Aggregating Bipolar Opinions (With Appendix)
AAggregating Bipolar Opinions (With Appendix)
Stefan Lauren
Imperial College London, [email protected]
Francesco Belardinelli
Imperial College London, UKUniversite d’Evry, [email protected]
Francesca Toni
Imperial College London,[email protected]
ABSTRACT
We introduce a novel method to aggregate Bipolar Argumentation(BA) Frameworks expressing opinions by different parties in debates.We use Bipolar Assumption-based Argumentation (ABA) as anall-encompassing formalism for BA under different semantics. Byleveraging on recent results on judgement aggregation in SocialChoice Theory, we prove several preservation results, both positiveand negative, for relevant properties of Bipolar ABA.
KEYWORDS
Bipolar Argumentation; Judgement Aggregation; Social Choice
ACM Reference Format:
Stefan Lauren, Francesco Belardinelli, and Francesca Toni. 2021. Aggregat-ing Bipolar Opinions (With Appendix). In
Proc. of the 20th InternationalConference on Autonomous Agents and Multiagent Systems (AAMAS 2021),Online, May 3–7, 2021 , IFAAMAS, 9 pages.
There is a long and well-established tradition in knowledge repre-sentation and reasoning to formally describe debates as exchangesof opinions by the parties involved through attacks [9] and sup-ports [21] between arguments understood abstractly as in
AbstractArgumentation (AA) [13] or as in
Bipolar Argumentation (BA) [7, 8].When these debates emerge in multi-agent systems [15, 21], a keyquestion concerns opinion aggregation, namely how we can obtaina collective consensus from several opinions expressed as argu-mentation frameworks, in such a way that the agents’ opinions arewell-portrayed in the collective outcome [4]. Recently, Chen andEndriss [9, 10] applied aggregation procedures from Social ChoiceTheory [18] to AA Frameworks (AAFs), by leveraging especiallyon judgement aggregation [17], and proving the preservation ofinteresting properties of AAFs, such as conflict-freeness, acyclicityand extensions according to several semantics. Some efforts hadbeen made to provide procedures for the aggregation of
Quanti-tative Argumentation Debate Frameworks [1, 21, 22] (a form of
BAFrameworks (BAFs) incorporating both attacks and supports butequipped with gradual, rather than extension-based, semantics),but the aggregation of BAFs is an open problem, and a challengingone mainly because there are different semantic interpretations ofsupport in BAFs (e.g., as deductive or necessary [5, 8, 20]), whichmight generate “inconsistencies” in the aggregated framework.Moving from the considerations above, we advance the stateof the art on the application of Social Choice Theory to opinionaggregation in computational argumentation by focusing on BA.To address the problem that different parties may adopt different
Proc. of the 20th International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7, 2021, Online interpretations of support in their opinions (BAFs), we use
BipolarAssumption-based Argumentation (ABA) frameworks [12] for repre-senting opinions. Bipolar ABA is a restricted (but “non-flat”) formof ABA providing a unified formalism to accommodate differentinterpretations of support [12]. Thus, by adopting Bipolar ABA, welet parties choose their interpretation of support before aggregationtakes place. Then, our contribution is twofold. Firstly, we defineaggregation procedure for Bipolar ABA frameworks based on So-cial Choice Theory. Our investigations mainly focus on quota andoligarchic rules [17]. Secondly, we conduct a study on the preser-vation of properties using the defined aggregation procedures. Insome cases, restrictions need to be placed, for example, towardsspecific aggregation rules.
Structure of the Paper.
In Section 2 we provide the necessarybackground on Bipolar ABA. Section 3 sets the ground, by formu-lating the aggregation problem, aggregation rules and preservationproperties in our Bipolar ABA setting. Section 4 gives the maintheoretical contribution of the paper, providing preservation resultsfor various properties of Bipolar ABA frameworks. Finally, Section5 concludes and elaborates on several promising directions for fu-ture works. Because of space limit, we omit some proofs: these canbe found in the Appendix.
Bipolar Assumption-based Argumentation [12] (Bipolar ABA) is aform of structured argumentation, where arguments and attacks arederived from assumptions, rules, and a contrary map from assump-tions. Note that contrary should not be confused with negation,which may or may not occur in the underlying language [23].Definition 1. [12] A
Bipolar ABA framework is a quadruple ⟨L , R , A , Ď ⟩ , where • ( L , R ) is a deductive system with L a language (i.e. a setof sentences) and R a set of rules of the form 𝜙 ← 𝛼 where 𝛼 ∈ A and either 𝜙 ∈ A or 𝜙 = s 𝛽 for some 𝛽 ∈ A ; 𝜙 is the head and 𝛼 the body of rule 𝜙 ← 𝛼 ; • A ⊆ L is a non-empty set of assumptions ; • Ď : A → L is a total map; for 𝛼 ∈ A , s 𝛼 is the contrary of 𝛼 . Then, a deduction for 𝜙 ∈ L supported by 𝐴 ⊆ A and 𝑅 ⊆ R ,denoted 𝐴 ⊢ 𝑅 𝜙 , is a finite tree with the root labelled by 𝜙 ; leaveslabelled by assumptions, with 𝐴 the set of all such assumptions;and each non-leaf node 𝜓 has a single child labelled by the body ofsome 𝜓 -headed rule in R , with 𝑅 the set of all such rules.Note that in Bipolar ABA rules are of a restricted kind, in com-parison with generic ABA [6]: their bodies amount to a single as-sumption, and thus, in particular, there are no rules with an emptybody; also, their heads are either assumptions or contraries thereof.Because assumptions may be “deducible” from rules in Bipolar ABA a r X i v : . [ c s . A I] F e b able 1: Bipolar ABA Semantics (for extension 𝐴 ⊆ A ).Semantics Conditions Admissible 𝐴 is closed, conflict-free and for every 𝐵 ⊆ A ,if 𝐵 is closed and attacks 𝐴 , then 𝐴 attacks 𝐵 .Preferred 𝐴 is ⊆ -maximally admissible.Complete 𝐴 is admissible and 𝐴 = { 𝛼 ∈ A : 𝐴 defends 𝛼 } where 𝐴 defends 𝛼 ∈ A iff for all closed 𝐵 ⊆ A :if 𝐵 attacks 𝛼 then 𝐴 attacks 𝐵 .Set-stable 𝐴 is closed, conflict-free, and attacks 𝐶𝑙 ( 𝛽 ) foreach 𝛽 ∈ A \ 𝐴 .Well-founded 𝐴 is the intersection of all complete extensions.Ideal 𝐴 is ⊆ -maximal such that it is admissible and 𝐴 ⊆ 𝐵 for all preferred extensions 𝐵 ⊆ A .frameworks, though, these frameworks may be non-flat in gen-eral, thus lacking some of the properties that flat ABA frameworks(where assumptions are not “deducible” from rules) exhibit [11].In Bipolar ABA, 𝐴 ⊆ A attacks 𝛽 ∈ A iff ∃ 𝐴 ′ ⊢ 𝑅 s 𝛽 , such that 𝐴 ′ ⊆ 𝐴 ; 𝛼 ∈ A attacks 𝛽 ∈ A iff { 𝛼 } attacks 𝛽 ; 𝐴 ⊆ A attacks 𝐵 ⊆ A iff ∃ 𝛽 ∈ 𝐵 such that 𝐴 attacks 𝛽 . Then, 𝐴 is conflict-free iff 𝐴 does not attacks 𝐴 . Let the closure of 𝐴 ⊆ A be 𝐶𝑙 ( 𝐴 ) = { 𝛼 ∈ A : ∃ 𝐴 ′ ⊢ 𝑅 𝛼, 𝐴 ′ ⊆ 𝐴, 𝑅 ⊆ R} . Then, 𝐴 is closed iff 𝐴 = 𝐶𝑙 ( 𝐴 ) .Several conditions can be imposed on Bipolar ABA frameworks,which characterise different sets of assumptions (also called exten-sions ) according to as many semantics . Table 1 gives the semanticsfor Bipolar ABA frameworks we will analyse in the paper, in addi-tion to properties of conflict-freeness and closedness. As a simpleillustration of these semantics, consider a Bipolar ABA frameworkwith L = { 𝛼, 𝛽,𝛾, s 𝛼, s 𝛽, s 𝛾 } , A = { 𝛼, 𝛽,𝛾 } and R = { s 𝛽 ← 𝛾, 𝛾 ← 𝛼 } . Then, { 𝛼 } and { 𝛼, 𝛽 } are not closed (and thus not admissible etc.), { 𝛽 } is closed and conflict-free but not admissible etc., and { 𝛼, 𝛾 } isclosed, conflict-free, admissible (etc.). Note that, in Bipolar ABA(as in flat ABA, but not in general ABA [11]), admissible, preferredand ideal extensions are guaranteed to exist: in particular, sincerules cannot have an empty body, the empty set of assumption isclosed, and thus admissible. Instead, complete, well-founded andset-stable extensions may not exist [12].Bipolar ABA provides an all-encompassing framework for cap-turing different interpretations of support (under admissible, pre-ferred and set-stable semantics) [12], as illustrated in the followingexample, adapted from [21]. Example 1.
