Strong Equivalence for Argumentation Semantics Based on Conﬂict-Free Sets

(1)

Strong Equivalence for Argumentation Semantics Based on Conflict-Free Sets

Sarah Alice Gaggl

Institute of Informationsystems, Vienna University of Technology

Joint work with Stefan Woltran

ECSQARU Belfast — June 30, 2011

(2)

Motivation

Argumentation is adynamic reasoning process.

During the process the participants come up with new arguments.

Whicheffectscausesadditional informationwrt. a semantics?

Which information doesnot contributeto the results?

Sarah A. Gaggl, TU Vienna Strong EQ for Argu. Sem. based on cf Sets 1

(3)

Motivation

Two AFsFandGarestrongly equivalent(wrt. a semanticsσ) iff F∪HandG∪Hhave the sameσ-extensions foreachAFH.

One can savelyreplacean AF by a strongly equivalent one without changing its extensions.

(4)

Motivation

Two AFsFandGarestrongly equivalent(wrt. a semanticsσ) iff F∪HandG∪Hhave the sameσ-extensions foreachAFH.

One can savelyreplacean AF by a strongly equivalent one without changing its extensions.

In anegotiationbetween two agents: SE allows to characterize situations where the two agents have anequivalent view of the worldwhich is moreoverrobust to additional information.

(5)

Motivation ctd.

Example

AFsFandGare equivalent (wrt. stable semantics).

(6)

Motivation ctd.

Example

stable(F∪H) =stable(G∪H) ={{b,d}}.

(7)

Motivation ctd.

Example

We identify thestable kernelof a frameworkF= (A,R)which removesredundant attacks:

F^sk= (A,R^sk)whereR^sk=R\ {(a,b)|a6=b,(a,a)∈R}.

(8)

Motivation ctd.

Identification ofredundant attacksis important in choosing an appropriate semantics.

Strong equivalence has been analyzed for many semantics in [Oikarinen and Woltran, 2010].

In this paper: naive, stageandcf2semantics.

(9)

Overview

1 Background

2 Strong Equivalence

3 Relations between Semantics wrt. Strong Equivalence

4 Summary

(10)

Argumentation Framework

Argumentation Framework [Dung, 1995]

Anargumentation framework (AF)is a pairF= (A,R), whereAis a finite set of arguments andR⊆A×A. Then(a,b)∈Rifaattacksb.

Example

F= (A,R),A={a,b,c,d},R={(a,b),(b,a),(b,b),(b,c),(c,d),(d,b)}, directed graph

(11)

Semantics

Semantics for AFs

LetF= (A,R)andS⊆A, we say

Sisconflict-freeinF, i.e.S∈cf(F), if there are noa,b∈S, s.t.

(a,b)∈R;

Sismaximal conflict-freeornaive, i.e.S∈naive(F), ifS∈cf(F)and for eachT ∈cf(F),S6⊂T.

Example

cf(F) ={∅,{a},{c},{d},{a,c},{a,d}},naive(F) ={{a,c},{a,d}}.

(12)

cf2 Semantics

Thecf2semantics is one of the SCC-recursive semantics introduced in [Baroni et al., 2005]

Separation

An AFF= (A,R)is calledseparatedif for each(a,b)∈R, there exists a path frombtoa. We define[[F]] =S

C∈SCCs(F)F|C and call[[F]]the separationofF.

Example

(13)

cf2 Semantics

Thecf2semantics is one of the SCC-recursive semantics introduced in [Baroni et al., 2005]

Separation

An AFF= (A,R)is calledseparatedif for each(a,b)∈R, there exists a path frombtoa. We define[[F]] =S

C∈SCCs(F)F|_C and call[[F]]the separationofF.

Example

(14)

cf2 Semantics ctd.

Reachability

LetF= (A,R)be an AF,Ba set of arguments, anda,b∈A. We say that bisreachableinFfromamoduloB, in symbolsa⇒^B_F b, if there exists a path fromatobinF|_B.

Definition (∆

_F,S

)

For an AFF= (A,R),D⊆A, and a setSof arguments,

∆F,S(D) ={a∈A| ∃b∈S:b6=a,(b,a)∈R,a6⇒^A\D_F b}, and∆_F,S be the least fixed-point of∆_F,S(∅).

(15)

cf2 Semantics ctd.

Reachability

LetF= (A,R)be an AF,Ba set of arguments, anda,b∈A. We say that bisreachableinFfromamoduloB, in symbolsa⇒^B_F b, if there exists a path fromatobinF|_B.

