Some Considerations on Epistemic and PracticalReasoning in Abstract Argumentation

Pietro Baroni and Massimiliano Giacomin

Dep. of Information Engineering, University of Brescia, Brescia, Italy{massimiliano.giacomin,pietro.baroni}@unibs.it

Abstract. In this short paper we discuss two approaches for integratingepistemic and practical reasoning in abstract argumentation, showingtheir commonalities and differences both from a technical and conceptualperspective.

Keywords: Abstract Argumentation, Semantics, Epistemic and Prac-tical Reasoning

1 Introduction

Abstract argumentation provides a general account of argumentation where ar-guments are simply represented as nodes of a directed graph AF = (A,→),called abstract argumentation framework [7], and binary attacks between themcorrespond to the edges of the graph, i.e. an argument a attacks an argumentb if (a, b) ∈→, also denoted as a → b. The issue is then to determine a justi-fication status for the arguments in A. To this purpose, several argumentationsemantics identify for each argumentation framework a number of extensions,each intuitively conceived as a set of arguments that can survive the conflicttogether. Most semantics embed a notion of defense, i.e. admissible sets of argu-ments are required to counterattack the arguments against them, and extensionsare defined as admissible sets satisfying additional constraints that reflect theintuitions and the degree of skepticism specific to the semantics [3]. More specif-ically, a set of arguments Args is admissible if ¬∃a, b ∈ Args : a→ b, and for anyb such that b→ a with a ∈ Args, there is an argument c ∈ Args such that c→ b.A complete extension is then an admissible set Args including all arguments itdefends, i.e. if ∀b such that b → a there is c ∈ Args such that c → b, thena ∈ Args. For the sake of the present paper, we need to recall grounded (GR)and preferred (PR) semantics only. Given an AF , grounded semantics identifiesas its unique extension, denoted as GRAF , the least (w.r.t. ⊆) complete exten-sion, while preferred semantics identifies as its extensions the maximal (w.r.t.⊆) complete extensions, denoted as EPR(AF ).

Abstract argumentation frameworks capture a variety of reasoning situa-tions, typically based on conflicting information affected by uncertainty andincompleteness. In particular, a distinction has been advocated in [9] betweenepistemic and practical arguments. While epistemic arguments concern reason-ing about what to believe, practical arguments concern reasoning about what

2 P. Baroni, M. Giacomin

to do, involving goals, desires and intentions. This distinction impacts on thesemantics adopted to evaluate argument justification.

On the grounds that conflicts between epistemic arguments mainly arisesfrom uncertainty and incompleteness of information, grounded semantics is pro-posed in [9] to deal with epistemic arguments, since it enforces a skeptical behav-ior ensuring that any indecision about arguments prevents their justification. Toexemplify this idea, a case concerning how to reach a remote small town is de-scribed in [9]. Suppose Mary warns that there will be a railway strike, while Bobdoes not believe there will be such a strike. Two conflicting arguments estrikeand e¬strike then arise and none of them can be justified, thus the questionconcerning whether the train will be available remains undecided.

Turning to practical arguments, in this case conflicts between them arisefrom the fact that distinct goals cannot all be fulfilled. The decision can then bebased on meta-level considerations, e.g. in the framework of value-based argu-mentation [6] each argument supports a value and an attack succeeds only if thevalue supported by the attacked argument is not strictly preferred to the valueassociated to the attacking argument. Continuing the example above, assumethat John has to give a talk in the remote town and, in order to reach it, hecan either take the train or drive the car. This situation can be modeled bytwo mutually conflicting practical arguments ptrain and pcar. If the train, differ-ently from the car, allows John to work on a paper he has to submit, then thevalue supported by ptrain (i.e. reaching the destination and working) is strictlypreferred 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 justified.This outcome corresponds to a general result in the context of value-based ar-gumentation frameworks, i.e. if values are totally and strictly ordered and thereare no attack cycles between arguments supporting the same value, then the re-sulting framework has a unique grounded and preferred extension [6]. However,when values do not allow to discriminate between conflicting arguments, one isled to make an arbitrary choice between them. On this basis, a very credulousapproach is advocated in [9] for practical arguments, i.e. selecting a preferredextension at random. For instance, if the available alternatives for John are thecar and a crowded bus (where working on the paper is impossible), then we havetwo mutually conflicting arguments pcar and pbus, but differently from the caseof estrike and e¬strike either of them should be justified.

The question then arises as to how to manage epistemic and practical ar-guments combined together. For instance, in the running example John mayevaluate 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 sectionwe review two approaches for integrating epistemic and practical reasoning, i.e.one introduced by Prakken in [9] and an approach first proposed in [10] andthen recasted, as sketched in [8], as an instance of a framework for combiningargumentation semantics based on decomposability [1]. The paper then providesa comparison between these approaches and finally outlines some perspectivesfor further research.

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(a) (b)Fig. 1. Representation of the running example and a variation.

2 Two approaches for epistemic and practical arguments

2.1 Prakken’s approach

As in [9], we assume that the set of arguments A of the AF is partitioned intoa set of epistemic arguments Ae and a set of practical arguments Ap, whereepistemic arguments can attack the practical ones but not viceversa.

The viable practical options are identified in [9] as follows. First, the groundedextension GRAF is computed, then all arguments that are attacked by epistemicarguments that are not in turn attacked by (arguments of) GRAF are removedfrom A. Finally, each practical option is derived from a preferred extension ofthe resulting argumentation framework.

Figure 1(a) depicts the argumentation framework modeling John’s prob-lem where the options of taking the train, taking the crowded bus and drivingthe car are considered, and the contradictory epistemic arguments concerningthe strike are taken into account. In this case, GRAF = ∅, leading to removeestrike, e¬strike and ptrain. Thus, the resulting preferred extensions are {pbus}and {pcar}, i.e. either the bus or the car should be chosen. Note that withoutargument estrike, the unique preferred extension would be {e¬strike, ptrain}, i.e.the train should be chosen since it allows John to work while traveling.