The UK public may hold a range of views on Brexit:A: The UK should leave the EU.B: The UK staying in the EU is good for its economy.C: The EU’s immigration policies are bad for the UK’s economy.D: EU membership fees are too high.E: The UK staying in the EU is good for world peace.Here 𝐴 may be deemed to be attacked by 𝐵 and 𝐸 , but supported by 𝐶 and 𝐷 . A Bipolar ABA representation for the deductive interpretation of support [8] is ⟨L , R , A , Ď ⟩ , where With an abuse of notation, we use s 𝑥 to denote the L -sentence amounting to thecontrary of 𝑥 ∈ A and omit to specify the contrary map explicitly. The formal definitions of how different interpretations of support are captured inBipolar ABA are in [12], and outside the scope of this paper. • L = { 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, s 𝐴, s 𝐵, s 𝐶, s 𝐷, s 𝐸 } ; • R = { s 𝐴 ← 𝐵, s 𝐴 ← 𝐸, 𝐴 ← 𝐶, 𝐴 ← 𝐷 } ; • A = { 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 } .Instead, a Bipolar ABA representation for the necessary interpreta-tion of support [20] is ⟨L , R ′ , A , Ď ⟩ with R ′ = { s 𝐴 ← 𝐵, s 𝐴 ← 𝐸, 𝐶 ← 𝐴, 𝐷 ← 𝐴 } . (Other interpretations of support can alsobe represented but are not illustrated here for lack of space). Note that Bipolar ABA can naturally capture supported attacks under the deductive interpretation of support [8], for example thesupported attack from 𝛼 to 𝛽 in a (standard) BAF where 𝛼 supports 𝛾 and 𝛾 attacks 𝛽 is matched by 𝛼 attacking 𝛽 in a Bipolar ABAframework where { s 𝛽 ← 𝛾, 𝛾 ← 𝛼 } ⊆ R . Social choice theory mainly focuses on how to aggregate (peo-ple’s or agents’) opinions into a single collective decision. Broadlyspeaking, there are mainly two types of aggregation: preferenceaggregation [2, 18] and judgement aggregation [18] (other aggrega-tion types exist but are omitted here). Here we focus on the latter,and adapt notions given by [9] to accommodate opinions drawnfrom Bipolar ABA frameworks.Hereafter we assume a set of agents 𝑁 = { , . . . , 𝑛 } ( 𝑛 > s 𝛼 ← 𝛼 (for 𝛼 ∈ A ) will never belong to R ;further elaboration on agents’ rationality, when they are definedargumentatively, can be found in [21].To collectively combine the opinions of all agents, i.e., the agents’Bipolar ABA frameworks, we need aggregation rules.Definition 2. Let ℱ be the set of all Bipolar ABA frameworkswith the same language L , set of assumptions A and contrary map-ping Ď . A Bipolar ABA aggregation rule is a mapping 𝐹 : ℱ 𝑛 → ℱ from 𝑛 Bipolar ABA frameworks into a single Bipolar ABA frame-work. Given 𝑛 (as opinions of agents in 𝑁 = { , . . . , 𝑛 } ) Bipolar ABAframeworks (cid:10) L , R , A , Ď (cid:11) , . . . , (cid:10) L , R 𝑛 , A , Ď (cid:11) , 𝐹 returns a single aggregated Bipolar ABA framework (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) . Inspired by graph aggregation [14], we restrict attention in thispaper to aggregation rules in the form of either quota rules oroligarchic rules, defined below in our setting.
Quota Rules.
These set a quota 𝑞 ∈ 𝑁 as a threshold to acceptsome set of rules 𝑅 ⊆ ℛ = (cid:208) 𝑖 ∈ 𝑁 R 𝑖 , i.e., there should be at least 𝑞 agents that accept the rules in 𝑅 .Definition 3. The quota rule 𝐹 𝑞 , for 𝑞 ∈ 𝑁 , is a Bipolar ABAaggregation rule such that 𝐹 𝑞 ( (cid:10) L , R , A , Ď (cid:11) , . . . , (cid:10) L , R 𝑛 , A , Ď (cid:11) ) = { 𝑟 ∈ ℛ : 𝑟 ∈ (cid:209) 𝑗 ∈ 𝑁 ′ R 𝑗 for 𝑁 ′ ⊆ 𝑁, | 𝑁 ′ | ≥ 𝑞 } . The quota 𝑞 can be any number, but there are several specialquotas that are commonly used: weak majority has a quota 𝑞 = ⌊ 𝑛 ⌋ ; trict majority has a quota 𝑞 = ⌈ 𝑛 ⌉ ; nomination accepts all rulesaccepted by at least 1 agent, i.e., 𝑞 =
1; and unanimity requires allagents to accept the same rules, i.e., 𝑞 = 𝑛 .For example, assume that 𝑛 = R = { s 𝐴 ← 𝐵 } , R = { 𝐴 ← 𝐶 } , and R = { s 𝐴 ← 𝐵, 𝐴 ← 𝐷 } .Using weak majority, the aggregated Bipolar ABA framework hasset of rules R 𝑎𝑔𝑔 = { s 𝐴 ← 𝐵, 𝐴 ← 𝐶, 𝐴 ← 𝐷 } ; with strictmajority, R 𝑎𝑔𝑔 = { s 𝐴 ← 𝐵 } ; nomination gives R 𝑎𝑔𝑔 = { s 𝐴 ← 𝐵, 𝐴 ← 𝐶, 𝐴 ← 𝐷 } , while unanimity returns R 𝑎𝑔𝑔 = {} . Oligarchic Rules.
These give agents the power to veto the ac-cepted rule sets. Clearly, oligarchic rules are not fair in that theopinions of agents without veto power are disregarded. However, insome cases they are necessary to avoid conflicts among the agents.Definition 4.
