Comparing argumentation semantics with respect to skepticism

Comparing argumentation semantics with respect to skepticism Pietro Baroni and Massimiliano Giacomin Dip. Elettronica per l’Automazione, University of Brescia, via Branze 38, 25123 Brescia, Italy baroni,[email protected]

Abstract. The issue of formalizing skepticism relations between argumentation semantics has been considered only recently in the literature. In this paper, we contribute to this kind of analysis by providing a systematic comparison of a significant set of literature semantics (namely grounded, complete, preferred, stable, semi-stable, ideal, prudent, and CF 2 semantics) using both a weak and a strong skepticism relation. Key words: Argumentation semantics, Skepticism

1

Introduction

The increasing variety of argumentation semantics proposed in the literature raises the issue of carrying out systematic principle-based comparisons between different approaches. While limitations of example-based comparisons have been pointed out earlier by several authors (see for instance [1, 2]), studies on general evaluation principles for argumentation semantics are appearing in the literature only in very recent years. For instance, in [3] general rationality postulates for argumentation systems are introduced, showing that there are argumentation systems where they are violated. At the more abstract level of Dung’s argumentation frameworks [4], in [5] several semantics evaluation criteria have been introduced and exploited for a systematic assessment of both “traditional” and more recent proposals. In this work we consider another aspect of this kind of systematic comparison, namely the issue of (partially) ordering argumentation semantics with respect to their skepticism. After recalling the necessary background concepts in Section 2, we review in Section 3 the definitions of the weak and strong skepticism relations between semantics, first introduced in [6]. Argumentation semantics considered in this paper are quickly described in Section 4, then Section 5 shows how they are partially ordered according to the weak and strong skepticism relations. A final discussion and conclusions are provided in Section 6.

2

Basic concepts

The present work lies in the frame of the general theory of abstract argumentation frameworks proposed by Dung [4].

2

P. Baroni, M. Giacomin

Definition 1. An argumentation framework is a pair AF =
A, →, where A is a set, and →⊆ (A × A) is a binary relation on A, called attack relation. In the following we will always assume that A is finite. Since we will frequently consider properties of sets of arguments, it is useful to extend to them the notations defined for the nodes. Definition 2. Given an argumentation framework AF =
A, →, a node α ∈ A and two sets S, P ⊆ A, we define S → α
∃β ∈ S : β → α; α → S
∃β ∈ S : α → β; S → P
∃α ∈ S, β ∈ P : α → β. In Dung’s theory, an argumentation semantics is defined by specifying the criteria for deriving, given a generic argumentation framework, the set of all possible extensions, each one representing a set of arguments considered to be acceptable together. Accordingly, a basic requirement for any extension E is that it is conflict-free, namely
α, β ∈ E : α → β. All argumentation semantics proposed in the literature satisfy this fundamental conflict-free property. Given a generic argumentation semantics S, the set of extensions prescribed by S for a given argumentation framework AF =
A, → is denoted as ES (AF). If ∀AF |ES (AF)| = 1, then the semantics S is said to follow the unique-status approach, otherwise it is said to follow the multiple-status approach. I-maximality is a relevant property of sets of extensions used in the following. Definition 3. A set of extensions E is I-maximal iff ∀E1 , E2 ∈ E, if E1 ⊆ E2 then E1 = E2 . A semantics S satisfies the I-maximality criterion if and only if ∀AF, ES (AF) is I-maximal. Note that I-maximality is a property of the set of extensions E per se and does not imply that maximality is prescribed by the semantics-specific definition of what an extension is. For instance any unique-status semantics necessarily satisfies Imaximality according to Definition 3, independently of the fact that the unique extension prescribed by the semantics is a maximal set in any sense. It is also worth noting that it may be the case that ES (AF) = ∅, i.e. that a semantics S is unable to prescribe any extension for some argumentation frameworks AF. We adopt as a standpoint that such argumentation frameworks lie outside the domain of definition of S and therefore have not to be considered in the evaluation of its properties. Formally, for a generic semantics S let DS be the set of argumentation frameworks where S admits at least one extension, namely DS = {AF : ES (AF) = ∅}. In the following, whenever we will refer to the comparison of two semantics S1 and S2 with respect to a generic argumentation framework AF we will implicitly assume that AF ∈ DS1 ∩ DS2 . In fact, one (or even both) of the terms of comparison would be undefined otherwise.

