We present a general framework to rank assumptions in assumption-based argumentation frameworks (ABA frameworks), relying on their relationship to other assumptions and the syntactical structure of the ABA framework. We define general principles for assessing the suitability of ranking-based semantics and propose a new family of semantics for ABA frameworks that is using reductions to the abstract argumentation setting and leveraging existing ranking-based semantics for abstract argumentation. We show that this family complies with many of our principles.
works induce a binary classification of arguments resp. assumptions: an argument or assumption is either acIn recent years, formal argumentation [1] has gained cepted or not. This can be considered too limiting in real attention as a rational decision-making model. Formal world scenarios like online debates [8]. For AFs, rankingargumentation is concerned with the representation of based semantics [9, 10] were introduced to overcome this arguments and their relationships. One important ap- limitation, where a ranking over the arguments based on proach is the abstract argumentation framework (AF) their individual strength is established. Hence, we can by Dung [2]. This framework uses directed graphs to not only state that an argument is part of an acceptable represent arguments as nodes and attacks between two set or not, but infer that one argument is “better” than arguments as edges between these two arguments, where another one. the source of an edge attacks the target. One way to rea- The ranking-based approach does not only allow us son with AFs is by using extension-based semantics, which to establish whether one assumption is “better” than anare relying on functions allowing us to state when a set other one, but additionally we can use it to refine other of arguments is jointly acceptable. reasoning methods. Let us assume we have two sets of as
In addition to AFs, other models of rational decision- sumptions 1 and 2 which are acceptable with respect making using argumentative reasoning were defined in to some extension-based semantics. Say these two sets the literature. One of them are assumption-based ar- are in a conflict with each other, i. e., we cannot accept gumentation frameworks (ABA frameworks) [3, 4, 5, 6]. both sets at the same time. With the help of the individual These frameworks are based on deductive systems over strength of each assumption, we can identify the “better” a formal language and rules. One particular part of the set of assumptions between them. For example 1 might formal language are the so-called assumptions, which contain assumptions ranked higher than assumptions of are used as the basis for deriving further pieces of in- 2, hence we can consider 1 to be better than 2. formation. Similar to AFs, one reasoning method for In this paper, we introduce ranking-based semantics ABA frameworks are extension-based semantics that for the ABA setting to rank assumptions based on their state when a set of assumptions is acceptable. Abstract strength. With the help of these semantics we can state argumentation frameworks and ABA frameworks are if an assumption is stronger than another one. To evalclosely related, the standard approach for reasoning with uate diferent ranking-based semantics approaches, we ABA frameworks includes the derivation of an AF and a propose principles each describing diferent desirable translation for the other direction exists as well [7]. behaviours for concrete approaches. With this principleThe classical semantics of both AFs and ABA frame- based approach we can compare diferent ranking-based semantics based solely on their behaviour. Additionally, 9th Workshop on Formal and Cognitive Reasoning, September 27- we present a family of ranking-based semantics based S$epkteemnnbeerth2.9sk,2ib0a2@3, fBeerrnluinn,iG-hearmgeann.dye (K. Skiba); on ideas for AFs. For an ABA framework, we look at matthias.thimm@fernuni-hagen.de (M. Thimm); the induced AF and calculate a ranking over arguments wallner@ist.tugraz.at (J. P. Wallner) to then lift the resulting ranking back to ABA and then © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License re-evaluate the result in the context of ABA. CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutRion W4.0oInrtekrnsahtioonpal (PCCroBYce4.0e).dings (CEUR-WS.org) This paper is organised as follows. We recall the neces
Definition 2 (Extension-based Semantics). Given = sary background information about AFs, ranking-based (, ), an admissible set ⊆ is
An abstract argumentation framework ( ) is a directed graph = (, ) where is a finite set of arguments and ⊆ ×
is an attack relation [2]. An argument say that an argument is defended by a set ⊆ is said to attack an argument if (, ) ∈ . We if every argument ∈ that attacks is attacked by some ∈ . For ∈ we define − = { | (, ) ∈ } and + = { | (, ) ∈ }, so the sets of attackers of and the set of arguments attacked by in . For a set of arguments ⊆ − and + via − = ⋃︀ we extend these definitions to ∈ − and + = ⋃︀ ∈ + , respectively. If the AF is clear in the context, we will omit the index.
