A Note on Rich Incomplete Argumentation Frameworks
aa r X i v : . [ c s . A I] N ov A Note on Rich Incomplete ArgumentationFrameworks
Jean-Guy MaillyLIPADE, Universit´e de Paris, [email protected]
Abstract
Recently, qualitative uncertainty in abstract argumentation has re-ceived much attention. The first works on this topic introduced uncer-tainty about the presence of attacks, then about the presence of argu-ments, and finally combined both kinds of uncertainty. This results in theIncomplete Argumentation Framework (IAFs). But another kind of un-certainty was introduced in the context of Control Argumentation Frame-works (CAFs): it consists in a conflict relation with uncertain orientation, i.e. we are sure that there is an attack between two arguments, but theactual direction of the attack is unknown. Here, we formally define RichIAFs, that combine the three different kinds of uncertainty that were pre-viously introduced in IAFs and CAFs. We show that this new model,although strictly more expressive than IAFs, does not suffer from a blowup of computational complexity. Also, the existing computational ap-proach based on SAT can be easily adapted to the new framework.
Abstract argumentation [16] is an important topic in the Knowledge Repre-sentation and Reasoning community. Intuitively, an abstract argumentationframework (AF) is a directed graph where nodes are arguments and edges arerelations (usually attacks) between these arguments. The outcome of such anAF is an evaluation of the arguments’ acceptance (through extensions [16, 3],labellings [7] or rankings [1]). In such an AF, the assumption of complete in-formation is made: an argument that appears in the graph is sure to actuallyexist, and similarly, an edge (or the absence of an edge) in the graph means thatthe attack between arguments certainly exists (or certainly does not).The question of how to incorporate uncertainty in AFs has then arisen. Twokinds of approaches have been proposed. If a quantitative evaluation of the un-certainty is available, it seems natural to use it in the definition of reasoningmechanisms. This corresponds ( e.g. ) to Probabilistic Argumentation Frame-works [20]. But such a quantitative information about uncertainty may not beavailable. The other approach is then the Incomplete Argumentation Frame-works (IAFs) [9, 5, 4], where the uncertainty is only qualitative. In an IAF,some arguments are identified as uncertain, i.e. there is a doubt whether theargument actually appears in the framework. Similarly, attacks may be un-certain. However, another form of uncertainty in AFs has been defined in the1iterature. Control Argumentation Frameworks [13] integrate uncertainty andargumentation dynamics [15] in a single framework. Besides the two aforemen-tioned forms of uncertainty, a third one has been proposed: a symmetric conflictrelation is defined, such that there is an uncertainty about the actual directionof the attack: either it appears in one direction, or in the other one, or in bothdirections at the same time. We investigate how this third kind of uncertaintycan be added to IAFs.The report is organized as follows. Section 2 describes the background no-tions on abstract argumentation and Incomplete Argumentation Frameworks(IAFs). In Section 3, we introduce Rich Incomplete Argumentation Frame-works (RIAFs), that generalize IAFs by adding a new kind of uncertainty overthe attacks. Section 4 concludes the report by mentioning several interestingresearch tracks about (R)IAFs.
Abstract argumentation was introduced in [16], where arguments are abstractentities whose origin or internal structure are ignored. The acceptance of argu-ments is purely defined from the relations between them.
Definition 1 (Abstract AF) . An abstract argumentation framework (AF) is adirected graph F = h A, R i , where A is a set of arguments , and R ⊆ A × A isan attack relation . We say that a attacks b when ( a, b ) ∈ R . If ( b, c ) ∈ R also holds, then a defends c against b . Attack and defense can be adapted to sets of arguments: S ⊆ A attacks (respectively defends) an argument b ∈ A if ∃ a ∈ S that attacks(respectively defends) b . Example 1.
