Introduction
The general argumentation theory [Bondarenko et al.1997, Dung1995b] has proved to be a powerful framework for representing nonmonotonic formalisms in general, and semantics for normal logic programs, in particular. Thus, it has been shown that the main semantics for the latter, suggested in the literature, are naturally representable in this framework (see, e.g., [Dung1995a, Kakas and Toni1999]).
In this report we suggest a certain extension of the general argumentation theory in which the attack relation is defined directly among sets of arguments. In other words, we will permit situations in which a set of arguments ‘collectively’ attacks another set of arguments in a way that is not reducible to attacks among particular arguments from these sets. It turns out that this extension is suitable for providing semantics for disjunctive logic programs in which the rules have multiple heads. In addition, it suggests a natural setting for studying kinds of argumentation in which the opponents could provisionally share their arguments. Moreover, the original argumentation theory can be ‘reconstructed’ in this framework by requiring, in addition, that the attack relation should be local in the sense that a set of arguments can attack another set of arguments only if it attacks a particular argument in this set.
The plan of the paper is as follows. After a brief description of the abstract argumentation theory, we suggest its generalization in which the attack relation is defined on sets of arguments. It is shown that the suggested collective argumentation theory is adequate for representing practically any semantics for disjunctive logic programs. As an application of the general theory, we consider two special cases of the general framework in which the opponents can share their arguments. The semantics obtained in this way will correspond to some familiar proposals, given in the literature.
1 Abstract Argumentation Theory
We give first a brief description of the argumentation theory from [Dung1995b].
Definition 1.1.
An abstract argumentation theory is a pair , where is a set of arguments, while a binary relation of an attack on .
A general task of the argumentation theory consists in determining ‘good’ sets of arguments that are safe in some sense with respect to the attack relation. To this end, we should extend first the attack relation to sets of arguments: if , are sets of arguments, then is taken to hold iff , for some , .
An argument will be called allowable for the set of arguments , if does not attack . For any set of arguments , we will denote by the set of all arguments allowable by , that is
An argument will be called acceptable for the set of arguments , if attacks any argument against . As can be easily checked, the set of arguments that are acceptable for coincides with .
Using the above notions, we can give a quite simple characterization of the basic objects of an abstract argumentation theory.
Definition 1.2.
A set of arguments will be called

conflictfree if ;

admissible if it is conflictfree and ;

a complete extension if it is conflictfree and ;

a preferred extension if it is a maximal complete extension;