Example 1. Consider the argumentation framework = (, ) depicted as a directed graph in Figure 1, with the nodes corresponding to arguments, and the edges corresponding to attacks = {, , , , , } = {(, ), (, ), (, ), (, ), (, ), (, )}.
Most semantics [11] for abstract argumentation are relying on two basic concepts: conflict-freeness and admissibility.
Definition 1 (Conflict-freeness, Admissibility) . Given = (, ), a set ⊆ is • admissible if it is conflict-free, and every element of is defended by .
We use ( ) and ( ) for denoting the sets of conflict-free and admissible sets of an argumentation framework , respectively. The intuition behind these concepts is that a set of arguments may be accepted only if it is internally consistent (conflict-freeness) and able to defend itself against potential threats (admissibility). The semantics proposed by Dung [2] are then defined as follows.
• a complete extension () if it contains every argument that it defends; complete extension; • a preferred extension () if it is a ⊆ -maximal • the unique grounded extension () if it is the ⊆ -minimal complete extension; • a stable extension () if + = ∖ .
The sets of extensions of an argumentation framework , for these four semantics, are denoted (respectively) ( ), ( ), ( ) and ( ).
Instead of only reasoning based on the acceptance of sets of arguments, ranking-based semantics [10] were introduced to focus on the strength of a single argument with respect to the other arguments. Note that the order returned by a ranking-based semantics is not necessarily total, i. e., not every pair of arguments is comparable.
a preorder1 ⪰ on .
A ranking-based semantics is a function, which maps an argumentation framework = (, ) to and ⪰ .
Intuitively, ⪰ means that is at least as strong as in . We define the usual abbreviations as follows; ≻ denotes strictly stronger, i. e. ⪰ and ̸⪰ . Moreover, ≃ denotes equally strong, i. e. ⪰
One example for ranking-based semantics is the Burden-based semantics [10]. This semantics calculates in each step a burden number for each argument based on the burden number of the attackers in the previous step and then lexicographically compares these numbers to establish a ranking.
[10] Let = (, ) be an AF. The burden number () for argument ∈ is denoted as () = (0(), 1(), 2(), ...) whereby the 1A preorder is a (binary) relation that is reflexive and transitive. values as follows: functions for ∈ N are mapping arguments to : → Q, () := s.t. for each , ∈ , it holds that ⪰ if () ⪰ (), where ⪰ is the lexicographical order, i. e. for two (possibly infinite) real number vectors = (1, 2, . . .) and ′ = (1′, 2′, . . .) we say ≻ ′ if ∃ s.t. < ′ and ∀ < , = ′ and Example 2. Given the AF from Example 1. We calculate for each argument the burden number. Argument is unattacked, hence () = (1, 1, 1, ...). Based on the value of , we can calculate the remaining burden numbers: considered stronger. () = (1, 2, , ...); () = (1, 2, 2, ...); () = (1, 2, , ...); () = (1, 2, , ...); () = (1, 3, 2.5, ...).
4 3 2 3 4 3 Since and have the same attacker , they receive in each step the same value. So, these burden numbers result in the following ranking: ≻ ≃ ≻ ≻ ≻ .
Argument is ranked highest, then and are equally strong, then followed by , and finally the least ranked argument is .
We recall some of the most fundamental principles [9] that guide the development of ranking-based semantics for abstract argumentation. The first and most basic principle states, that the names of the arguments should not influence the ranking.
Definition 5 (Isomorphism). An isomorphism between two argumentation frameworks = (, ) and ′ = all , ∈ , (, ) ∈ if ( (), ()) ∈ ′. (′, ′) is a bijective function : → ′ such that for
: → () ⪰ ′ (). isfies
Abstraction (short Abs) if for every pair of AFs = (, ), ′ = (′, ′) and every isomorphism ′, for all , ∈ , we have ⪰ if
The next principle states, that unattacked arguments should be ranked better than any attacked argument. holds that for all , ∈ with (, ) ∈/ and (, ) ∈ ,
The principle Cardinality Precedence focuses on the number of attackers. If an argument has fewer attackers than another argument, the first argument can be ≻ .