Let 𝑁 𝑣 ⊆ 𝑁 be the agents with veto power .The oligarchic rule 𝐹 𝑜 is a Bipolar ABA aggregation rule such that 𝐹 𝑜 ( (cid:10) L , R , A , Ď (cid:11) , . . . , (cid:10) L , R 𝑛 , A , Ď (cid:11) ) = { 𝑟 ∈ ℛ : 𝑟 ∈ (cid:209) 𝑗 ∈ 𝑁 𝑣 R 𝑗 } . If | 𝑁 𝑣 | = then the oligarchic rule is called dictatorship . If all agents have veto powers, then the oligarchic rule coincideswith unanimity. As an example, assume, as above, that 𝑛 = R = { s 𝐴 ← 𝐵 } , R = { 𝐴 ← 𝐶 } , and R = { s 𝐴 ← 𝐵, 𝐴 ← 𝐷 } . If agents 1 and 3 are given veto powers,then the aggregated Bipolar ABA framework has set of rules R 𝑎𝑔𝑔 = { s 𝐴 ← 𝐵 } . On the other hand, if all agents have veto powers, then R 𝑎𝑔𝑔 = {} , as with unanimity.We will study whether the properties of the agents’ BipolarABA frameworks, including semantics, conflict-freeness and closed-ness, and other “graph” properties, are preserved in the aggregatedBipolar ABA framework. To produce stronger preservation results,we assume that a property under consideration for an aggregatedBipolar ABA framework (obtained by applying some Bipolar ABAaggregation rule) needs to be satisfied by all agents.Definition 5. Let 𝑃 be a property of Bipolar ABA frameworks. If Δ ⊆ A is 𝑃 in each agents’ Bipolar ABA framework (cid:10) L , R 𝑖 , A , Ď (cid:11) (with 𝑖 ∈ 𝑁 ), then 𝑃 is preserved in the aggregated Bipolar ABAframework F = (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) if and only if Δ is 𝑃 in F . In this section, we present preservation results for several proper-ties 𝑃 (see Definition 5), specifically for 𝑃 equal to conflict-freenessand closedness of sets of assumptions, 𝑃 any of the semantics inTable 1, 𝑃 amounting to assumption acceptability under these se-mantics, and 𝑃 amounting to (implicative and disjunctive) “graph”properties adapted from [14]. Throughout, we will assume that F = (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) is the aggregated Bipolar ABA frameworkresulting from the Bipolar ABA aggregation rules as considered,and, for any such rule, we will say that the rule preserves a property 𝑃 to mean that 𝑃 is preserved in F , as specified in Definition 5. Conflict-freeness is a basic property in argumentation, always pre-served in our setting.Theorem 4.1.
Every quota rule and oligarchic rule preserves con-flict-freeness.
Proof. Assume that Δ ⊆ A is conflict-free in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 . By contradiction, assume that Δ is not conflict-freein F . Then ∃ 𝛼, 𝛽 ∈ Δ such that 𝛼 attacks 𝛽 , i.e., ∃ 𝑅 = { s 𝛽 ← 𝛾 , . . . , 𝛾 𝑚 − ← 𝛾 𝑚 } ⊆ R 𝑎𝑔𝑔 for 𝑚 ≥ { 𝛾 , . . . ,𝛾 𝑚 } ⊆ A and 𝛾 𝑚 = 𝛼 . By definition of quota and oligarchic rules, there has tobe at least one agent 𝑖 ∈ 𝑁 such that 𝑅 ⊆ R 𝑖 and thus 𝛼 attacks 𝛽 in (cid:10) L , R 𝑖 , A , Ď (cid:11) , thus contradicting our assumption that Δ isconflict-free in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 . □ Note that Theorem 4.1 is a direct extension of Theorem 2 in [9]because conflict-freeness in Bipolar ABA frameworks holds underthe same conditions as in AAFs.
Closedness is an important property in Bipolar ABA frameworks,made so by the presence of “support” between assumptions (in theform of rules whose head and body are assumptions). Instead, it isnot meaningful in AA, and indeed it is not studied in [9].Theorem 4.2.
Every quota and oligarchic rule preserves closedness.
Proof. Assume that Δ ⊆ A is closed in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 . By contradiction, assume that Δ is not closed in F . Then ∃ 𝛼 ∈ Δ and 𝛽 ∉ Δ such that 𝛽 ∈ 𝐶𝑙 ({ 𝛼 }) , i.e. there exists 𝑅 = { 𝛽 ← 𝛾 , . . . , 𝛾 𝑚 − ← 𝛾 𝑚 } ⊆ R 𝑎𝑔𝑔 for 𝑚 ≥ { 𝛾 , . . . ,𝛾 𝑚 } ⊆ A and 𝛾 𝑚 = 𝛼 . By definition of quota and oligarchic rules, there has to beat least one agent 𝑖 ∈ 𝑁 such that 𝑅 ⊆ R 𝑖 and thus 𝛽 ∈ 𝐶𝑙 ({ 𝛼 }) in (cid:10) L , R 𝑖 , A , Ď (cid:11) , thus contradicting our assumption that Δ is closedin (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 . □ The preservation result below (Theorem 4.3) for admissibility ex-tends Theorem 3 in [9] by considering also support. It assumesconstraints on the number of assumptions. Theorem 4.4 belowinstead analyses the preservation of admissibility for corner cases.Theorem 4.3.
For |A| ≥ , nomination is the only quota rule thatpreserves admissibility. Proof. First we prove that nomination preserves admissibility.Assume that Δ ⊆ A is admissible in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all agents 𝑖 ∈ 𝑁 . By contradiction, assume that Δ is not admissible in F .Then ∃ 𝛼 ∈ Δ that is attacked by 𝛽 ∈ A \ Δ , i.e., 𝑅 = { s 𝛼 ← 𝛾 , . . . , 𝛾 𝑚 − ← 𝛾 𝑚 } ⊆ R 𝑎𝑔𝑔 , for 𝑚 ≥ 𝛾 𝑚 = 𝛽 , and (cid:154) 𝛾 ∈ Δ such that 𝛾 attacks 𝛽 in F . By definition of nomination rule, 𝑅 ⊆ R 𝑖 for some 𝑖 ∈ 𝑁 and 𝛽 attacks 𝛼 in the Bipolar ABA framework F 𝑖 of agent 𝑖 . Then, given that Δ is admissible in F 𝑖 , ∃ 𝛾 ∈ Δ such that 𝛾 attacks 𝛽 in F 𝑖 , i.e., ∃ 𝑅 ′ = { s 𝛽 ← 𝛿 , . . . , 𝛿 𝑙 − ← 𝛿 𝑙 } ⊆ R 𝑖 , for 𝑙 ≥ 𝛿 𝑙 = 𝛾 . But, by definition of nomination rule, 𝑅 ′ ⊆ R 𝑎𝑔𝑔 ,and thus 𝛾 attacks 𝛽 in F : contradiction. To complete the proof,we need to show that for |A| ≥
4, other quota rules except fornomination do not preserve admissibility. If 𝑁 − 𝑞 agents chooserule R = {} , 𝑞 − R = { s 𝐷 ← 𝐵, s 𝐶 ← 𝐷 } ,and 1 agent chooses rule R = { s 𝐷 ← 𝐴, s 𝐶 ← 𝐷, 𝐴 ← 𝐵 } ,as illustrated in Figure 1, then Δ = { 𝐴, 𝐵, 𝐶 } is admissible in allframeworks. Using quota rules with 𝑞 > R 𝑎𝑔𝑔 = { s 𝐶 ← 𝐷 } and Δ is not admissible anymore as assumption 𝐶 is attacked by 𝐷 and itis undefended by other assumptions in Δ . □ igure 2: Counter Example of the Preservation of Set-stableExtensions (for Theorem 4.5). Here, we use BAFs as a graph-ical representation of Bipolar ABA frameworks (assumingthe deductive interpretation of support). Single-lined edgesare attacks, double-lined edges are supports.Figure 1: Counter Example for the Preservation of Admis-sibility (for Theorem 4.3). Here, we use BAFs as a graphicalrepresentation of Bipolar ABA frameworks (assuming thedeductive interpretation of support). Single-lined edges areattacks, double-lined edges are supports. We remark that for at most three assumptions, quota and oli-garchic rules are guaranteed to preserve admissibility.Theorem 4.4.
For |A| ≤ , every quota rule and oligarchic rulepreserves admissibility. Proof. If |A| =
1, the result holds vacuously. If |A| =
2, assumethat A = { 𝛼, 𝛽 } and Δ = { 𝛼 } is admissible in (cid:10) L , R 𝑖 , A , Ď (cid:11) forall 𝑖 ∈ 𝑁 . Then 𝑟 = s 𝛼 ← 𝛽 ∉ R 𝑖 for all 𝑖 ∈ 𝑁 , and thus 𝑟 ∉ R 𝑎𝑔𝑔 .Then, every quota and oligarchic rule yields { 𝛼 } as an admissibleextension in F . The other cases ( Δ = { 𝛽 } , Δ = {} or Δ = A ) can beproven similarly. If |A| =
3, assume that A = { 𝛼, 𝛽,𝛾 } . Considerthe case where Δ = { 𝛼 } is admissible in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 .By contradiction, assume that Δ is not admissible in F . Then, giventhat Δ is conflict-free and closed in F no matter which aggregationrule, by Theorems 4.1 and 4.2, there are 𝑅 ⊆ R 𝑎𝑔𝑔 and 𝐴 ⊆ { 𝛽, 𝛾 } such that 𝐴 ⊢ 𝑅 s 𝛼 in F . By quota and oligarchic rules, there mustbe 𝑖 ∈ 𝑁 such that 𝑅 ⊆ R 𝑖 ; thus Δ is not admissible for agent 𝑖 :contradiction. The other cases can be proven similarly. □ Note that the restriction to consider at most three assumptionsmay be useful in some settings, e.g., when at most three optionsare up for debate.