3

Skepticism relations

The notion of skepticism has often been used in informal ways to discuss semantics behavior, e.g. by observing that a semantics is “more skeptical” than

Comparing argumentation semantics with respect to skepticism

3

another one. Intuitively, a semantics is more skeptical than another if it makes less committed choices about the justification of the arguments. A comparison of skepticism between semantics can be based on a relationship E between the sets of extensions they prescribe. Given two sets of extensions E1 , E2 of an argumentation framework AF, E1 E E2 will denote that E1 is at least as skeptical as E2 in some sense. Then a relation of skepticism S between semantics, induced by E , can be defined. Definition 4. Let E be a skepticism relation between sets of extensions. The skepticism relation between argumentation semantics S induced by E is defined as follows: for any argumentation semantics S1 and S2 , S1 S S2 ⇔ for any argumentation framework AF, ES1 (AF) E ES2 (AF). We will consider two actual skepticism relations between sets of extensions: E a weak relation, denoted as E W , and a strong relation, denoted as S . These relations have been introduced in [6] to which the reader is referred for more extensive explanations, not reported here due to space limitation. As a starting point, we recall that to compare a single extension E1 with a set of extensions E2 , the relation ∀E2 ∈ E2 E1 ⊆ E2 has often be used in the literature (for instance to verify that the unique extension prescribed by grounded semantics is more skeptical than the set of extensions prescribed by preferred semantics). A direct generalization to the comparison of two sets of extensions is represented by the following weak skepticism relation E W. Definition 5. Given two sets of extensions E1 and E2 of an argumentation framework AF, E1 E W E2 iff ∀E2 ∈ E2 ∃E1 ∈ E1 : E1 ⊆ E2 . Relation E W is in a sense unidirectional, since it only constrains the extensions of E2 , while E1 may contain additional extensions unrelated to those of E2 . One may consider also a more symmetric (and stronger) relationship E S , where it is also required that any extension of E1 is included in an extension of E2 . Definition 6. Given two sets of extensions E1 and E2 of an argumentation E framework AF, E1 E S E2 iff E1 W E2 and ∀E1 ∈ E1 ∃E2 ∈ E2 : E1 ⊆ E2 . By definition, given two sets of extensions E1 and E2 it holds that E1 E S E E E2 ⇒ E1  E W E2 . Instantiating Definition 4 with W and S gives rise to two corresponding skepticism relations between semantics, denoted as S1 and S2 , ordered by the same implication: S1 SS S2 ⇒ S1 SW S2 . It is worth noting that the skepticism relations introduced above are not total orders, since in general there can be two sets of extensions (and therefore two semantics) which are not comparable. We recall some properties (proved in [6]) of the skepticism relations between sets of extensions (which are of course “inherited” by the relations between semantics). E Proposition 1. Relations E W and S are preorders, i.e. they are reflexive and E E transitive. Relations W and S are also partial orders when the considered sets of extensions are I-maximal, namely given two I-maximal sets of extensions E1 E and E2 , if E1 E W E2 and E2 W E1 then E1 = E2 .

4

4

P. Baroni, M. Giacomin

A review of extension-based argumentation semantics

We review the definition of several argumentation semantics which will be compared according to the skepticism relations defined in previous section. 4.1

