Instantiated Abduction in Argumentation

In this section we show that an abductive argumentation framework (F, I) can be instantiated on the basis of an abductive logic program (P, U ). The idea is that every possible hypothesis (∆+_{, ∆}−_{) maps to an abducible argumentation}

framework generated by the logic program (P ∪ ∆+) \ ∆−. The hypotheses for a query Q then correspond to the abducible argumentation frameworks that explain the observation B consisting of all arguments with conclusion Q. The construction of (F, I) on the basis of (P, U ) is defined as follows.

Definition 5.4.6. Let ALP = (P, U ) be an abductive logic program. Given a hypothesis (∆+, ∆−), we denote by F(∆+_,∆−₎ the argumentation framework

(A(P ∪∆+_)\∆−, _{(P ∪∆}+_)\∆−). The abductive argumentation framework gener-

ated by ALP is denoted by MALP and defined by MALP = (FP, IALP), where

IALP= {F(∆+_,∆−₎| ∆+, ∆−⊆ U, ∆+∩ ∆−= ∅}.

The following theorem states the correspondence between the explanations of a query Q in an abductive logic program ALP and the explanations of an obser- vation B in the abductive argumentation framework MALP.

Theorem 5.4.2. Let ALP = (P, U ) be an abductive logic program, Q ∈ AtP a

query and (∆+_{, ∆}−_{) a hypothesis. Let M}

ALP = (FALP, IALP). We denote by

XQ the set {(C, R, N ) ∈ AP | C = Q}. It holds that (∆+, ∆−) sceptically (resp.

credulously) explains Q iff F(∆+_,∆−₎sceptically (resp. credulously) explains XQ.

Proof of theorem 5.4.2. Follows directly from theorem 5.4.1 and definitions 5.4.5 and 5.4.6.

This theorem shows that our model of abduction in argumentation can indeed be seen as an abstraction of abductive logic programming.

Example 5.4.2. Let ALP = (P, U ) be the ALP as defined in example 5.4.1. All arguments generated by ALP are:

a = (p, {(p ← ∼s, r), r}, {s}) d = (r, {r}, ∅)

b = (q, {(q ← ∼p)}, {p}) e = (s, {s}, ∅)

c = (p, {(p ← ∼s, ∼q)}, {s, q})

Given these definitions, the abductive argumentation framework in example 5.2.1 is equivalent to MALP. In example 5.4.1 we saw that the query q is sceptically ex-

plained by the hypotheses ({s}, ∅) and ({s}, {r}), while (∅, {r}) only credulously explains it. Indeed, looking again at example 5.2.1, we see that G1 = F({s},∅)

and G3 = F({s},{r}) explain sceptical support for the observation {b} = Xq,

while G2= F(∅,{r}) only explains credulous support.

This method of instantiation shows that, on the abstract level, hypotheses can- not be represented by independently selectable abducible arguments. The run- ning example shows e.g. that a and d cannot be added or removed indepen- dently.

5.5

Related Work

Some of the ideas we applied also appear in the model of Sakama [83]. In his model of abduction in argumentation, both additions and removals of ar- guments from an abstract argumentation framework act as explanations for the observation that an argument is accepted or rejected. The main difference between Sakama’s model of abduction in abstract argumentation and the one presented here, is that he takes an explanation to be a set of independently selectable abducible arguments, while we take it to be a change to the argumen- tation framework that is applied as a whole. We have demonstrated, however,

that this is necessary when applying the abstract model in an instantiated set- ting. Furthermore, Sakama did address computation in his framework, but his method was based on translating abstract argumentation frameworks into logic programs. Sakama did not explore the instantiation of his model.

Some of the ideas we applied also appear in work by Wakaki et al. [90]. In their model, an ALP generates an instantiated argumentation framework and each hypothesis yields a different division into active/inactive arguments. Unlike our model, as well as Sakama’s [83], Wakaki et al. do not consider removal of arguments as explanation.