a stable extension if .
A set of arguments is conflictfree if it does not attack itself. A conflictfree set is admissible iff any argument from is also acceptable for , and it is a complete extension if it coincides with the set of arguments that are acceptable with respect to it. Finally, a stable extension is a conflictfree set of arguments that attacks any argument outside it. Clearly, any stable extension is also a preferred extension, any preferred extension is a complete extension, and any complete extension is an admissible set. Moreover, as has been shown in [Dung1995b], any admissible set is included in some complete extension. Consequently, preferred extensions coincide with maximal admissible sets. In addition, the set of complete extensions forms a complete lower semilattice: for any set of complete extensions, there exists a unique greatest complete extension that is included in all of them. In particular, there always exists a least complete extension of an argumentation theory.
As has been shown in [Dung1995a], under a suitable translation, the above objects correspond to wellknown semantics suggested for normal logic programs. Thus, stable extensions correspond to stable models (answer sets), complete extensions correspond to partial stable models, preferred extensions correspond to regular models, while the least complete extension corresponds in this sense to the wellfounded semantics (WFS). These results have shown, in effect, that the abstract argumentation theory successfully captures the essence of logical reasoning behind normal logic programs.
Unfortunately, the above argumentation theory cannot be extended directly to disjunctive logic programs. The reasons for this shortcoming, as well as a way of modifying the argumentation theory are discussed in the next section.
2 Collective Argumentation
We begin with pointing out a peculiar discrepancy between the abstract argumentation theory, on the one hand, and the general abductive framework used for interpreting semantics for logic programs, on the other hand (see, e.g., [Bondarenko et al.1997, Dung1995a, Kakas and Toni1999]). The main objects of the abductive argumentation theory are sets of assumptions (abducibles) of the form that play the role of arguments in the associated argumentation theory. In addition, the attack relation is defined in this framework as a relation between sets of abducibles and particular abducibles they attack. For example, the program rule is interpreted as saying that the set of assumptions attacks the assumption .
The above attack relation is employed for defining the basic objects (such as extensions) of the source abductive framework. The abstract argumentation theory defines its main objects, however, as sets of arguments. Consequently, they should correspond to sets of sets of assumptions in the abductive framework. The abductive theory defines such objects, however, as certain plain sets of assumptions! In other words, we have a certain discrepancy between the levels of representations of intended objects in these two theories.
The above discrepancy will disappear once we notice that all the basic objects of the abstract argumentation theory are definable, in effect, in terms of the derived attack relation between sets of arguments and particular arguments; only the latter was used in defining the above operator . As a result, the abductive argumentation theory can be constructed in the same way as the abstract theory, with the only distinction that the attack relation between sets of arguments and particular arguments is not reducible to the attack relation among individual arguments.
The above construction of abductive argumentation naturally suggests that assumptions, or abducibles, can be considered as fullfledged arguments, while the attack relation is best describable as a relation among sets of arguments. Indeed, once we allow for a possibility that a set of arguments can produce a nontrivial attack that is not reducible to attacks among particular arguments, it is only natural to allow also for a possibility that a set of arguments could be attacked in such a way that we cannot single out a particular argument in the attacked set that could be blamed for it. In a quite common case, for example, we can disprove some conclusion jointly supported by the disputed set of arguments. The following generalization of the abstract argumentation framework reflects this idea.
Definition 2.1.
A collective argumentation theory is a pair , where is a set of arguments, and is an attack relation on finite subsets of satisfying the following monotonicity condition:
(Monotonicity) If , then .
Though the attack relation is defined above only on finite sets of arguments, it can be naturally extended to arbitrary such sets by imposing the following compactness property:
(Compactness) only if , for some finite , .
As the reader may notice, the suggested argumentation framework has many properties in common with ordinary sequent calculus, or consequence relations. Moreover, we will use in what follows the same agreements for the attack relation as that commonly accepted for consequence relations. Thus, will have the same meaning as . Similarly, will be an alternative notation for , etc.
The argumentation theory from [Dung1995a] satisfies all the above properties. Moreover, the above modification of the abstract argumentation theory has already been suggested, in effect, in [Kakas and Toni1999]. However, the attack relation defined in the latter paper satisfied also a couple of further properties described in the following definition.
Definition 2.2.
A collective argumentation theory will be called

affirmative if no set of arguments attacks the empty set ;

local if it satisfies the following condition:
(Locality) If , then either or .

normal if it is both affirmative and local.
If a collective argumentation theory is normal, then it can be easily shown that will hold if and only if , for some . Consequently, the attack relation in such argumentation theories is reducible to the relation between sets of arguments and single arguments, and the resulting argumentation theory will coincide, in effect, with that given already in [Dung1995a].
It turns out, however, that the general, nonlocal framework of collective argumentation is precisely what is needed in order to represent semantics of disjunctive logic programs.
2.1 Collective Argumentation and Disjunctive Programs
Despite an obvious success, the abstract argumentation theory is still not abundant with intuitions and principles that could guide its development independently of applications. In this respect, logic programming and its semantics constitute one of the crucial sources and driving forces behind development of argumentation theories. Consequently, as a first step in studying collective argumentation, we consider its representation capabilities in describing semantics for disjunctive logic programs.
In what follows, given a set of propositional atoms , we will denote by the complement of in the set of all atoms. In addition, will denote the set of all negative literals (abducibles) , for .
By the general correspondence between normal logic programs and abductive argumentation frameworks, a set of abducibles attacks an abducible in the abductive theory associated with a normal logic program if , taken together with as a set of additional assumptions, allows to derive .
The above description immediately suggests a generalization according to which any disjunctive logic program determines an attack relation among sets of abducibles as follows:
attacks iff derives .
As can be easily verified, the above defined attack relation satisfies all the properties of collective argumentation. However, it is in general not local: may support without supporting either or . Still, it will be affirmative for disjunctive logic programs without constraints.
The appropriateness of the original argumentation theory for representing semantics of normal logic programs was based, ultimately, on the fact that these semantics are completely determined by rules of the form that are derivable from a program. A similar principle, called the principle of partial deduction, or evaluation is valid also for the majority of semantics suggested for disjunctive logic programs. According to this principle, semantics of such programs should be completely determined by rules without positive atoms in bodies that are derivable from the source program. See also [Bochman1998] for the role of this principle in determining semantics of logic programs of a most general kind.
The above considerations indicate that practically all ‘respectable’ semantics for disjunctive programs should be expressible in terms of collective argumentation theories associated with such programs.
It turns out, however, that the actual semantics suggested for disjunctive programs do not fit easily into the general constructions of Dung’s argumentation theory. A most immediate reason for this is that the operator of the abstract argumentation theory is no longer suitable for capturing the main content of the collective attack relation, since the latter is defined as holding between sets of arguments. Accordingly, it seems reasonable to generalize it to an operator that outputs a set of sets of arguments:
As in the abstract argumentation theory, will collect argument sets that are allowable with respect to . Notice that, due to monotonicity of the attack relation, will be closed with respect to subsets, that is, if and , then . Consequently, any set will be completely determined by maximal argument sets belonging to it. As can be easily verified, such maximal sets will always exist due to compactness of the attack relation.
2.2 Stable and Partial Stable Argument Sets
Using the above generalized operator of allowability, we can give a rather simple description of stable and partial stable models for disjunctive programs (see, e.g., [Gelfond and Lifschitz1991, Przymusinski1991]) in terms of collective argumentation.
Definition 2.3.