Traditional semantics

Stable semantics relies on the idea that an extension should be able to reject the arguments that are outside the extension itself [4]. Definition 7. Given an argumentation framework AF =
A, →, a set E ⊆ A is a stable extension of AF if and only if E is conflict-free ∧ ∀α ∈ A : α ∈ / E, E → α. Stable semantics will be denoted as ST , and, accordingly, the set of all the stable extensions of AF as EST (AF). Stable semantics suffers by a significant limitation since there are argumentation frameworks where no extensions complying with Definition 7 exist. No other semantics considered in this paper is affected by this problem except the prudent version of stable semantics. The requirement that an extension should attack all other external arguments can be relaxed by imposing that an extension is simply able to defend itself from external attacks. This is at the basis of the notions of acceptable argument and admissible set [4]. Definition 8. Given an argumentation framework AF =
A, →, an argument α ∈ A is acceptable with respect to a set E ⊆ A if and only if ∀β ∈ A : β → α, E → β. Given an argumentation framework AF =
A, →, a set E ⊆ A is admissible if and only if E is conflict-free and ∀β ∈ A : β → E, E → β. The set of the arguments acceptable with respect to a set E is traditionally denoted using the characteristic function FAF (E): Definition 9. Given an argumentation framework AF =
A, →, the function FAF : 2A → 2A which, given a set E ⊆ A, returns the set of the acceptable arguments with respect to E, is called the characteristic function of AF. Building on these concepts, the notion of complete extension can be introduced, which plays a key role in Dung’s theory, since all semantics encompassed by his framework select their extensions among the complete ones. Definition 10. Given an argumentation framework AF =
A, →, a set E ⊆ A is a complete extension if and only if E is admissible and every argument of A which is acceptable with respect to E belongs to E. The notion of complete extension is not associated to a notion of complete semantics in [4], however, the term complete semantics has subsequently gained acceptance in the literature and will be used to refer to the properties of the set of complete extensions. Complete semantics will be denoted as CO. The well-known grounded semantics belongs to the unique-status approach and its unique extension, denoted as GE(AF), can be defined as the least fixed point of the characteristic function.

Comparing argumentation semantics with respect to skepticism

5

Definition 11. Given an argumentation framework AF =
A, →, the grounded extension of AF, denoted as GE(AF), is the least fixed point (with respect to set inclusion) of FAF . Preferred semantics, denoted as PR, is obtained by simply requiring the property of maximality along with admissibility. Definition 12. Given an argumentation framework AF =
A, →, a set E ⊆ A is a preferred extension of AF if and only if it is a maximal (with respect to set inclusion) admissible set. 4.2

CF 2 semantics

CF 2 semantics, first introduced in [7], is a SCC-recursive semantics [8] which features the distinctive property of treating in a “symmetric” way odd- and evenlength cycles while belonging to the multiple-status approach. SCC-recursiveness is related to the graph-theoretical notion of strongly connected components (SCCs) of AF, namely the equivalence classes of nodes under the relation of mutual reachability, denoted as SCCSAF . Due to space limitations, we can not examine in detail the definition of CF 2 semantics: the interested reader may refer to [7] and [8]. Definition 13. Given an argumentation framework AF =
A, →, a set E ⊆ A is an extension of CF 2 semantics iff – E ∈ MCF AF if |SCCSAF | = 1 – ∀S ∈ SCCSAF (E ∩ S) ∈ ECF 2 (AF↓UPAF (S,E) ) otherwise where MCF AF denotes the set of maximal conflict-free sets of AF, and, for any set S ⊆ A, AF↓S denotes the restriction of AF to S, namely AF↓S =
S, → ∩(S × S), and U PAF (S, E) = {α ∈ S |
β ∈ E : β ∈ / S, β → α}. CF 2 semantics can be roughly regarded as selecting its extensions among the maximal conflict free sets of AF, on the basis of some topological requirements related to the decomposition of AF into strongly connected components. In particular it turns out that when AF consists of exactly one strongly connected component, the set of extensions prescribed by CF 2 semantics exactly coincides with the set of maximal conflict free sets of AF. 4.3

Semi-stable semantics

Semi-stable semantics [9], denoted in the following as SST , aims at guaranteeing the existence of extensions in any case (differently from stable semantics) while coinciding with stable semantics (differently from preferred semantics) when stable extensions exist. The definition of extensions satisfying these desiderata is ingeniously simple. Definition 14. Given an argumentation framework AF =
A, → a set E ⊆ A is a semi-stable extension if and only if E is a complete extension such that (E ∪ {α | E → α}) is maximal with respect to set inclusion.

