Strong admissibility for abstract dialectical frameworks

Definition 1 ([34]).

An abstract argumentation framework (AF) is a pair ( A , R ) in which A is a set of arguments and R ⊆ A × A is a binary relation representing attacks among arguments.

Let F = ( A , R ) be an AF. A conflict-free set S is

admissible in F if S ⊆ Γ F ( S ) ;

Let F = ( { a , b , c } , { ( a , b ) , ( b , c ) } ) be an AF. In F, ( a , b ) means that argument a attacks b, and ( b , c ) means that b attacks c. Here, argument c is defended by set { a } (alternatively, c is acceptable with respect to { a } ), since a attacks the attacker of c, namely b. The set of conflict-free sets of F is cf ( F ) = { ∅ , { a } , { b } , { c } , { a , c } } . Since Γ F 2 ( ∅ ) = { a , c } , the unique grounded extension of F is { a , c } . The intuition is that a is not attacked by any argument, thus no one has any doubt to accept argument a. Argument c is attacked by b, however, it is defended by a, which was accepted by everyone. Thus, { a , c } is the unique grounded extension of F.

Given an argumentation framework F = ( A , R ) , a ∈ A and S ⊆ A , it is said that a is strongly defended by S if and only if each attacker c ∈ A of a is attacked by some s ∈ S ∖ { a } such that s is strongly defended by S ∖ { a } .

Definition 4.

Given an AF ( A , R ) and set S ⊆ A , it is said that S is a strongly admissible extension of S if every s ∈ S is strongly defended by S.

In Example 1, sets S 1 = ∅ , S 2 = { a } , and S 3 = { a , c } are strongly admissible extensions of F; all of them are subsets of the grounded extension of F. However, set S ′ = { c } is not a strongly admissible extension of F, since c ∈ S ′ is not strongly defended by S ′ . Because argument c is attacked by b, however, no argument in S ′ ∖ { c } attacks b.

In [25], the concept of strongly admissible semantics of AFs is defined without having to refer to strong defence; we rephrase it in Definition 5, as follows.

Definition 5 (from [25]).

Let F = ( A , R ) be an argumentation framework. We say that S ⊆ A is a strongly admissible extension of F if and only if every a ∈ S is defended by some S ′ ⊆ S ∖ { a } which in its turn is a strongly admissible extension.

Theorem 1 (from [8]).

Let F be an AF. The set of strongly admissible semantics of F forms a lattice with the empty set as the least element and the grounded extension as the maximum element.

Proposition 2 (based on [8,25]).

Let F be an AF. The following properties hold for semantics of AF:

Each strongly admissible extension in F is an admissible extension and a conflict-free extension, however, the other direction does not hold.

Proof.

Let F be an AF.

In Theorem 4 in [25] it has been proved that each strongly admissible extension in F is an admissible extension and a conflict-free extension. We provide a proof that the other direction does not hold, by giving a counter-example. Let F = ( { a , b } , { ( a , b ) , ( b , a ) } ) . Now the set S = { a } is a conflict-free and admissible extension of F, however, it is not a strongly admissible extension in F. This is because there is no S ′ ⊆ S ∖ { a } that defends a against the attack of b.

It has been proved in [8] that the strongly admissible semantics of an AF form a lattice with the grounded extension as the maximum element. However, it does not hold that any admissible extension of a given AF that is a subset of the grounded extension is a strongly admissible extension. We provide a proof by giving a counter-example. Let F = ( { a , b , c } , { ( a , b ) , ( b , c ) , ( c , b ) } ) . The grounded extension of F is { a , c } ; furthermore, S = { c } is an admissible extension of F. However, S is not a strongly admissible extension of F, since there is no S ′ ⊆ S ∖ { c } that defends c against the attack of b. □

The notion of an ideal extension is presented in Definition 2. In Definition 3.48 in [8], first the notion of ideal set is defined, then the notion of ideal semantics is presented; we rephrase this definition here. An admissible extension e is called ideal iff it is a subset of each preferred extension of F. The ideal extension of F is a maximal (with respect to set-inclusion) ideal set. We show that the strongly admissible extensions differ from the ideal sets and the ideal extension of a given AF.

Proposition 3.

The notion of strongly admissible semantics of AFs differs from the notion of ideal semantics of AF.

Proof.

