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Comparing the Expressiveness of Argumentation Semantics COMMA 2012 - - PowerPoint PPT Presentation

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Comparing the Expressiveness of Argumentation Semantics COMMA 2012 Wolfgang Dvok, Christof Spanring Database and Artificial Intelligence Group Institut fr Informationssysteme Technische Universitt Wien September 11, 2012

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SLIDE 1

Comparing the Expressiveness of Argumentation Semantics⋄

COMMA 2012

Wolfgang Dvořák, Christof Spanring

Database and Artificial Intelligence Group Institut für Informationssysteme Technische Universität Wien

September 11, 2012

⋄ Supported by the Vienna Science and Technology Fund (WWTF) under grant ICT08-028. Comparing the Expressiveness of Argumentation Semantics Slide 1

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SLIDE 2
  • 1. Motivation

Motivation

“Plethora” of Argumentation Semantics

Comparison of semantics still relates to basic properties, computational aspects, but do not provide satisfying answers about expressiveness.

Comparing the Expressiveness of Argumentation Semantics Slide 2

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SLIDE 3
  • 1. Motivation

Motivation

“Plethora” of Argumentation Semantics

Comparison of semantics still relates to basic properties, computational aspects, but do not provide satisfying answers about expressiveness.

Intertranslatability

A translation function transforms Argumentation Frameworks s.t. one can switch from one semantics to another. Intertranslatability w.r.t. efficiency has been studied for several semantics and gives a clear hierarchy [Dvořák and Woltran, 2011]. Considering expressiveness we no longer care about efficiency.

Comparing the Expressiveness of Argumentation Semantics Slide 2

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SLIDE 4
  • 1. Motivation

Outlook

We consider 9 semantics: conflict-free, naive, grounded, admissible, stable, complete, preferred, semi-stable and stage. We present consider two kinds of translations (faithful and exact), and provide full hierarchies of expressiveness. Semi-stable and preferred are of same expressiveness (although they have different complexity).

Comparing the Expressiveness of Argumentation Semantics Slide 3

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SLIDE 5
  • 2. Background

Argumentation Frameworks

Definition

An argumentation framework (AF) is a pair (A, R) where A is a non-empty set of arguments R ⊆ A × A is a relation representing “attacks” (“defeats”)

Example

F=( {a,b,c,d,e} , {(a,b),(c,b),(c,d),(d,c),(d,e),(e,e)} )

a b c d e

Comparing the Expressiveness of Argumentation Semantics Slide 4

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SLIDE 6
  • 2. Background

Translations

Definition

A Translation Tr is a function mapping (finite) AFs to (finite) AFs.

a b c d e

Comparing the Expressiveness of Argumentation Semantics Slide 5

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SLIDE 7
  • 2. Background

Translations

Definition

A Translation Tr is a function mapping (finite) AFs to (finite) AFs.

a b c d e a∗ b∗ c∗ d ∗ e∗

Comparing the Expressiveness of Argumentation Semantics Slide 5

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SLIDE 8
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 9
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Example (An exact translation: cf ⇒ adm)

a b c d e {b, d} ∈ cf (F)

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 10
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Example (An exact translation: cf ⇒ adm)

a b c d e {b, d} ∈ cf (F) {b, d} ∈ adm(Tr (F))

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 11
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 12
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Example (A faithful translation: comp ⇒ stable)

a b c d e {a} ∈ comp(F)

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 13
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|.

Example (A faithful translation: comp ⇒ stable)

a b c d e a∗ b∗ c∗ d∗ e∗ {a} ∈ comp(F) {a, a∗, c∗, d∗, e∗} ∈ stable(Tr (F))

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 14
  • 2. Background

Translations

“Levels of Faithfulness” (for semantics σ, σ′)

exact: for every AF F, σ(F) = σ′(Tr (F)) faithful: for every AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F))} and |σ(F)| = |σ′(Tr (F))|. weakly exact: there is a fixed S of sets of arguments, such that for any AF F, σ(F) = σ′(Tr (F)) \ S; weakly faithful: there is a fixed S of sets of arguments, such that for any AF F, σ(F) = {E ∩ AF | E ∈ σ′(Tr (F)) \ S} and |σ(F)| = |σ′(F) \ S|. We further consider translations w.r.t. the properties efficient, covering, embedding, monotone, and modular.

Comparing the Expressiveness of Argumentation Semantics Slide 6

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SLIDE 15
  • 3. Contribution

State of the Art

Table: Faithful / exact intertranslatability (efficient).

cf naive ground adm stable comp pref semi stage cf

  • naive
  • ground
  • / -

/ - / - / ? / ? / ? adm –

  • / -
  • / -

/ - / - stable –

  • comp

– / - / -

  • / -

/ - / - pref – – – –

  • ? / -

semi – – – – –

  • ? / -

stage – – – – –

  • Comparing the Expressiveness of Argumentation Semantics

Slide 7

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SLIDE 16
  • 3. Contribution

State of the Art

Table: Faithful / exact intertranslatability (inefficient).

cf naive ground adm stable comp pref semi stage cf

  • naive
  • ground
  • / ?

