Handling ignorance in argumentation: semantics of partial ...

Handling ignorance in argumentation: semantics of partial argumentation frameworks C. Cayrol1 , C. Devred23 , and M.C. Lagasquie-Schiex1 1

3

IRIT, Universit´e Paul Sabatier, Toulouse, ccayrol, lagasq @irit.fr 2 CRIL, Universit´e d’Artois, Lens, [email protected] LERIA, Universit´e d’Angers, [email protected]

Abstract. In this paper we propose semantics for acceptablity in partial argumentation frameworks (PAF). The PAF is an extension of Dung’s argumentation framework and has been introduced in [1] for merging argumentation frameworks. It consists in adding a new interaction between arguments representing the ignorance about the existence of an attack. The proposed semantics are built following Dung’s method, so that they generalize Dung’s semantics without increasing the temporal complexity.

1 Introduction Argumentation has become an influential approach to treat AI problems including defeasible reasoning and some forms of dialogue between agents (see e.g. [2–6]). Argumentation is basically concerned with the exchange of interacting arguments. Usually, the interaction takes the form of a conflict, called attack. For example, a logical argument can be a pair hset of assumptions, conclusioni, where the set of assumptions entails the conclusion according to some logical inference schema. Then a conflict occurs for instance if the conclusion of an argument contradicts an assumption of another argument. The main issue for any theory of argumentation is the selection of acceptable sets of arguments, based on the way arguments interact. Intuitively, an acceptable set of arguments must be in some sense coherent and strong enough (e.g. able to defend itself against all attacking arguments). It is convenient to explore the concept of acceptability through argumentation frameworks, and especially Dung’s framework ([7]), which abstracts from the nature of the arguments, and represents interaction under the form of a binary relation “attack” on a set of arguments. Such an argumentation framework fits well with situations where the knowledge about the interactions can be assumed to be complete. That is, for each pair (a, b) of considered arguments, it is possible to prove that there is an attack, or not. There are numerous proposals by numerous researchers for capturing forms of argumentation in logic. Many of these logic-based argumentation systems are promising for simulating argumentation forms which arise in the real-world. Nevertheless, it remains some cases in which it is difficult to formalize logically the structure of the arguments. For instance, professionals such as lawyers, journalists, politicians for instance use arguments for analyzing situations before presenting some information to a given audience. An argument in that case can be a piece of text or of a discourse, used for

convincing the audience. In other cases, arguments are used for analyzing situations before making some decisions. Such arguments can be just positions that can be advanced for or against options or other positions. More generally, argumentation can be based on any kind of information coming from heterogeneous sources, objective information such as measured or observed values for instance, or subjective information such as beliefs. So, arguments may have no explicit internal logical structure. Consider an agent for which different kinds of information are available. Some kinds of information enable the agent to build logical arguments (for instance a, b, c), while other kinds of information only permit to advance informal arguments (for instance d, e) for trying to defeat other arguments. So, the agent can be sure that a attacks b and a does not attack c, but the agent also wants to take into account an attack of e by d even if nothing can be said about an attack of d by e. Moreover, the generation of arguments often takes place in a resource-bounded environment, and the agent may not have enough time for computing all the possible attacks. So, there is a need for the representation of partial knowledge about the interactions between arguments. In Dung’s framework, there is no space for ignorance. If a given pair of arguments (a, b) does not appear in the graph of the attack relation, it means that certainly a does not attack b. Another interpretation could be given, according to which there is some uncertainty, and even ignorance about an attack of b by a. This interpretation enables to complete the knowledge later on. In this paper, we propose to investigate a new kind of abstract argumentation framework in order to handle ignorance in argumentation, and to distinguish between certainty of non-attack and ignorance of attack. This new framework, called Partial Argumentation Framework (or PAF), has recently been introduced in ([1, 8]) for the particular purpose of merging argumentation systems. However, no formal study of a partial argumentation framework has been made, especially concerning the key concept of acceptability. Such a study is the topic of this paper. The paper is organized as follows : Dung’s abstract framework is recalled in Section 2. Partial argumentation frameworks are defined in Section 3. Section 4 presents the links between classical frameworks and partial frameworks. It is shown that a classical framework can be given different interpretations in terms of a partial framework. On the other hand, it is shown that a partial framework represents a set of classical frameworks. Acceptability in a partial framework is studied in Section 5. The main properties of the proposed semantics are given in Section 6 (in particular some results about temporal complexity).