A set of arguments will be said to be stable with respect to a collective argumentation theory if it is a maximal set in .

A pair of sets will be called pstable if , is a maximal set in , and is a maximal set in .
The following lemmas give more direct, and often more convenient, descriptions of the above objects. The proofs are immediate, so we omit them.
Lemma 2.1.
is a stable set iff .
The above equation says that a stable set is a set consisting of all arguments such that does not attack . A similar description can be given for partial stable sets:
Lemma 2.2.
is pstable iff , , and .
Recall that normal collective argumentation theories could be identified with abstract Dung’s argumentation theories. Moreover, the above descriptions can be used to show that if a collective argumentation theory is normal, then stable argument sets will coincide with stable extensions, while pstable pairs will correspond exactly to complete extensions of the abstract argumentation theory. These facts could also be obtained as a byproduct of the correspondence between such objects and relevant semantics of disjunctive programs stated below.
The correspondence between the above descriptions and (partial) stable models of disjunctive logic programs is established in the following theorem.
Theorem 2.3.
If is a collective argumentation theory corresponding to a disjunctive program , then

is a stable model of iff is a stable set in .

is a pstable model of iff is pstable in .
Pstable models have been introduced in [Bochman1998] as a slight modification of Przymusinski’s partial stable models for disjunctive programs from [Przymusinski1991]; the reason for the modification was that the original Przymusinski’s semantics violated the abovementioned principle of partial deduction. In our present context, this means that it is not definable directly in terms of the collective argumentation theory associated with a disjunctive program. Note, however, that the modification does not change the correspondence with partial stable models for normal logic programs.
The above results could serve as an instance of our earlier claim that semantics of disjunctive programs are representable in the framework of collective argumentation. These results reveal, however, that the relevant objects are significantly different from the corresponding objects of the abstract argumentation theory. In order to get a further insight on the differences, we will consider now various notions of admissibility for argument sets that are definable in the framework of collective argumentation.
2.3 Argument sharing
In ordinary disputation and argumentation the parties can provisionally accept some of the arguments defended by their adversaries in order to disprove the latter. Two basic cases of such an ‘argument sharing’ in attacking the opponents are described in the following definition (see also [Bondarenko et al.1997]).
Definition 2.4.

positively attacks (notation ) if ;

negatively attacks (notation ) if .
In a positive attack, the proponent temporarily accepts opponent’s arguments in order to disprove the latter, while in a negative attack she shows that her arguments are sufficient for challenging an addition of opponent’s arguments. Clearly, if attacks directly, then it attacks the latter both positively and negatively. The reverse implications do not hold, however.
Note that the above defined notion of a stable argument set was formulated, in effect, in terms of negative attacks. Indeed, it is easy to see that a set is stable iff it negatively attacks any argument outside it:
As can be seen, the above definition is equivalent to the definition of stable extensions in the abstract argumentation theory, given earlier, so stable extensions and stable sets of collective argumentation are indeed close relatives.
Recall now that admissible argument sets in Dung’s argumentation theory are definable as conflictfree sets that counterattack any argument against them. Given the above proliferation of the notion of an attack in collective argumentation, however, we can obtain a number of possible definitions of admissibility by allowing different kinds of attack and/or counterattack among sets of arguments. Three such notions turns out to be of special interest.
Definition 2.5.
A conflictfree set of arguments will be called

admissible if whenever ;

positively admissible if whenever ;