6

P. Baroni, M. Giacomin

4.4

Ideal semantics

Ideal semantics [10] provides an alternative unique-status approach which is less skeptical than grounded semantics, i.e. for any argumentation framework the (unique) ideal extension is a (sometimes strict) superset of the grounded extension. Also in this case the definition is quite simple. Definition 15. Given an argumentation framework AF =
A, → a set E ⊆ A is ideal if and only if E is admissible and ∀P ∈ EPR (AF) E ⊆ P . The ideal extension is the maximal (with respect to set inclusion) ideal set. We will use the symbol ID to refer to the ideal semantics, and denote the ideal extension of an argumentation framework AF as ID(AF). 4.5

Prudent semantics

Prudent semantics [11, 12] emphasizes the role of indirect attacks: forbidding them leads to the definition of p(rudent)-admissible sets. Definition 16. Given an argumentation framework AF =
A, →, an argument α indirectly attacks another argument β, denoted as α → β, if there is an oddlength path from α to β in the defeat graph corresponding to AF. A set S is without indirect conflicts, denoted as icf (S), if and only if ∃α, β ∈ S : α → β. Definition 17. Given an argumentation framework AF =
A, →, a set of arguments S ⊆ A is p(rudent)-admissible if and only if ∀α ∈ S α is acceptable with respect to S and icf (S). On this basis, the prudent version of several traditional notions of extensions (and then of the relevant semantics) has been defined. Definition 18. Given an argumentation framework AF =
A, →, a set of arguments S ⊆ A is: – a preferred p-extension if and only if S is a maximal (with respect to set inclusion) p-admissible set; – a stable p-extension if and only if icf (S) and ∀α ∈ (A \ S) S → α; – a complete p-extension if and only if S is p-admissible and there is no argument α ∈ / S such that α is acceptable with respect to S and icf (S ∪ {α}). Definition 19. Given an argumentation framework AF =
A, →, the function FpAF : 2A → 2A which, given a set S ⊆ A, returns the set of the acceptable arguments with respect to S such that icf (S ∪ {α}) is called the p-characteristic function of AF. Let j be the lowest integer such that the sequence Fp,i AF (∅) is stationary for i ≥ j: Fp,j (∅) is the grounded p-extension of AF, denoted as AF GPE(AF). The prudent versions of grounded, complete, preferred and stable semantics will be denoted as GRP, COP, PRP and ST P, respectively.

Comparing argumentation semantics with respect to skepticism SST

PR

ID

CO

7

CF2

GR

PRP

GRP

COP

Fig. 1.
S S relation for any argumentation framework. α

β

Fig. 2. A mutual attack.

5

Skepticism comparison of argumentation semantics

In this section we examine comparability between semantics according to the relations SW and SS . Before entering the matter, a result which will be used in the following and whose proof is immediate needs to be stated. Lemma 1. Given two argumentation semantics S1 and S2 , if for any argumentation framework AF ES2 (AF) ⊆ ES1 (AF), then S1 SW S2. Note also from Proposition 1 that, given two distinct semantics S1 , S2 (i.e. such that ∃AF : ES1 (AF) = ES2 (AF)) satisfying the I-maximality criterion, if S1 SW S2 then S2 SW S1 (and also S2 SS S1). Since all semantics reviewed in Section 4 are distinct and, with the exception of CO and COP, also I-maximal, this fact will be implicitly exploited for relations not involving CO and COP. To begin our comparison, we examine SS whose Hasse diagram is shown in Figure 1. Starting from the bottom, let us show that GRP SS GR, which, both belonging to the unique-status approach, is equivalent to the inclusion relation proved in Proposition 2. Proposition 2. For any argumentation framework AF, GPE(AF) ⊆ GE(AF).
Proof. Recall from [4] that GE(AF) = i≥1 FiAF (∅) and GPE(AF) = Fp,j AF (∅), p,i where j is the lowest integer such that the sequence FAF (∅) is stationary for i ≥ j. Now, obviously F1AF (∅) = Fp,1 AF (∅) (since both coincide with the set of unattacked i arguments in AF). Assume inductively that Fp,i AF (∅) ⊆ FAF (∅), then it holds that p,i+1 p,i+1 i+1 FAF (∅) ⊆ FAF (∅). In fact, any α ∈ FAF (∅) is defended by Fp,i AF (∅), but then i+1 it is also defended by FiAF (∅) ⊇ Fp,i (∅), and therefore α ∈ F (∅). AF AF Going up in the diagram, since, as well-known, the grounded extension is included in any complete extension, GR SS CO. On the other hand, CO SS GR, as