2 Argumentation frameworks (AF) In [7], Dung has proposed an abstract framework for argumentation in which he focuses only on the definition of the status of arguments. For that purpose, he assumes that a set of arguments is given, as well as the different conflicts among them. We briefly recall that abstract framework: Definition 1. An argumentation framework (AF) is a pair hA, Ri of a set A of arguments and a binary relation R on A called the attack relation. ai Raj means that ai attacks aj (or aj is attacked by ai ). An argumentation framework may be represented

by a directed graph, called the interaction graph, whose nodes are arguments and edges represent the attack relation. In Dung’s framework, the acceptability of an argument depends on its membership to some sets, called extensions. These extensions are characterised by particular properties. It is a collective acceptability. Let AF = hA, Ri be an argumentation framework, let S ⊆ A, the main characteristic properties are: Definition 2. S is conflict-free for AF iff there exist no ai , aj in S such that ai Raj . An argument a is acceptable w.r.t. S for AF iff ∀b ∈ A such that bRa, ∃c ∈ S such that cRb. S is acceptable for AF iff ∀a ∈ S, a is acceptable w.r.t. S for AF. Then several semantics for acceptability have been defined in [7]. For instance: Definition 3. S is an admissible set for AF iff S is conflict-free and acceptable for AF. S is a preferred extension of AF iff S is maximal for ⊆ among the admissible sets for AF.

3 Partial argumentation frameworks (PAF) In [1, 8], an extension of Dung’s argumentation framework has been proposed in order to take into account the possible ignorance about the attack between arguments. Definition 4. A partial4 argumentation framework (PAF) is a tuple hA, R, I, N i where A is a set of arguments and R, I and N are three binary relations on A making a partition of A×A. R represents the attack relation, N represents the non-attack relation and I represents the ignorance relation5 . Intuitively, (a, b) ∈ R means that “a certainly attacks b”, (a, b) ∈ N means that “a certainly does not attack b”, (a, b) ∈ I means that “the agent does not know the nature of the interaction between a and b”. More generally, (a, b) ∈ I may be interpreted as “the agent is not certain of the existence of an attack from a to b, but it is possible that this attacks exists”. Note that the set of arguments is not assumed finite in the above definition. In the following, we have chosen to represent a PAF graphically as in [1, 8]: an attack (resp. ignorance) from a to b is represented by a plain (resp. dotted) edge from a to b, and the non-attack relation is not explicitly represented6. Example 1. Consider PAF = h{a, b, c}, {(a, b), (b, c), (a, c)}, {(c, a)}, {(a, a), (b, b), (c, c), (b, a), (c, b)}i represented by: 4

5

6

a

b

c

A notion of “partiality” has already been introduced in [9], but with a very different viewpoint (this “partiality” refers to a notion of approximate arguments in order to solve computational issues), clearly not related to our partial argumentation framework. Though R, I and N make a partition of A × A, these 3 relations must appear in the tuple because, a priori, we do not know which relation can be deduced by the complementation of the two other ones. Such a representation is judicious in the case there are many non-attacks.

4 From AF to PAF, and vice-versa 4.1 Viewing an AF as a particular PAF In Dung’s framework, (a, b) 6∈ R is classically interpreted as “there is no attack from a to b”. This interpretation can be said “closed” because there is no doubt about the interaction between a and b. There is an analogy with the well-known Closed World Assumption. So, the closed interpretation considers that an AF is a PAF where I is the empty set, i.e. hA, R, ∅, A × A \ Ri. This is the more complete interpretation: for each pair (a, b) of arguments, either one knows that a attacks b, or one knows that a does not attack b. A more cautious interpretation, called “open”, considers that there is some ignorance about the interaction from a to b when (a, b) 6∈ R and a and b are different. However, intuitively it seems sound to remove the self-attacks7 . So, (a, a) 6∈ R will always be interpreted by (a, a) ∈ N . So, the open interpretation considers that an AF is a PAF where N is reduced to the set of pairs (a, a) which are not in R, i.e. hA, R, {(a, b)|a 6= b and (a, b) 6∈ R}, {(a, a)|(a, a) 6∈ R}i. So, a given AF can be interpreted by two PAFs according to the agent’s attitude: PAFc (closed attitude) or PAFo (open attitude). Example 2. a