Kontarinis et al. [61] use term rewriting logic to compute changes to an abstract argumentation framework with the goal of changing the status of an argument. There are two similarities between their approach and ours. Firstly, we use production rules to generate dialogues and these rules can be seen as a kind of term rewriting rules. Secondly, their approach amounts to rewriting goals into statements to the effect that certain attacks in the argumentation framework are enabled or disabled. These statements resemble the moves PRO: x +_{y and}

PRO: x − y in our system. However, they treat attacks as entities that can be enabled or disabled independently. As discussed, different arguments (or in this case attacks associated with arguments) cannot be regarded as independent entities, if the abstract model is instantiated.

Other work dealing with the change of an argumentation framework with the goal of changing the status of arguments include Baumann [10], Baumann and Brewka [12], Bisquert et al. [17] and Perotti et al. [20]. Furthermore, Booth et al. [23] and Coste-Marquis et al. [37] frame it as a problem of belief revision. None of these works, however, make a connection between change of abstract argumentation and abduction.

5.6

Conclusion and Future Work

We developed a model of abduction in abstract argumentation, in which changes to an argumentation framework act as explanations for sceptical/credulous sup- port for observations. We presented sound and complete dialogical proof proce- dures for the main reasoning tasks, i.e., finding explanations for sceptical/cre- dulous support. In addition, we showed that our model of abduction in abstract argumentation can be seen as an abstract form of abduction in logic program- ming.

As a possible direction for future work, we consider the incorporation of addi- tional criteria for the selection of good explanations, such as minimality with respect to the added and removed arguments/attacks, as well as the use of preferences over different abducible argumentation frameworks. An interesting question is whether the proof theory can be adapted so as to yield only the preferred explanations.

Chapter 6

Change in Preference-Based

Argumentation

6.1

Introduction

Many works have recognized the importance of preferences in argumentation. Preferences over arguments may be derived, e.g., from their relative specificity or from the relative strength of the beliefs with which they are built. On the abstract level preferences can be represented by preference-based argumentation frameworks, which instantiate argumentation frameworks with a preference re- lation over the set of arguments [2, 86]. An attack of an argument x on y then succeeds only if y is not strictly preferred over x. Value-based argumentation frameworks provide yet another account of how preferences are derived [14]. The idea here is that arguments promote certain values and that different au- diences have different preferences over values, from which the preferences over arguments are derived.

An underexposed aspect in these models is change of preferences. Preferences are usually assumed to be fixed and no account is provided of how or why they may change. We address this aspect by applying Dietrich and List’s recently in- troduced model of property-based preference [41, 40]. In this model, preferences over alternatives are derived from preferences over sets of properties satisfied by the alternatives. Furthermore, agents are assumed to have a motivational state, consisting of the properties on which the agent focuses in a given situation, when forming preferences over alternatives. The authors present an axiomatic characterization of their model, in terms of a number of reasonable constraints on the relationship between motivational states and preferences.

Our contribution is a new, dynamic model of preferences in argumentation, cen- tering on what we call property-based argumentation frameworks. It is based on the model of Dietrich and List and provides an account of how and why pref- erences in argumentation may change. Our model generalizes preference-based argumentation frameworks as well as value-based argumentation frameworks, if properties are used to represent values. We look at two types of acceptance,

a

_b

c

d

a

_b

c

d

Figure 6.1: Two argumentation frameworks.

called weak and strong acceptance (i.e., acceptance in some or all motivational states). We also provide a dialogical proof theory that establishes whether an argument is weakly accepted. It is based on the grounded game [68] and extends it with dialogue moves consisting of properties.

The outline of this chapter is as follows. In section 6.2 we first give a brief outline of preference-based and value-based abstract argumentation. Then we give in section 6.3 an overview of the relevant parts of Dietrich and List’s model of property-based preferences. We move on to our own work in section 6.4, where we present our model of property-based argumentation frameworks, followed by a dialogical proof procedure for weak acceptance in section 6.5. We discuss some related work in section 6.6 and we conclude in section 6.7.

The results presented in this chapter are based on joint work with Richard Booth and Souhila Kaci [22].