Let F = h A, R i be the AF depicted at Figure 1, with A = { a, b, c, d, e } and R = { ( b, a ) , ( c, a ) , ( c, d ) , ( d, b ) , ( d, c ) , ( e, a ) } . Each arrow repre- a bc de Figure 1: The AF F sents an attack. d defends a against both b and c , since these are attackers of a that are, in turn, both attacked by d . In [16], Dung introduces different semantics to evaluate the acceptabilityof arguments. They are based on two basic concepts: conflict-freeness and admissibility . Definition 2 (Conflict-freeness and Admissibility) . Given F = h A, R i , a set ofarguments S ⊆ A is: conflict-free iff ∀ a, b ∈ S , ( a, b ) R ; • admissible iff it is conflict-free, and defends each a ∈ S against each of itsattackers. We use cf( F ) and ad( F ) for denoting the sets of conflit-free and admissiblesets of an argumentation framework F .The intuition behind these principles is that a set of arguments may beaccepted only if it is internally consistent (conflict-freeness) and able to defenditself against potential threats (admissibility). The semantics proposed by Dungare then defined as follows [16]. Definition 3 (Extension Semantics) . Given F = h A, R i , an admissible set S ⊆ A is: • a complete extension iff it contains every argument that it defends; • a preferred extension iff it is a ⊆ -maximal complete extension; • the unique grounded extension iff it is the ⊆ -minimal complete extension; • a stable extension iff it attacks every argument in A \ S . The sets of extensions of an AF F , for these four semantics, are denoted(respectively) co( F ), pr( F ), gr( F ) and st( F ).Based on these semantics, we can define the status of any (set of) argu-ment(s), namely skeptically accepted (belonging to each σ -extension), credu-lously accepted (belonging to some σ -extension) and rejected (belonging to no σ -extension). Given an AF F and a semantics σ , we use (respectively) sk σ ( F ),cr σ ( F ) and rej σ ( F ) to denote these sets of arguments. Example 2.
We consider again F given at Figure 1. Its extensions for the dif-ferent semantics, as well as the sets of accepted arguments, are given at Table 1. σ σ ( F ) cr( F ) sk( F )co { e } , { d, e } , { b, c, e } { b, c, d, e } { e } pr { d, e } , { b, c, e } { b, c, d, e } { e } gr { e } { e } { e } st { d, e } , { b, c, e } { b, c, d, e } { e } Table 1: Extensions and Accepted Arguments of F for σ ∈ { co , pr , gr , st } For more details about argumentation semantics, we refer the interestedreader to [16, 3].Now, we introduce Incomplete Argumentation Frameworks [9, 5, 4], i.e.
AFswith qualitative uncertainty about the presence of some arguments or attacks.
Definition 4 (Incomplete AF) . An Incomplete Argumentation Framework (IAF)is a tuple I = h A, A ? , R, R ? i , where A and A ? are disjoint sets of arguments,and R, R ? ⊆ ( A ∪ A ? ) × ( A ∪ A ? ) are disjoint sets of attacks. A and R are certain arguments and attacks, i.e. the agent issure that they appear in the framework. On the opposite, A ? and R ? representuncertain arguments and attacks. For each of them, there is a doubt about theiractual existence. Example 3.
Let us consider I = h A, A ? , R, R ? i given at Figure 2. We useplain nodes and arrows to represent certain arguments and attacks, i.e. A = { a, b, c, d, e } and R = { ( b, a ) , ( c, a ) , ( d, b ) , ( d, c ) } . Uncertain arguments are rep-resented as dashed square nodes ( i.e. A ? = { f } ) and uncertain attacks arerepresented as dotted arrows ( i.e. R ? = { ( e, a ) , ( f, d ) } ). a bc de f Figure 2: The IAF I The notion of completion in abstract argumentation was first defined in [9]for Partial AFs ( i.e.
IAFs with A ? = ∅ ), and then adapted to IAFs. Intuitively,a completion is a classical AF which describes a situation of the world coherentwith the uncertain information encoded in the IAF. Definition 5 (Completion of an IAF) . Given I = h A, A ? , R, R ? i , a completionof I is F = h A ′ , R ′ i , such that • A ⊆ A ′ ⊆ A ∪ A ? ; • R | A ′ ⊆ R ′ ⊆ R | A ′ ∪ R ? | A ′ ;where R | A ′ = R ∩ ( A ′ × A ′ ) (and similarly for R ? | A ′ ). The set of completions of an IAF I is denoted comp( I ). Example 4.
