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Resolution-basedargumentation semantics

Pietro Baroni, Massimiliano Giacomin

{baroni, giacomin}@ing.unibs.it

DEA - Dipartimento di Elettronica per l’Automazione

Università degli Studi di Brescia (Italy)

COMMA 2008, Toulouse, France

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The context: Dung’s Argumentation Framework

Defeat graph

Semantics S

AF = <A, →>

arguments

attack relation

Set of extensions ℰS(AF) Defeat Status

[Justified arguments: belong to all extensions]

(Justification Status)

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Previous results

Several argumentation semantics

• Grounded Semantics

• Preferred Semantics

• Recent proposals, e.g. prudent, semistable, CF2 semantics

How to evaluate and compare them [without examples]?

• General criteria considering conflict definition (Amgoud, Caminada 07)

• General criteria focusing on semantics definition (Baroni, Giacomin 06)

- admissibility

- [weak, CF] reinstatement

- Directionality

- I-maximality

- Skepticism-Adequacy

- Resolution-Adequacy

- strong admissibility

NONE OF TRADITIONAL

AND RECENT SEMANTICS

SATISFY ALL OF THEM

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Aim of the work and presentation plan

CAN DESIRABLE CRITERIA

BE SATISFIED ALTOGETHER?

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Aim of the work and presentation plan

CAN DESIRABLE CRITERIA

BE SATISFIED ALTOGETHER?

YES!!!RESOLUTION-BASED SCHEME

OF SEMANTICS DEFINITION:

Resolution-based grounded semantics

Resolution-based preferred semantics

From S to S*

(see also

Modgil 06)

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Aim of the work and presentation plan

CAN DESIRABLE CRITERIA

BE SATISFIED ALTOGETHER?

YES!!!RESOLUTION-BASED SCHEME

OF SEMANTICS DEFINITION:

Resolution-based grounded semantics

Resolution-based preferred semantics

PRESENTATION PLAN

1) Review of desirable criteria definitions

2) Definition and evaluation of resolution-based semantics

From S to S*

(see also

Modgil 06)

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I-maximality principle

I-maximality principle

A semantics S satisfies the “I-maximality principle” iff

∀∀∀∀ AF, ∀∀∀∀ E1,E2∈∈∈∈ℰS(AF) if E1⊆ E2 then E1=E2

it is not the case

• Grounded and preferred (as well as prudent, semistable, ideal

and CF2-semantics) satisfy I-maximality

• Complete semantics do not

E1E2

E3

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Admissibility and reinstatement principles

Admissibility

Reinstatement

E∀∀∀∀ AF, ∀∀∀∀ E ∈∈∈∈ℰS(AF)

• E is conflict-free

• E defends all of its arguments

∀∀∀∀ AF, ∀∀∀∀ E∈∈∈∈ℰS(AF)

• if E “defends” α then α∈∈∈∈E

Eα

[weak and CF reinstatement entailed]

Admissibility + reinstatement = complete extensions

• Grounded and preferred (as well as complete-based) semantics: YES

• CF2 semantics: NO (admissibility is not satisfied)

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Directionality principle

Basic idea

Extension membership of an argument is determined by its ancestors,

while it is not affected by the arguments it defeats

Definition

∀∀∀∀ AF, ∀∀∀∀ U “unattacked set” of AF,

{(E ∩ U) | E ∈∈∈∈ℰS(AF)} = ℰS(AF )U

Semantics behavior

• Both preferred and grounded semantics satisfy directionality

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Directionality: example with preferred semantics

δα β

δα β

δα β

δα β γ

δα β γ

δα β γ

δα β γ

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Skepticism related criteria

The informal notion of skepticism

Making “less|more committed choices” for arguments,

i.e. assigning to them “less|more decided” justification states.

Two kinds of skepticism relations

A basic skepticism relation between sets of extensions:E

E1

E2

Edenotes that E

1is “at least as skeptical as”

(or “not more committed” than) E2

A skepticism relation between argumentation frameworks:A

AF1 AF2 denotes that AF1 is “at least as skeptical as” AF2

A

Adequacy criteria

Skepticism relations between AFi Skepticism relations between Ei

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Skepticism relations between sets of extensions

Comparison between two extensions E1 and E2: E1⊆ E2

E1’

E1’’

E2’

E2’’

E2’’’

∀∀∀∀ E2 ∈∈∈∈E2, ∃∃∃∃ E1 ∈∈∈∈E1 : E1⊆⊆⊆⊆ E2:E

WE2E1

Comparison between an extension E1 and a set of extensions E2 :

∀∀∀∀ E2∈∈∈∈E2 E1⊆ E2 (e.g. GEAF contained in any pref. extension)

A direct generalization: the relation

E1

⊆⊆⊆⊆ E2 :

E

∩∩∩∩E2E1

U U

A finer generalization: the relation

E

∩∩∩∩

E

W

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Skepticism relation between argumentation frameworks

The Basic idea

α β α βVS.