Consider AF = h{a, b}, {(a, b)}i with its PAFc = h{a, b}, {(a, b)}, {}, {(a, a), (b, b), (b, a)}i and its PAFo = h{a, b}, {(a, b)}, {(b, a)}, {(a, a), (b, b)}i represented by:

b AF

a

b PAFo

a

b PAFc

More generally, many other PAFs can be obtained from a given AF, keeping the same set of arguments A, the same attack relation R, and taking I as a subset of {(a, b)|a 6= b and (a, b) 6∈ R}. 4.2 Viewing a PAF as a set of AFs Using the notion of completion, introduced in [1, 8], a PAF compactly represents a set of AFs. Definition 5. Let PAF = hA, R, I, N i. Let AF = hA, R′ i. AF is a completion of PAF if and only if R ⊆ R′ ⊆ R∪I. The set of all completions of PAF is denoted by C(PAF). Note that if I is the empty set, the PAF has only one completion. This is the case for instance when PAF is the closed PAF associated with AF = hA, Ri. AF is the unique completion of its closed PAF. 7

In order to avoid the derivability of the arguments which could be self-attacked: in Dung’s approach, whatever the semantics, an argument a which is derivable is such that (a, a) 6∈ R.

Example 3. Consider PAF = hA = {a, b, c}, R = {(a, b)}, I = {(a, c), (b, c)}, A × A \ (R ∪ I)i. PAF has the following completions: b

b

a

b

a c

b

a c

a c

c

5 Semantics for PAF As for the AFs, we are interested in a collective acceptability. So, we have to restate the notions of conflict-free set and acceptable set in the context of PAF. Let PAF = hA, R, I, N i. Let S ⊆ A. Definition 6. S is R-conflict-free for PAF iff there exist no ai , aj in S such that ai Raj . S is RI-conflict-free for PAF iff there exist no ai , aj in S such that ai Raj or ai Iaj . The R-conflict-free notion corresponds exactly to Dung’s conflict-free notion (see Property 3). The RI-conflict-free notion is a more cautious one. If S is RI-conflict-free, we are sure that for any pair (a, b) of arguments in S, a does not attack b. Definition 7. An argument a is R-acceptable w.r.t. S for PAF iff ∀b ∈ A such that bRa, ∃c ∈ S such that cRb. S is R-acceptable for PAF iff ∀a ∈ S, a is R-acceptable w.r.t. S for PAF. An argument a is RI-acceptable w.r.t. S for PAF iff ∀b ∈ A such that bRa or bIa, ∃c ∈ S such that cRb. S is RI-acceptable for PAF iff ∀a ∈ S, a is RI-acceptable w.r.t. S for PAF. The R-acceptability corresponds exactly to Dung’s acceptability (see Property 3). The RI-acceptability is a very cautious acceptability: in order to accept a, first we consider not only known attacks on a but also potential attacks on a, and secondly S must defend a only with certainly known attacks. Following Dung’s method, several semantics for admissibility can be defined. In particular such semantics can be obtained by the combination of the above defined notions of conflict-free and acceptability. However, among the four possible combinations, two of them are equivalent because of the following property: Property 1. If S is R-conflict-free and RI-acceptable for PAF then S is RI-conflictfree for PAF. Proof. Consider (a, b) ∈ S × S such that aIb. S is RI-acceptable so ∃c ∈ S such that cRa which is contradictory with S R-conflict-free. So S is also RI-conflict-free.

Definition 8. S is an admissible set for PAF iff S is R-conflict-free and R-acceptable for PAF. S is a R-admissible set for PAF iff S is RI-conflict-free and R-acceptable for PAF. S is a RI-admissible set for PAF iff S is RI-conflict-free and RI-acceptable for PAF.