The set-stable semantics for Bipolar ABA frameworks generalisesthe stable semantic for AAFs to accommodate supports (see [12]).This generalisation though does not affect preservation. Thus, The-orem 4.5 extends Proposition 5 in [9].Theorem 4.5.
Nomination is the only quota rule that preservesset-stable extensions.
Proof. Assume that Δ ⊆ A is set-stable in (cid:10) L , R 𝑖 , A , Ď (cid:11) forall 𝑖 ∈ 𝑁 . By Theorem 4.1 and 4.2, nomination preserves closednessand conflict-freeness. Therefore, Δ is both closed and conflict-free in F . To be set-stable, Δ has to attack the closure of every assumption 𝛽 not Δ , i.e., ∃ 𝑅 ⊆ R 𝑎𝑔𝑔 such that { 𝛼 } ⊢ 𝑅 s 𝛾 for some 𝛾 ∈ 𝐶𝑙 ({ 𝛽 }) .This is trivially the case if Δ = A . Otherwise, as Δ is set-stable inall agents’ frameworks, then 𝑅 𝑖 ⊆ R 𝑖 must exist, for all 𝑖 ∈ 𝑁 , suchthat { 𝛼 } ⊢ 𝑅 𝑖 s 𝛾 for some 𝛾 ∈ 𝐶𝑙 ({ 𝛽 }) in (cid:10) L , R 𝑖 , A , Ď (cid:11) . Thus, byusing nomination, 𝐶𝑙 ({ 𝛽 }) , for 𝛽 ∈ A \ Δ , is attacked also in F .Other quota rules do not preserve set-stable extensions because,while preserving conflict-freeness and closedness, they do not guar-antee that the closure of every assumption not in the extension isattacked. A counter example follows: assume three Bipolar ABAframeworks with rules R = { s 𝐷 ← 𝐵, 𝐵 ← 𝐴 } , R = { s 𝐷 ← 𝐶 } ,and R = { s 𝐷 ← 𝐴, s 𝐶 ← 𝐷, 𝐴 ← 𝐵 } , as illustrated in Figure 2.In each framework, the set of assumptions { 𝐴, 𝐵, 𝐶 } is set-stable.Using other quota rules with 𝑞 > R 𝑎𝑔𝑔 = {} , and { 𝐴, 𝐵, 𝐶 } is notset-stable anymore as the assumption 𝐷 is not included in it and itis not attacked either ( 𝐶𝑙 ({ 𝐷 }) = { 𝐷 } in this example). □ Assumption acceptability concerns the preferred, complete, set-stable, well-founded, and ideal semantics but at the level of singleassumption rather than full extensions. If an assumption is accept-able in one of those semantics (by belonging to a set of assumptionsaccepted by the semantics) in each of the agents’ framework, thenpreservation amounts to that assumption being still acceptable un-der the same semantics in the aggregated Bipolar ABA framework.Definition 6. (Acceptability of Assumptions) An assumption 𝛼 ∈ A is acceptable (in a Bipolar ABA framework) under preferred,complete, set-stable, well-founded, or ideal semantic iff there is Δ ⊆ A with 𝛼 ∈ Δ such that Δ is (respectively) a preferred, complete, set-stable,well-founded, or ideal extension (in the Bipolar ABA framework). The proof of preservation results regarding acceptability useadaptations of results from [14] on implicative and disjunctive prop-erties to represent impossibility results with dictatorship. We castthese properties for Bipolar ABA frameworks as follows:Definition 7. (Implicative Properties). A Bipolar ABA frameworkproperty 𝑃 is implicative in (cid:10) L , R , A , Ď (cid:11) iff there exist three rules 𝑅 , 𝑅 , 𝑅 ∉ R such that 𝑃 holds in (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) for R 𝑎𝑔𝑔 = R ∪ S , for all
S ⊆ { 𝑅 , 𝑅 , 𝑅 } , except for S = { 𝑅 , 𝑅 } . Intuitively, if the aggregated Bipolar ABA framework includes 𝑅 and 𝑅 as additional rules in S , then it should adopt 𝑅 as wellto preserve property 𝑃 .Definition 8. (Disjunctive Properties). A Bipolar ABA frameworkproperty 𝑃 is disjunctive in (cid:10) L , R , A , Ď (cid:11) iff there exist two rules 𝑅 , 𝑅 ∉ R , such that 𝑃 holds in (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) for R 𝑎𝑔𝑔 = R ∪ S ,for all
S ⊆ { 𝑅 , 𝑅 } , except for S = {} . Intuitively, the aggregated Bipolar ABA framework has to in-clude at least one of 𝑅 or 𝑅 to preserve the property 𝑃 . Definitionsof implicative and disjunctive properties lead us to prove two lem-mas on preservation. igure 3: Acceptability of an Assumption: Implicative case(Left) and Disjunctive case (Right) (for the proof of Theo-rem 4.6). Here, we use BAFs as a graphical representation ofBipolar ABA frameworks (assuming the deductive interpre-tation of support). Single-lined and hyphenated edges areattacks, double-lined edges are supports. Lemma 1.
Let a Bipolar ABA framework property 𝑃 be implicativein (cid:10) L , R 𝑖 , A , Ď (cid:11) , for each 𝑖 ∈ 𝑁 . Then, unanimity preserves 𝑃 . Proof. Let R ⊇ S , . . . , R 𝑛 ⊇ S 𝑛 and let S 𝑖 ⊆ { 𝑅 , 𝑅 , 𝑅 } for all 𝑖 ∈ 𝑁 . Let S 𝑖 ≠ { 𝑅 , 𝑅 } for all 𝑖 ∈ 𝑁 . Then, unanimity preserves 𝑃 because it is impossible to get S 𝑎𝑔𝑔 = { 𝑅 , 𝑅 } , with R 𝑎𝑔𝑔 ⊇ S 𝑎𝑔𝑔 . □ Note that, even if 𝑃 is implicative, nomination and majority donot preserve 𝑃 in general, as it is possible to get S 𝑎𝑔𝑔 = { 𝑅 , 𝑅 } .For example, let S = { 𝑅 } , and S = { 𝑅 } . Using nomination andmajority, R 𝑎𝑔𝑔 ⊇ S 𝑎𝑔𝑔 = { 𝑅 , 𝑅 } ; hence, 𝑃 is not guaranteed to bepreserved.Lemma 2. Let a Bipolar ABA framework property 𝑃 be implicativeand disjunctive in (cid:10) L , R 𝑖 , A , Ď (cid:11) , for each 𝑖 ∈ 𝑁 . Then, the onlyBipolar ABA aggregation rule that preserves 𝑃 is dictatorship. Proof. The proof for the implicativeness can be found in Lemma1. For the disjunctiveness, let R ⊇ S , . . . , R 𝑛 ⊇ S 𝑛 and let S 𝑖 ⊆{ 𝑅 , 𝑅 } for all 𝑖 ∈ 𝑁 . Let S 𝑖 ≠ {} for all 𝑖 ∈ 𝑁 . Then, nominationand majority preserve 𝑃 because it is impossible to get S 𝑎𝑔𝑔 = {} ,with R 𝑎𝑔𝑔 ⊇ S 𝑎𝑔𝑔 . As 𝑃 is implicative and disjunctive, 𝑃 is preservedonly with dictatorship. None of the quota rules preserve 𝑃 as usingnomination or majority rule, it is possible to get S 𝑎𝑔𝑔 = { 𝑅 , 𝑅 } and violating the implicativeness; and using unanimity rule, it ispossible to get S 𝑎𝑔𝑔 = {} and violating the disjunctiveness. □ Notice that, in the definition of implicative and disjunctive prop-erties, the rules 𝑅 , 𝑅 , and (if applicable) 𝑅 can only be in theform of s 𝛼 ← 𝛽 for some 𝛼, 𝛽 ∈ A , thus bringing attacks betweenassumptions. They cannot be in the form of 𝛼 ← 𝛽 that denotesupports between assumptions because then some agents may havedifferent closures of