More skeptical

(less committed)

Less skeptical

(more committed)

The General relation

AF1AF2

AAF1 AF2

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Skepticism relation between argumentation frameworks

A-maximal AF [resolutions in (Modgil 06)]

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a semantics S is -adequate iff for any AF1, AF2

Given a skepticism relation between sets of extensions,

Skepticism-related criteria definition

Skepticism-adequacy of a semantics

AF1 AF2

A⇒⇒⇒⇒ ℰS (AF1) ℰS (AF2)

Resolution-adequacy of a semantics

E

E

E

a semantics S is -resolution-adequate iff for any AF

Given a skepticism relation between sets of extensions,E

E

ℰS (AF)EℰS (AF’)UAF’ ∈∈∈∈ RES(AF)

Implication orders

•

•

E

∩∩∩∩

E

W -adequacy ⇒⇒⇒⇒ -adequacy

E

∩∩∩∩

E

W -resolution-adequacy ⇒⇒⇒⇒ -resolution-adequacy

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Adequacy criteria in action: grounded vs. preferred semantics

ββββ

αααα

γγγγ δδδδ

ββββ

αααα

γγγγ δδδδ

ββββ

αααα

γγγγ δδδδ

ℰS (AF’)=UAF’ ∈∈∈∈ RES(AF) {{α, δ}, {β, δ}}ℰGR (AF)={ø}

ℰPR (AF)= {{α, δ}, {β, δ}}

Preferred semantics: resolution adequate

Grounded semantics: NO

AF

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• Grounded semantics: skepticism adequate

GEAF↔(β, α)

⊆⊆⊆⊆ GEAF W - adequate

• Preferred semantics: NO

β

α

γ δ

β

α

γ δ

All arguments

provisionally defeatedα and β definitely justified

Adequacy criteria in action: grounded vs. preferred semantics

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Defining resolution-based semantics

The Basic idea

ℰS (AF) ℰS (AF’) E

More skeptical w.r.t.

extensions of a resolution

ℰS (AF)EℰS (AF’)UAF’

A semantics should be…

Less skeptical w.r.t.

extensions of all resolutions

The definition

Given a semantics S, its resolution-based version is S* such that

ℰS* (AF) = MIN ( )ℰS (AF’)UAF’ ∈∈∈∈ RES(AF)

[Grounded semantics: too much skeptical, preferred: too much committed]

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General desirable properties

I-maximality

• Achieved by definition: ℰS* (AF) = MIN (…)

Skepticism-adequacy

• For any semantics S, S* satisfies -skepticism-adequacyW

Resolution-adequacy

• S is I-maximal ⇒⇒⇒⇒ S* satisfies -resolution-adequacyW

Admissibility and reinstament

• S: complete extensions ⇒⇒⇒⇒ S* satisfies admissibility

and reinstatement

BOTH RESOLUTION-BASED GROUNDED AND PREFERRED

SEMANTICS SATISFY ALL OF THESE PRINCIPLES

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Examples

ββββ

αααα

γγγγ δδδδ

ℰPR* (AF)= ℰGR* (AF) = {{α, δ}, {β, δ}}

AF

β

α

γ δ

ℰPR* (AF)= ℰGR* (AF) = {ø}

β

α

γ δ

ββββ

αααα

γγγγ δδδδ

ββββ

αααα

γγγγ δδδδ

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And directionality?

• Resolution-based grounded semantics satisfies directionality

[see the paper for a –complicated– proof]

• Resolution-based preferred semantics does not

ℰPR* : {ø}

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And directionality?

• Resolution-based grounded semantics satisfies directionality

[see the paper for a –complicated– proof]

• Resolution-based preferred semantics does not

ℰPR* : {ø}

ℰPR* :

{{α, ε, η},{θ, ι, ζ}}

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Conclusions

• All desirable criteria (I-maximality, admissibility, reinstatement,

directionality, skepticism adequacy, resolution adequacy) can be

satisfied altogether

• Resolution-based version of traditional semantics:

- GR*: satisfies all desirable criteria

- PR*: does not satisfy directionality

• Future work:

- resolution-based version of other semantics

- is GR* useful in practice?

… algorithms and complexity…