Definition 9 (CP). A ranking-based semantics satisfies Cardinality Precedence (short CP) if for any AF = (, ), it holds that for all , ∈ with |− | < |− |,
The final principle states that any of two arguments should be comparable. , ∈ either ⪰ or ⪰ .
Definition 10 (Total). A ranking-based semantics satisfies
Total if for any AF = (, ), it holds that for all
Note that, this is not a complete list of principles, more principles can be found in the literature [9] and also these principles are not mandatory principles since there are incompatibilities between them, e.g., CP and SC are incompatible. satisfies Abs, VP, CP and Total and violates SC [9].
Assumption-based Argumentation (ABA) uses a deductive system (ℒ, ℛ), where ℒ is a formal language and ℛ a set of rules of the form = 0 ← 1, ..., with ∈ ℒ .
We say that 0 is the head of the rule (ℎ() = 0) and the set {1, ..., } is the body (() = {1, ..., }).
An ABA framework is a tuple a non-empty set of assumptions, and so-called contrary function. : → ℒ is a
We focus in this work on flat ABA frameworks, i. e., ℎ() ∈/ for each rule ∈ ℛ .
A sentence ∈ ℒ is derivable from a set of assumptions ⊆ and rules ⊆ ℛ , denoted by ⊢ ,
(Abs). A ranking-based semantics sat- (ℒ, ℛ, , ), where (ℒ, ℛ) is a deductive system, ⊆ ℒ
Cyras and Toni [7] showed that the acceptance coincides. So, if a set of assumptions is acceptable in the ABA framework , then is also acceptable in the corresponding AF (in the form of conclusions of an Definition 12. For = (ℒ, ℛ, , ) be an ABA frame- extension). work and a conflict-free set of assumptions ⊆ , we say is if there is a finite rooted labelled tree with the root being labelled with , the set of labels for the leaves of is equal to or ∪ {⊤}, and the internal nodes are labelled with ℎ() according to a rule ∈ s.t. the children are labelled with () or ⊤ if the body is empty. Each assumption ∈ has an associated leaf labelled with and each rule ∈ has an associated node in the tree. For a tree , we denote by ( ) the set of assumptions used to derive the conclusion denoted ( ) with rules ( ).
Similar to AFs, ABA frameworks can be used as a rational decision-making model. In order to reason with ABA frameworks, extension-based semantics were introduced to state when a set of assumptions is acceptable. A set of assumptions attacks a set of assumptions ⊆ if there is ′ ⊆ , ⊆ ℛ , s.t. ′ ⊢ for some ∈ . is conflict-free if does not attack . defends assumption if attacks each assumption set that attacks {}.
• admissible in ( ∈ ()) if defends itself, • complete in ( ∈ ()) if is admissible and
contains every assumptions set it defends, • grounded in ( ∈ ()) if is ⊆ -minimally
complete, • preferred in ( ∈ ()) if is ⊆ -maximally
complete, and • stable in ( ∈ ()) if attacks every as
sumption ∈ ∖ .
Example 3. Consider the ABA framework with assumptions = {, , } and rules: 1 : ← 2 : ← 3 : ← , ; ; , with = , = , = . Then for example, we can derive from {} with rules 2 and 3 and since = we see that {} attacks {}. Additionally, we see that {} and ∅ are the two admissible sets.
AFs and ABA frameworks are closely related [7], and we can define an AF as an instance of an ABA framework and the other way around.
Definition 13. The associated AF = (, ) of an ABA framework = (ℒ, ℛ, , ) is given by = { | is a tree for ∈ ℒ with ( ) = } and attack relation ( , ′) ∈ if there is ∈ ( ′) s.t. = ( ).
Definition 14. Let = (, ) be an AF. The associated ABA framework of is ( ) = (ℒ, ℛ, , ) with p c r a • = , • ℒ = ∪ {| ∈ }, • ℛ = { ← |(, ) ∈ }, • for all ∈ : = .
Example 4. Continuing Example 3, we can construct the corresponding AF = (, ) of , with = {a, b, c, p, q, r} where • a is a tree with (a) = {}, (a) = , and
(a) = ∅, • b is a tree with (b) = {}, (b) = , and
(b) = ∅, • c is a tree with (c) = {}, (c) = , and
(c) = ∅, • p is a tree with (p) = {}, (p) = , and
(p) = {3}, • q is a tree with (q) = ∅, (q) = , and
(q) = {2}, • r is a tree with (r) = {, }, (r) = , and
(r) = {1} and the attack relation
= {(q, b), (q, r), (r, a), (r, p), (p, r), (p, c)}.