assumptions from the other agents. As a con-sequence, some agents’ Bipolar ABA frameworks may satisfy 𝑃 ,while some others may not because of closedness.The preservation result on the acceptability of an assumptionin Theorem 4.6 below is an extension, within our more generalsetting, of Theorem 1 in [9]. This result is true for all five semantics:preferred, complete, set-stable, well-founded, or ideal. If the agents’Bipolar ABA frameworks have no (rules for) support, then thisTheorem 4.6 is the same as Theorem 1 in [9].Theorem 4.6. For |A| ≥ , the only Bipolar ABA aggregationrule that preserves the acceptability of an assumption under preferred,complete, set-stable, well-founded, or ideal semantic is dictatorship. Proof. Let 𝑃 be acceptability of an assumption under preferred,complete, set-stable, well-founded, or ideal semantics. We need toprove that for |A| ≥ 𝑃 is implicative and disjunctive. Then, byLemma 2, the theorem holds. The proof has the same structure foreach of the five semantics. Consider a set of at least four assumptions A = { 𝐴, 𝐵, 𝐶, 𝐷, . . . } .To show that 𝑃 is implicative, let 𝐵 be the accepted assumption.Let R = { s 𝐶 ← 𝐴, 𝐷 ← 𝐴 } , 𝑅 = { s 𝐵 ← 𝐶 } , 𝑅 = { s 𝐴 ← 𝐵 } ,and 𝑅 = { s 𝐶 ← 𝐷 } (see the left graph of Figure 3). Consider anaggregated framework with R 𝑎𝑔𝑔 = R ∪ S with
S ⊆ { 𝑅 , 𝑅 , 𝑅 } .If S = {} , { 𝑅 } , { 𝑅 } , or { 𝑅 , 𝑅 } then 𝐵 is unattacked. If S = { 𝑅 } , { 𝑅 , 𝑅 } , or { 𝑅 , 𝑅 , 𝑅 } then 𝐵 is defended by other assumptions.Therefore, 𝐵 is either unattacked or defended in all seven cases, and 𝐵 is acceptable under preferred, complete, set-stable, well-founded,and ideal semantics. However, if S = { 𝑅 , 𝑅 } , { 𝐴, 𝐵, 𝐶 } forms cyclicattacks so that the assumptions 𝐴, 𝐵 , and 𝐶 are not acceptable underpreferred, complete, set-stable, well-founded, and ideal semantics.Thus, we have identified a set of rules R and three rules 𝑅 , 𝑅 , 𝑅 such that 𝑃 holds in (cid:10) L , R ∪S , A , Ď (cid:11) iff S ≠ { 𝑅 , 𝑅 } . Accordingly, 𝑃 is implicative.To show that 𝑃 is disjunctive, let 𝐵 be the accepted assumption.Let R = { s 𝐵 ← 𝐴, 𝐷 ← 𝐶 } , 𝑅 = { s 𝐴 ← 𝐶 } , and 𝑅 = { s 𝐴 ← 𝐷 } ( see the right graph of Figure 3). Consider R 𝑎𝑔𝑔 = R ∪ S with
S ⊆ { 𝑅 , 𝑅 } . If S = { 𝑅 } , { 𝑅 } , or { 𝑅 , 𝑅 } then the assumption 𝐵 is defended. Therefore, 𝐵 is acceptable under the five semantics.However, if S = {} , the assumption 𝐵 is attacked by 𝐴 and is notdefended, thus 𝐵 is unacceptable under preferred, complete, set-stable, well-founded, and ideal semantics. Thus, we have identifieda set of rules R and two rules 𝑅 , 𝑅 such that 𝑃 holds in (cid:10) L , R ∪S , A , Ď (cid:11) iff S ≠ {} . Therefore, 𝑃 is disjunctive. □ Theorem 4.6 shows that it is not easy to even preserve the ac-ceptability of one assumption, as dictatorship is needed. We willsee, in Section 4.6 below, that it is even more difficult to preservewhole extensions. On the other hand, for |A| ≤
3, acceptability ofan assumption can be preserved with both quota and oligarchicrules.Theorem 4.7.
For |A| = , majority, unanimity, and oligarchicrules preserve assumption acceptability under preferred, complete,set-stable, well-founded, and ideal semantics. Proof. Let a Bipolar ABA framework property 𝑃 be the accept-ability of an assumption under preferred, complete, set-stable, well-founded, or ideal semantics. Assume that 𝑃 holds in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 , where A = { 𝛼, 𝛽,𝛾 } and assume that 𝛼 is accept-able under preferred, complete, set-stable, well-founded, and idealsemantics in all frameworks.By contradiction, assume 𝑃 does not hold in F . In other words, ∃ 𝑅 ⊆ R 𝑎𝑔𝑔 such that { 𝛿 } ⊢ 𝑅 s 𝛼 and not ∃ 𝑅 ⊆ R 𝑎𝑔𝑔 such that { 𝜃 } ⊢ 𝑅 s 𝛿 for some 𝜃 ∈ { 𝛽, 𝛾 } and 𝛿 ∈ { 𝛽, 𝛾 } , 𝜃 ≠ 𝛿 . As a result, 𝛼 isnot acceptable under preferred, complete, set-stable, well-founded,and ideal semantics. By definition of of majority rule, unanim-ity rule, and oligarchic rules, the deduction from rules { 𝛿 } ⊢ 𝑅 s 𝛼 must exist in the majority (majority rule), all (unanimity rule), orveto powered (oligarchic rules) agents’ frameworks, but there isat least one framework (cid:10) L , R 𝑖 , A , Ď (cid:11) for some 𝑖 ∈ 𝑁 without the igure 4: Preferred, Complete, Well-founded, and Ideal ex-tensions: Implicative case (Left) and Disjunctive case (Right)(for the proof of Theorem 4.9). Here, we use BAFs as a graph-ical representation of Bipolar ABA frameworks (assumingthe deductive interpretation of support). Single-lined andhyphenated edges are attacks, double-lined edges are sup-ports. deduction { 𝜃 } ⊢ 𝑅 s 𝛿 for 𝜃 ≠ 𝛿 , thus 𝛼 is not acceptable under pre-ferred, complete, set-stable, well-founded, and ideal semantics inthe agents’ frameworks as well. It contradicts the initial assumptionthat 𝛼 is acceptable in (cid:10) L , R 𝑖 , A , Ď (cid:11) for all 𝑖 ∈ 𝑁 .To show that for |A| =
3, nomination rule do not preserve theassumption acceptability under preferred, complete, set-stable, well-founded, or ideal semantics, a counter example is given. Take threeBipolar ABA frameworks with rules R = { s 𝐴 ← 𝐶, s 𝐵 ← 𝐶 } , R = { s 𝐵 ← 𝐴, s 𝐶 ← 𝐵 } , and R = { s 𝐶 ← 𝐴, s 𝐴 ← 𝐵 } . Letassumption 𝐶 be the accepted assumption in check. From the firstframework, a set of assumptions { 𝐶 } is preferred, complete, set-stable, well-founded, and ideal. On the second framework is { 𝐴, 𝐶 } and third framework is { 𝐵, 𝐶 } , both extensions are preferred, com-plete, set-stable, well-founded, and ideal as well. In all three frame-works, the assumption 𝐶 is acceptable. It is still acceptable usingunanimity rule as the aggregated rule is R = {} . However, usingnomination rule the preferred, complete, and set-stable extensionsare { 𝐴 } , { 𝐵 } , and { 𝐶 } ; while the well-founded and ideal extensionsare {} . Hence, { 𝐶 } is not acceptable in those five semantics. □ Theorem 4.8.