Note that the first admissibility corresponds exactly to Dung’s admissibility (see Property 3). And the following interesting property holds: Property 2. S is RI-admissible for PAF ⇒ S is R-admissible for PAF ⇒ S is admissible for PAF. The proof is obvious. Due to Definitions 6, 7, 8. Then using the maximality for ⊆, three kinds of preferred extensions can be defined. Definition 9. S is a preferred extension of PAF iff S is maximal for ⊆ among the admissible sets for PAF. S is a R-preferred extension of PAF iff S is maximal for ⊆ among the R-admissible sets for PAF. S is a RI-preferred extension of PAF iff S is maximal for ⊆ among the RI-admissible sets for PAF. Example 4. Consider the PAF represented by the following figure: {a, c} is R-conflict-free but it is not RI-conflict-free. c is R-acceptable w.r.t. {a} but it is not RI-acceptable b w.r.t. {d}. d {e, d, a, c} is a preferred extension but it is not a Ra preferred extension or a RI-preferred extension (because it is not RI-conflict-free). c e {e, a} is a R-preferred extension but it is not a RIpreferred extension. {e} is the RI-preferred extension.

6 Properties The first properties are about the inclusion links between the different types of extensions. Then, some properties about complexity of the derivability process are given. 6.1 Inclusion links The main property explicits the links between the less cautious semantics for a given PAF = hA, R, I, N i and Dung’s semantics for the associated AF = hA, Ri (same set of arguments and same attack relation). It also gives the particular links between the semantics for PAF and Dung’s semantics for the associated AF when PAF is the closed PAF of AF: Property 3. 1. 2. 3. 4.

The conflict-free sets for AF are exactly the R-conflict-free sets for PAF; Let S ⊆ A, a is acceptable w.r.t S for AF iff a is R-acceptable w.r.t. S for PAF; The admissible sets for AF are exactly the admissible sets for PAF; The preferred extensions of AF are exactly the preferred extensions of PAF.

In the particular case where I = ∅ (for instance, if PAF is the closed PAF of AF):

1. The conflict-free sets for AF are exactly the RI-conflict-free sets for PAF; 2. Let S ⊆ A, a is acceptable w.r.t S for AF iff a is RI-acceptable w.r.t. S for PAF; 3. The admissible sets for AF are exactly the R-admissible sets for PAF and the RIadmissible sets for PAF ; 4. The preferred extensions of AF are exactly the R-preferred extensions of PAF and the RI-preferred extensions of PAF. The proof is obvious due to Definitions 6, 7, 8 and 9 and to the fact that PAF and AF use the same set of arguments and the same attack relation. And for the particular case, the proof also uses the fact that I = ∅ when PAF is the closed PAF of AF. The following properties show the inclusion links between semantics for PAF. Property 4. 1. The set of all admissible (resp. R-admissible, RI-admissible) sets for PAF is a complete partially ordered set in (2A , ⊆). 2. For every admissible (resp. R-admissible, RI-admissible) set S for PAF, there exists at least one preferred extension (resp. R-preferred extension, RI-preferred extension) E of PAF s.t. S ⊆ E. 3. There always exists at least one preferred extension (resp. R-preferred extension, RI-preferred extension) of PAF. Proof. The case concerning the admissible sets and the preferred extensions is obtained directly of [7] and Property 3. The case concerning the R-admissible sets and the R-preferred extensions is given by (the proof is the same type for the RI-admissible sets and the RI-preferred extensions): 1. For proving that a set S is a complete partially ordered set (cpo), we must to prove that (i) S has a least element, and that (ii) each directed subset of S has a least upper bound in S. For the point (i), it is easy to see that the set of the R-admissible sets has a least element w.r.t. ⊆ since ∅ is always R-admissible for PAF. For the point (ii), let X = {S1 , . . . , Sn } be a directed set of R-admissible sets for PAF. Hence, every Sj is R-admissible and for all Sj and Sk , there W exists a R-admissible S set S such that S of X exists, then j ⊆ S and Sk ⊆ S. Clearly, if the upper bound X W S W W Sj Sj ≤ X (because the order considered is ⊆). Hence, j Sj = X and X exists, if j Sj is R-admissible. S Proof that j Sj is R-admissible. Every Sj is R-admissible, hence if a ∈ Sj , a is RS acceptable w.r.t. SS j and then a is R-acceptable w.r.t. j Sj . Now, assume that j Sj is not RI-conflict-free. Then, there exists a ∈ Si and b ∈ Sj such that aRb or aIb. In other words, Si ∪ Sj is not RI-conflict-free. But, as X is a directed set, there exists S ⊇ Si ∪ Sj such that S is SR-admissible, hence RI-conflict-free. This contradicts the fact that aRb or aIb. Hence, j Sj is RI-conflict-free. W 2. Direct of the previous point (take X the set of all R-admissible sets containing S; X is a R-preferred subset containing S). 3. Obvious because ∅ is a R-admissible set for PAF.