8

P. Baroni, M. Giacomin ε δ

γ

β

α

Fig. 3. Direct and indirect attacks.

shown, for instance, by the example of Figure 2 where ECO (AF) = {∅, {α}, {β}}, while EGR (AF) = {∅}. Furthermore, it is easy to see that CO SS PR since any preferred extension is also a complete extension and any complete extension, being an admissible set, is included in a maximal admissible set, i.e. in a preferred extension. On the other hand, PR SS CO (indeed PR SW CO): considering again Figure 2 we have EPR (AF) = {{α}, {β}}, ECO (AF) = {∅, {α}, {β}} and there is no preferred extension E1 such that E1 ⊆ ∅ ∈ ECO (AF). As to the upper-left part of the diagram, it is shown in [10] that GE(AF) ⊆ ID(AF), entailing that GR SS ID. Moreover, by definition, the ideal extension is included in any preferred extension. It follows that ID SS PR and, since any semi-stable extension is also a preferred extension [9], ID SS SST . As to CF 2 semantics, it is known [8] that ∀AF ∀E ∈ ECF 2 (AF) GE(AF) ⊆ E, which, since GE(AF) ∈ ECO (AF) entails CO SW CF 2. Moreover, it is known [7] that ∀E1 ∈ EPR (AF) ∃E2 ∈ ECF 2 (AF) : E1 ⊆ E2 . Since in turn any complete extension is included in a preferred extension, summing up we have that CO SS CF 2. On the other hand, since in the example of Figure 2 CF 2 semantics behaves as preferred semantics, it follows that CF 2 SS CO (indeed CF 2 SW CO). Turning to the right bottom part, it is easy to see that COP SS CO. COP SW CO follows from the fact that GPE(AF) ∈ ECOP (AF) and ∀E2 ∈ ECO (AF) GPE(AF) ⊆ GE(AF) ⊆ E2 (using Proposition 2 for the first inclusion). Moreover, any complete prudent extension is an admissible set and is therefore included in a preferred extension, which is also a complete extension of AF. On the other hand, CO SS COP (indeed CO SW COP): considering Figure 3, we have ECOP (AF) = {{δ, }, {α, }} (note in particular that α indirectly conflicts with δ), while ECO (AF) = {{δ, , α}}. Turning finally to preferred prudent semantics, since any preferred prudent extension is also a complete prudent extension and any complete prudent extension is included in a preferred prudent extension, it holds that COP SS PRP. Since in the example of Figure 2 complete prudent and preferred prudent behave as their traditional counterparts, it follows that PRP SS COP (indeed PRP SW COP). Let us consider now the weak skepticism relation SW , whose Hasse diagram is shown in Figure 4: we will comment only on edges not implied by the relations SS already examined. Starting from the bottom, since GPE(AF) ∈ ECOP (AF) Lemma 1 directly entails that COP SW GRP. On the other hand, considering Figure 3 and recalling that ECOP (AF) = {{δ, }, {α, }}, we have that GPE(AF) = {{δ, }}, which entails GRP SW COP. The next difference with re-

Comparing argumentation semantics with respect to skepticism

9

SST PR ID

CF2

PRP

GR = CO GRP COP Fig. 4.
S W relation for any argumentation framework. γ

α α

β

δ

β

δ

ε

γ AF1

AF2

Fig. 5. CF 2 semantics is not comparable with some others.

spect to SS concerns grounded and complete semantics. We already know that GR SW CO, we now note, by Lemma 1, that CO SW GR, since GE(AF) ∈ ECO (AF). Finally, again by Lemma 1, we have PR SW SST since it is shown in [9] that any semi-stable extension is also a preferred extension. While we have now proved the existence of all the edges shown in Figures 1 and 4, one might wonder whether additional relations hold. We prove that this is not the case, starting from SW relation. Consider first CF 2 semantics: it is not comparable with ideal, preferred and semi-stable semantics. In fact, referring to Figure 5, it holds that EPR (AF1 ) = ESST (AF1 ) = EID (AF1 ) = {{β, δ}} while ECF 2 (AF1 ) = {{γ, δ}, {β, δ}, {α}}, therefore letting S be any of the three considered semantics S SW CF 2. On the other hand, it holds that EPR (AF2 ) = ESST (AF2 ) = EID (AF2 ) = {∅}, while ECF 2 (AF2 ) = {{α, }, {β, }, {γ, }}, from which it follows that CF 2 SW S. Turning to preferred prudent semantics, by transitivity of SW it is sufficient to show GRP SW PRP, PRP SW SST and PRP SW CF 2. As to the first condition, in the example of Figure 3 it holds that EPRP (AF) = {{δ, }, {α, }} while GPE(AF) = {δ, }. Then, letting S be any of the semantics shown in Figure 4 (except the complete prudent semantics), it holds that S SW PRP. As to the second and third condition, in the example of Figure 6 taken from [12] it holds that EPRP (AF) = {{η}}, while ESST (AF) = {{α, γ, η}, {β, δ, ζ}, {β, δ, η}}