We provide a proof by an example. Let F = ( { a , b } , { ( a , b ) , ( b , a ) , ( b , b ) } ) , as depicted in Fig. 1. The unique grounded extension of F is the empty set. The set of strongly admissible extensions of F is { ∅ } . However, the set of ideal sets of F is { ∅ , { a } } . Thus, { a } is the ideal extension of F. As we see, the set of strongly admissible extensions of F is neither equal with the set of ideal sets of F nor is it equal with the ideal extension of F.  □ OPEN IN VIEWER

Fig. 1.

AF of Proposition 3.

It is also possible to define the semantics of AFs in terms of argument labellings [20,30,58].

Definition 6 ([25, Definition 4]).

Let F = ( A , R ) be an argumentation framework. An argument labelling is a function L : A ↦ { in , out , undec } . An argument labelling is called an admissible labelling if and only if L is a total function and for each a ∈ A it holds that:

if L ( a ) = in , then for each b that attacks a it holds that L ( b ) = out ,

Let L be an admissible labelling of argumentation framework F = ( A , R ) . A min-max numbering is a total function M M L : in ( L ) ∪ out ( L ) ↦ N ∪ ∞ such that for each a ∈ in ( L ) ∪ out ( L ) it holds that:

if L ( a ) = in , then M M L ( a ) = max ( { M M L ( b ) | ( b , a ) ∈ R and L ( b ) = out } ) + 1 (with max ( ∅ ) defined as 0);

Every admissible labelling has a unique min-max numbering.

Definition 8 ([25, Definition 10]).

A strongly admissible labelling is an admissible labelling whose min-max numbering yields natural numbers only (so no argument is numbered ∞).

Example 2.

Let F = ( { a , b , c , d } , { ( a , b ) , ( c , d ) , ( d , c ) } ) . By Definition 7, admissible labelling { a ↦ in , b ↦ out , c ↦ in , d ↦ out } has a unique min-max numbering { ( a : 1 ) , ( b : 2 ) , ( c : ∞ ) , ( d : ∞ ) } . However, this admissible labelling is not a strongly admissible labelling in F, since the M M L ( c ) = M M L ( d ) = ∞ . On the other hand, the admissible labelling { a ↦ in , b ↦ out , c ↦ undec , d ↦ undec } has a unique min-max numbering { ( a : 1 ) , ( b : 2 ) } , since both M M L ( a ) and M M L ( b ) are finite, so by Definition 8, it is a strongly admissible interpretation in F.

In [6], both extension-based and labelling-based approaches of semantics of AFs are presented. Moreover, two functions are defined to map extension-based semantics of a given AF F to labelling-based semantics and vice versa, to show that each extension-based semantics of AF has a labelling-based semantics reformulation and vice versa. Namely, Ext2Lab ( λ ) is used to present the extension form of labelling λ of F, and Lab2Ext ( e ) is used to present the labelling form of the extension e of F; we recall them in Definitions 9–10.

Definition 9 ([6, Definition 3.6]).

Let F = ( A , R ) and let S be an extension of F. For a ∈ A , we write a + = { b | ( a , b ) ∈ R } and S + = ∪ { a + | a ∈ S } . If S is a conflict-free set of F, then the corresponding labelling is defined as Ext2Lab ( S ) = { S , S + , A ∖ ( S ∪ S + ) } .

Definition 10 ([6, Definition 2.7]).

Given an argumentation framework F = ( A , R ) and a labelling λ, the corresponding set of arguments Lab2Ext ( λ ) is defined as Lab2Ext ( λ ) = in ( λ ) . That is, Lab2Ext ( λ ) is the set of all arguments that are labelled in in λ.

Proposition 5 ([6, Proposition 3.14]).

For any argumentation framework F = ( A , R ) , it holds that:

if S is a strongly admissible set of F, then Ext2Lab ( S ) is a strongly admissible labelling of F;

Definition 11.

An abstract dialectical framework (ADF) is a tuple F = ( A , L , C ) where:

A is a set of arguments (statements, positions), denoted by letters;

L ⊆ A × A is a set of links between arguments;

C = { φ a } a ∈ A is a collection of propositional formulas over arguments, called acceptance conditions.

An ADF can be represented by a graph in which nodes indicate arguments and links show the relation among arguments. Each argument a in an ADF is labelled by a propositional formula, called acceptance condition, φ a over par ( a ) such that par ( a ) = { b | ( b , a ) ∈ L } . An argument a is called an initial argument if par ( a ) = { } . The acceptance condition of an argument clarifies under which condition the argument can be accepted [14,17,18]. Furthermore, acceptance conditions indicate the set of links implicitly. Thus, in a concrete example of ADFs, we oftentimes only define acceptance conditions explicitly and implicitly define links via the variables of the propositional formulas.