The following property shows the cautiousness of the RI-preferred semantics w.r.t. the R-preferred semantics and the cautiousness of the R-preferred semantics w.r.t. the preferred semantics in a PAF.

Property 5. Let E be a RI-preferred extension (resp. R-preferred extension) of PAF. There always exists at least one R-preferred extension (resp. preferred extension) E ′ of PAF such that E ⊆ E ′ . Proof. Obvious because RI-admissibility for PAF implies R-admissibility for PAF which implies admissibility for PAF and, following Property 4, a RI-preferred extension of PAF is included in at least one R-preferred extension of PAF which is included in at least one preferred extension of PAF.

A consequence of the previous property (and of Property 3) is the fact that every Rpreferred extension (resp. RI-preferred extension) of PAF = hA, R, I, N i is included in at least one preferred extension of AF = hA, Ri. So, our semantics is more constrained than Dung’s semantics. The following property shows that an argument which is certainly unattacked is always in an extension of the PAF: Property 6. Let a ∈ A such that ∄b ∈ A and (bIa or bRa). 1. a belongs to each preferred extension of PAF; 2. a belongs to at least one R-preferred extension (but not always to each R-preferred extension) of PAF; 3. a belongs to each RI-preferred extension of PAF. Proof. The first point is obvious because a is non-attacked so we can use Dung’s results. The second point is given by: {a} is RI-conflict-free and R-acceptable. So {a} is R-admissible and is included in at least one R-preferred extension. c has two R-preferred extensions {a} and {b} b However, the PAF a and a does not belong to each one. For the last point, let E be a RI-preferred extension. If a 6∈ E, consider E ∪ {a}, which strictly contains E. E ∪ {a} is RI-conflict-free and also RI-acceptable since no argument in A attacks or ignores a. That contradicts the fact that E is a ⊆-maximal RI-admissible set. The next properties give some information about handling the cycles of attack or ignorance8 in a PAF. Property 7. Let a ∈ A such that aIa. a belongs neither to a R-preferred extension nor to a RI-preferred extension of PAF (but a may belong to a preferred extension of PAF). Let a ∈ A such that aRa. There is no preferred extension (resp. R-preferred extension, RI-preferred extension) of PAF which contains a. The proof is obvious due to the fact that {a} is not RI-conflict-free when aIa, and is not R-conflict-free when aRa. 8

A cycle {a0 , . . . , an−1 } of attack or ignorance is a set of n arguments defined by: ∀i = 0 . . . n − 1, ai Ra(i+1)modulo n or ai Ia(i+1)modulo n , and ∄T ⊆ {a0 , . . . , an−1 } such that T is a cycle; so our cycle is always elementary (it does not contain 2 edges with the same initial extremity or the same ending extremity). n is the length of the cycle.

