Collective Attacks in Assumption-Based Argumentation

Conflicts in argumentation-based frameworks are usually described in terms of attacks of arguments, or sets of arguments, on specific counter-arguments. In this paper we consider (assumption-based) argumentation frameworks, in which attacks have a more general form: they are performed on a collective of arguments that cannot stand together with the attacking arguments. We show that not only that this generalized form of attacks increases the expressive power of the argumentation frameworks, but in certain cases it also allows more sensible patterns of reasoning with conflicting considerations. Along the way, we also provide a novel characterization of the grounded semantics in prioritized argumentation frameworks.


INTRODUCTION
Assumption-based argumentation frameworks (ABFs, for short) [4] are deductive rule-based systems, where assumptions and their contraries are incorporated for capturing different paradigms of non-monotonic reasoning.It has been shown that these frameworks are adequate for modeling reasoning with conflicting and dynamic information (see, e.g., [9,14,24] for some relevant tutorials).The basic idea in assumption-based argumentation is that sets of formulas (in some language) may 'attack' each other, where -intuitively -one set attacks another set, if formulas in the attacking set entail (according to some underlying inference relation) the contrary of a formula in the attacked set.
The purpose of this paper is to generalize the concept of attacks in ABFs, for better capturing cases in which conflicts may arise, and so refine and improve the decision making process in various conflicting situations.The idea is, instead of attacking a particular

SIMPLE CONTRAPOSITIVE ABFS
Our starting point is a logic E = (L, 1-), where Lis a (propositional) language and I-is a consequence relation for L. Atomic formulas in L are denoted by p, q, r, compound formulas are denoted by if,,</>, u, and sets of formulas in Lare denoted by r, /1, e (all may be primed or indexed).The consequence relation I-is a binary relation between sets of formulas and formulas in L, satisfying the following conditions: • Reflexivity: if if, E f then f 1-if,.
As usual, the 1--transitive closure of a set r of L-formulas is Cnc (r) = {if, I r 1-if,}.We say that a formula if, is !--tautological if if, E Cnc(0), and that r is !--consistent ifCnc(f) * L.
We shall assume that the language L contains at least the following connectives and constant: !--negation-,, satisfying: p .I' -,p and -,p .I' p (for every atomic p).
We abbreviate {-,y I y E r} by -,f, and when r is finite we denote by /\f (respectively, by vr), the conjunction (respectively, the disjunction) of all the formulas in r.
Intuitively, this respectively means that every conclusion follows from a contradictory set of assumption, and that the formulas may be replaced by their negated formulas when switched between the two sides of the consequence relations. 2ow we can define assumption-based argumentation frameworks [4].The following family of these frameworks is shown in [15] to be a useful setting for argumentative reasoning.Definition 2.1.An assumption-based framework (ABF, for short) is a tuple ABF = (E, r,Ab, ~}, where: • E = ( L, 1-} is a propositional logic. • r (the strict assumptions) and Ab (the candidate/defeasible assumptions) are distinct (countable) sets of L-formulas, where the former is assumed to be !--consistent and the latter is assumed to be nonempty.• ~ : Ab -+ 2L is a contrariness operator, assigning a finite set of L-formulas to every defeasible assumption in Ab, such that for every consistent and non-tautological formula t/t E Ab\ {F} it holds that t/t .I' A ~ t/t and A ~ t/t .I' tjt.
A simple contrapositive ABF is an assumption-based framework ABF = (E, r,Ab, ~}, where Eis an explosive and contrapositive logic, and for every t/t E Ab it holds that ~ t/t = { ,tjt}.
Defeasible assertions in an ABF may be attacked in the presence of a counter defeasible information.This is described in the next definition.
• l'J. is complete (in ABF) iffit is admissible and contains every l'J.' k Ab that it defends.
• l'J. is well-founded (in ABF) iff /' J. = n{e k Ab I e is complete}.