negatively admissible if whenever .
Plain admissibility is a direct counterpart of the corresponding notion from the abstract argumentation theory. Unfortunately, in the context of collective argumentation it lacks practically all the properties it has in the latter. Notice, in particular, that stable sets as defined above need not be admissible in this sense.
As can be seen, positive and negative admissibility coincide with plain admissibility for respective ‘extended’ attack relations. The latter have some specific features that make the overall structure simpler and more regular. They will be described in the following sections.
3 Negative Argumentation
The definition below provides a general description of collective argumentation based on a negative attack. Such argumentation theories will be shown to be especially suitable for studying stable argument sets.
Definition 3.1.
A collective argumentation theory will be called negative if always implies .
As can be easily verified, any collective argumentation theory will be negative with respect to the negative attack relation . Moreover, the latter determines a least negative ‘closure’ of the source attack relation.
The following result gives an important alternative characterization of negative argumentation; it establishes a correspondence between negative argumentation and shift operations studied in a number of papers on disjunctive logic programming [Dix et al.1994, Schaerf1995, You et al.2000].
Lemma 3.1.
An argumentation theory is negative iff it satisfies:
(Importation) If , then .
Proof.
If the argumentation theory is negative and , then by monotonicity, and hence . The reverse implication is immediate. ∎
As an important special case of Importation, we have that if , then . Thus, any nontrivial negative argumentation theory is bound to be nonaffirmative. Furthermore, this implies that selfcontradictory arguments attack any argument:
These ‘classical’ properties indicate that negative attack is similar to a rule holding in a supraclassical consequence relation. Though the latter does not admit contraposition, we nevertheless have that if , then .
The connection between negative argumentation and stable argument sets is based on the following facts about general collective argumentation.
Theorem 3.2.

If is negatively admissible, and is a conflictfree set that includes , then is also negatively admissible.