Property 8. Let {a0 , . . . , an−1 } be an odd-length cycle of attack or ignorance which is certainly unattacked9. There is no RI-admissible set for PAF which can contain an element of the cycle. Proof. The proof is realized by reductio ad absurdum. Assume that ∃E RI-admissible set and ∃ai on the cycle such that ai ∈ E. If ai ∈ E then ai is RI-acceptable w.r.t. E. However, ai belongs to the cycle, so ai is certainly not unattacked and there must have in E arguments for defending it. Consider a new numbering of the arguments of the cycle defined by: a′0 is a(i+1)modulo n , . . . , a′n−1 is ai . With this numbering, one can identified the set of the arguments of the cycle (they must be on the cycle due to the assumption that the cycle is certainly unattacked) which can be mandatory for making ai RI-acceptable w.r.t. E: {a′n−3 , . . . , a′2 , a′0 }. So, ai and a(i+1)modulo n must belong to E if ai is RI-acceptable w.r.t. E. This is contradictory with the fact that E is RI-conflict-free, so the initial assumption is false and ai 6∈ E. This property does not hold in the case of even cycles. Thus, our approach departs from [10] who consider that odd-length and even-length cycles in an argumentation framework should be considered in the same way. We think that our approach is more cautious. Note that even if there is no cycle in a PAF, one can have several R-preferred extensions of this PAF (which is not possible with Dung’s approach). For instance, consider the c , with the two R-preferred extensions {a} and {b}. b PAF a The last properties of this section give some links between a PAF and its completions or between an AF and its closed or open PAFs. Property 9. Let S ⊆ A. S is RI-conflict-free for PAF iff S is conflict-free for any completion of PAF. Proof. ⇒ Assume there exists a completion AF′ = hA, R′ i of PAF such that S is not conflictfree for AF′ . There exists a pair (a, b) of arguments in S such that aR′ b. By definition of a completion, R′ ⊆ R ∪ I, so S is not RI-conflict-free for PAF. ⇐ If S is conflict-free for any completion of PAF, then S is conflict-free for the completion hA, R ∪ Ii, which means exactly that S is RI-conflict-free for PAF.

We also have a link between the conflict-free property for an AF and the conflict-free property for its open or closed PAF: Property 10. Let S ⊆ A. If S is conflict-free (in Dung’s sense) for AF, then S is Rconflict-free (resp. RI-conflict-free) for the open (resp. closed) PAF associated with AF. The proof is obvious. Property 11. Let S ⊆ A. S is RI-admissible for PAF ⇒ S is admissible for each completion of PAF. 9

i.e. such that there do not exist x ∈ A \ {a0 , . . . , an−1 } and 0 ≤ i ≤ n − 1 such that xRai or xIai

The proof follows Property 9 and uses the fact that if a is RI-acceptable w.r.t. S for PAF then a is acceptable w.r.t. S for hA, R′ i with R ⊆ R′ ⊆ R ∪ I. Property 12. Let AF′ = hA, R′ i a completion of PAF. Let E be a preferred extension of AF′ . The fact that a ∈ E does not imply that there exists a R-preferred extension or a RI-preferred extension of PAF which contains a. Proof. Consider PAF represented by: PAF

AF1

AF2

(completions) a

a

a

b

b

b

c

c

c

The R-preferred extensions of PAF are {a} and {b}. The RI-preferred extension of PAF is {a}. The preferred extension of AF1 is {a, c} and the preferred extension of AF2 is {a, b}. c belongs to the extension of AF1 but it belongs neither to a R-preferred extension of PAF, nor to the RIpreferred extension of PAF.

Property 13. If E is a RI-preferred extension of PAF then for each completion AFi of PAF, i = 1 . . . n, there exists a preferred extension of AFi denoted by Ei such that E ⊆ E1 ∩ . . . ∩ En . Proof. E is RI-admissible for PAF. From Property 11, E is admissible in each completion AFi of PAF. So, from [7], for each AFi , there exists a preferred extension Ei of AFi containing E. So, E ⊆ E1 ∩ . . . ∩ En .

6.2 Complexity results Now, let us consider some complexity issues. Indeed, in an AI perspective, it is important to determine how hard are the new inference relations we pointed out w.r.t. the computational point of view. We assume the reader acquainted with basic notions of complexity theory, especially the complexity classes P, NP, coNP and the polynomial hierarchy (see e.g. [11]), and with complexity results in Dung’s framework (see [12]). [13] have shown that considering sets of arguments (instead of single arguments) as input queries for the inference problem10 does not lead to a complexity shift when Dung’s inference relations are considered. As to inference relations concerning the PAFs, the same conclusion can be drawn. First of all, it is easy to show that, given a finite partial argumentation framework PAF, deciding whether a given argument R-interacts or I-interacts with a given argument is in P, and deciding whether a set of arguments is R-conflict-free for PAF and deciding whether a set of arguments is RI-conflict-free for PAF are in P. Accordingly, deciding whether a given set of arguments is admissible for PAF, deciding whether a given set of arguments is R-admissible for PAF and deciding whether a given set of arguments is RI-admissible for PAF are in P. Besides, deciding whether a set of arguments S is a preferred extension of PAF, deciding whether a set of arguments S is a R-preferred extension of PAF and deciding 10