Remark.

ADF D = ( A , L , C ) is called a finite abstract dialectical framework if A is a finite set of arguments. Note that in this work we assume that all ADFs are finite.

Example 3.

An example of an ADF D = ( A , L , C ) is shown in Fig. 2, which contains four arguments, i.e., A = { a , b , c , d } . Dependencies between arguments are shown by the directed edges in the associated graph, and acceptance conditions are shown as propositional formulas attached to each node. The acceptance condition of an argument clarifies the set of parents of the argument, thus, there is no need of presenting L in D explicitly. Furthermore, an acceptance condition of an argument indicates under which condition the argument can be accepted/denied. For instance, the acceptance condition of c, namely φ c : ¬ b ∧ d , says that c depends on b and d, i.e., par ( c ) = { b , d } , and it states that c can be accepted if b is denied and d is accepted. In this ADF, the arguments a and d are initial arguments, since par ( a ) = par ( d ) = { } . The acceptance condition of a, namely φ a : ⊤ , says that a is always accepted and the acceptance condition of d, namely φ d : ⊥ , says that d is always denied.

OPEN IN VIEWER

Fig. 2.

ADF of Examples 3, 5 and 7.

An interpretation v (for D) is a function v : A ↦ { t , f , u } that maps arguments to one of the three truth values true (t), false (f), or undecided (u). The interpretation v is called trivial, and v is denoted by v u , if v ( a ) = u for each a ∈ A . Moreover, v is called a two-valued interpretation if for each a ∈ A either v ( a ) = t or v ( a ) = f .

Truth values can be ordered via the information ordering relation < i given by: u < i t and u < i f and no other pair of truth values are related by < i . Relation ⩽ i is the reflexive and transitive closure of < i . The pair ( { t , f , u } , ⩽ i ) is a complete meet-semilattice with the meet operator ⊓ i , such that t ⊓ i t = t and f ⊓ i f = f , while it returns u otherwise. The meet of two interpretations v and w is then defined as ( v ⊓ i w ) ( a ) = v ( a ) ⊓ i w ( a ) for all a ∈ A .

Let V be the set of all interpretations for an ADF D. It is said that an interpretation v is an extension of another interpretation w, if w ( a ) ⩽ i v ( a ) for each a ∈ A , denoted by w ⩽ i v . Furthermore, if both v ⩽ i w and w ⩽ i v hold, then v and w are equivalent, denoted by v ∼ i w . Interpretations v and w are incomparable if neither w ≰ i v nor v ≰ i w , denoted by w ≁ i v .

For reasons of brevity, we will sometimes shorten the notion of three-valued interpretation v = { a 1 ↦ t 1 , … , a m ↦ t m } with arguments a 1 , … , a m and truth values t 1 , … , t m as follows: v = { a i | v ( a i ) = t } ∪ { ¬ a i | v ( a i ) = f } . For instance, v = { a ↦ f , b ↦ t } = { ¬ a , b } . Furthermore, in the current work we say that the truth value of a is presented in v, if v ( a ) ∈ { t , f } also denoted as v ( a ) = t / f . Interpretations can be ordered via ⩽ i with respect to their information content.

Semantics for ADFs can be defined via the characteristic operator Γ D , which maps interpretations to interpretations. Given an interpretation v (for D), the partial valuation of φ a by v is v ( φ a ) = φ a v = φ a [ b / ⊤ : v ( b ) = t ] [ b / ⊥ : v ( b ) = f ] , for b ∈ par ( a ) .

Definition 12.

Let D be an ADF and let v be an interpretation of D. Applying Γ D on v leads to v ′ such that for each a ∈ A , v ′ is as follows: Γ D ( v ) ( a ) = v ′ ( a ) = t if φ a v is irrefutable (i.e., φ a v is a tautology) , f if φ a v is unsatisfiable (i.e., φ a v is a contradiction) , u otherwise.

Intuitively, the characteristic operator Γ D assigns an argument a to t (f) if the partial evaluation of the acceptance condition of a by a given interpretation is irrefutable (unsatisfiable, respectively). Note that the operator Γ D is ⩽ i -monotonic, that is, when v ⩽ i w for interpretations v and w, then Γ D ( v ) ⩽ i Γ D ( w ) .

Example 4.