COLLECTIVE ATTACKS
Let's reconsider Example 2.5.The grounded and the well-founded semantics in this case look overly cautious, since they allow only tautological inferences, while the inference of q (or any other formula in Cn~ ( { q})) is forbidden, although there is no indication whatsoever that q is related to the inconsistency in Ab.One way to overcome this weakness is to extend the notion of attacks, and apply it to sets of formulas.The extended notion, called collective attacks, is defined next (cf.Definition 2.2).Definition 3.1.Let AB F = (E, r, Ab, -, } be a simple-contrapositive ABF.Then /' J. collectively attacks {t/t1, ... , t/tn} ifr, /' J. 1-•"?=t o/i• This notion is carried on to supersets: /' J. collectively attacks a set e k Ab, if /' J. collectively attacks some { t/t1, ... , t/tn} k e. Example 3.2.Consider again the assumption-based framework of Example 2.5, but now attacks are replaced by collective attacks.The attack diagram that is obtained is similar to that in Figure 1, with two additional collective attacks: One, from{} on {p, ,p, q} (since I--,(p A -,p)) and the other one from {q} on {p, -,p, q} (since q I--,(p A -,p)).As a consequence, {q} defends itself from any (collective) attacker, which implies that this set is the grounded as well as the well-founded extension of the assumption-based framework with collective attacks.In particular, any formula in Cn~ ( { q}) is now inferred also by the grounded and the well-founded semantics in this case.
The fact that the transition in the last example to collective attacks provides more intuitive conclusions is not a coincidence.Indeed, in [15] the following interesting facts about simple contrapositive frameworks with collective attacks are shown: ( 1) Preferential and stable semantics coincide: Prf ( AB F) = Stb ( AB F) (2) Grounded and well-founded semantics equals to the intersection of the maximally consistent sets: Grd(ABF) = WF(ABF) = {nMCS(ABF)}, where MCS(ABF) = {t\ k Ab I r, ,'\ l' F and r, L\ 1 I-F for all,'\ <;; t\' k Ab}.
Our goal in this paper is to examine to what extent these and other properties carry on to more expressive ABFs, in particular those in which preferences among the defeasible assumptions are introduced.In the next section we define such extended settings, and in Section 5 we study the properties of the induced entailment relations.We show, in particular, that the two facts above carry on to the prioritized case (with some obvious adjustments).In particular, the extension of the second fact to preferential argumentation frameworks provides, to the best of our knowledge, a novel characterization of the grounded semantics in such frameworks.

ADDING PREFERENCES
We now extend simple contrapositive ABFs with priorities, expressing the relative strengths (or reliability) of assumptions.Adding quantitative measurements to qualitative information is a common approach in argumentation-based reasoning in general (see e.g.[17,18]) and in assumption-based argumentation frameworks in particular (ABA + [8,10]) is a notable example).In our case, the extension is made by allocating to every defeasible assumption a preference value (where, intuitively, lower values indicate higher preferences), and then making sure that assumptions with higher preferences cannot be attacked by assumptions with strictly lower preferences.This is formalized next.
Definition 4.1.[2] Let Ab be a set of formulas (which, in our case, will be the defeasible assumptions of an ABF).
• An allocation function on Ab is a total function g : Ab ---t N.
• A numeric aggregation function f is a total function that maps multisets of natural numbers into non-negative real numbers, suchthatVx E Nf({x}) = x. 4 Wealsoassumethatanaggregation function is k-coherent in its values, namely, it is either nondecreasing with respect to the subset relation (f(X') :;; f(X) whenever X' k X) or non-increasing with respect to the subset relation (f (X') ~ f (X) whenever X' k X). • A preference setting on Ab is a pair P = (g,f), where g is an allocation function on Ab and f is a numeric aggregation function.
An allocation function makes preferences among the defeasible information.The sets Abi = {tj, E Ab I g(tj,) = i} form a partition of Ab, which in turn may be viewed as a stratified set.This is sometimes denoted by Ab = Ab1 EB ... EB Abn.Aggregation functions are then used for aggregating the preferences.The maximum, minimum, and the summation functions are common aggregation functions.We shall write f(g(L\)) instead off( {9(8) I 8 E t\}).Some useful properties of preference settings are defined next.Definition 4.2.[2] Let P = (g,f) be a preference setting for Ab and ,'\ k Ab an arbitrary non-empty set of assumptions.
The selection property assures that f(g(L\)) is a selection of values in {f(g( 8)) I 8 E t\}, i.e., f(g(t\)) does not introduce 'new' values other than those that are assigned to the elements in t\.Preference settings where f =minor f = max are clearly selecting.In the latter case, the setting is also max-lower-bounded and conservative under union.
Prioritized ABFs are defined now as follows: A prioritized assumption-based framework (prioritized ABF, or pABF, for short) is a pair pABF = (ABF, P}, where ABF is a (simple contrapositive) assumption-based argumentation framework and P is a preference setting.
A prioritized ABF is called selecting, max-lower-bounded, or conservative under union, if so is its preference setting.
The semantics of the structures in Definition 4.4 are defined just as before (Definition 2.3), where 'attacks' are replaced by 'collectivep-attacks'.The corresponding entailment relations are defined just as in Definition 2.4.
Example 4.5.Consider the pABF that is obtained from the ABF of Examples 2.5 and 3.2, together with the allocation function g(p) = 1, g(-,p) = 2, g( q) = 3, and the aggregation function f = max.The diagram for this case is shown in Figure 2 (cf. Figure 1).In the figure, dashed lines denote collective attacks that are not standard (pointed) attacks.5Note that this time, since p is strictly preferred over -,p, the grounded extension is {p, q} and not just { q} as in the non-prioritized case.Thus, intuitively, p and q belong to the grounded extension in this case for two different reasons: p due to its high priority, and q since it is not related to the inconsistency in Ab.In the next section we shall prove that this is not a coincidence.

