF Representation

Section 3

slide-15
SLIDE 15

Uwe Bubeck NBFs as Models for QBFs 19

Nested Boolean Functions

A Nested Boolean Function (NBF) [Cook/Soltys 1999] is a sequence of functions 𝐺 = 𝑔

0, … , 𝑔 𝑙 with

  • initial functions 𝑔

0, … , 𝑔 𝑒 given as propositional formulas

  • compound functions 𝑔

𝑗 𝑦𝑗 ≔ 𝑔 π‘˜0 𝑔 π‘˜1 𝑦1 𝑗 , … , 𝑔 π‘˜π‘  𝑦𝑠 𝑗

for previously defined functions 𝑔

π‘˜0, … , 𝑔 π‘˜π‘ .

Example: parity of Boolean variables 𝑔

0 π‘ž1, π‘ž2 ≔ Β¬π‘ž1 ∧ π‘ž2 ∨ (π‘ž1 ∧ Β¬π‘ž2)

𝑔

1 π‘ž1, π‘ž2, π‘ž3, π‘ž4 ≔ 𝑔 0 𝑔 0 π‘ž1, π‘ž2 , 𝑔 0 π‘ž3, π‘ž4

𝑔

2 π‘ž1, … , π‘ž16 ≔ 𝑔 1 𝑔 1 π‘ž1, … , π‘ž4 , … , 𝑔 1 π‘ž13, … , π‘ž16

slide-16
SLIDE 16

Quantifier Encoding in NBF 1/2

Uwe Bubeck NBFs as Models for QBFs 20

QBF: Ξ¦ π’œ ≔ βˆƒπ‘¦ 𝜚(𝑦, π’œ) NBF: 𝐺0 𝑦, π’œ ≔ 𝜚 𝑦, π’œ 𝐺

1 π’œ

≔ 𝐺0(𝐺0 1, π’œ , π’œ) = 1 if x = 1 is a satisfying choice = 𝐺0 1, π’œ = 1 Ξ¦ π’œ β‰ˆ 𝐺

1(π’œ)

slide-17
SLIDE 17

Quantifier Encoding in NBF 1/2

Uwe Bubeck NBFs as Models for QBFs 21

QBF: Ξ¦ π’œ ≔ βˆƒπ‘¦ 𝜚(𝑦, π’œ) NBF: 𝐺0 𝑦, π’œ ≔ 𝜚 𝑦, π’œ 𝐺

1 π’œ

≔ 𝐺0(𝐺0 1, π’œ , π’œ) = 0 if x = 1 is not satisfying = 𝐺0 0, π’œ

slide-18
SLIDE 18

Quantifier Encoding in NBF 2/2

Uwe Bubeck NBFs as Models for QBFs 22

QBF: Ξ¦ π’œ ≔ βˆ€π‘§βˆƒπ‘¦ 𝜚 𝑦, 𝑧, π’œ NBF: 𝐺0 𝑦, 𝑧, π’œ ≔ 𝜚 𝑦, 𝑧, π’œ 𝐺

1 𝑧, π’œ

≔ 𝐺0 𝐺0 1, 𝑧, π’œ , 𝑧, π’œ 𝐺2 π’œ ≔ 𝐺

1 𝐺 1 0, π’œ , π’œ

= 0 if y = 0 is not satisfying = 𝐺

1 0, π’œ = 0

slide-19
SLIDE 19

Quantifier Encoding in NBF 2/2

Uwe Bubeck NBFs as Models for QBFs 23

QBF: Ξ¦ π’œ ≔ βˆ€π‘§βˆƒπ‘¦ 𝜚 𝑦, 𝑧, π’œ NBF: 𝐺0 𝑦, 𝑧, π’œ ≔ 𝜚 𝑦, 𝑧, π’œ 𝐺

1 𝑧, π’œ

≔ 𝐺0 𝐺0 1, 𝑧, π’œ , 𝑧, π’œ 𝐺2 π’œ ≔ 𝐺

1 𝐺 1 0, π’œ , π’œ

= 1 if y = 0 is satisfying = 𝐺

1 1, π’œ

slide-20
SLIDE 20

Quantifier Encoding in NBF 2/2

Uwe Bubeck NBFs as Models for QBFs 24

QBF: Ξ¦ π’œ ≔ βˆ€π‘§βˆƒπ‘¦ 𝜚 𝑦, 𝑧, π’œ NBF: 𝐺0 𝑦, 𝑧, π’œ ≔ 𝜚 𝑦, 𝑧, π’œ 𝐺

1 𝑧, π’œ

≔ 𝐺0 𝐺0 1, 𝑧, π’œ , 𝑧, π’œ 𝐺2 π’œ ≔ 𝐺

1 𝐺 1 0, π’œ , π’œ

β†’ Concise representation of QDPLL branching: innermost call of 𝐺𝑗 is the first branch,