The corresponding graph representation can be found in Figure 2. So, for each derivable sentence in an ABA framework, we create an argument in the corresponding AF. We know that is derivable from {} by rules 2 and 3, hence p ∈ and additionally the attacks in the AF are representing the attacks from one set of assumptions to another set of assumptions. For example, the attack (p, r) ∈ is representing the fact, that {} attacks {}.
Note that in the following, we call argument a, based on a tree of the form (a) = {}, (a) = and (a) = ∅, where is an assumption, the assumption argument of .
Up to this point, reasoning in ABA is focused on sets of assumptions being acceptable with respect to a semantics like admissible, complete, or preferred semantics. So, we able set of assumptions or not, however this reasoning can state that an assumption is contained in an accept- = ℎ ← approach does not give us any insight into the strength of individual assumptions. In this paper, we shift the focus to the strength of the individual assumptions. For that purpose, we develop a general framework for ranking assumptions in ABA. This framework allows us to state that an assumption is stronger than another one.
preorder ⪰ on .
A ranking-based semantics is a function that maps an ABA framework = (ℒ, ℛ, , ) to a
Intuitively, ⪰ means, that assumption is at least as strong as in . We define the usual abbreviations as follows;
̸⪰ .
⪰ . • ≻ denotes strictly stronger, i.e. ⪰ and • ≃ denotes equally strong, i.e. ⪰ and
() ⪰ ′ ().
Definition 17 (Isomorphism). An isomorphism between two ABA frameworks
= (ℒ, ℛ, , ) and ′ = (ℒ′, ℛ′, ′, ′) is a bijective function : →
′ (extended to ℒ ′ with for all ∈
ℒ ∖ : () = and ℛ ′ = { ()| ∈ ℛ}, where for all ∈
ℛ with 1, ..., : (ℎ ←
1, ..., ) = (ℎ) ← (1), ..., ()) and = ()′ for all ∈ .
Definition 18 (P2). Ranking-based semantics satisfies P2 if for every pair of ABA frameworks = (ℒ, ℛ, , ) and ′ = (ℒ′, ℛ′, ′, ′) and for all isomorphisms
s.t. ′ = (), for all , ∈ , we have ⪰ if
The next principle is focusing on the addition of rules, where the head is a contrary of an assumption. In a sense, these rules can be considered as attacking rules.
The addition of such a rule for an assumption should not raise the strength of that assumption. added, i. e., − = (ℒ, ℛ ∪ {−}, , ).
Definition 19 (P3). Let = (ℒ, ℛ, , ) be an ABA framework and ∈ . Let − be a rule with − ∈/ ℛ and ℎ(−
) = . − is a copy of with − Ranking-based semantics satisfies
P3 if for all ABA frameworks = (ℒ, ℛ, , ) it holds for all , ∈
with ̸= that ⪰ − implies ⪰ .
In order to evaluate diferent ranking-based approaches for ABA frameworks, we will follow a principle-based
So, the addition of rules, which in a sense can lower the strength of an assumption, should at least not raise approach, like typical in the area of argumentation [12, 9]. the strength of that assumption.
These principles will give us insight into the behaviour of the diferent ranking-based semantics, hence allowing us to compare these diferent approaches. Note that all these principles are not mandatory and should be selected depending on the application. Some application favour
Cyras and Toni [7] have shown that the acceptance of extension-based semantics coincides for ABA frameworks and their corresponding AFs. However, the transformation from an ABA framework to an AF and back to an ABA framework does add new rules and theresome principles over others. In some other applications, fore changes the framework. The next principle ensures we want to avoid satisfying particular principles, since these principles are obstructing or counter-intuitive in these scenarios. Next, we will propose a number of principles for ranking-based semantics for ABA frameworks.
The first principle states that an assumption for which we cannot derive the contrary should be ranked higher than any assumption for which we can derive the contrary.