REASONING WITH COLLECTIVE ATTACKS
In this section we consider some basic properties of the semantics and the entailment relations that are induced by pABFs with collective attacks.

Preferred and Stable Semantics
First, we check the correspondence between preferred and stable extensions in such frameworks.This is shown in Proposition 5.2.For this proposition, we need the following lemma.LEMMA 5.1.Let pABF = (ABF, P), be a selecting prioritized ABF with collective attacks that is conservative under union, and let /' J. be a conflict-free set in Ab.Then /' J. is maximally admissible iff it collectively attacks any ifr E Ab \ l'J.. PROOF.One direction is clear: as already shown in [13] (for regular attacks), if a conflict-free set /' J. p-attacks any ifr E Ab \ /' J. it must be maximally admissible.Since /' J. p-attacks a formula ifr iff it collectively p-attacks {ifr}, we are done.
By [C3] and [C4] we get a contradiction to the maximal admissibility of /'l.

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By the last lemma, we conclude the following result: PROPOSITION 5.2.Let pABF be a prioritized ABF with collective attacks, which is both selecting and conservative under union.Then the stable extensions and the preferred extensions of pAB F coincide.
In [2] it is shown that when f = max (i.e., when the comparison criterion is with respect to the least preferred formulas, also known as the 'weakest link principle'), the stable extensions of a prioritized ABF (with regular attacks) are the same as the Cg-preferred maximally consistent subsets of Ab, where Cg is Brewka's preference order [5].Formally: • LetAbi = {1/!E Ab I g(ifr) = i} for i = 1, .. .,n be a partition of Ab, and let /1, e !;;; Ab.We denote /1 Cg e (intuitively read as: '/1 is preferred over 0'), if there is an 1 ~ i ~ n such that Abj n /1 = Abj n 0 for every 1 ~ j < i, andAbi n /1;?Abi n 0.

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The last corollary allows us to easily prove some properties of the induced entailment relations.The following properties were introduced by Kraus, Lehmann and Magidor in [19] and [20], and their formulations are adjusted to our setting.Some of the properties (CM, CC, and LLE) take into account also the priority setting.In such cases, the original formulation in [19] is obtained just by ignoring the conditions about the allocation function.
• We denote by Atoms(r) the set of all atoms occurring in r.
• We say that ABF1 and ABF2 are syntactically disjoint if so are f1 u Ab1 and f2 u Ab2.
For extending non-interference to the prioritized case, we further suppose that there are priority settings Pi = (gi, f) over Abi (i = 1, 2).When ABF1 and ABF2 are syntactically disjoint, we can define a priority setting P = (g,f) over Ab1 U Ab2, where g coincides with 9i on Abi.In such a case, non-interference is defined as in the non-prioritized case, except that now we require that pABF 1 f-, ifr iff p(ABF1 U ABF2) f-, if!, where p(ABF1 U ABF2) = (ABF1 U ABF2, P). for some allocation junction g.Then both f-, ~em and f-, rem satisfy non-interferenceforSem E {Prf,Stb}.
Propositions sintilar to the the last two results, but for standard (pointed) attacks, are shown in [2], based on Proposition 5.3.By Corollary 5.4, their proofs are carried on to Propositions 5.7 and 5.8, incorporating collective attacks.
What about preferentiality?It turns out that skeptical reasoning ( f-, rem) is preferential, while credulous reasoning ( f-, ~em) is not.PROPOSITION 5.9.Let pABF = (ABF, P) be a prioritized simple contrapositive ABF with collective attacks, in which P = (g, max) for some allocation function g.Then, for every Sem E {Prf, Stb} it holds that f-, rem is a preferential logic, while f-, ~em is not a preferential logic.

