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1
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
2
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)
3
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
5
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)
6
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)
7
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
8
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)
9
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
11
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
12
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
13
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
15
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
16
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
17
• 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
18
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]
19
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
20
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
21
And directionality?
• Resolution-based grounded semantics satisfies directionality
[see the paper for a –complicated– proof]
• Resolution-based preferred semantics does not
ℰPR* : {ø}
22
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}}
23
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…