22/07/11IJCAI 2011 Barcelona Relating the Semantics of Abstract Dialectical Frameworks and Standard AFs Gerd Brewka (II, Leipzig) Paul E. Dunne (DCS, Liverpool)

22/07/11 IJCAI 2011 Barcelona

Relating the Semantics of Abstract Dialectical Frameworks

and Standard AFs

Gerd Brewka (II, Leipzig)

Paul E. Dunne (DCS, Liverpool)

Stefan Woltran (DBAI, Vienna)


Argumentation Frameworks

• Introduced in Dung (AIJ, 1995)• Arguments: X• AttacksAXX• Acceptability concept: : 2X {,T} • E(<X,A>)={S X :(S)}• Examples: Grounded, Preferred, Stable• S is stable if conflict-free (SSA=)

and for each yS we have some xS with <x,y>A.

22/07/11 IJCAI 2011 Barcelona 3

Problematic AspectsProblematic Aspects

• Approach is extremely abstract, so can complicate modelling “real-world” cases.

• Incompatibility of arguments, p and q, can only be (directly) expressed through a binary attack relation, <p,q>A, so that “p is acceptable if q is not”.

• But, we may often want to describe more sophisticated interactions.


Extending from Binary Attacks

• Amgoud, Cayrol et al. (2005, 2008) propose bipolar frameworks, whereby an additional (binary) support relation, R, is used: <p,q>R expresses “q is acceptable if p is so”.

• Brewka & Woltran (KR2010) develop this notion of describing more complex argument interaction by introducing Abstract Dialectical Frameworks.


Abstract Dialectical Frameworks (ADFs)

sr5

r1

r2

r3r4Conditions for s to be

acceptable expressedvia acceptability ofits parents – {r1,r2,…,}

That is, as a propositionalfunction, over the acceptanceconditions controlling each r


Abstract Dialectical Frameworks (ADFs) (continued)

• Formally, an ADF is a triple (S,L,C) with S a set of arguments, LSS a set of links, (cf <X,A> in AFs) and C a set of acceptance conditions, Cs, the acceptance condition for sS being a predicate

Cs: 2par(s) {,T}• Hence, “s is acceptable if an

appropriate configuration of its attackers as given through Cs is acceptable”


Examples

a. Dung-style standard AF: C = ({r : rpar(s))

b. All links are supporting:C = ({r : rpar(s)}

c. s is acceptable if exactly one of its parents is

({r : rpar(s)}) {(r t) : {r,t}par(s)}


Models in ADFs

• The most basic semantics for “acceptable sets” in ADFs are models.

• For (S,L,C) and MS, M is conflict-free if for each s in S, Cs[Mpar(s)]=T; M is a model if M is conflict-free and should Cs[Mpar(s)]=T then sM.


AFs to ADFs (and back again?)

• From Example (a) it is easy to transform an AF <X,A> to an ADF (SX,LA,C) so that stable extensions map to models.

• This translation has |SX|=|X|.• In going from an ADF (S,L,C) to an AF

<XS,AL> with models mapping to stable extensions a naïve translation gives |XS|2|S| .

• Is this exponential increase needed?


Polynomial size simulations

• We say <XS,AL> model simulates (S,L,C) if S XS and

A. For every model M of (S,L,C) there is a subset Y of XS with MY a stable extension of <XS,AL>.

B. For every stable extension P of <XS,AL>, PS is a model of (S,L,C).

• A model simulation is polynomial if |XS| is polynomially bounded in the “size” of (S,L,C).


What is the “size” of an ADF?

• Defining the size of D=(S,L,C) to be |S| fails to acknowledge that the conditions given in C may be very intricate.

• In addition, for computation, some formal description of C must be used.

• We should, therefore, include the “cost” of such descriptions in defining size.

• e.g. if each Cs is presented as a propositional formula, s then size(D) is the sum of |s|, ie operations defining .


Main Results

1. Let D=(S,L,C) be an ADF. There is an AF, <XS,AL> that model simulates D and has |XS| =O(size(D)).

2. <XS,AL> may be constructed in time polynomial in size(D).

3. Both (1) & (2) continue to hold if “propositional formula” is replaced by “Boolean combinational network” as the representational formalism for Cs.


Outline of Proof

• Translate each Cs to an AF, <Xs,As> containing par(s) and s amongst its arguments.

• Each subset R of par(s) for which Cs[R]=T induces a stable extension of <Xs,As>.

• Each stable extension, P, of <Xs,As> has Cs[P par(s)]=T.

• Combine individual <Xs,As> (respecting L) to complete simulation.


Some Issues

• Models are a very limited solution concept.• The notions of stable and well-founded model

are far more useful.• The former, defined for bipolar ADFs, B, are

the least models of an ADF, BM, obtained by a translation similar to the Gelfond-Lifschitz rewriting of logic programs.

• The latter is the least fixed point of a particular binary operator on S.

• How do these relate to structures within AFs?


Well-founded & Stable Models

1. If G is the grounded extension of the model simulating AF for (S,L,C) then GS is the well-founded model of (S,L,C).

2. If B is a BADF, we may construct in polynomial time, an ADF, D*, whose models define exactly the stable models of B.

3. The construction in (2) is rather indirect and exploits ideas originating in the treatment of “loop formulae” and “level mappings”.


Summary

• Several basic solution concepts for ADFs may be “easily” mapped to extensions in a corresponding AF.

• ADFs are a more natural modelling technique, however, there is a significant body of work on algorithms in AFs.

• Motivates modelling scenarios as ADFs and computation via the related AF (cf HLL to machine-level compilation).

• Potential realistic application is given through the Carneades frameworks of (Gordon et al., 2007) and the reconstruction of these as ADFs (Brewka & Gordon, 2010).

Documents

22/07/11IJCAI 2011 Barcelona Relating the Semantics of Abstract Dialectical Frameworks and Standard AFs Gerd Brewka (II, Leipzig) Paul E. Dunne (DCS, Liverpool)