Definition 16 (P1). Ranking-based semantics satisfies P1 if for every ABA framework = (ℒ, ℛ, , ) it holds that for every assumption ∈ s.t. is not derivable from any set of assumptions ⊆ ∈ s.t. is derivable it holds that ≻ .
and for every assumption that these transformations between frameworks do not change the resulting ranking.
Definition 20 (P4). Ranking-based semantics satisfies P4 if for every ABA framework
= (ℒ, ℛ, , ) and the corresponding AF to , and () the cor
, ∈ that we have ⪰ if ⪰ () . responding ABA framework to , it holds for any pair
The next principle focuses on assumptions, for which we can derive the contrary by only using the assumption itself. These assumptions are in a sense self-attacking and should be ranked worse than any non-self-attacking assumption.
In other words, if there is no reason to lower the
Definition 21 (P5). Ranking-based semantics satisfies strength of an assumption, then the strength of that as- P5 if for every ABA framework = (ℒ, ℛ, , ) the sumption should not be lowered. sumptions do not influence the ranking.
One simple principle states that the names of the as- and {} ⊢ℛ then ≻ .
following holds for every assumptions , ∈ , if {} ̸⊢ℛ
We define a family of ranking-based semantics for ABA frameworks that relies on the reduction of an ABA framework to its corresponding AF, an application of a rankingunattacked arguments then correspond to assumptions for which we can not derive the contrary. based semantics for AFs on this derived AF, and a re- Theorem 1. If satisfies VP, then ABA- satisfies P1. interpretation of the resulting ranking over arguments in terms of assumptions.
Let = (ℒ, ℛ, , ) be an ABA frame- corresponding assumptions arguments, and a rankingHence, is the strongest assumption, then , and is the thus it is intuitive that is the strongest assumption. While weakest assumption. The preferred extension of is {}, satisfies Abs, we know that for any pair of arguments
In the remainder of this section, we look at the com- Proof. Let = (ℒ, ℛ, , ) be an flat ABA framework, ranking-based semantics for AFs. The ranking-based se- able. Since is not derivable, we know that a can not be work, = (, ) the corresponding AF, , ∈ a, b the corresponding assumptions arguments, and a
, mantics ABA- returns ⪰
ABA- if a ⪰
In other words, assumption is at least as strong in if the corresponding assumption argument a is at least as strong as b in the corresponding AF of .
Example 5. In the following example, we use the Burdenbased semantics as an example ranking-based semantics.
Other semantics can be applied equivalently. Consider the ABA framework from Example 3 and its corresponding AF constructed in Example 4.
Similar to Example 2, if we apply the Burden-based semantics to , the resulting ranking is:
q ≻ p ≃ a ≻ c ≻ b ≻ r.
Restricting the ranking to only assumption arguments gives us
a ≻ c ≻ b.
We can project this ranking back to ABA: ≻
ABA- ≻
ABA- is attacked by a fact ← strong and therefore should be ranked below .
meaning that is not really
So, the corresponding AF of an ABA framework gives us insight into the relationship between each assumption.
We see that if the corresponding argument is strong or highly ranked in the corresponding AF, then the assumption will also be strong in the ABA framework as well.
Additionally, we can compare two assumptions and with each other, which is not possible by using solely extension-based semantics since both assumptions are not acceptable. pliance of this family of ranking-based semantics with respect to our principles from before. For that we look at the principles the underlying ranking-based semantics for AFs satisfies. With the help of these principles, we can show that principles in the ABA setting are satisfied.
Proof. Let = (ℒ, ℛ, , ) be an ABA framework, = (, ) the corresponding AF, , ∈ , a, b the
Assume satisfies VP, is not derivable and is derivattacked in , because we do not have any argument x in with (x) = . Hence, a− = ∅. Additionally, we know that b is attacked at least once, because is derivable in , so there has to be an argument x′ s.t. (x′) = . Hence, b− ̸
= ∅. Since satisfies VP, we know that a ≻ b and therefore also ≻
We see that the names of assumptions do not influence the ranking, despite translating the ABA framework into an AF.
Theorem 2. If satisfies Abs, then ABA- satisfies P2.
Proof. Let = (ℒ, ℛ, , ) and ′ = (ℒ′, ℛ′, ′, ′) be two ABA frameworks and be an isomorphism s.t. ′ = (). Let = (, ) resp. ′ = (′, ′) be the corresponding AFs for and ′. Let be an ranking-based semantics for AFs.