  • utermost call of 𝐺𝑗 is the second branch, or a

repetition of the first one if it is already conclusive.

slide-21
SLIDE 21

Complete Equiv. Model in NBF 1/2

Uwe Bubeck NBFs as Models for QBFs 25

First branch determines which branch is conclusive. β†’ this is our witness, i.e. (counter)model.

slide-22
SLIDE 22

Complete Equiv. Model in NBF 1/2

Uwe Bubeck NBFs as Models for QBFs 27

First branch determines which branch is conclusive. β†’ this is our witness, i.e. (counter)model. QBF: Ξ¦ π’œ ≔ βˆ€π‘§βˆƒπ‘¦ 𝜚 𝑦, 𝑧, π’œ NBF: 𝐺0 𝑦, 𝑧, π’œ ≔ 𝜚 𝑦, 𝑧, π’œ 𝐺

1 𝑧, π’œ

≔ 𝐺0 𝐺0 1, 𝑧, π’œ , 𝑧, π’œ 𝐺2 π’œ ≔ 𝐺

1 𝐺 1 0, π’œ , π’œ

β„Žπ‘§(π’œ) β„Žπ‘¦(π’œ)

slide-23
SLIDE 23

Complete Equiv. Model in NBF 1/2

Uwe Bubeck NBFs as Models for QBFs 28

QBF: Ξ¦ π’œ ≔ βˆ€π‘§βˆƒπ‘¦ 𝜚 𝑦, 𝑧, π’œ NBF: 𝐺0 𝑦, 𝑧, π’œ ≔ 𝜚 𝑦, 𝑧, π’œ 𝐺

1 𝑧, π’œ

≔ 𝐺0 𝐺0 1, β„Žπ‘§(π’œ), π’œ , 𝑧, π’œ 𝐺2 π’œ ≔ 𝐺

1 𝐺 1 0, π’œ , π’œ

β„Žπ‘§(π’œ) β„Žπ‘¦(π’œ)

slide-24
SLIDE 24

Complete Equiv. Model in NBF 2/2

Uwe Bubeck NBFs as Models for QBFs 29

In general: Ξ¦ π’œ ≔ π‘…π‘œπ‘€π‘œ … 𝑅1𝑀1 𝜚 𝑀1, … , π‘€π‘œ, π’œ Complete equivalence model: β„Žπ‘— π’œ ≔ πΊπ‘—βˆ’1 0, β„Žπ‘—+1 π’œ , … , β„Ž1 π’œ , π’œ , if 𝑅𝑗 = βˆ€ πΊπ‘—βˆ’1 1, β„Žπ‘—+1 π’œ , … , β„Ž1 π’œ , π’œ , if 𝑅𝑗 = βˆƒ Clearly polynomial size, which is not possible with a propositional encoding if Ξ£2

𝑄 β‰  Ξ 2 𝑄.

Admittedly more difficult to evaluate. But:

  • Equiv. model checking PSPACE-hard even if β„Žπ‘— π’œ ∈ {0,1}.
slide-25
SLIDE 25

Conclusion

Section 4

slide-26
SLIDE 26

Conclusion

Uwe Bubeck NBFs as Models for QBFs 31

  • Complete equivalence models as a generalization of

Skolem/Herbrand (counter)models parameterized by free variables.

  • Concise characterization of QDPLL branching and thus

polynomial space by nested Boolean functions with

  • ne initial function and special recursive instantiation.
slide-27
SLIDE 27

Future Work

Uwe Bubeck NBFs as Models for QBFs 32

  • Restrictions on the model structure for subclasses of

QBF, e.g. Horn, 2-CNF, etc.

  • Build a NBF solver.

The End