For |A| ≤ , every quota and oligarchic rule pre-serves the acceptability of an assumption under preferred, complete,set-stable, well-founded, and ideal semantics. The proof of preservation for the preferred, complete, well-founded,and ideal semantics uses the concept of implicative and disjunctiveproperties from Section 4.5. The preservation result is an extensionof Theorem 4 in [9] with the addition of ideal semantics.Theorem 4.9.
For |A| ≥ , the only Bipolar ABA aggregation rulethat preserves preferred, complete, well-founded, and ideal semanticsis dictatorship. Proof. Let 𝑃 be that an extension Δ ⊆ A is preferred, complete,well-founded, or ideal. We need to prove that for |A| ≥ 𝑃 isimplicative and disjunctive. Then by Lemma 2, the theorem holds.The proof has the same structure for all four semantics. It uses ageneric A = { 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, . . . } with at least five assumptions.To show that 𝑃 is implicative, let Δ = { 𝐵, 𝐷, 𝐸 } . Let R = { s 𝐶 ← 𝐷, s 𝐴 ← 𝐵, 𝐸 ← 𝐷 } , 𝑅 = { s 𝐵 ← 𝐶 } , 𝑅 = { s 𝐷 ← 𝐴 } , and 𝑅 = { s 𝐴 ← 𝐸 } (see the left graph of Figure 4). Consider R 𝑎𝑔𝑔 = R ∪ S with
S ⊆ { 𝑅 , 𝑅 , 𝑅 } . If S = {} , { 𝑅 } , { 𝑅 } , { 𝑅 } , { 𝑅 , 𝑅 } , { 𝑅 , 𝑅 } ,or { 𝑅 , 𝑅 , 𝑅 } . Then Δ is preferred, complete, well-founded, andideal. However, if S = { 𝑅 , 𝑅 } , { 𝐴, 𝐵, 𝐶, 𝐷 } forms cyclic attackssuch that Δ is not preferred, complete, well-founded, or ideal. Thus,we have identified a set of rules R and three rules 𝑅 , 𝑅 , 𝑅 suchthat 𝑃 holds in (cid:10) L , R ∪ S , A , Ď (cid:11) iff S ≠ { 𝑅 , 𝑅 } . Hence, 𝑃 isimplicative.To show that 𝑃 is disjunctive, let Δ = { 𝐵, 𝐷, 𝐸 } . Let R = { s 𝐶 ← 𝐷, s 𝐵 ← 𝐶, s 𝐴 ← 𝐵, s 𝐷 ← 𝐴, 𝐷 ← 𝐸 } , 𝑅 = { s 𝐶 ← 𝐸 } ,and 𝑅 = { s 𝐴 ← 𝐸 } (see the right graph of Figure 4). Consider R 𝑎𝑔𝑔 = R ∪ S with
S ⊆ { 𝑅 , 𝑅 } . If S = { 𝑅 } , { 𝑅 } , or { 𝑅 , 𝑅 } ,then Δ is preferred, complete, well-founded, and ideal. However, if S = {} , { 𝐴, 𝐵, 𝐶, 𝐷 } forms cyclic attacks such that Δ is not preferred,complete, well-founded, or ideal. Therefore, we have identified aset of rules R and two rules 𝑅 , 𝑅 such that 𝑃 holds in (cid:10) L , R ∪S , A , Ď (cid:11) iff S ≠ {} . Thus, 𝑃 is disjunctive. □ Although preferred and complete semantics may accept multipleextensions, as long as all agents agree on the extensions, then The-orem 4.9 still holds. The only restriction is the presence of supportsin the agents’ frameworks. If all agents agree on the supports, i.e.,the supports are included in R 𝑖 for all 𝑖 ∈ 𝑁 , then support does notaffect the preservation. Otherwise, some agents will have differentsets of closed assumptions from the other agents. This may lead intodifferent preferred, complete, well-founded, and ideal extensions.The corner cases show an impossibility result for |A| = |A| =
4. Thus, preserving whole extensions is more difficult thanpreserving the acceptability of single assumptions as in Section 4.5.Theorem 4.10.
For |A| = and |A| = , quota and oligarchicrules do not preserve preferred, complete, well-founded, and idealsemantics. Theorem 4.11.
For |A| ≤ , every quota and oligarchic rulepreserves preferred, complete, well-founded, and ideal semantics. The well-founded extension is guaranteed to exist in a Bipolar ABAframework. However, to make sure that the well-founded extensionis not empty, then the framework must have at least one unattackedassumption. This way, the unattacked assumptions are includedin all complete extensions, and the intersection always has theunattacked assumptions in it. The preservation of non-emptiness ofthe well-founded extension guarantees the existence of unattackedassumption with a concept called k-exclusivity [9].Definition 9. (k-exclusivity). Let 𝑃 be a property of Bipolar ABAframework. 𝑃 is k-exclusive if there exist rules S = { 𝑅 , . . . , 𝑅 𝑘 } suchthat if R ⊇ S then 𝑃 does not hold, but if R ⊂ S then 𝑃 holds. Thus, to preserve 𝑃 , the rules S cannot be adopted together, butonly a subset of them. It leads us to the lemma for the preservation.Lemma 3. Let 𝑃 be a k-exclusive property of Bipolar ABA frame-work. For 𝑘 ≥ 𝑁 , where 𝑁 is the number of agents, 𝑃 is preserved ifat least one of the 𝑁 agents has veto power. Proof. It needs to be showed that if an aggregation rule pre-serves 𝑃 , then it has to give at least one agent with veto powers. igure 5: Graphical illustration of the k-exclusivity property(for the proof of Theorem 4.12) Notice that if all agents accept a rule 𝑟 , then it must be accepted inthe aggregated rules, i.e., 𝑟 ∈ R 𝑎𝑔𝑔 iff 𝑟 ∈ R 𝑖 for all 𝑖 ∈ 𝑁 .For some agents 𝑖 ∈ 𝑁 to have veto powers means that R 𝑎𝑔𝑔 = ( (cid:209) R 𝑖 ) . In other words, some agents have veto power, if the inter-section of the agents’ rules in (cid:209) R 𝑖 are all accepted in R 𝑎𝑔𝑔 . Then,take any rule 𝑟 ∈ R 𝑎𝑔𝑔 ; as 𝑟 is accepted in the aggregated frame-work, then all agents with veto powers must accept 𝑟 as well suchthat the intersection of the set of rules (cid:209) R 𝑖 is not empty.Thus, the next step is to show that if an aggregation rule pre-serves 𝑃 , then the intersection of 𝑘 set of rules must be non-empty,i.e., R ∩ . . . ∩ R 𝑘 ≠ {} . To prove by contradiction, assume thereexist a profile of set of rules {R ∪ . . . ∪ R 𝑘 } ⊆ R 𝑎𝑔𝑔 such that R ∩ . . . ∩ R 𝑘 = {} . Then, it means that for every 𝑗 ∈ { , . . . , 𝑘 } , ex-actly (the agent with rule set) R 𝑗 accepts a rule 𝑟 𝑗 . As no rule exist inall R 𝑖 for 𝑖 ∈ 𝑁 , no agents accept all 𝑘 rules. However, as each of the 𝑘 rules is accepted by an agent and {R ∪ . . . ∪R 𝑘 } ⊆ R 𝑎𝑔𝑔 , they areall accepted in the aggregated framework, i.e., { 𝑟 , . . . , 𝑟 𝑘 } ⊆ R 𝑎𝑔𝑔 ,such that 𝑃 does not hold due to it being an k-exclusive property.This contradicts the initial assumption that the aggregation rulepreserves 𝑃 .Therefore, as it can be showed that the intersection of the agents’rules is not empty, then some agents must have veto powers. □ Theorem 4.12.
