Dialogical logic and abstract argumentation frameworks

Two weeks ago I was back in Lisbon for ECAI, and after having attended quite a few talks, tutorials, and workshops on dialogues and argumentation. This has caused me to start thinking about some alternative approaches to dialogical logic, which may or may not turn out to be fruitful in the long run, and I started writing up a blogpost about them. However, I got sidetracked and didn't finish the post until now. Argumentation frameworks (AFs) were introduced in Dung 1995. In their most basic form, an argumentation framework is a pair $\langle X,R\rangle$, where X is a set of arguments and R is a binary relation on X called an attack relation, specifying which arguments are attacks on other arguments. Both the arguments and the attack relation are considered atomic/basic; in an AF, there is no further structure to either, they are just given. AFs have been extended in various ways; one such way is by allowing arguments to attack not just other arguments but also the attack relations themselves. AFs with this functionality are called Extend Argumentation Frameworks. One can think of an attack on an attack relation as a defense against that attack. The notion of attacks and defense in the AF and EAFs parallels, at least superficially, the notion of attack and defense in dialogical logic, so the natural question that arises is whether EAFs could be used to model dialogical logic, and, if so, whether this would give us any new insights. To be more precise: For a given formula $\phi$ we can construct a directed graph $D_\phi$ where the nodes of $D_\phi$ are the subformulae of $\phi$ along with the symbolic attacks ?L, ?R, and ?, and there is an edge from node $x$ to node $x'$ if $x$ is an attack of $x'$, according to the set of particle rules in place. We can then extend this graph to one with edges between nodes and other edges, where there is an edge between node $x$ and edge $e$ iff $x$ is a defense of the attack represented by $e$. Once we've represented a dialogical formula this way, we need to answer the following questions to determine whether this representation is helpful or not: - Such a representation does not keep track of either the player making the move or the round in which it occurs. Do we need to add this information? - Does representing a dialogue in this way differ at all from the tree representation of winning strategies given by Felscher? - Are there properties of these graphs that are worth exploiting; that is, are there any correspondents between properties of the graph and the existence of a winning strategy for P? - If we do introduce labels for players and rounds, then how do we represent the structural rules in terms of constraints on the graphs? For example, D10 would say that for any propositional letter, the label for P must be larger than the label for O. Put this way, it makes me think that if we build in player and round information into the labeling of the edges, we're going to end up with something that is just isomorphic to Felscher's trees, and hence this will not provide us with any new insights. - Even so, it would be interesting to ask if all the (known) structural rules can be expressed as constraints on the labeling. References:

• Dung, P. M. 1995. "On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and $n$-person games", Artificial Intelligence 77: 321--357.
• Modgil, S. 2009. "Reasoning about preferences in argumentation frameworks", Artificial Intelligence 173, nos. 9/10: 901-934.