Skepticism-Adequacy. - Resolution-Adequacy. - strong admissibility. NONE OF TRADITIONAL. AND RECENT SEMANTICS. SATISFY ALL OF THEM ...
Resolution-based argumentation 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
AF = attack relation arguments Defeat graph
Semantics S Set of extensions ℰS(AF)
Defeat Status (Justification Status)
[Justified arguments: belong to all extensions] 2
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) - strong admissibility -
admissibility [weak, CF] reinstatement Directionality I-maximality Skepticism-Adequacy
NONE OF TRADITIONAL AND RECENT SEMANTICS SATISFY ALL OF THEM
- Resolution-Adequacy 3
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:
(see also Modgil 06)
From S to S* Resolution-based grounded semantics Resolution-based preferred semantics
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Aim of the work and presentation plan
CAN DESIRABLE CRITERIA BE SATISFIED ALTOGETHER?
YES!!!
RESOLUTION-BASED SCHEME OF SEMANTICS DEFINITION:
(see also Modgil 06)
From S to S* Resolution-based grounded semantics Resolution-based preferred semantics PRESENTATION PLAN 1) Review of desirable criteria definitions 2) Definition and evaluation of resolution-based semantics 6
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 E2
E1
E3
it is not the case
• Grounded and preferred (as well as prudent, semistable, ideal and CF2-semantics) satisfy I-maximality • Complete semantics do not
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Admissibility and reinstatement principles Admissibility ∀ AF, ∀ E
∈ℰS(AF)
E
• E is conflict-free • E defends all of its arguments
Reinstatement ∀ AF, ∀ E∈ℰS(AF)
E
α
• if E “defends” α then α∈ ∈E [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) 8
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
α
α
β
β
α
β
δ
α
β
δ
δ
δ
γ
α
β
δ
γ
α
β
δ
γ
α
β
δ
γ 10
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
E1
E
E2
E
between sets of extensions:
denotes that
E1 is “at least as skeptical as”
(or “not more committed” than)
A skepticism relation AF1
A
A
E2
between argumentation frameworks:
AF2 denotes that AF1 is “at least as skeptical as” AF2
Adequacy criteria Skepticism relations between AFi
Skepticism relations between
Ei 11
Skepticism relations between sets of extensions Comparison between two extensions E1 and E2: E1⊆ E2 E
A direct generalization: the ∩
E2 :
E1 ⊆ E2
U
E
relation
U
E1
∩
A finer generalization: the
E W
relation
Comparison between an extension E1 and a set of extensions
E2 :
∀ E2∈E2 E1⊆ E2 (e.g. GEAF contained in any pref. extension)
E1
E W
E2 :
∀ E2 ∈E2, ∃ E1 ∈E1 : E1⊆ E2’’E2 E1
E1’’
E2’’’
’
E2’
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Skepticism relation between argumentation frameworks The Basic idea
β
α
More skeptical (less committed)
VS.
α
β
Less skeptical (more committed)
The General relation
AF1
AF1
A
AF2
AF2 13
Skepticism relation between argumentation frameworks A
-maximal AF [resolutions in (Modgil 06)]
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Skepticism-related criteria definition Skepticism-adequacy of a semantics Given a skepticism relation
between sets of extensions,
E
a semantics S is AF1
E
-adequate iff for any AF1, AF2 A
AF2
⇒
ℰS (AF1)
E
ℰS (AF2)
Resolution-adequacy of a semantics Given a skepticism relation a semantics S is
E
between sets of extensions,
E
-resolution-adequate iff for any AF
UAF’ ∈ RES(AF) ℰS (AF’)
E
ℰS (AF)
Implication orders E
•
W -adequacy
•
E
⇒
E
-adequacy
∩
-resolution-adequacy W
⇒
E
-resolution-adequacy
∩
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Adequacy criteria in action: grounded vs. preferred semantics β β
γ
δ
γ
δ
α γ
δ β
α AF
α
ℰGR (AF)={ø} ℰPR (AF)= {{α, δ}, {β, δ}}
UAF’ ∈ RES(AF) ℰS (AF’)={{α, δ}, {β, δ}}
Preferred semantics: resolution adequate Grounded semantics: NO 16
Adequacy criteria in action: grounded vs. preferred semantics • Grounded semantics: skepticism adequate
GE AF↔(β, α) ⊆
GE AF
W
- adequate
• Preferred semantics: NO
β
β γ
α All arguments provisionally defeated
δ
γ
δ
α α
and
β definitely justified 17
Defining resolution-based semantics The Basic idea A semantics should be…
ℰS (AF)
E
ℰS (AF’)
More skeptical w.r.t. extensions of a resolution
UAF’ ℰS (AF’)
E
ℰS (AF)
Less skeptical w.r.t. extensions of all resolutions
[Grounded semantics: too much skeptical, preferred: too much committed]
The definition Given a semantics S, its resolution-based version is S* such that
ℰS* (AF) = MIN ( UAF’ ∈ RES(AF) ℰS (AF’) ) 18
General desirable properties I-maximality • Achieved by definition:
ℰS* (AF) = MIN (…)
Skepticism-adequacy • For any semantics S, S* satisfies
W
-skepticism-adequacy
Resolution-adequacy • S is I-maximal ⇒ S* satisfies
-resolution-adequacy
W
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 β β γ
γ
δ
γ
δ
α
δ
β α
AF α
ℰPR* (AF)= ℰGR* (AF) = {{α, δ}, {β, δ}} β
β γ
δ
α
γ
δ
α
ℰPR* (AF)= ℰGR* (AF) = {ø} 20
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* : {{α, ε, η},{θ, ι, ζ}}
ℰPR* : {ø}
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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…
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