10

P. Baroni, M. Giacomin δ γ

β

ζ

α

η

ε

Fig. 6. Preferred prudent semantics is not comparable with many others. δ α

β

γ ε

Fig. 7. A case showing that COP S S SST .

and ECF 2 (AF) = {{α, γ, η}, {β, δ, ζ}, {β, δ, η}, {, β, η}, {, β, ζ}}. By transitivity, second condition implies PRP SW S for S ∈ {GRP, GR, CO, ID, PR}. Turning to non-existence of edges in Figure 1, first note that S1 SW S2 ⇒ S1 SS S2 . In particular, nothing remains to be said about CF 2 and PRP semantics. As to COP semantics, the example of Figure 7 shows that COP SS SST since ESST (AF) = {{β, δ}}, while there is a complete prudent extension, namely {α}, which is not included in any semi-stable extension. By transitivity of SS , this also entails COP SS S for any S ∈ {GRP, GR, ID}. On the other hand, for any semantics S ∈ {GRP, GR, ID, SST } we already know from Figure 4 that S is not comparable with PRP, which entails S SS COP. As to preferred semantics, we already know that SST SS PR, while PR SS SST holds since otherwise, by transitivity, it would be the case that COP SS SST . Similarly, from Figure 4 we have ID SS CO and SST SS CO. Furthermore, COP SS SST implies CO SS ID and CO SS SST . Having completed the analysis concerning argumentation semantics able to prescribe extensions in any case, in Figure 8 we provide (without comments and proofs due to space limitation) the Hasse diagrams restricted to the case of argumentation frameworks where stable extensions exist (relations concerning stable prudent semantics, when its extensions exist, are shown dashed).

6

Conclusions

We have provided a systematic skepticism comparison concerning a significant range of both “traditional” and more recent argumentation semantics, using both a weak and a strong comparison criterion. The weak criterion gives rise to an almost linear ordering not including just CF 2 and preferred prudent semantics. Semantics related to the notion of stable extension, namely semi-stable, stable and stable prudent turn out to be the least skeptical, while the notions of grounded and complete extension (more so their prudent counterpart) provide a bottom reference for skepticism. Ideal and preferred semantics lie orderly

Comparing argumentation semantics with respect to skepticism

11

STP SST = ST PR ID GR = CO

CF2

PRP

SST = ST STP

ID

PR

CF2

CO

GRP

GR

COP

GRP

PRP COP

S Fig. 8.
S W and
S for argumentation frameworks where stable extensions exists.

between grounded and stable-related semantics and can be regarded as “intermediate”. So does CF 2 semantics, while being not comparable with ideal and preferred. It is also interesting to note that prudent versions of stable, grounded and complete semantics tend to make “more extreme” the skepticism properties of their traditional counterparts, while preferred prudent semantics shows a sort of singularity being comparable only with complete prudent and stable prudent semantics. The strong relation gives rise to a more complicated situation, where grounded and complete semantics are not equivalent any more while grounded prudent and complete prudent semantics are still the most skeptical in some sense (but are incomparable each other). At the “top level”, stable-related, preferred, and CF 2 semantics are not comparable and turn out to be less skeptical than any other semantics they are comparable with. Ideal and complete semantics play a sort of intermediate role between grounded and other less skeptical semantics, while preferred prudent semantics is still somehow isolated. While all the above remarks are interesting in some respect, one may be led to conclude that it is probably the case that the strong relation is actually too demanding (as also observed when applying these criteria to a different kind of analysis in [13]), while the weak relation is more reasonable and gives rise, as a consequence, to a more useful picture. Different pictures would be obtained considering alternative notions of skepticism, which is a topic for future work. It has however been proved in [14] that comparing the intersection of all extensions gives rise to the same partial order as SW . Finally, it has to be remarked that skepticism can be regarded as an attitude rather than an evaluation criterion for semantics: a more (or less) skeptical semantics is not preferable per se. In fact, characterizing the appropriate level of skepticism with respect to the requirements of a specific reasoning context is an interesting open problem: an example of this kind of investigation, concerning epistemic vs. practical reasoning, is given in [15].