Assume satisfies Abs. We know that for every assumption ∈ there is an isomorphic assumption ′ ∈ ′, hence for every argument a ∈ there is an isomorphic assumption argument a′ in ′ . Similar can be reason for any other element in . Therefore, there has to be an isomorphism ′ for s.t. ′() = ′ . Since
(a, b) it holds that if a ⪰ therefore P2 is satisfied.
Hence, it holds that () ⪰ ′
b then ′(a) ⪰ ′
′(b).
ABA- () if ⪰
ABA- and
In the following, we show that if the underlying ranking-based semantics for AFs satisfies CP and Total, then we know that the addition of rules, with which we can derive the contrary of an assumption, do not raise the strength of that assumption.
Theorem 3. If satisfies CP and Total, then ABA- satisifes P3. and let − = (, ) the corresponding AF and a rankingbased semantics for AFs. Let − is a new rule for ∈ , where − ∈/ ℛ and ℎ(−
) = and − is a copy of with − added, i.e. − = (ℒ, ℛ ∪ {−}, , ) be the corresponding AF.
Assume satisfies CP, Total and ⪰ −
ABA- for ∈ and the corresponding assumption arguments a and b. First, we look at the case that − can not be activated, so there is no tree x s.t. − ∈ (x) meaning that, (−
) ̸⊆ of rules (1, ..., , −) from ⋃︀
=1 ℎ() ∪ . Then the addition of − change the corresponding AF, i.e. = ℛ s.t. (− ) ⊆ does not
and and there is no sequence therefore ⪰
ABA- − implies ⪰
Next, we look at the case, where − can be activated.
The addition of any attack into an AF can only raise the number of attackers for an argument and can not lower the number of attackers. Similar hold for ABA frameworks, the addition and activation of a new rule does not yield to deactivation of other rules. Hence, it holds that |x− | ≤ | x−− corresponding assumption
ifes CP and Total and ⪰ −
| for any ∈ and its argument x. Since satis holds, we know that
− |a− − | ≤ | b− − | .
If |b− | = |b−
|, then it is clear that |a− | ≤ − − a ⪰ − b and therefore also ⪰ |b− | and since satisfies CP and Total it holds that
For |b− | < |b−
| we know that we can derive in and this activates a rule ′ with ∈ (′) and
− This implies that can not be derived in otherwise we could activate ′ in as well and that means that |b− | < |b−
| could not hold. Since can not be derived this implies |a− | = 0 and therefore |a− | ≤
|b− | and also a ⪰ b, which implies ⪰ in is also derivable in (). This implies that the number of attacker for any assumption argument a in is equal to the number of attacker for the corresponding assumption argument in (). Since satisfies CP and Total and ⪰
− , we know |a− | ≤ | and since the number of attacker is the same in
b− | and (), i.e. |()− | = |()−() |, we have |a−() | ≤ | b−() |. CP and Total then imply a ⪰ () and therefore also ⪰ () .
ABA-
If the underlying ranking-based semantics does satisfy SC, then we know that in the ABA setting assumptions for which we can derive the contrary by only using the assumption should be ranked worst.
Theorem 5. If satisfies SC, then ABA- satisfies P5.
Proof. Let = (ℒ, ℛ, , ) be an ABA framework, = (, ) the corresponding AF, , ∈ , the corresponding assumptions arguments a, b, and a rankingbased semantics for AFs.
Assume satisfies SC and can not attack it self. Because a ≻ b and this implies ≻ {} ⊢ℛ . This implies that (b, b) ∈ and (a, a) ∈/ .