Grounded and Well-Founded Semantics
Let us turn now to the grounded and the well-founded semantics.As Examples 2.5 and 3.2 show, a transition from pointed attacks to collective attacks may affect the grounded extension.Yet, the following proposition carries on to frameworks with collectives attacks: PROPOSITION 5.10.Let pABF be a prioritized ABF with collective attacks.Then WF(pABF) = n Grd(pABF).

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Hence, the grounded and the well-founded semantics of a pABF with collective attacks coincide iff the pABF has a unique grounded extension.Such a case is assured by the characterization of the grounded extensions in terms of minimally inconsistent subsets that we provide below (Proposition 5.16).To the best of our knowledge, this is the first time that such a characterization has been provided for logic-based argumentation with prioritized knowledge-bases.
Ofer Arieli and Jesse Heyninck The idea behind this construction is the following: we proceed iteratively, starting from the assumptions with the best (lowest) priority, and select all the free formulas there (those that do not belong to any minimally inconsistent set).Then, we use these free formulas as strict premises in the next step, where we construct MICi+1 as the sets that are minimally conflicting in view of the strict premises r and the free formulas Free; ( pAB F) obtained in the previous step.All the formulas not involved in any such conflict are then designated as free on the (i + 1)th level.NoTE 3.An equivalent way of defining Freei(pABF) is by the (union of the intersections of the) maximally consistent subset of Abj U ~ i) w.r.t.rand Free;-1 (pABF), namely: and there is no /1 ~ 11' such that r, 11' .I' F}.Note that the definition above is different than the notion of prioritized MCS w.r.t.preferred subtheories (MCSc: 9 (ABF)), introduced in Section 5.1.This difference is illustrated in Example 5.15 bellow.
• MIC2(pABF) = {{-,s}, {s A-,r,r}}, Free2(pABF) = {s}.So in this case the only free formula of the prioritized framework is s, as expected.NoTE 4. One might wonder whether the intersection of preferred subtheories coincides with the free formulas, since in the nonprioritized case it holds that Free(ABF) = n MCS(ABF).As is observed in [11], this is not the case when taking into account priorities.Here is a counter-example: Thus, nMCSi:::y{pABF) = {c} -t,.0 = Free(pABF).
We can now show the following characterization of the grounded extension for prioritized argumentation frameworks with collective attacks and the weakest-link principle (max-based aggregations).PROPOSITION 5.16.Let pABF = (ABF, P) be a prioritized ABF with collective attacks, in which P = (g, max) and n is the maximal number in the image of g.Then: Grd(pABF) = Freen(pABF).
PROOF.We first show that Freen(pABF) is contained in every complete extension.We do this by induction on the maximal value of g, showing that Freei(pABF)) is contained in every complete extension for every i ;::-: 1.For the base case, suppose that e k Free1 (pABF) and some ti collectively p-attacks 8. Without loss of generality, let ti beak-minimal set attacking 8. Thenf(g(E>)) is not strictly JP>-stronger than valf,g(ti, 8).As f = max, and g(ij,) = 1 for every tp E e (since 8 k Free1 (pABF)), this means that g(8) = g(</J) = 1 for every 8 E ti and ¢ E e.But then there is ti' k ti U 8 E MIC1 (pABF) with ti' n E> -t,.0 or f, ti 1-F.The first case constitutes a contradiction against e k Free1 (pABF).Suppose that r, ti 1-F.Then 0 collectively attacks ti (since r 1--,j\ti) and thus e is defended by 0. Hence, e is included in any complete extension.
For the inductive step, suppose that Freei(pABF) is contained in a complete extension <I> of pABF and let e k Freei+1 (pABF)) \ Freei(pABF)).Suppose that some ti collectively p-attacks e.Again, without loss of generality, let ti be a k-minimal set attacking e.
Then f(g(E>)) is not strictly JP>-stronger than valf,g(ti, 8), which means, in our case, that valf, 9 (ti, E>) ~ f(g( 8)) = max(g(E>)) ~ i + 1 (since e k Abi+1)-Thus, either r, ti 1-F (in which case we have already shown in the base case that e is defended by 0, and so it is in <I> as the latter is complete), or there is ti' k ti U e E MICi+l (pABF) with ti' n e * 0, a contradiction to the assumption that e k Freei+l (pABF).We have thus shown that <I>~ Freen(pABF).
We now show that Freen(pABF) is complete.We first make the following observation: ( t): if e collectively p-attacks ti, then g( 0) ~ g ( 8) for every 0 E e and 8 E ti.
This follows immediately from the fact that f = max.(Indeed, for that attack to take place, for every 8 E ti we require that f (g( 8)) should not be preferred over f(g(E>)).In our case, and since the preference order is linear, this means that max(g(E>)) should be less than or equal to g (8) for every 8 E ti.Thus V8 E ti, V0 E 8, g(0) ~ g(8).) We now show that Freen (pABF) is conjlic-free.Suppose towards a contradiction that there are some ti 1, ti2 k Freen (pABF) s.t.ti1 collective p-attacks ti2.Then (ti1 U ti2) n MICi(pABF) * 0 for i = max0e& 1 u& 2 g (8), thus Freen(pABF) n MICi(pABF) if,.0, a contradiction to the definition of Freen(pABF).
We now show that Freen(pABF) defends all of its elements.We show by induction on i that Freei(pABF) defends all of its elements.For the base case, suppose that ti1 k Ab attacks some ti2 k Free1 (pABF).Then with (t), ti1 k Ab1 and thus f, ti1 U ti2 1-F.
Suppose now towards a contradiction that en MICi(pABF) * 0, i.e., there is some ti k U}=l Abi and some 0 * e' k e s.t.