/ ? / ? / ? / ? / ? adm –

  • / -
  • / -

/ - / - stable –

  • comp

– / - / -

  • / -

/ - / - pref –

  • ? / -

semi –

  • ? / -

stage –

  • Comparing the Expressiveness of Argumentation Semantics

Slide 7

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SLIDE 17
  • 3. Contribution

Summarized Results

Table: Faithful / exact intertranslatability

cf naive ground adm stable comp pref semi stage cf

  • / -

–

  • / -
  • / -

/ - / - naive –

  • –

/ - / - / -

  • ground

–

  • / -

/ -

  • adm

– – –

  • / -
  • / -

/ - / - stable – – –

  • comp

– – – / - / -

  • / -

/ - / - pref – – – / - / - / -

  • / -

semi – – – / - / - / -

  • / -

stage – – – / - / - / -

  • Comparing the Expressiveness of Argumentation Semantics

Slide 7

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SLIDE 18
  • 3. Contribution

Main Contributions

The Paper

For the 9 Semantics under our considerations we provide exact / faithful translations whenever possible, and prove that no such translation exists otherwise.

Comparing the Expressiveness of Argumentation Semantics Slide 8

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SLIDE 19
  • 3. Contribution

Main Contributions

The Paper

For the 9 Semantics under our considerations we provide exact / faithful translations whenever possible, and prove that no such translation exists otherwise.

The Talk

In the following we give examples for both kind of results. Translation 8: exact for semi-stable to stage semantics. Theorem 3: There is no weakly faithful translation for preferred to naive semantics.

Comparing the Expressiveness of Argumentation Semantics Slide 8

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SLIDE 20
  • 3. Contribution

Definition

For F = (A, R) an Argumentation Framework and a set S ⊆ A we call S+ = S ∪ {a ∈ A | ∃b ∈ A, b ֌ a} the range of S.

Definition

Let F = (A, R) be an Argumentation Framework. For S ⊆ A it holds that S ∈ cf (F) if there are no a, b ∈ S, such that (a, b) ∈ R; S ∈ adm(F), if each a ∈ S is defended by S; S ∈ pref (F), if S ∈ adm(F) and there is no T ∈ adm(F) with T ⊃ S; S ∈ semi(F), if S ∈ adm(F) and there is no T ∈ adm(F) with T +

R ⊃ S+ R .

Comparing the Expressiveness of Argumentation Semantics Slide 9

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SLIDE 21
  • 3. Contribution

Translation 8, semi ⇒ pref

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 22
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E} a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 23
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 24
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 25
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 26
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 27
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 28
  • 3. Contribution

Translation 8, semi ⇒ pref

Definition

Tr (A, R) = (A′, R′) A′ = A ∪ {E | E ∈ pref (F) \ semi(F)} R′ = R ∪ {(a, E), (E, E), (E, b) | a ∈ A \ E, b ∈ E}

{a, c}F

a b c d e

Example

pref (F) = {{a, c}, {a, d}} semi(F) = {{a, d}} pref (Tr (F)) = {{a, d}}

Comparing the Expressiveness of Argumentation Semantics Slide 10

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SLIDE 29
  • 3. Contribution

Definition

Let F = (A, R) be an Argumentation Framework. For S ⊆ A it holds that S ∈ cf (F) if there are no a, b ∈ S, such that (a, b) ∈ R; S ∈ naive(F), if there is no T ∈ cf (F) with T ⊃ S; S ∈ adm(F), if each a ∈ S is defended by S; S ∈ pref (F), if S ∈ adm(F) and there is no T ∈ adm(F) with T ⊃ S;

Comparing the Expressiveness of Argumentation Semantics Slide 11

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SLIDE 30
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 31
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 32
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3 a1 b2 b3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}}

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 33
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3 b1 a2 b3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}}

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 34
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3 b1 b2 a3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}}

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 35
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}} ⊆ naive(Tr (F))

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 36
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for pref ⇒ naive.

Counterexample

a1 b1 a2 b2 a3 b3 b1 b2 b3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}} ⊆ naive(Tr (F)) ⇒ {b1, b2, b3} ∈ cf (Tr (F))

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 37
  • 3. Contribution

Theorem 3, pref ⇒ naive

Theorem

There is no weakly faithful translation for {stage, stable, semi, pref , comp, adm} ⇒ {cf , naive} .

Counterexample

a1 b1 a2 b2 a3 b3

pref (F) = {{a1, b2, b3}, {b1, a2, b3}, {b1, b2, a3}} ⊆ naive(Tr (F)) ⇒ {b1, b2, b3} ∈ cf (Tr (F))

Comparing the Expressiveness of Argumentation Semantics Slide 12

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SLIDE 38
  • 4. Conclusion

Results

(weakly) exact cf stable adm comp ground naive stage semi, pref (weakly) faithful stage, stable, semi, pref , comp, adm cf ground naive

Comparing the Expressiveness of Argumentation Semantics Slide 13

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SLIDE 39
  • 4. Conclusion

Almost finished. . .

Achievments

Full hierarchy of expressiveness for the selected semantics. Extended existing investigations on intertranslatability

to naive extensions and conflict-free sets, and to the case of inefficient translations.

Improved an existing translation w.r.t. size of transformed Argumentation Frameworks.

Open Questions

More semantics for investigation Labeling-preserving translations

Comparing the Expressiveness of Argumentation Semantics Slide 14

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SLIDE 40
  • 4. Conclusion

Finished.

Achievments

Full hierarchy of expressiveness for the selected semantics. Extended existing investigations on intertranslatability

to naive extensions and conflict-free sets, and to the case of inefficient translations.

Improved an existing translation w.r.t. size of transformed Argumentation Frameworks.

Open Questions

More semantics for investigation Labeling-preserving translations

Comparing the Expressiveness of Argumentation Semantics Slide 14