Definition (∆

_F,S

)

For an AFF= (A,R),D⊆A, and a setSof arguments,

∆F,S(D) ={a∈A| ∃b∈S:b6=a,(b,a)∈R,a6⇒^A\D_F b}, and∆_F,S be the least fixed-point of∆_F,S(∅).

(16)

cf2 Semantics ctd.

cf2

Extensions [Gaggl and Woltran, 2010]

Given an AFF= (A,R). A setS⊆Ais acf2-extensionofF, if Sis conflict-free inF

andS∈naive([[F−∆_F,S]]).

Example

S={c,f,h},S∈cf(F).

(17)

cf2 Semantics ctd.

cf2

Extensions [Gaggl and Woltran, 2010]

andS∈naive([[F−∆_F,S]]).

Example

S={c,f,h},S∈cf(F).

(18)

cf2 Semantics ctd.

cf2

Extensions [Gaggl and Woltran, 2010]

andS∈naive([[F−∆F,S]]).

Example

S={c,f,h},∆F,S(∅) ={d,e}.

(19)

cf2 Semantics ctd.

cf2

Extensions [Gaggl and Woltran, 2010]

Example

S={c,f,h},∆F,S({d,e}) ={d,e}.

(20)

cf2 Semantics ctd.

cf2

Extensions [Gaggl and Woltran, 2010]

Example

S={c,f,h},∆F,S={d,e},S∈naive([[F−∆F,S]]).

(21)

Strong Equivalence (SE)

Strong Equivalence [Oikarinen and Woltran, 2010]

Two AFsFandGarestrongly equivalentto each other wrt. a semantics σ, in symbolsF≡^σ_s G, iff for each AFH,σ(F∪H) =σ(G∪H).

By definitionF≡^σ_s Gimpliesσ(F) =σ(G).

(22)

SE wrt. naive Semantics

naive(F) =naive(G) ={{a}}

(23)

SE wrt. naive Semantics

naive(F∪H) =naive(G∪H) ={{d},{a,e}}

(24)

SE wrt. naive Semantics

naive(F∪H) =naive(F) ={{a}}but naive(G∪H) ={{a,b}}.

(25)

SE wrt. naive Semantics

naive(F∪H) =naive(F) ={{a}}but naive(G∪H) ={{a,b}}.

Theorem

The following statements are equivalent:

1 F≡^naive_s G;

2 naive(F) =naive(G)andA(F) =A(G);

3 cf(F) =cf(G)andA(F) =A(G).

(26)

SE wrt. cf2 Semantics

(27)

SE wrt. cf2 Semantics

H = (A∪ {d,x,y,z},

{(a,a),(b,b),(b,x),(x,a),(a,y),(y,z),(z,a), (d,c)|c∈A\ {a,b}}).

(28)

SE wrt. cf2 Semantics

LetE={d,x,z},E∈cf2(F∪H)butE6∈cf2(G∪H).

(29)

SE wrt. cf2 Semantics

LetE={d,x,z},E∈cf2(F∪H)butE6∈cf2(G∪H).

No matter which AFsF6=G, one can always construct anHs.t.

cf2(F∪H)6=cf2(G∪H);

The missing attack in one AF leads to different SCCs and therefore to differentcf2extensions.

(30)

SE wrt. cf2 Semantics

No matter which AFsF6=G, one can always construct anHs.t.

cf2(F∪H)6=cf2(G∪H);

The missing attack in one AF leads to different SCCs and therefore to differentcf2extensions.

Theorem

For any AFsFandG,F≡^cf2_s GiffF =G.

(31)

Comparing Semantics wrt. SE

(32)

Summary

We provide characterizations for strong equivalence wrt. stage, naiveandcf2semantics.

cf2semantics is the only one whereno redundant attacksexist.

cf2semanticstreats self-loopsin amore sensitive waythan other semantics.

We analyzedlocalandsymmetricequivalence.

(33)

Baroni, P., Giacomin, M., and Guida, G. (2005).

SCC-Recursiveness: A General Schema for Argumentation Semantics.

Artif. Intell., 168(1-2):162–210.

Dung, P. M. (1995).

On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.

Artif. Intell., 77(2):321–358.

Gaggl, S. A. and Woltran, S. (2010).

cf2 Semantics Revisited.

In Baroni, P., Cerutti, F., Giacomin, M., and Simari, G. R., editors, (COMMA 2010), volume 216, pages 243–254. IOS Press.

Oikarinen, E. and Woltran, S. (2010).

Characterizing Strong Equivalence for Argumentation Frameworks.

In Lin, F., Sattler, U., and Truszczynski, M., editors,(KR 2010), pages 123–133. AAAI Press.