For |A| ≥ 𝑁 , at least one agent must have vetopower to preserve the non-emptiness of the well-founded extension. Proof. Let a Bipolar ABA framework property 𝑃 be the non-emptiness of the well-founded extension. We need to show that 𝑃 is k-exclusive . Let 𝑘 = |A| and { 𝐴 , . . . , 𝐴 𝑘 } ⊆ A . Assume that S consists of all rules Ě 𝐴 𝑖 + ← 𝐴 𝑖 for 𝑖 < |A| as well as Ď 𝐴 ← 𝐴 𝑘 , il-lustrated in Figure 5. This S fits the definition of k-exclusive . Indeed,if S ⊆ R , then in the case of S = R , the well-founded extensionis empty due to the cyclic attacks. However, if only a subset of itis adopted, R ⊂ S , the well-founded extension is not empty as atleast one assumption is not attacked. Thus, 𝑃 is preserved when atleast one agent has veto power to prevent cyclic attacks. □ Supports in Bipolar ABA framework do not affect the preser-vation of non-emptiness of the well-founded extension becausesupports between assumptions do not affect the unattacked as-sumption: if there is a rule s 𝛼 ← 𝛽 for 𝛼, 𝛽 ∈ A and 𝛽 is unattacked,then supports from and into 𝛽 do not change the fact that 𝛽 isunattacked; and supports from and into 𝛼 also leave 𝛽 unattacked. It is clear that k-exclusivity deals with cyclic attacks. A BipolarABA framework is acyclic if there does not exist any cyclic attacks among the assumptions. Corollary 1 is extended from Theorem 8in [9] in the way that supports are also considered.Definition 10. (Cyclic Attacks) The rule set R in (cid:10) L , R , A , Ď (cid:11) contain cyclic attacks if there exist a chained connection betweensome of the assumptions in A , such that R ⊇ { s 𝛼 ← 𝛼 , s 𝛼 ← 𝛼 , . . . , Ď 𝛼 𝑘 ← 𝛼 } for 𝛼 𝑖 ∈ A and 𝑘 ≥ . The preservation result for acyclicity has a similar proof struc-ture as the preservation of the non-emptiness of the well-foundedextension in Theorem 4.12. Thus, it is presented as a corollary.Corollary 1.
For |A| ≥ 𝑁 , at least one agent must have vetopower to preserve acyclicity. Proof. Let 𝑃 be acyclicity. We need to show that 𝑃 is k-exclusive .To get a cycle, a minimum number of two assumptions are required.Thus, let 𝑘 = |A| ≥ { 𝐴 , . . . , 𝐴 𝑘 } ⊆ A . Assume that therule set S consists of Ě 𝐴 𝑖 + ← 𝐴 𝑖 for 𝑖 < |A| as well as Ď 𝐴 ← 𝐴 𝑘 ,illustrated in Figure 5. This S fits the definition of k-exclusivity .Indeed, if S ⊆ R , then in the case of S = R , the cyclic attacksremain in the framework. However, if only a subset of S is adopted( R ⊂ S ), then the cyclic attacks are broken because at least onerule that connects the cycle disappears. Therefore, 𝑃 is preservedwhen at least one agent has veto power. □ The presence of supports does not make an acyclic frameworkto become cyclic, but instead may break any existing cycle. Let 𝑘 = |A| with |A| ≥ { 𝐴 , . . . , 𝐴 𝑘 } ⊆ A , and S = { Ě 𝐴 𝑖 + ← 𝐴 𝑖 : 𝑖 < |A|} . The rules in S are acyclic and if a support 𝐴 ← 𝐴 𝑘 or 𝐴 𝑘 ← 𝐴 is added, for example, then they will remain acyclic. Onthe contrary, if there exist cyclic attacks, then support may breakthe cycle due to closedness. Coherence amounts to two or more semantics coinciding (in otherwords, given a Bipolar ABA framework, two or more semantics giveidentical extensions thereof). For example, if a set of assumptions isset-stable, then it is preferred as well. Our next preservation resultextends Theorem 9 in [9] and shows that, in order to preservecoherence, the aggregation rule must be dictatorial. The proof forthe result uses the concept of implicativeness and disjunctivenessintroduced in Section 4.5.Theorem 4.13.
For |A| ≥ , the only aggregation rule preservingcoherence is dictatorship. Proof. Let 𝑃 be coherence. We need to prove that, for |A| ≥ 𝑃 is implicative and disjunctive. Take a Bipolar ABA frameworkwith at least four assumptions A = { 𝐴, 𝐵, 𝐶, 𝐷, . . . } .To show that 𝑃 is implicative, let R = { s 𝐶 ← 𝐴, 𝐷 ← 𝐴 } , 𝑅 = { s 𝐵 ← 𝐶 } , 𝑅 = { s 𝐴 ← 𝐵 } , and 𝑅 = { s 𝐶 ← 𝐷 } , as illustrated in theleft graph of Figure 6. Consider an aggregated framework with S ⊆{ 𝑅 , 𝑅 , 𝑅 } . If S = {} , { 𝑅 } , { 𝑅 } , or { 𝑅 , 𝑅 } , the only preferredextension is { 𝐴, 𝐵, 𝐷 } , which is set-stable as well. If S = { 𝑅 } ,the set of assumptions { 𝐵, 𝐶, 𝐷 } is both preferred and set-stable. If S = { 𝑅 , 𝑅 } or { 𝑅 , 𝑅 , 𝑅 } ; then the set of assumptions { 𝐵, 𝐷 } isboth preferred and set-stable as well. However, if S = { 𝑅 , 𝑅 } , theonly preferred extension is { 𝐷 } and it is not set-stable as the otherassumptions are not attacked. Thus, there exists a set of rules R igure 6: Coherence: Implicative case (Left) and Disjunctivecase (Right) (for the proof of Theorem 4.13). Here, we useBAFs as a graphical representation of Bipolar ABA frame-works (assuming the deductive interpretation of support).Single-lined and hyphenated edges are attacks, double-linededges are supports. and three rules 𝑅 , 𝑅 , 𝑅 such that 𝑃 holds in (cid:10) L , R ∪ S , A , Ď (cid:11) iff S ≠ { 𝑅 , 𝑅 } . Accordingly, 𝑃 is an implicative property.To show that 𝑃 is disjunctive, let R = { s 𝐴 ← 𝐷, s 𝐵 ← 𝐴, s 𝐷 ← 𝐵, 𝐶 ← 𝐴 } , 𝑅 = { s 𝐷 ← 𝐶 } , and 𝑅 = { s 𝐵 ← 𝐶 } , as illustratedin the right graph of Figure 6. Consider an aggregated framework (cid:10) L , R 𝑎𝑔𝑔 , A , Ď (cid:11) , where R 𝑎𝑔𝑔 = R ∪ S with
S ⊆ { 𝑅 , 𝑅 } . If S = { 𝑅 } or { 𝑅 , 𝑅 } , the set of assumptions { 𝐴, 𝐶 } is both preferredand set-stable. If S = { 𝑅 } , the set of assumptions { 𝐶, 𝐷 } is alsopreferred and set-stable. However, if S = {} , the preferred extensionis { 𝐶 } and it is not set-stable because the other assumptions arenot attacked. Therefore, there exists a set of rules R and two rules 𝑅 , 𝑅 such that 𝑃 holds in (cid:10) L , R ∪ S , A , Ď (cid:11) iff S ≠ {} . Hence, 𝑃 is a disjunctive property.As 𝑃 is proven to be both implicative and disjunctive, then byLemma 2, for 𝑃 to be preserved, the aggregation rule must be dicta-torial. □ Note that Theorem 4.13 also works for other semantics (indeed,in the proof, the accepted sets of assumptions may be complete andwell-founded, rather than just preferred and set-stable).The presence of supports is acceptable in the preservation ofcoherence only if the supports are adopted by each agent, such thatall agents have the same closure of assumptions. If supports jointhe additional rules in S as either 𝑅 , 𝑅 , or 𝑅 , then coherence isnot preserved in the aggregated framework as some agents havedifferent set of closures than the other agents.For corner cases, it is easier to preserve coherence, as indeedunanimity preserves it for |A| ≤
3. Moreover, admittedly lessinterestingly, both quota and oligarchic rules preserve coherencewhen there is one assumption.Theorem 4.14.
For |A| = or |A| = , unanimity rule is the onlyquota rule that preserves coherence. Theorem 4.15.