Stable sets coincide with maximal negatively admissible sets.
Proof.
(1) Assume that is negatively admissible, and . Then , and hence by Importation. Since is negatively admissible, we obtain , and hence by Importation. But the latter amounts to , which shows that is also negatively admissible.
(2) It is easy to check that any stable set is negatively admissible. Moreover, any superset of a stable set will not already be conflictfree. Consequently stable sets will be maximal negatively admissible sets. In the other direction, if is a maximal negatively admissible set and , then will not be conflictfree by the previous claim, and hence . Consequently , and therefore (since is negatively admissible). But the latter implies , which shows that is actually a stable set. ∎
Recall now that negatively admissible sets are precisely admissible sets with respect to the negative attack . Moreover, in negative argumentation theories admissible sets will coincide with negatively admissible ones, while any conflictfree set will already be positively admissible. Accordingly, all nontrivial kinds of admissibility in such theories will boil down to (negative) admissibility; furthermore, maximal admissible sets in such argumentation theories will coincide with stable sets.
It can also be easily verified that any collective argumentation theory has the same stable sets as its negative closure. So, Importation is an admissible rule for argumentation systems based on stable sets. As a result, negative argumentation theories suggest themselves as a natural framework for describing stable sets.
Though the above results demonstrate that negatively admissible sets behave much like logically consistent sets, there is a crucial difference: the empty set is not, in general, negatively admissible. Moreover, an argumentation theory may be ‘negatively inconsistent’, that is, it may have no negatively admissible sets at all; this happens precisely when it has no stable sets.
Unfortunately, the above considerations indicate also that negative argumentation is inappropriate for studying argument sets beyond stable ones. Recall that one of the main incentives for introducing partial stable and wellfounded models for normal programs was the desire to avoid taking stance on each and every literal and argument. However, negativity implies that selfcontradictory arguments attack any argument whatsoever, so any admissible set is forced now to counterattack any such argument. In particular, if , then any admissible set should attack . This means that complete extensions (and partial stable models) are no longer a viable alternative for such argumentation systems. This means as well that Importation (and corresponding shift operations in logic programming) is an appropriate operation only for describing stable models^{1}^{1}1despite some attempts made in this direction – see, e.g., [You et al.2000]. Actually, the same difficulty plagues attempts to define partial stable semantics for default logic..
4 Positive Argumentation
The following definition provides a description of argumentation based on a positive attack.
Definition 4.1.
A collective argumentation theory will be called positive if always implies .
Any collective argumentation theory will be positive with respect to the positive attack relation . Moreover, the latter determines a least positive extension of the source attack relation.
Example.
Consider an argumentation theory containing only two arguments and such that and . As can be seen, this argumentation theory has no extensions, while the corresponding positive theory has a unique extension .
Similarly to negative argumentation, positive argumentation can be characterized by the ‘exportation’ property described in the lemma below:
Lemma 4.1.
An argumentation theory is positive iff it satisfies:
(Exportation) If , then .
The above characterization implies, in particular, that selfconflicting arguments are attacked by any argument:
So, in positive argumentation we are relieved, in effect, from the obligation to refute selfcontradictory arguments. In particular, no allowable argument will be selfcontradictory.
It is interesting to note that positive and negative argumentation are, in a sense, incompatible on pain of trivialization. Namely, if we combine positive and negative argumentation, we obtain a symmetric attack relation:
Lemma 4.2.
If an argumentation theory is both positive an negative, then always implies .
Proof.
If , then by monotonicity. Consequently, by negativity and hence by positivity. ∎
In this case we can consider the attack relation as expressing plain incompatibility of the arguments and in a classical logical sense. In other words, we could treat arguments as propositions and define as .
4.1 Local Positive Argumentation
As a matter of fact, positively admissible sets were introduced for normal programs in [Kakas and Mankarella1991] (see also [Kakas and Toni1999]) under the name weakly stable sets; maximal such sets were termed stable theories. Accordingly, a study of such objects will amount to a study of admissible sets in collective argumentation theories that are positive closures of normal argumentation theories.
Note first that a positive closure of a local argumentation theory need not, in general, be local. Still, the following property provides a characterization of positive theories arising from local argumentation theories.
Definition 4.2.
A collective argumentation theory will be called lpositive if it is positive and satisfies
(Semilocality) If , then either or
The following basic result shows that lpositive argumentation theories are precisely positive closures of local theories. Though the proof is not trivial, we omit it due to the lack of space.
Theorem 4.3.
An argumentation theory is lpositive iff it is a positive closure of some local argumentation theory.
Due to the above result, stable theories from [Kakas and Mankarella1991] are exactly representable as maximal admissible sets in lpositive argumentation theories. It turns out that many properties of stable theories can be obtained in this abstract setting; the corresponding descriptions will be given in an extended version of this report. Still, we should mention that, since lpositive argumentation theories are not local, the corresponding structure of admissible sets is more complex than in the local case. For example, the set of admissible sets no longer forms a lower semilattice.
5 Preliminary Conclusions
Collective argumentation suggests itself as a natural extension of the abstract argumentation theory. It allows, in particular, to represent and study semantics for disjunctive logic programs. Speaking generally, it constitutes an argumentation framework in which the attack relation has structural properties allowing to represent cooperation and sharing of arguments among the parties.
The present report is very preliminary, however; it only barely scratches the surface of the vast number of problems and issues that arise in this setting. One of such issues consists in extending the approach to other, weaker semantics suggested for disjunctive programs. A more general task amounts, however, to determining general argumentation principles underlying collective argumentation. This is a subject of an ongoing research.
References
 [Bochman1998] A. Bochman. A logical foundation for logic programming, I and II. Journal of Logic Programming, 35:151–194, 1998.
 [Bondarenko et al.1997] A. Bondarenko, P. M. Dung, R. A. Kowalski, and F. Toni. An abstract, argumentationtheoretic framework for default reasoning. Artificial Intelligence, 93:63–101, 1997.
 [Dix et al.1994] J. Dix, G. Gottlob, and V. Marek. Causal models for disjunctive logic programs. In P.Van Hentenryck, editor, Proc. 11th Int. Conf. On Logic Programming, S. Margherita Ligure, pages 290–302. MIT Press, 1994.
 [Dung1995a] P. H. Dung. An argumentationtheoretic foundation for logic programming. J. of Logic Programming, 22:151–177, 1995.
 [Dung1995b] P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and npersons games. Artificial Intelligence, 76:321–358, 1995.
 [Gelfond and Lifschitz1991] M. Gelfond and V. Lifschitz. Classical negation and disjunctive databases. New Generation Computing, 9:365–385, 1991.
 [Kakas and Mankarella1991] A. C. Kakas and P. Mankarella. Stable theories for logic programs. In V. Saraswat and K. Ueda, editors, Proc. Int. Symp. On Logic Programming, pages 85–100. MIT Press, 1991.
 [Kakas and Toni1999] A. C. Kakas and F. Toni. Computing argumentation in logic programming. J. of Logic and Computation, 9:515–562, 1999.
 [Przymusinski1991] T. C. Przymusinski. Stable semantics for disjunctive programs. New Generation Computing, 9:401–424, 1991.
 [Schaerf1995] M. Schaerf. Negation and minimality in disjunctive databases. J. of Logic Programming, 23:63–86, 1995.
 [You et al.2000] J.H. You, L. Y. Yuan, and R. Goebel. An abductive apprach to disjunctive logic programming. J. of Logic Programming, 44:101–127, 2000.