22/07/11IJCAI 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)

22/07/11IJCAI 2011 Barcelona Argumentation Frameworks Introduced in Dung (AIJ, 1995) Arguments: X AttacksA  X  X Acceptability concept:  : 2 X  { ,T} E  ( )={S  X :  (S)} Examples: Grounded, Preferred, Stable S is stable if conflict-free (S  S  A=  ) and for each y  S we have some x  S with  A.

22/07/11IJCAI 2011 Barcelona3 Problematic 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,  A, so that “p is acceptable if q is not”. But, we may often want to describe more sophisticated interactions.

22/07/11IJCAI 2011 Barcelona Extending from Binary Attacks Amgoud, Cayrol et al. (2005, 2008) propose bipolar frameworks, whereby an additional (binary) support relation, R, is used:  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.

22/07/11IJCAI 2011 Barcelona Abstract Dialectical Frameworks (ADFs) s r5 r1 r2 r3 r4 Conditions for s to be acceptable expressed via acceptability of its parents – {r1,r2,…,} That is, as a propositional function, over the acceptance conditions controlling each r

22/07/11IJCAI 2011 Barcelona Abstract Dialectical Frameworks (ADFs) (continued) Formally, an ADF is a triple (S,L,C) with S a set of arguments, L  S  S a set of links, (cf in AFs) and C a set of acceptance conditions, C s, the acceptance condition for s  S being a predicate C s : 2 par(s)  { ,T} Hence, “s is acceptable if an appropriate configuration of its attackers as given through C s is acceptable”

22/07/11IJCAI 2011 Barcelona Examples a.Dung-style standard AF: C =  ({  r : r  par(s)) b.All links are supporting: C =  ({r : r  par(s)} c.s is acceptable if exactly one of its parents is (  {r : r  par(s)})   {(  r   t) : {r,t}  par(s)}

22/07/11IJCAI 2011 Barcelona Models in ADFs The most basic semantics for “acceptable sets” in ADFs are models. For (S,L,C) and M  S, M is conflict-free if for each s in S, C s [M  par(s)]=T; M is a model if M is conflict-free and should C s [M  par(s)]=T then s  M.

22/07/11IJCAI 2011 Barcelona AFs to ADFs (and back again?) From Example (a) it is easy to transform an AF to an ADF (S X,L A,C) so that stable extensions map to models. This translation has |S X |=|X|. In going from an ADF (S,L,C) to an AF with models mapping to stable extensions a naïve translation gives |X S |  2 |S|. Is this exponential increase needed?

22/07/11IJCAI 2011 Barcelona Polynomial size simulations We say model simulates (S,L,C) if S  X S and A.For every model M of (S,L,C) there is a subset Y of X S with M  Y a stable extension of. B.For every stable extension P of, P  S is a model of (S,L,C). A model simulation is polynomial if |X S | is polynomially bounded in the “size” of (S,L,C).

22/07/11IJCAI 2011 Barcelona 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 C s is presented as a propositional formula,  s then size(D) is the sum of |  s |, ie operations defining .

22/07/11IJCAI 2011 Barcelona Main Results 1.Let D=(S,L,C) be an ADF. There is an AF, that model simulates D and has |X S | =O(size(D)). 2. 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 C s.

22/07/11IJCAI 2011 Barcelona Outline of Proof Translate each C s to an AF, containing par(s) and s amongst its arguments. Each subset R of par(s) for which C s [R]=T induces a stable extension of. Each stable extension, P, of has C s [P  par(s)]=T. Combine individual (respecting L) to complete simulation.

22/07/11IJCAI 2011 Barcelona 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, B M, 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?

22/07/11IJCAI 2011 Barcelona Well-founded & Stable Models 1.If G is the grounded extension of the model simulating AF for (S,L,C) then G  S 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”.

22/07/11IJCAI 2011 Barcelona 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).