Consider ADF D = ( { a , b , c , d } , { φ a : ⊤ , φ b : a ∧ ¬ c , φ c : ¬ b ∧ d , φ d : ⊥ } ) from Example 3 and the trivial interpretation v u of D. We calculate v ′ = Γ D ( v u ) over all the arguments of D. Consider argument a; since φ a : ⊤ , it holds that φ a v u : ⊤ , that is, φ a v u is irrefutable. Thus, a is assigned to t in v ′ . However, since both of the parents of b, namely, a and c are assigned to u in v u , it holds that φ b v u = φ b , which is neither a tautology nor unsatisfiable. Thus, v ′ ( b ) = Γ D ( v u ) ( b ) = u . By the same reason, it holds that v ′ ( c ) = Γ D ( v u ) ( c ) = u . However, it holds that φ d v u : ⊥ , that is, φ d v u is unsatisfiable. Thus, v ′ ( d ) = Γ D ( v u ) ( d ) = f . Hence, v ′ = { a ↦ t , b ↦ u , c ↦ u , d ↦ f } .

If we apply the characteristic operator Γ D over v ′ , then since Γ D is a monotonic operator, the truth values of a and d in Γ D ( v ′ ) are equal to v ′ ( a ) and v ′ ( d ) , respectively. Since φ c v ′ = ¬ b ∧ ⊥ ≡ ⊥ , it holds that Γ D ( v ′ ) ( c ) = f . However, since φ b v ′ = ⊤ ∧ ¬ c = ¬ c , it holds that Γ D ( v ′ ) ( b ) = u . Thus, Γ D ( v ′ ) = { a ↦ t , b ↦ u , c ↦ f , d ↦ f } .

The semantics of ADFs, as defined by [15], are based on (collections of) three-valued interpretations and can be defined via the characteristic operator as follows.

Definition 13.

Given an ADF D, an interpretation v is:

conflict-free iff v ( s ) = t implies φ s v is satisfiable and v ( s ) = f implies φ s v is unsatisfiable;

admissible in D iff v ⩽ i Γ D ( v ) ;

preferred in D iff v is ⩽ i -maximal admissible;

complete in D iff v = Γ D ( v ) ;

the grounded interpretation of D iff v is the least fixed point of Γ D .

The set of all σ interpretations for an ADF F is denoted by σ ( F ) , where σ ∈ { cf , adm , prf , com , grd } abbreviate the different semantics in the obvious manner. In any ADF D, it holds that prf ( D ) ⊆ com ( D ) ⊆ adm ( D ) ⊆ cf ( D ) and grd ( D ) ⊆ com ( D ) .

The intuitions behind the semantics of ADFs are as follows. In ADFs an interpretation is called admissible if it does not contain any unjustifiable information. An interpretation is called preferred if it represents maximum information about arguments without losing admissibility. Thus, each admissible interpretation is contained in a preferred interpretation. That is, to answer the credulous decision problem under preferred semantics (i.e., to investigate whether there is a preferred interpretation that contains the truth value of a given argument), it is sufficient to answer the problem under admissible semantics. An interpretation is complete if it exactly contains the relevant justifiable information. Finally, an interpretation is grounded if it collects all the information that is beyond any doubt.

Example 5.

Let us consider again ADF D = ( { a , b , c , d } , { φ a : ⊤ , φ b : a ∧ ¬ c , φ c : ¬ b ∧ d , φ d : ⊥ } ) from Example 3. Furthermore, consider several three-valued interpretations { v 0 , v 1 , v 2 , v 3 , v , v ′ } as shown in Table 1. We investigate whether they are part of certain semantics. Since for each i with 0 ⩽ i ⩽ 3 it holds that v i ⩽ i Γ D ( v i ) , it holds that each v i for i with 0 ⩽ i ⩽ 3 is an admissible interpretation of D. In general, each admissible interpretation is a conflict-free interpretation. Thus, each v i for i with 0 ⩽ i ⩽ 3 is a conflict-free interpretation of D. The interpretation v 3 is a complete interpretation of D, since Γ D ( v 3 ) = v 3 . Furthermore, v 3 is the grounded interpretation of D, since it the least fixed point of Γ D . In addition, v 3 is a maximal admissible interpretation of D, therefore, v 3 is a preferred interpretation of D.

Note that the interpretation v is not a(n) admissible/preferred/complete/the grounded interpretation of D, since Γ D ( v ) = { a ↦ t , b ↦ u , c ↦ f , d ↦ f } , that is, v ≰ i Γ D ( v ) . However, v is a conflict-free interpretation of D. The truth value of b is assigned to t in v. Thus, to show that v is a conflict-free interpretation of D, it is enough to check whether φ b v is satisfiable. Since φ b v is indeed satisfiable, it holds that v is a conflict-free interpretation of D.