We consider again the IAF from Figure 2. Its set of completionsis described at Figure 3. a bc de a bc de f a bc de fa bc de a bc de f a bc de f
Figure 3: The Completions of I The number of completions of an IAF I = h A, A ? , R, R ? i is bounded by 2 n ,with n = | A ? | + | R ? | . However, this upper bound may not be reached, as itis the case in the previous example. Indeed, the uncertain attack ( f, d ) cannotappear in completions where the uncertain argument f does not appear.4 .2 Reasoning with Incomplete AFs To conclude this section, let us introduce the different reasoning problems forIAFs that have been studied in the literature, as well as their complexity. They are the adaptation to IAFs of three classical reasoning problems forAFs: • Verification: given an AF, a set of arguments, and a semantics, is the setan extension of the AF under the chosen semantics? • Credulous acceptance: given an AF, an argument, and a semantics, is theargument a member of some extension under the chosen semantics? • Skeptical acceptance: given an AF, an argument, and a semantics, is theargument a member of each extension under the chosen semantics?Adapting these problems to IAFs requires to take into account the set ofcompletions. Indeed, an argument being accepted in one completion is muchless demanding than being accepted in all the completions. This is why thereare two variants of these problems for IAFs: the possible and the necessaryvariant. The definition of the possible variant quantifies existentially over theset of completions, while the necessary variant quantifies universally.Verification for IAFs was first studied in [6]: σ - IncPV
Given I = h A, A ? , R, R ? i an IAF and S ⊆ A ∪ A ? , is there a completion F = h A ′ , R ′ i such that S ∩ A ′ is a σ -extension of F ? σ - IncNV
Given I = h A, A ? , R, R ? i an IAF and S ⊆ A ∪ A ? , for each completion F = h A ′ , R ′ i , is S ∩ A ′ a σ -extension of F ?In [19], a set of arguments for which the answer to σ - IncPV (respectively σ - IncNV ) is called a possible (respectively necessary) i -extension. The authorsidentify some issues with this definition (for instance, a set of arguments could beidentified as an i -extension even if it is not conflict-free). To remedy this issue,they define so-called i ∗ -extensions, and the corresponding verification problems: σ - IncPV ∗ Given I = h A, A ? , R, R ? i an IAF and S ⊆ A ∪ A ? , is there a comple-tion F = h A ′ , R ′ i such that S is a σ -extension of F ? σ - IncNV ∗ Given I = h A, A ? , R, R ? i an IAF and S ⊆ A ∪ A ? , for each completion F = h A ′ , R ′ i , is S a σ -extension of F ?We refer the interested reader to [19] for a detailled discussion of the differencebetween i -extensions and i ∗ -extensions.Finally, the (possible and necessary) variants of credulous and skeptical ac-ceptance are studied in [4]: σ - PCA
Given I = h A, A ? , R, R ? i an IAF and a ∈ A , is there a completion F = h A ′ , R ′ i such that a is credulously accepted in F under σ ? σ - NCA
Given I = h A, A ? , R, R ? i an IAF and a ∈ A , for each completion F = h A ′ , R ′ i , is a a credulously accepted in F under σ ? We suppose that the reader is familiar with basic concepts of computational complexity,like (non-)deterministic polynomial algorithms, and the classes of the polynomial hierarchy:P, NP , coNP , Σ Pk , Π Pk , where k ∈ N . Otherwise, we refer the interested reader to, e.g. , [2]. - PSA
Given I = h A, A ? , R, R ? i an IAF and a ∈ A , is there a completion F = h A ′ , R ′ i such that a is skeptically accepted in F under σ ? σ - NSA
Given I = h A, A ? , R, R ? i an IAF and a ∈ A , for each completion F = h A ′ , R ′ i , is a a skeptically accepted in F under σ ?Now, let us give the complexity of these problems under several classicalsemantics. C -c means that the problem is complete for the complexity class C ,under the given semantics. σ IncPV IncNV IncPV ∗ IncNV ∗ PCA NCA PSA NSA ad NP -c P P P NP -c Π P -c trivial trivialst NP -c P P P NP -c Π P -c Σ P -c coNP -cco NP -c P P P NP -c Π P -c NP -c coNP -cgr NP -c P P P NP -c coNP -c NP -c coNP -cpr Σ P -c coNP -c Σ P -c coNP -c NP -c Π P -c Σ P -c Π P -cTable 2: Complexity of IAFs for Various Problems under σ ∈ { ad , st , co , gr , pr } Now, we enrich the definition of IAFs.