2.2 Decomposability-based approach

The formal treatment of this approach requires a labelling-based definition ofargumentation semantics, where each extension E is replaced by a correspondinglabelling, i.e. a total function from A to {in, out, undec}, such that an argumenta is labelled in iff a ∈ E; it is labelled out iff ∃ b ∈ E such that b → a; itis labelled undec if neither of the above conditions holds. Thus, a semantics Sreturns for any AF a set of labellings LS(AF ) instead of a set of extensions.

Given an AF = (A,→), the general approach to combine semantics intro-duced in [8] considers a partition P = {P1, . . . , Pn} of A with associated seman-tics S(Pi). Each set Pi identifies the sub-framework AF↓Pi

= (Pi,→ ∩(Pi×Pi)),where the semantics S(Pi) is locally applied. The local computation of S(Pi) isexpressed by a local function FS(Pi), which assigns to the arguments in Pi a setof labellings on the basis of AF↓Pi

, of the set Piinp = {a /∈ Pi | ∃b ∈ Pi : a→ b}

including the external arguments attacking Pi (playing the role of input ar-guments), of the labels externally assigned to them1 (

⋃j=1...n,j 6=i LabPj

)↓Piinp ,

and of the attack relation PiR =→ ∩(Pi

inp × Pi) from the input arguments ofPi

inp to Pi. Then the resulting labelling are {LabP1∪ . . . ∪ LabPn

| LabPi∈

FS(Pi)(AF↓Pi, Pi

inp, (⋃

j=1...n,j 6=i LabPj)↓Pi

inp , PiR)}.

1 More precisely, given a labelling L and a set of arguments Args, L↓Args , L∩(Args×{in, out, undec}).

4 P. Baroni, M. Giacomin

In order to apply this general schema to epistemic and practical arguments,we consider the partition {Ae,Ap} with S(Ae) = GR and S(Ap) = PR (notehowever that the framework can easily be generalized to other semantics). Asshown in [1], the relevant local functions can be identified by applying eachsemantics S to a standard argumentation framework where the input argumentsare added to AF↓Pi and the input labelling is enforced through the additionof arguments attacking out-labelled arguments and self-attacks for all undec-labelled arguments.

For instance, Figure 1(b) depicts an argumentation framework represent-ing a variation of the running example where also the train is crowded (andthus does not allow John to work), and as a consequence the three options areequally preferable. The application of GR to AF↓{estrike,e¬strike} (with emptyinput arguments) yields both epistemic arguments undecided. Thus, the localapplication of PR to AF↓{pcr−train,pbus,pcar} yields the two preferred labellings{(pcr−train, undec), (pbus, in), (pcar, out)} and {(pcr−train, undec), (pbus, out),(pcar, in)}. It can be checked that Prakken’s approach gives the same outcome,identifying as extensions the sets {pbus} and {pcar}. Intuitively, since the avail-ability of the train is uncertain, either the train or the car should be considered.

3 Comparing the two approaches

In order to analyze the relationship between the two approaches, it is convenientto partition the arguments of A on the basis of the labelling assigned by GR ap-plied 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 toGR. We also partition Ap into inp, outp, undec←e

p and undec∼p , where inp andoutp have the same meaning as above, undec←e

p includes those arguments of Ap

that are labelled undec by GR and are attacked by an argument of undece, andundec∼p includes the remaining arguments of Ap that are labelled undec.

The following observations concerning Prakken’s approach are relatively easyto prove: i) oute, undece, outp and undec←e

p are removed from A, thus theydo not belong to any preferred extension; ii) ine and inp belong to all pre-ferred extensions. A direct consequence of the first observation is that afterremoval undec∼p does not receive attacks from external arguments, and dueto the property of directionality of PR [4] the set of preferred extensions is{ine ∪ inp ∪ E | E ∈ EPR(AF↓undec∼p )}.

As to the decomposability-based approach, using the results in [1] it is pos-sible to prove that a corresponding outcome is yielded for all arguments butundec∼p . In particular, any obtained labelling assigns the label in to the argu-ments in ine and inp, the label out to the arguments in oute and outp, and thelabel undec to the arguments in undece and undec←e

p . Differences only concernthe arguments of undec∼p . Taking into account decomposability of PR [1], itcan be shown that the labellings restricted to undec∼p can be obtained by lo-cally applying PR into the argumentation framework AF↓undec∼p by taking intoaccount the input arguments of undec∼p in undec←e

p . Due to a monotony prop-

On Epistemic and Practical Reasoning 5

erty of the local function of preferred semantics, it can be shown that for anylabelling obtained by the decomposability-based approach there is an extensionof Prakken’s approach which (not necessarily strictly) includes all arguments la-belled in. Intuitively, the decomposability-based approach is more skeptical wrtPrakken’s proposal.

From a conceptual point of view, this difference can be appreciated in thecase of Figure 1(a), where the decomposability-based approach returns a uniquelabelling with all arguments undecided. While this may seem undesirable, webelieve 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 andthus it is perfectly sensible to choose between the car and the bus, in the caseof Figure 1(a) leaving the train out of consideration would prevent a possiblealternative which is stricly more preferred than the car and the bus.

4 Perspectives for further research

We believe the considerations pointed out in this short paper open the way toseveral interesting investigations. A first direction is applying different semanticsbesides GR and PR, such as [2]. Another interesting question is how to handlea gradual evaluation of practical arguments given different degrees of strengthfor epistemic arguments, possibly exploiting a probabilistic approach [5].

References

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