Furthermore, for v ′ it holds that Γ D ( v ′ ) = { a , ¬ d } . Thus, since v ′ ≰ i Γ D ( v ′ ) , it holds that v ′ is not a(n) admissible/preferred/complete/the grounded interpretation of D. To check whether v ′ is a conflict-free interpretation, we have to check whether φ d v ′ is satisfiable. Since φ d v ′ ≡ ⊥ is unsatisfiable, it holds that v ′ is not a conflict-free interpretation of D.

OPEN IN VIEWER

Table 1

Interpretations for the ADF from Example 5

Interpretation	a	b	c	d	Γ D	Part of semantics
v 0 = v u	u	u	u	u	Γ D ( v u ) = v 1	cf , adm
v 1	t	u	u	f	Γ D ( v 1 ) = v 2	cf , adm
v 2	t	u	f	f	Γ D ( v 2 ) = v 3	cf , adm
v 3	t	t	f	f	Γ D ( v 3 ) = v 3	cf , adm , prf , grd , com
v	u	t	u	u	Γ D ( v ) = v 2	cf
v ′	u	u	u	t	Γ D ( v ′ ) = v 1	−

The semantics for ADFs generalize the semantics for AFs. Definition 14 presents the associated ADF for a given AF.

Definition 14.

For an AF F = ( A , R ) , define the ADF associated with F as D F = ( A , R , C ) with C = { φ a } a ∈ A such that for each a ∈ A the acceptance condition is as follows: φ a = ⋀ ( b , a ) ∈ R ¬ b

For instance, the associated ADF D F to the AF F presented in Example 1, i.e., F = ( { a , b , c } , { ( a , b ) , ( b , c ) } ) , is D F = ( A , R , { φ a : ⊤ , φ b : ¬ a , φ c : ¬ b } ) . In [17, Theorem 2], it is shown that: an extension is admissible, complete, preferred, grounded for F iff it is admissible, complete, preferred, grounded for D F . The notion of an argument being acceptable and the symmetric notion of an argument being deniable in an interpretation are defined as follows.

Definition 15.

Let D = ( A , L , C ) be an ADF and let v be an interpretation of D.

An argument a ∈ A is called acceptable with respect to v if φ a v is irrefutable.

An argument a ∈ A is called deniable with respect to v if φ a v is unsatisfiable.

For instance, considering ADF D of Example 3, it holds that a is acceptable with respect to v = { a ↦ t , b ↦ u , c ↦ u , d ↦ f } .

One of the main decision problems of ADFs is whether an argument is credulously acceptable (deniable) under a type of semantics, presented in Definition 16.

Definition 16.

Given an ADF D = ( A , L , C ) , an argument a ∈ A and a semantics σ ∈ { adm , prf , com , cf , grd } . Argument a is credulously acceptable (deniable) under σ if there exists a σ interpretation v of D such that v ( a ) = t ( v ( a ) = f , respectively). This reasoning task is denoted by Cred σ ( a ↦ t / f , D ) .

Considering ADF of Example 3, it holds that b is credulously acceptable under σ for σ ∈ { adm , prf , com , cf , grd } in D. This is because v 3 , as it is shown in Table 1, is a σ-interpretation for σ ∈ { adm , prf , com , cf , grd } with v 3 ( b ) = t . However, b is not credulously deniable under σ for σ ∈ { adm , prf , com , grd } in D. The only preferred/complete/grounded interpretation of D is v 3 , which assigns b to t. Thus, b is not credulously deniable under σ-semantics, for σ ∈ { prf , com , grd } . Since each admissible interpretation must be equally or less informative than a preferred interpretation, b is not credulously deniable under admissible semantics of D.

Thanks to the fact that in an ADF D each preferred interpretation is a ⩽ i -maximal admissible (complete) interpretation of D, it holds that the credulous decision problems under admissible, complete, and preferred semantics coincide.

In ADFs, relations between arguments can be classified into four types, reflecting the relationship of attack and/or support that exists between the arguments. These are listed in Definition 17. Further, we denote the update of an interpretation v with a truth value x ∈ { t , f , u } for an argument b by v | x b , i.e. v | x b ( b ) = x and v | x b ( a ) = v ( a ) for a ≠ b . Definition 17.