12

P. Baroni, M. Giacomin

Acknowledgments. We thank the referees for their helpful comments.

References 1. Vreeswijk, G.A.W.: Interpolation of benchmark problems in defeasible reasoning. In: Proc. of the 2nd World Conf. on the Fundamentals of Artificial Intelligence (WOCFAI 95), Paris, France (1995) 453–468 2. Prakken, H.: Intuitions and the modelling of defeasible reasoning: some case studies. In: Proc. of the 9th Int. Workshop on Non-Monotonic Reasoning (NMR 2002), Toulouse, France (2002) 91–102 3. Caminada, M., Amgoud, L.: An axiomatic account of formal argumentation. In: Proc. of the Twentieth National Conf. on Artificial Intelligence (AAAI-05), Menlo Park, CA, AAAI Press (2005) 608–613 4. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n-person games. Artificial Intelligence 77(2) (1995) 321–357 5. Baroni, P., Giacomin, M.: Evaluation and comparison criteria for extension-based argumentation semantics. In: Proc. of the 1st Int. Conf. on Computational Models of Arguments (COMMA 2006), Liverpool, UK, IOS Press (2006) 157–168 6. Baroni, P., Giacomin, M., Guida, G.: Towards a formalization of skepticism in extension-based argumentation semantics. In: Proc. of the 4th Workshop on Computational Models of Natural Argument (CMNA 2004), Valencia, E (2004) 47–52 7. Baroni, P., Giacomin, M.: Solving semantic problems with odd-length cycles in argumentation. In: Proc. of the 7th European Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2003), Aalborg, Denmark, LNAI 2711, Springer-Verlag (2003) 440–451 8. Baroni, P., Giacomin, M., Guida, G.: SCC-recursiveness: a general schema for argumentation semantics. Artificial Intelligence 168(1-2) (2005) 165–210 9. Caminada, M.: Semi-stable semantics. In: Proc. of the 1st Int. Conf. on Computational Models of Arguments (COMMA 2006), Liverpool, UK, IOS Press (2006) 121–130 10. Dung, P.M., Mancarella, P., Toni, F.: A dialectic procedure for sceptical, assumption-based argumentation. In: Proc. of the 1st Int. Conf. on Computational Models of Arguments (COMMA 2006), Liverpool, UK, IOS Press (2006) 145–156 11. Coste-Marquis, S., Devred, C., Marquis, P.: Prudent semantics for argumentation frameworks. In: Proc. of the 17th IEEE Int. Conf. on Tools with Artificial Intelligence (ICTAI 2005), Hong Kong, China, IEEE Computer Society (2005) 568–572 12. Coste-Marquis, S., Devred, C., Marquis, P.: S´emantiques prudentes pour les syst`emes d’argumentation. In: Proc. of the 15th Congres AFRIF-AFIA Reconnaissance des Formes et Intelligence Artificielle (RFIA 2006), Tours, F (2006) 13. Baroni, P., Giacomin, M.: Evaluating argumentation semantics with respect to skepticism adequacy. In: Proc. of the 8th European Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2005), Barcelona, E (2005) 329–340 14. Baroni, P., Giacomin, M.: On principle-based evaluation, comparison, and design of extension-based argumentation semantics. Tech. Rep., Univ. of Brescia, I (2007) 15. Prakken, H.: Combining sceptical epistemic reasoning with credulous practical reasoning. In: Proc. of the 1st Int. Conf. on Computational Models of Arguments (COMMA 2006), Liverpool, UK, IOS Press (2006) 311–322