Definition 6 (Rich IAF) . A Rich Incomplete Argumentation Framework (RIAF) is a tuple R = h A, A ? , R, R ? , ↔ ? i , where A and A ? are disjoint setsof arguments, and R, R ? , ↔ ? ⊆ ( A ∪ A ? ) × ( A ∪ A ? ) are disjoint sets of attacks,such that ↔ ? is symmetric. The new relation ↔ ? is borrowed from Control Argumentation Frameworks[13]. It is a symmetric (uncertain) conflict relation: if ( a, b ) ∈↔ ? , then we aresure that there is a conflict between a and b , but not of the direction of theattack. This new relation impacts the definition of completions. Definition 7 (Completion of a RIAF) . Given R = h A, A ? , R, R ? , ↔ ? i , a com-pletion of R is F = h A ′ , R ′ i , such that • A ⊆ A ′ ⊆ A ∪ A ? ; • R | A ′ ⊆ R ′ ⊆ R | A ′ ∪ R ? | A ′ ∪ ↔ ? | A ′ ; • if ( a, b ) ∈↔ ? | A ′ , then ( a, b ) ∈ R ′ or ( b, a ) ∈ R ′ (or both);where R | A ′ = R ∩ ( A ′ × A ′ ) (and similarly for R ? | A ′ and ↔ ? | A ′ ). Again, we use comp( R ) to denote the set of completions of a RIAF R . Example 5.
We present a slight modification of the IAF from Example 3, wherethe (certain) attack ( b, a ) is replaced by a symmetric uncertain conflict between a and b . The resulting RIAF is given at Figure 4. bc de f Figure 4: The RIAF R a bc de f a bc de f a bc de f Figure 5: Three Completions of R We do not give here the full set of completions of R . Let us focus on oneoption for each of ( e, a ) , ( f, d ) and f , and we only illustrate the three optionsfor ( a, b ) . These three completions are given at Figure 5.Similarly, for each other configuration of ( e, a ) , ( f, d ) and f ( i.e. each com-pletion at Figure 3), there are three options for the conflict between a and b ,leading to three different completions. Observation 1.
Let us notice that ↔ ? is not defined as a symmetric relationin [13]. However, its meaning imposes the relation to be symmetric. Indeed, ( a, b ) ∈↔ ? means that there is a conflict between a and b whose direction isuncertain. Formally, it means that ( ceteris paribus ) there are three completionswith (respectively) ( a, b ) ∈ R ′ , or ( b, a ) ∈ R ′ , or both. This is obviously equiv-alent to “there is a conflict between b and a whose direction is uncertain”, i.e.( b, a ) ∈↔ ? . A non-symmetric relation can be used as a more compact represen-tation of its symmetric counterpart. Now, we prove that RIAFs are strictly more expressive than IAFs. Saidotherwise, it means that the new relation ↔ ? cannot be equivalently representedwith a combination of fixed and uncertain attacks. Proposition 2 (Relative Expressivity of IAFs and RIAFs) . • For any IAF I , there exists a RIAF R such that comp( I ) = comp( R ) . • There exists a RIAF R such that there is no IAF I with comp( I ) =comp( R ) .Proof. The first item is straightforward: any IAF is a RIAF with ↔ ? = ∅ . Forthe second item, consider R = h{ a, b } , ∅ , ∅ , ∅ , { ( a, b ) , ( b, a ) }i . This RIAF and itsthree completions are given at Figure 6. a b a ba b a b Figure 6: A RIAF and its Completions7ow, let us prove that there is no IAF with the same set of completions.Reasoning towards a contradiction, suppose that such a IAF I = h A, A ? , R, R ? i exists. Since all the completions have the same set of arguments { a, b } , therecannot be uncertain argument, i.e. A ? = ∅ .Let us now consider the different options for R and R ? . If ( a, b ) ∈ R (re-spectively ( b, a ) ∈ R ), then there is an attack from a to b (respectively from b to a ) in every completion. This is not the case. Similarly, there cannot be anyself attack in R (since there is no such attack in any completion). Thus R = ∅ .In the case where only ( a, b ) (respectively ( b, a )) belongs to R ? , then thecompletions with ( b, a ) (respectively ( a, b )) do not belong to comp( I ). On thecontrary, if both ( a, b ) and ( b, a ) belong to R ? , then a fourth completion wherethere is no attack between a and b belongs to comp( I ). Of course, self-attacksin R ? are not possible, since they would yield addition completions (with thesame self-attack appearing in them).So we can conclude that I does not exist. While we have shown that RIAFs are strictly more expressive than IAFs, nowwe prove that this expressivity is not at the price of a complexity blow up. Let usrecall that the complexity results for IAFs [6, 4, 19] are summarized at Table 2.The fact that any IAF is a RIAF with ↔ ? = ∅ is enough to prove that reasoningwith RIAFs is at least as hard as reasoning with IAFs. But also, we notice thatthe upper bounds of the complexity coincides with the upper bound for IAFs.Roughly speaking, this can be explained by the fact that non-deterministicallyguessing a completion is not different for RIAFs than for IAFs. Possible Verification