The purpose is to determine whether such sets are derivable from a given finite argumentation framework AF.

whether a set of arguments S is a RI-preferred extension of PAF are in coNP (in order to show that the complementary problem is in NP, it is sufficient to guess a proper superset S ′ of S and to check in polynomial time that S ′ is admissible for PAF, Radmissible for PAF or RI-admissible for PAF). Property 14. Let PAF = hA, R, I, N i be a finite partial argumentation framework and S ⊆ A. 1. Deciding whether S is included in a preferred extension of PAF (resp. in a Rpreferred extension of PAF, in a RI-preferred extension of PAF) is NP-complete; 2. Deciding whether S is included in all preferred extensions of PAF (resp. in all R-preferred extensions of PAF, in all RI-preferred extensions of PAF) is Π2p complete. Proof. For the first problem: Membership: Deciding whether a given set of arguments S is included in a preferred extension of PAF is in NP (according to Property 4, it is sufficient to guess a set E ⊆ A and to check in polynomial time if E is an admissible set for PAF and to check that S is included in E). Hardness: We build in polynomial time a polynomial functional reduction f from the problem to decide whether a given set of arguments S is included in a preferred extension of AF = hA, Ri. Let AF = hA, Ri be a finite argumentation framework and S ⊆ A. f : hAF = hA, Ri, S ⊆ Ai 7→ hPAFc = hA, R, ∅, (A × A) \ Ri, S ′ ⊆ Ai with S ′ = S. According to Property 3, S is included in a preferred extension of AF iff S ′ is included in a preferred extension of PAFc . As, according to [13], deciding whether a given set of arguments S is included in a preferred extension of AF is in NP-complete, deciding whether a given set of arguments S is included in a preferred extension of AF is in NP-complete. For the second problem: Membership: Deciding whether a given set of arguments S is included in every preferred extension of PAF is in Π2p (in order to show that the complementary problem is in Σ2p , it is sufficient to guess a set E ⊆ A and to check in polynomial time using an NP oracle that E is a preferred extension of AF and that S is not included in E). Hardness: We build in polynomial time a polynomial functional reduction f from the problem to decide whether a given set of arguments S is included in all preferred extensions of AF = hA, Ri. Let AF = hA, Ri be a finite argumentation framework and S ⊆ A. f : hAF = hA, Ri, S ⊆ Ai 7→ hPAFc = hA, R, ∅, (A × A) \ Ri, S ′ ⊆ Ai with S ′ = S. According to Property 3, S is included in all preferred extensions of AF iff S ′ is included in all preferred extensions of PAFc . As, according to [13], deciding whether a given set of arguments S is included in all preferred extensions of AF is in Π2p -complete, deciding whether a given set of arguments S is included in all preferred extensions of AF is in Π2p -complete. The proof is similar for the R-preferred extensions and the RI-preferred extensions.

7 Conclusion In this paper, we have studied an extension of Dung’s argumentation framework (AF) introduced in [1, 8]. This extension, called “partial argumentation framework” (PAF), consists in introducing another interaction between arguments which represents some ignorance about the existence of an attack between two arguments. In [1, 8], the PAFs have been used in order to merge different AFs. Here, we have proposed a more formal study of acceptability semantics in a PAF.

So we have generalized the basic notions proposed by Dung (conflict-free set, acceptability of an argument, admissibility) to the PAF and we have identified several interesting semantics. The main properties of these semantics concern the inclusion links between them; a taxonomy of these semantics can be found where the more general case is given by Dung’s semantics which take into account only the attacks, and the less general case is given by the semantics, proposed in this paper, which take into account attacks and ignorance in the same time in the notion of conflict-free and in the notion of acceptability of an argument. Another fundamental property concerns the complexity of the problems related to our new semantics: the generalisation to a PAF does not imply an increasing cost of the temporal complexity in the worst case (the complexity results are exactly the same as those given for the AF). Future works concern the development of algorithms in order to compute efficiently the extensions of our semantics. A more theoretical issue is to apply our semantics directly on the merging process proposed in [1, 8] in order to reduce the cost of this merging.

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