Abstract argumentation frameworks

The purpose of an abstract argumentation framework is to determine if some claim is true by ana- lyzing the attack relations between the arguments put forward at a given point in an argumentation process. Such frameworks do not concern the invention of arguments or their internal structure, in fact, for them the content of the argument per se is not relevant. The only information that is necessary for building an argumentation framework is a set of arguments and a binary relation of attacks between those arguments. Therefore, nothing is assumed about the form of the argument, other than that it may attack or be attacked by other arguments. Such a framework can be repre- sented by a directed graph where the nodes are arguments and each edge is an attack between a pair of arguments (the direction of the attack being coincident with the direction of the edge).

Dung [Dun95] proposed the first abstract argumentation framework in a seminal paper in 1995, which has become one of the most important foundations for many developments in argumenta- tion. He starts by defining the concept of argumentation framework in a manner similar to the pre- vious paragraph, then defines different semantics for determining the acceptability of arguments, concluding with a demonstration of the correctness of his theory in light of other approaches to that date. What follows is a selection of some of the most important definitions in Dung’s work, which are essential to its comprehension.

Argumentation framework "An argumentation framework is a pair AF = hAR, attacksi, where ARis a set of arguments, and attacks is a binary relation on AR, i.e. attacks ⊆ AR × AR." attack(A, B) represents an attack of argument A on argument B.

Conflict-free set "A set S of arguments is said to be conflict-free if there are no arguments A and Bin S such that A attacks B", i.e. if S has no internal attacks.

Acceptable argument "An argument A ∈ AR is acceptable w.r.t. a set S of arguments iff for each argument B ∈ AR: if B attacks A then B is attacked by [an argument in] S."

Admissible extension "A conflict-free set of arguments S is an admissible extension iff each ar- gument in S is acceptable w.r.t. S", i.e. iff S defends all of its elements (by attacking all attackers).

Complete extension "A conflict-free set of arguments S is a complete extension iff it is admissible and contains all the arguments it defends".

Grounded extension "A conflict-free set of arguments S is a grounded extension iff it is a minimal (w.r.t. set inclusion) complete extension of AF".

Preferred extension "A conflict-free set of arguments S is a preferred extension iff it is a maximal (w.r.t. set inclusion) admissible extension of AF", i.e. iff S defends all of its elements and is as large as possible. All preferred extensions are complete.

Stable extension "A conflict-free set of arguments S is a stable extension iff S attacks each argu- ment which does not belong to S [even if some of those arguments do not attack S]", i.e. iff Sis a preferred extension that attacks any argument not included in itself.

It is expected that participants are rational and agree about what arguments have been put forward and what attacks exist between those arguments (naturally this does not implicate agree- ment). In Dung’s framework, however, it is also assumed that arguments have the same strength, i.e. that each argument attacks other arguments with the same force and that participants agree about that fact. This may be sufficient for some uses, but it’s obviously not a good model of what happens in argumentation between humans. In real life, people may recognize that an opposing argument is valid in its essence (structure and logic) but still disagree about its relevance.

This limitation in Dung’s abstract framework has led to the development of many extensions to address the problem of different argument strengths. Bourguet [BAT10] provides a summary of some of those extensions, such as PAFs [AC02] and CPAFs [APP00] (which are briefly introduced in chapter4), and a new framework unifying their benefits. It is particularly interesting to analyze Bench-Capon’s [BC03] extension in the context of this thesis because he incorporates the concept of values (as in social or personal values, not economical or statistical values), which can be used as inspiration for modeling the preferences of different expert agents.

Bench-Capon theorizes that the perception of the strength of an argument is closely related to the concept of social or personal values. He exemplifies this way:

"(...) two arguments can conflict, and yet both be accepted. For an example sup- pose that Trevor and Katie need to travel to Paris for a conference. Trevor offers the argument "we should travel by plane because it is quickest". Katie replies with the argument "we should travel by train because it is much pleasanter". Trevor and Katie may continue to disagree as to how to travel, but they cannot deny each other’s argu- ments. The conclusion will be something like "we should travel by train because it is much pleasanter, even though traveling by plane is quicker."[BC03]

He draws attention to the fact that the persuasive impact of an argument is largely determined by the audience to which it is addressed. So an argument can be sound from a logical perspective but still be perceived as of little importance. Such an audience consists of one or more people with the same notion of values, or to be more specific, the same preferential order of personal values. In a different work [ABC07], Bench-Capon and Atkinson also indicate a shortcoming in Walton’s practical inference scheme (see section2.2.1) because it does not explain why G is a goal for agent A or how important it is for agent A to bring about goal G, therefore recommending a

more elaborate scheme in the lines of "In the circumstances R, the agent should perform action A to achieve new circumstances S, which will realize some goal G which will promote some value V of the agent". Value V is thus clearly stated as the reason why the agent desires to achieve goal G. Each agent can independently choose the order of importance of his values. Nevertheless, during the argumentative process, it may sometimes be impossible to satisfy the values of both agents simultaneously. In such cases, it’s necessary for the agents to conciliate their views on the importance of their values in order to agree on a course of action. Bench-Capon also accounts for the fact that some arguments are taken to be true regardless of the preferences of the agents (in the context of the broader example quoted above, he gives the example of a train strike, which makes it impossible to decide in favor of taking the train). Such arguments become associated to the value Truth, which is respected by all audiences above all other values.

Bench-Capon’s framework, besides providing semantics for acceptability, includes two impor- tant definitions:

Value based framework "A value-based argumentation framework (VAF) is a 5-tuple: VAF = hA, R,V, val, Pi", where A and R are equivalent to AR (set of arguments) and attacks (binary relation) in Dung’s framework respectively, V is a non-empty set of values, val is a mapping of each action to the value it promotes and P is the set of possible audiences (or partial orders on V ).

Audience specific VAF "An audience specific value-based argumentation framework (AVAF) is a 5-tuple: VAFa= hA, R,V, val,Val pre fai", where A, R, V and val are the same as for the

definition of VAF, a is an audience (from set P of the VAF) and Val pre fais an irreflexive,

assimetric and transitive preference relation Val pre fa⊆ V × V , which represents the order

of preference of values from the perspective of audience a.

One of the limitations of Bench Capon’s theory is that arguments can’t simultaneously support multiple values. This problem has been addressed in an extension of his work in [KvdT08]. How- ever, neither framework accounts for the fact that the relationship between preferences or values, in human discourse, is not necessarily linear. Extending Bench-Capon’s example of Trevor and Katie, imagine that the only train ride they can afford to take is very slow, such that they would need a couple of weeks to arrive at their destination. Even if the conference they wish to attend is only to be held within a month (therefore taking the train would not implicate missing the con- ference), would they still prefer taking the train instead of the plane? Probably not, for various reasons. One could argue that this is a matter of inserting more values in the argumentation frame- work related to the inconveniences or risks associated to taking the slow train. Nevertheless, that would require a difficult process of enumeration. Typical human reasoning would take those fac- tors into account implicitly by saying something like "I prefer to travel by train because it is more comfortable, unless the train is too slow, in which case I prefer to take the plane instead". This isn’t possible to represent directly using Bench-Capon’s framework or its extension, in part be- cause they don’t allow the explicit indication of negative impacts on values, or the representation

of the fact that when two arguments promote the same value, one may be preferred to the other if it promotes the value better or with more likelihood of success.