Let us start with
IncPV and
IncPV ∗ . The problemfor RIAFs can be solved by the non-deterministic guess of a completion, andthen checking whether the queried set of arguments S is a σ -extension of thegiven completion (for IncPV ∗ ), or S ∩ A ′ where A ′ is the set of arguments thatappear in the completion (for IncPV ). Since verification of an extension in AFs ispolynomial for σ ∈ { ad , st , co , gr } and coNP -complete for σ = pr, the results for IncPV from Table 2 are valid for RIAFs, as well as the result for
IncPV ∗ underthe preferred semantics. For IncPV ∗ under the other semantics, the reasoningfrom [19] applies. First, if there are ( a, b ) , ( b, a ) ∈↔ ? such that a, b ∈ S , then S is not conflict-free in R , thus it is not an extension (for σ = ad , st , gr , co).Otherwise, for ( a, b ) , ( b, a ) ∈↔ ? with a ∈ S and b S , include only ( a, b ) in thecompletion built by the algorithm. Necessary Verification
Then, let us focus on
IncNV ∗ , and σ = pr. A nega-tive instance for this problem can be identified by non-deterministically guessinga completion and a superset S ′ of the queried set of arguments S . Then, poly-nomially checking whether S ′ is admissible allows to conclude that S is notnecessary a preferred i ∗ -extension. Thus IncNV ∗ ∈ coNP . For IncNV , the rea-soning is the same, except that S ′ must be a superset of S ∩ A ′ instead of asuperset of S . 8or the other semantics, the reasoning from [6, 19] holds for RIAFs: IncNV and
IncNV ∗ can be solved polynomially by reducing the problem to reasoningwith Argument-Incomplete AFs, or by constructing the adequate completion.If there are ( a, b ) , ( b, a ) ∈↔ ? such that a, b ∈ S , then S is not conflict-free in R ,thus it is not an extension (for σ = ad , st , gr , co). Otherwise, for ( a, b ) , ( b, a ) ∈↔ ? with a ∈ S and b S , include only ( a, b ) in the Argument-Incomplete AF orcompletion built in the proof. Credulous Acceptance
Now, we look at acceptance problems. For
PCA ,the problem is solved by non-deterministically guessing a completion C and a setof arguments S that contains the queried argument a . Then, it can be checkedpolynomially whether S is a σ -extension, for σ ∈ { ad , st , co , gr } . Moreover, if S is an admissible extension, then a also belongs to some preferred extension(since each admissible set is included in some preferred extension).For NCA , let us non-deterministically guess a completion and check whetherthe queried argument is credulously accepted in it. This check is doable inpolynomial time for σ = gr, and with a NP oracle for the other semantics underconsideration, hence the result. Skeptical Acceptance
For
PSA and
NSA , the admissible semantics remainsa trivial case: since the empty set is admissible for any AF, there is no skepti-cally accepted argument under ad. For
PSA , we can non-deterministically guessa completion, and check whether a is skeptically accepted in it. This check ispolynomial for σ ∈ { gr , co } , in coNP for σ = st, and in Π P for σ = pr, so we ob-tain the result. For NSA , the reasoning described in [4] still holds: checking thenecessary skeptical acceptance of an argument can be represented by a universalquantification over the set of completions and the set of sets of arguments ( S )that do not contain the queried argument a . The universal quantifiers are fol-lowed by a (deterministic) polynomial check for σ ∈ { st , co , gr } . For σ = pr, theuniversal quantifiers are followed by an existential quantifier over the supersetsof S , and finally a (deterministic) polynomial check. Hence the results.So we can conclude that the complexity of reasoning with RIAFs, for thevarious problems introduced in Section 2.2 and for σ ∈ { ad , st , co , gr , pr } , is thesame as in the case of IAFs. Proposition 3.
The complexity results for IAFs given at Table 2 also hold forRIAFs.
