In this short paper we discuss two approaches for integrating epistemic and practical reasoning in abstract argumentation, showing their commonalities and di erences both from a technical and conceptual perspective.
to do, involving goals, desires and intentions. This distinction impacts on the semantics adopted to evaluate argument justi cation.
On the grounds that con icts between epistemic arguments mainly arises from uncertainty and incompleteness of information, grounded semantics is proposed in [9] to deal with epistemic arguments, since it enforces a skeptical behavior ensuring that any indecision about arguments prevents their justi cation. To exemplify this idea, a case concerning how to reach a remote small town is described in [9]. Suppose Mary warns that there will be a railway strike, while Bob does not believe there will be such a strike. Two con icting arguments estrike and e:strike then arise and none of them can be justi ed, thus the question concerning whether the train will be available remains undecided.
Turning to practical arguments, in this case con icts between them arise from the fact that distinct goals cannot all be ful lled. The decision can then be based on meta-level considerations, e.g. in the framework of value-based argumentation [6] each argument supports a value and an attack succeeds only if the value supported by the attacked argument is not strictly preferred to the value associated to the attacking argument. Continuing the example above, assume that John has to give a talk in the remote town and, in order to reach it, he can either take the train or drive the car. This situation can be modeled by two mutually con icting practical arguments ptrain and pcar. If the train, di erently from the car, allows John to work on a paper he has to submit, then the value supported by ptrain (i.e. reaching the destination and working) is strictly preferred w.r.t. the value supported by pcar (i.e. reaching the destination only), thus ptrain attacks pcar but not vice versa, and ptrain turns out to be justi ed. This outcome corresponds to a general result in the context of value-based argumentation frameworks, i.e. if values are totally and strictly ordered and there are no attack cycles between arguments supporting the same value, then the resulting framework has a unique grounded and preferred extension [6]. However, when values do not allow to discriminate between con icting arguments, one is led to make an arbitrary choice between them. On this basis, a very credulous approach is advocated in [9] for practical arguments, i.e. selecting a preferred extension at random. For instance, if the available alternatives for John are the car and a crowded bus (where working on the paper is impossible), then we have two mutually con icting arguments pcar and pbus, but di erently from the case of estrike and e:strike either of them should be justi ed.
The question then arises as to how to manage epistemic and practical arguments combined together. For instance, in the running example John may evaluate the three options corresponding to the train, the car and the (crowded) bus, taking into account the possibility of the railway strike. In the next section we review two approaches for integrating epistemic and practical reasoning, i.e. one introduced by Prakken in [9] and an approach rst proposed in [10] and then recasted, as sketched in [8], as an instance of a framework for combining argumentation semantics based on decomposability [1]. The paper then provides a comparison between these approaches and nally outlines some perspectives for further research.
In order to apply this general schema to epistemic and practical arguments, we consider the partition fAe; Apg with S(Ae) = GR and S(Ap) = PR (note however that the framework can easily be generalized to other semantics). As shown in [1], the relevant local functions can be identi ed by applying each semantics S to a standard argumentation framework where the input arguments are added to AF #Pi and the input labelling is enforced through the addition of arguments attacking out-labelled arguments and self-attacks for all undeclabelled arguments.
For instance, Figure 1(b) depicts an argumentation framework representing a variation of the running example where also the train is crowded (and thus does not allow John to work), and as a consequence the three options are equally preferable. The application of GR to AF #festrike;e:strikeg (with empty input arguments) yields both epistemic arguments undecided. Thus, the local application of PR to AF #fpcr train;pbus;pcarg yields the two preferred labellings f(pcr train; undec); (pbus; in); (pcar; out)g and f(pcr train; undec); (pbus; out); (pcar; in)g. It can be checked that Prakken's approach gives the same outcome, identifying as extensions the sets fpbusg and fpcarg. Intuitively, since the availability of the train is uncertain, either the train or the car should be considered. 3
In order to analyze the relationship between the two approaches, it is convenient to partition the arguments of A on the basis of the labelling assigned by GR applied to the whole AF . In particular, we partition Ae into ine, oute and undece, where le denotes the subset of Ae including arguments labelled l according to GR. We also partition Ap into inp, outp, undecp e and undecp , where inp and outp have the same meaning as above, undecp e includes those arguments of Ap that are labelled undec by GR and are attacked by an argument of undece, and undecp includes the remaining arguments of Ap that are labelled undec.
The following observations concerning Prakken's approach are relatively easy to prove: i) oute, undece, outp and undecp e are removed from A, thus they do not belong to any preferred extension; ii) ine and inp belong to all preferred extensions. A direct consequence of the rst observation is that after removal undecp does not receive attacks from external arguments, and due to the property of directionality of PR [4] the set of preferred extensions is fine [ inp [ E j E 2 EPR(AF #undecp )g.
As to the decomposability-based approach, using the results in [1] it is possible to prove that a corresponding outcome is yielded for all arguments but undecp . In particular, any obtained labelling assigns the label in to the arguments in ine and inp, the label out to the arguments in oute and outp, and the label undec to the arguments in undece and undecp e. Di erences only concern the arguments of undecp . Taking into account decomposability of PR [1], it can be shown that the labellings restricted to undecp can be obtained by locally applying PR into the argumentation framework AF #undecp by taking into account the input arguments of undecp in undecp e. Due to a monotony property of the local function of preferred semantics, it can be shown that for any labelling obtained by the decomposability-based approach there is an extension of Prakken's approach which (not necessarily strictly) includes all arguments labelled in. Intuitively, the decomposability-based approach is more skeptical wrt Prakken's proposal.
From a conceptual point of view, this di erence can be appreciated in the case of Figure 1(a), where the decomposability-based approach returns a unique labelling with all arguments undecided. While this may seem undesirable, we believe the outcome more closely corresponds to the direction of the attacks. In particular, while in Figure 1(b) there is no preference towards the train and thus it is perfectly sensible to choose between the car and the bus, in the case of Figure 1(a) leaving the train out of consideration would prevent a possible alternative which is stricly more preferred than the car and the bus. 4
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