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The last result provides another justification for switching to collective attacks, since as shown in [2, Example 18] non-interference is not satisfied by entailment relations that are induced by the grounded semantics ofpABFs with standard attacks.

CONCLUSION AND RELATED WORKS
The primary goal of this paper is to demonstrate the usefulness of incorporating collective attacks in (prioritized) assumption-based frameworks and to investigate some of the properties of the resulting argumentation frameworks.In passing, we have also obtained some further interesting results, such as the characterization of grounded extensions in prioritized ABFs.To the best of our knowledge, this is the first such characterization.Indeed, it has been observed before that in prioritized logic-based argumentation, the grounded extension does not always coincide with the intersection of preferred subtheories [11].We now give a precise characterization of what is included in the grounded extension.This also allows us to derive further properties of the grounded extension, such as non-interference, which is not guaranteed for the grounded extension in logic-based argumentation [1] and prioritized ABFs with standard attacks [2, Example 18].
As noted previously, some preliminary results concerning collective attacks in ABFs have already been introduced in [15].Those results are carried on in this paper to the prioritized case.Using uniform allocation functions (namely, those that assign the same preference value to all the defeasible assumptions) brings us back to the results in [15], thus some results in this paper are conservative extensions of those in [15], and some others are new.
Attacks of sets of arguments on other sets of arguments have recently been considered also for other frameworks for argumentative Ofer Arieli and Jesse Heyninck reasoning.For such a work in the context of abstract argumentation frameworks, we refer to [12].In sequent-based argumentation [3] collective attacks are enabled by attack rules on subsets of the arguments' supports.
In future work, it would be interesting to extend the notion of collective attacks to other structured argumentation settings, such as rule-based assumption-based argumentation [4] and systems allowing for structured argumentation with defeasible rules [ 16,21].
• /1 is a maximally consistent set (MCS) in ABF, if (a) r, 11 l' F and (b) r, 11' t-F for every /1 ~ 11' !;;; Ab.The set of the maximally consistent sets in ABF is denoted MCS(ABF).• /1 is a preferred maximally consistent set (pMCS) in pABF, if /1 E MCS(ABF) and there is no 0 E MCS(ABF) that is Cgpreferred over /1.The set of the preferred maximally consistent sets in pABF is denoted MCSc 9 (ABF).