Now we show how to adapt the SAT-based algorithms for reasoning with IAFs[21] to RIAFs.Roughly speaking, the encoding is made of one part that represents thestructure of the IAF, i.e. the existence of (uncertain or not) arguments andattacks; and one part that maps this structure with the arguments acceptanceevaluation (with respect to a chosen semantics). A slight modification to takeinto account ↔ ? is enough to reason with RIAFs instead of IAFs. Following thedefinition of encodings in [21], we define, for a RIAF R = h A, A ? , R, R ? , ↔ ? i the Boolean variables: • y a is true if and only if a ∈ A ′ ; 9 r a,b is true if and only if ( a, b ) ∈ R ′ ; • x a is true if and only if a ∈ S , for some S ∈ σ ( F );where F = h A ′ , R ′ i is a completion of R corresponding to the y a and r a,b variables. While y a and r a,b are necessarily true for a ∈ A and ( a, b ) ∈ R , theymay be true or false for a ∈ A ? and ( a, b ) ∈ R ? . We also need to specify thateither r a,b or r b,a is true if ( a, b ) , ( b, a ) ∈↔ ? . Finally, an argument a must be inthe completion ( i.e. y a is true) in order to appear in an extension ( i.e. x a istrue) or in some attacks ( i.e. r a,b or r b,a is true for some b ). This correspondsto the following formula, for R = h A, A ? , R, R ? , ↔ ? i : φ ? ( R ) = ( V a ∈ A y a ) ∧ ( V ( a,b ) ∈ R r a,b ) ∧ ( V ( a,b ) ∈↔ ? r a,b ∨ r b,a ) ∧ ( V a ∈ A ? ( ¬ y a → ( ¬ x a ∧ V ( a,b ) ∈ R ? ¬ r a,b ∧ V ( b,a ) ∈ R ? ¬ r b,a )))Then, a formula encoding the semantics is given, for instance: φ cf ( R ) = ^ ( a,b ) ∈ R ∪ R ? ∪↔ ? ( y a ∧ y b ∧ r a,b ) → ( ¬ x a ∨ ¬ x b )encodes conflict-freeness: if both arguments appear in the completion, as wellas the attack between them, then they cannot be both accepted. Similarly, φ ad and φ st are provided in [21], we show their adaptation to RIAFs: • φ ad ( R ) = φ cf ( R ) ∧ V a ∈ A ∪ A ? V ( b,a ) ∈ R ∪ R ? ∪↔ ? (( x a ∧ y a ∧ y b ∧ r b,a ) → z b ), • φ st ( R ) = φ cf ( R ) ∧ V a ∈ A ∪ A ? ( ¬ x a ∧ y a → z a ),where z a is a newly introduced variable for each argument a ∈ A ∪ A ? , meaningthat a is defeated. This is formally encoded by z a → W ( b,a ) ∈ R ∪ R ? ∪↔ ? ( x b ∧ y b ∧ r b,a ). φ ad and φ st can be used directly for solving problems at the first level of thepolynomial hierarchy, or as NP abstraction in CounterExample Guided AbstractRefinement algorithms. We refer the interested reader to [21] for more details. This report introduces Rich Incomplete Argumentation Frameworks, that gener-alize IAFs by adding a new kind of uncertainty. We have shown that this modelis strictly more expressive than IAFs, but not at the price of an increase ofcomplexity. Moreover, a slight modification of existing logical encodings allowsto use the algorithms described in the literature.While complexity of reasoning with (R)IAFs has been well studied, thereare still many open questions regarding this formalism. For instance, as far aswe know, the only algorithms proposed (and implemented) for IAFs concernthe acceptance problems, as mentioned in the Section 3.2.2. Other problemsmentioned in Section 2.2 have not been tackled yet. Also, this study onlyconsiders the initial semantics defined by Dung, but other semantics have notreceived interest in the context of (R)IAFs, e.g. semi-stable [8], stage [22] orideal [17] semantics.Several works about the revision [10, 11], the update [14] or the merging [9, 12]10f AFs have faced the difficulty to represent the uncertainty of the result as asingle AF, and chose to return a set of AFs as output. Let us recall that PartialAFs (that are a special case of RIAFs with A ? = ∅ and ↔ ? ) were defined as apart of the process for computing the merging of AFs [9], but did not appear inthe result of the operation. RIAFs might provide an interesting solution in orderto have a more compact output for these operations. Related to these questions,the issue of realizability of extension sets [18] in the context of (R)IAFs is alsointeresting. References [1] Leila Amgoud and Jonathan Ben-Naim. Ranking-based semantics for ar-gumentation frameworks. In
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