For |A| = , every quota rule and oligarchic rulepreserves coherence. We have considered Bipolar ABA Frameworks [12] to account forboth attack and support relationships between arguments in BipolarArgumentation, as it allows to capture uniformly different interpre-tations of support. The aggregation of Bipolar ABA Frameworkscombines the rules of all agents into a collective set of rules. Wemade use of the aggregation rules from judgement aggregation [17, 18], specifically, quota and oligarchic rules, to combine theseagents’ rules and extended results (on Abstract Argumentation)from [9]. Generally, the preservation results show that most prop-erties can be preserved, but with significant restrictions sometimes.The results assume agreement among the agents on language, as-sumptions, contraries, and assume that agents accept the sameproperties (Definitions 2 and 5). We observe that, when the notionof agreement comes into play, the presence of supports does notgreatly affect the performance of the aggregation rules towardspreservation.As regards positive results, conflict-freeness and closedness arepreserved by any quota and oligarchic rule (Theorems 4.1 and 4.2).We proved positive results for admissibility and set-stability aswell, albeit with some restrictions, such as limiting the number ofassumptions or the choice of aggregation rules. Admissibility ispreserved by nomination for at least four assumptions, else it ispreserved by every quota and oligarchic rule; while the set-stablesemantics is preserved by nomination (Theorems 4.3, 4.4, and 4.5).We show that some properties can only be preserved by oli-garchic rules or dictatorship. These particular aggregation rules areactually not ideal, as they ignore most opinions. However, we stilldeem this better than not being able to preserve the properties at all.For the properties of acceptability of an assumption, and coherencewhen the number of assumptions is at least four, dictatorship isthe only preserving rule (Theorems 4.6 and 4.13). The same holdsfor preferred, complete, well-founded, and ideal semantics, but byassuming at least five assumptions (Theorem 4.9). Unsurprisingly,in the corner cases, these properties can be preserved with otherquota rules (Theorems 4.7, 4.8, 4.10, 4.11 , 4.14, and 4.15).Preservation results also involve the non-emptiness of the well-founded extension and acyclicity. We proved that both propertiesare preserved when at least one agent has veto power and thenumber of assumptions is greater or equal than the number ofagents (Theorem 4.12 and Corollary 1). This unique constraint ismeant to avoid cyclic relationships.To conclude, our preservation study produces stronger results tofill the gaps in [9] since we consider more properties, some relevantto Bipolar Argumentation only (closedness) others also relevant toAbstract Argumentation (ideal semantics); we also provide preser-vation results for corner cases.There are several possible directions for future work. First of all,here the preservation of properties relies on the agreement of allagents. However, in real applications it is likely that some agentshave different opinions, i.e., some of them might disagree on theproperties. Thus, it would be interesting to study preservation whena number of agents disagree. Another path to work on in the futureis to have agents with different knowledge about the environment,meaning that they might have different languages, assumptions,or contraries. A further promising direction for future work isto expand the choice of aggregation rules with a more complexformalisation. Finally, it would be worth to generalise this study forthe more general ABA Frameworks of [6, 11], as well as for otherforms of structured argumentation, such as ASPIC [19], DeLP [16]or logic-based argumentation [3]. In particular, the possibility ofhaving rules with empty body might need specific attention whenit comes to aggregation.
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APPENDIX
Proof of Theorem 4.8. If |A| =
1, the result holds vacuouslyas 𝛼 ∈ A must be accepted under all five semantics. If |A| = 𝛼 ∈ A is accepted under those semantics, then 𝛼 must be acceptedin the aggregated framework as there cannot exist a deduction { 𝛽 } ⊢ 𝑅 s 𝛼 for 𝛽 ∈ A \ { 𝛼 } with 𝑅 ⊆ R 𝑎𝑔𝑔 . By definition of quotarules and oligarchic rules, then no such 𝑅 can exist in the agents’rule sets either. Thus, 𝛼 is accepted by all agents. □ Proof of Theorem 4.10. Let 𝑃 be any property amongst beinga preferred, complete, well-founded, or ideal extension. To showthat 𝑃 is not preserved by quota rules, we give counter examples.For |A| =
3, assume four Bipolar ABA frameworks with two frame-works having rules R , = { s 𝐵 ← 𝐴, s 𝐶 ← 𝐵 } and the other twoframeworks having rules R , = { s 𝐴 ← 𝐵, s 𝐵 ← 𝐶 } . The setof assumptions Δ = { 𝐴, 𝐶 } is preferred, complete, well-founded,and ideal in each framework. However, using the unanimity rule( 𝑞 =
4) or oligarchic rule with veto powers given to all frame-works, { 𝐴, 𝐵, 𝐶 } is well-founded and ideal, as well as the only exten-sion to be preferred, complete, in the aggregated framework, giventhat R 𝑎𝑔𝑔 = {} . With majority or nomination rule, the extensions { 𝐴, 𝐶 } and { 𝐵 } are preferred, { 𝐴, 𝐶 } , { 𝐵 } , and {} are complete, and {} is well-founded and ideal in the aggregated framework, with R 𝑎𝑔𝑔 = { s 𝐵 ← 𝐴, s 𝐶 ← 𝐵, s 𝐴 ← 𝐵, s 𝐵 ← 𝐶 } . Thus, Δ (and so 𝑃 ) is not preserved using quota rules and oligarchic rules.For |A| =
4, assume three Bipolar ABA frameworks with rules R = { s 𝐴 ← 𝐷, s 𝐷 ← 𝐵, s 𝐶 ← 𝐷 } , R = { s 𝐴 ← 𝐷, s 𝐵 ← 𝐷, s 𝐷 ← 𝐶 } , and R = { 𝐷 ← 𝐴 } . Δ = { 𝐴, 𝐵, 𝐶 } is preferred, com-plete, well-founded, and ideal in each framework. However, usingunanimity ( 𝑞 =
3) or oligarchic rules with veto powers given to allagents, the well-founded and ideal extension, as well as the only pre-ferred, complete extension, is { 𝐴, 𝐵, 𝐶, 𝐷 } in the aggregated frame-work, as R 𝑎𝑔𝑔 = {} . With majority ( 𝑞 = { 𝐵, 𝐶, 𝐷 } is preferred,complete, well-founded, and ideal in the aggregated framework,with R 𝑎𝑔𝑔 = { s 𝐷 ← 𝐴 } . Lastly, nomination gives { 𝐴, 𝐵, 𝐶 } and { 𝐷 } as the preferred extensions, { 𝐴, 𝐵, 𝐶 } , { 𝐷 } , and {} as the completeextensions, and {} as the well-founded and ideal extension. Thus, Δ (and so 𝑃 ) is not preserved using quota or oligarchic rules. □ Proof of Theorem 4.11. If |A| =
1, the result holds vacuouslyas the only extension is Δ = { 𝛼 } for 𝛼 ∈ A . If |A| =
2, if Δ = { 𝛼 } is preferred, complete, well-founded, or ideal in all the agents’frameworks, then, for each 𝑖 ∈ 𝑁 , there must be a deduction { 𝛼 } ⊢ 𝑅 𝑖 s 𝛽 . Thus, any quota and oligarchic rules preserve 𝑃 . The other cases( Δ = { 𝛽 } and Δ = { 𝛼, 𝛽 } ) can be proven similarly. □ Proof of Theorem 4.14. Assume that coherence holds for eachagent, for an extension Δ . By contradiction, assume that Δ is notcoherent in the aggregated F . By unanimity, R 𝑖 ⊇ R 𝑎𝑔𝑔 for all 𝑖 ∈ 𝑁 . Thus, Δ is not coherent in the agents’ frameworks, leadingto a contradiction. □ Proof of Theorem 4.15. If |A| = R 𝑖 = {} (for all 𝑖 ∈ 𝑁 ) and { 𝛼 } is coherent as it is closed, conflict-free, admissible,preferred, complete, set-stable, well-founded, and ideal. Therefore,using any quota or oligarchic rule, coherence also holds for theaggregated framework.is coherent as it is closed, conflict-free, admissible,preferred, complete, set-stable, well-founded, and ideal. Therefore,using any quota or oligarchic rule, coherence also holds for theaggregated framework.