So, b attacks itself and also an assumption argument x for ∈ can only attack it self if {} ⊢ℛ , hence a
satisfies SC, we know The principles the underlying ranking-based semantics
{} ̸⊢ℛ and with proposed for the ABA setting are simple. A number of diferent ranking-based semantics for AFs are suitable to be used, since they satisfy a good number of principles like the Burden-based semantics satisfies Abs, VP, CP and Total and therefore ABA- satisfies P1, P2, P3 and P4. However, Besnard et al. [13] have shown that a few principles for the AF setting are incompatible with each other, in particular CP and SC are incompatible. Hence, there is no ranking-based semantics, which satisfies CP and SC. Therefore, we have to check the principles in the ABA setting by hand. For ABA- we know, that SC is violated, therefore we have to check P5. By adapting the counterexample used to show the incompatibility of CP and SC, we can show that ABA- does violate P5 ing AF is Example 6. Let 1 = (ℒ, ℛ, , ), with = {, , }, rules 1 : → , 2 : → , and = . The correspond
1 = ({a, b, c, r1, r2}, {(r1, a), (r2, a), (b, b)}), where
() = ∅, • a is a tree with (a) = {}, (a) = and this rule is needed to activate rule ′′ with ℎ(′′) = . for AFs need to satisfy in order to satisfy every principle Proof. Let = (ℒ, ℛ, , ) be a flat ABA framework, = (, ) the corresponding AF, () the cor- [13, 14]. responding ABA framework of , () the corresponding AF to () and a ranking-based semantics for AFs. Let , ∈ , a be the corresponding assumptions argument of and b be the corresponding assumption argument of .
Assume satisfies CP, Total and ⪰ tence is derivable in , then there is a corresponding argument in and every argument in is an assumption in () and since assumptions are always derivable, we know that everything, which is derivable
ABA- . If a sen• b is a tree with (b) = {}, (b) = and
() = ∅, • c is a tree with (c) = {}, (c) = and
() = ∅, • r1 is a tree with (r1) = {}, (r1) = and
((1) = {1}, • r2 is a tree with (r2) = {}, (r2) = and
((2) = {2}.
Depicted in Figure 3.
So, implies its contrary {} ⊢∅ and does not imply its contrary. When we apply the Burdenbased semantics, we see that (a) = (1, 3, 3, ...) and
a and there(b) = (1, 2, 2, ...), this implies b ≻ 1 fore ≻ 1 ABA-
ABA- . However, this contradicts ≻ 1 like P5 implies. calculating the preorder based on the language and rules given.
In ASPIC+ and ABA+ preferences are used to disable or reverse attacks. If the target of an attack is considered better than the attacker, this attack is discarded or reversed, so the attacker is the attacked.
One interesting idea with ABA+ is to use the underlying ranking over assumptions to construct the corresponding ABA+ framework for an ABA framework. So, we take an ABA framework and calculate a ranking over the assumption with any ranking-based semantics like ABA- to then construct an ABA+ framework using our ranking as a preference order. An ABA+ framework constructed in such a way has similarities with the underlying ABA framework for example the conflict-free sets are the same. Hence, we can transform any ABA framework into an ABA+ framework without additional information like a preference order.
_ABA uses preferences to discredit sets of assumptions. Wakaki [21] proposes preorders over sets of assumptions. However, their approach has two big diferences: first, in _ABA preferences are part of the input and, secondly, they can only diferentiate sets of assumptions, which belong to an extension-based semantics.
In the literature, ranking-based semantics are used to refine extension-based reasoning in the area of abstract argumentation. For example Bonzon et al. [22] uses the aggregated strength values of each argument of a set to compare two sets. While Konieczny, Marquis, and Vesic [23] are comparing two sets of arguments using a pairwise comparison based on a criterion like the number of arguments inside the first set not attacked by the second set. So, the presented ranking-based semantics for ABA frameworks are the first step towards refining extensionbased reasoning inside structured argumentation.
One of the most discussed topics in structured
argumentation are preferences over uncertain information. These
preferences state that information is better or more
believable than information . A number of frameworks
working with preferences can be found the literature
like ASPIC+ [
In these frameworks the preferences are preorders We also defined a family of ranking-based semantics for over rules and ordinary premises (ASPIC+), assumptions ABA based on approaches and ideas for AFs. For an ABA (ABA+) or sentences (_ABA). Hence, these preferences framework we construct the corresponding AF then apare similar to our rankings over assumptions. All these ply known ranking-based semantics in order to rank arpreferences can be seen as a strength notion, if an as- guments in the corresponding AF to finally re-interpret sumption is preferred over assumption in an ABA+ this ranking in the ABA setting. We have shown that if framework, then this relationship between and can the underlying ranking-based semantics for AFs satisfies be seen as is better than . However, all these frame- certain principles, then this family of semantics satisfies works receive their preferences as an input rather than our proposed principles for the ABA setting as well.