Let D = ( S , L , C ) be an ADF. A relation ( b , a ) ∈ L is called

supporting (in D) if for every two-valued interpretation v, v ( φ a ) = t implies v | t b ( φ a ) = t ;

attacking (in D) if for every two-valued interpretation v, v ( φ a ) = f implies v | t b ( φ a ) = f ;

redundant (in D) if it is both attacking and supporting;

dependent (in D) if it is neither attacking nor supporting.

Example 6.

Consider the acceptance condition of φ a : b ∨ ¬ c for argument a. By φ a the set of parents of a is { b , c } . Thus, we clarify the type of ( b , a ) and ( c , a ) . There are three satisfying two-valued interpretations, i.e., v 1 = { b ↦ t , c ↦ t } , v 2 = { b ↦ t , c ↦ f } and v 3 = { b ↦ f , c ↦ f } , and one that does not satisfy the formula, i.e., v 4 = { b ↦ f , c ↦ t } . By the definition of supporting links we have to check that whether v i ( φ a ) = t for i with 1 ⩽ i ⩽ 3 implies v i | t b ( φ a ) = t . Since for i with 1 ⩽ i ⩽ 3 , v i ( φ a ) = t implies v i | t b ( φ a ) = t , it holds that ( b , a ) is a supporting link. Furthermore, since v 4 ( φ a ) = f but v 4 | t b ( φ a ) = t , link ( b , a ) is not an attack link.

Moreover, since v 3 ( a ) = t but v 3 | t c ( φ a ) = f , it holds that ( c , a ) is not a support link. However, it holds that v 4 ( φ a ) = f and v 4 | t c ( φ a ) = f . Thus, ( c , a ) is only an attacking link.

As an example for a link that is both attacking and supporting, consider φ a : b ∨ ¬ b . There are two satisfying two-valued interpretations for the formula, i.e., v 1 = { b ↦ t } and v 2 = { b ↦ f } . Since for i with 1 ⩽ i ⩽ 2 it holds that v i ( φ a ) = t implies v i | t b ( φ a ) = t , it holds that ( b , a ) is a supporting link. Furthermore, since there is no two-valued interpretation that does not satisfy the formula, the link ( b , a ) is also an attacking link. Thus, ( b , a ) is a redundant link in φ a : b ∨ ¬ b .

As an example for a link that is neither an attacking nor supporting, consider φ a : ( ¬ c ∨ b ) ∧ ( c ∨ ¬ b ) . Let v = { b ↦ f , c ↦ f } be a two-valued interpretation that satisfies the formula. However, v | t b ( φ a ) = f . Thus, ( b , a ) is not a support link. Further, let v = { b ↦ f , c ↦ t } be a two-valued interpretation that does not satisfy the formula. However, it holds that v | t b ( φ a ) = t . Thus, ( b , a ) is not an attacking. Hence ( b , a ) is a dependent link.

3. The strongly admissible semantics for ADFs

In the following, we present the concept of strong admissibility semantics for ADFs. As we discussed in the introduction, we are aiming to generalize the notion of strong admissibility semantics of AFs, so that the concept of strong admissibility semantics of ADFs relates to grounded semantics of ADFs in a similar way as the concept of admissible semantics of ADFs relates to preferred semantics of ADFs. As we mentioned in the introduction, following the definition by Baroni and Giacomin [8], Caminada showed in [21,23] that strong admissibility plays a critical role in discussion games for AFs under grounded semantics [21,25]. In [40], we introduced a discussion game to answer the credulous decision problem of ADFs under grounded semantics without constructing the full grounded interpretation of the given ADF. However, the concept of strong admissibility semantics of ADFs has not been introduced in the literature so far. This was a motivation for us to present the notion of strong admissibility semantics for ADFs and study the characteristics of this concept. Similarly to the AF case, a strongly admissible interpretation of an ADF may be used not only to answer the credulous decision problem, but also to explain why an argument is justified in the grounded interpretation.

In ADFs, beside an argument being acceptable in an interpretation, there is a symmetric notion of an argument being deniable. In contrast with Definition 4, in which the concept of strong admissibility semantics of AFs is defined based on the concept of strongly defended arguments, in ADFs we define the concept of strong admissibility semantics based on the concept of strongly acceptable/deniable arguments. To this end, in Definition 18 we introduce the notion of strong justification (i.e., strongly accepted/denied) of an argument in an ADF in a given interpretation.

Note that in the following, v | P is equal to v ( p ) for any p ∈ P ; however, it assigns all other arguments that do not belong to P to u, i.e., v | P = v u | v ( p ) p ∈ P .